Subgroups — C₈₉×#62916 #36440764

Order 22784.

7938 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3546	C₂	1	12	yes	maximal, complemented
2	C₈₉×#3543	C₂	1	12	yes	maximal, complemented
3	C₈₉×#3537	C₂	1	12	yes	maximal, complemented
4	C₈₉×#3517	C₂	1	4	yes	maximal, complemented
5	C₈₉×#3498	C₂	1	6	yes	maximal, complemented
6	C₈₉×#3498	C₂	1	4	yes	maximal, complemented
7	C₈₉×#3496	C₂	1	3	yes	maximal, complemented
8	C₈₉×#3488	C₂	1	4	yes	maximal, complemented
9	C₈₉×#3480	C₂	1	2	yes	maximal, complemented
10	C₂×#550×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3460	C₂	1	3	yes	maximal, complemented
12	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	4	yes	complemented
13	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	6	yes	complemented
14	C₂²×#41×C₈₉	C₂²	1	2	yes	complemented
15	#560×C₈₉	C₂²	1	48	yes	complemented
16	#559×C₈₉	C₂²	1	6	yes	complemented
17	#557×C₈₉	C₂²	1	2	yes	complemented
18	#556×C₈₉	C₂²	1	6	yes	complemented
19	#553×C₈₉	C₂²	1	12	yes	complemented
20	#553×C₈₉	C₂²	1	24	yes	complemented
21	#550×C₈₉	C₂²	1	4	yes	complemented
22	#550×C₈₉	C₂²	1	12	yes	complemented
23	#550×C₈₉	C₂²	1	2	yes	complemented
24	#550×C₈₉	C₂²	1	8	yes	complemented
25	#550×C₈₉	C₂²	1	12	yes	complemented
26	#550×C₈₉	C₂²	1	12	yes	complemented
27	#548×C₈₉	C₂²	1	4	yes	complemented
28	#547×C₈₉	C₂²	1	12	yes	complemented
29	#546×C₈₉	C₂²	1	24	yes	complemented
30	#546×C₈₉	C₂²	1	24	yes	complemented
31	D₄²×C₈₉	C₂²	1	24	yes	complemented
32	D₄²×C₈₉	C₂²	1	12	yes	complemented
33	D₄²×C₈₉	C₂²	1	12	yes	complemented
34	D₄²×C₈₉	C₂²	1	24	yes	complemented
35	D₄²×C₈₉	C₂²	1	24	yes	complemented
36	D₄²×C₈₉	C₂²	1	12	yes	complemented
37	#544×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	#542×C₈₉	C₂²	1	12	yes	complemented
39	#540×C₈₉	C₂²	1	12	yes	complemented
40	#540×C₈₉	C₂²	1	24	yes	complemented
41	#538×C₈₉	C₂²	1	12	yes	complemented
42	#538×C₈₉	C₂²	1	24	yes	complemented
43	#537×C₈₉	C₂²	1	3	yes	complemented
44	#537×C₈₉	C₂²	1	12	yes	complemented
45	#535×C₈₉	C₂²	1	6	yes	complemented
46	#535×C₈₉	C₂²	1	12	yes	complemented
47	#535×C₈₉	C₂²	1	24	yes	complemented
48	#535×C₈₉	C₂²	1	24	yes	complemented
49	#534×C₈₉	C₂²	1	12	yes	complemented
50	#534×C₈₉	C₂²	1	24	yes	complemented
51	#532×C₈₉	C₂²	1	12	yes	complemented
52	#532×C₈₉	C₂²	1	6	yes	complemented
53	#532×C₈₉	C₂²	1	12	yes	complemented
54	C₂×#127×C₈₉	C₂²	1	3	yes	complemented
55	C₂×#127×C₈₉	C₂²	1	12	yes	complemented
56	#529×C₈₉	C₂²	1	12	yes	complemented
57	#529×C₈₉	C₂²	1	12	yes	complemented
58	#525×C₈₉	C₂²	1	12	yes	complemented
59	C₂×#121×C₈₉	C₂²	1	6	yes	complemented
60	C₂×#121×C₈₉	C₂²	1	12	yes	complemented
61	#520×C₈₉	C₂²	1	4	yes	complemented
62	C₄×#41×C₈₉	C₂²	1	6	yes	complemented
63	C₄×#41×C₈₉	C₂²	1	4	yes	complemented
64	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
65	C₂×C₄×D₄×C₈₉	C₂²	1	3	yes	complemented
66	H(5,𝔽₂)×C₈₉	—	4	24	no	—
67	H(5,𝔽₂)×C₈₉	—	4	16	no	—
68	C₂×#41×C₈₉	—	4	8	no	—
69	C₂×#41×C₈₉	C₂³	1	6	yes	—
70	C₂×#41×C₈₉	C₂³	1	4	yes	—
71	C₂×#41×C₈₉	C₂³	1	4	yes	—
72	C₂×#41×C₈₉	C₂³	1	12	yes	—
73	C₂×#41×C₈₉	C₂³	1	24	yes	—
74	C₂×#41×C₈₉	C₂³	1	12	yes	—
75	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂²×D₄×C₈₉	C₂³	1	3	yes	—
77	C₂²×D₄×C₈₉	C₂³	1	6	yes	—
78	C₂²×D₄×C₈₉	C₂³	1	24	yes	—
79	C₂²×D₄×C₈₉	C₂³	1	12	yes	—
80	C₂²×D₄×C₈₉	C₂³	1	24	yes	—
81	C₂²×D₄×C₈₉	C₂³	1	24	yes	—
82	C₂²×D₄×C₈₉	C₂³	1	6	yes	—
83	C₂²×D₄×C₈₉	C₂³	1	24	yes	—
84	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
85	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
86	C₄⋊Q₈×C₈₉	C₂³	1	6	yes	—
87	#127×C₈₉	C₂³	1	12	yes	—
88	#127×C₈₉	C₂³	1	6	yes	—
89	#127×C₈₉	C₂³	1	12	yes	—
90	#127×C₈₉	C₂³	1	12	yes	—
91	#127×C₈₉	C₂³	1	24	yes	—
92	#127×C₈₉	C₂³	1	24	yes	—
93	#126×C₈₉	C₂³	1	24	yes	—
94	#126×C₈₉	C₂³	1	24	yes	—
95	#125×C₈₉	C₂³	1	4	yes	—
96	#125×C₈₉	C₂³	1	12	yes	—
97	#124×C₈₉	C₂³	1	12	yes	—
98	#124×C₈₉	C₂³	1	12	yes	—
99	#124×C₈₉	C₂³	1	24	yes	—
100	#124×C₈₉	C₂³	1	12	yes	—
101	#123×C₈₉	C₂³	1	24	yes	—
102	#123×C₈₉	C₂³	1	24	yes	—
103	#123×C₈₉	C₂³	1	24	yes	—
104	#123×C₈₉	C₂³	1	48	yes	—
105	C₂²⋊Q₈×C₈₉	C₂³	1	4	yes	—
106	C₂²⋊Q₈×C₈₉	C₂³	1	12	yes	—
107	C₂²⋊Q₈×C₈₉	C₂³	1	12	yes	—
108	C₂²⋊Q₈×C₈₉	C₂³	1	12	yes	—
109	#121×C₈₉	C₂³	1	12	yes	—
110	#121×C₈₉	C₂³	1	12	yes	—
111	#121×C₈₉	C₂³	1	24	yes	—
112	#121×C₈₉	C₂³	1	24	yes	—
113	#121×C₈₉	C₂³	1	24	yes	—
114	#121×C₈₉	C₂³	1	24	yes	—
115	#121×C₈₉	C₂³	1	12	yes	—
116	#121×C₈₉	C₂³	1	12	yes	—
117	#121×C₈₉	C₂³	1	12	yes	—
118	#121×C₈₉	C₂³	1	12	yes	—
119	#121×C₈₉	C₂³	1	24	yes	—
120	#121×C₈₉	C₂³	1	12	yes	—
121	#121×C₈₉	C₂³	1	12	yes	—
122	#121×C₈₉	C₂³	1	24	yes	—
123	#121×C₈₉	C₂³	1	24	yes	—
124	#121×C₈₉	C₂³	1	48	yes	—
125	#121×C₈₉	C₂³	1	48	yes	—
126	#120×C₈₉	C₂³	1	24	yes	—
127	#120×C₈₉	C₂³	1	48	yes	—
128	#120×C₈₉	C₂³	1	48	yes	—
129	#120×C₈₉	C₂³	1	48	yes	—
130	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
131	C₄×Q₈×C₈₉	C₂³	1	6	yes	—
132	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
133	C₄×Q₈×C₈₉	C₂³	1	6	yes	—
134	C₄×Q₈×C₈₉	C₂³	1	6	yes	—
135	C₄×D₄×C₈₉	C₂³	1	12	yes	—
136	C₄×D₄×C₈₉	C₂³	1	12	yes	—
137	C₄×D₄×C₈₉	C₂³	1	12	yes	—
138	C₄×D₄×C₈₉	C₂³	1	12	yes	—
139	C₄×D₄×C₈₉	C₂³	1	6	yes	—
140	C₄×D₄×C₈₉	C₂³	1	12	yes	—
141	C₄×D₄×C₈₉	C₂³	1	12	yes	—
142	C₄×D₄×C₈₉	C₂³	1	6	yes	—
143	C₄×D₄×C₈₉	C₂³	1	24	yes	—
144	C₄×D₄×C₈₉	C₂³	1	24	yes	—
145	C₄×D₄×C₈₉	C₂³	1	12	yes	—
146	C₄×D₄×C₈₉	C₂³	1	24	yes	—
147	#117×C₈₉	C₂³	1	2	yes	—
148	#117×C₈₉	C₂³	1	12	yes	—
149	#117×C₈₉	C₂³	1	12	yes	—
150	#117×C₈₉	C₂³	1	12	yes	—
151	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
152	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
153	C₂×C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
154	C₂×C₂²⋊C₄×C₈₉	C₂³	1	6	yes	—
155	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	—
156	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
157	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
158	C₂⁴×C₈₉	C₂⁴	1	6	yes	—
159	C₂⁴×C₈₉	C₂⁴	1	24	yes	—
160	#41×C₈₉	—	2	16	no	—
161	#41×C₈₉	—	2	16	no	—
162	#41×C₈₉	—	2	24	no	—
163	#41×C₈₉	—	4	24	no	—
164	#41×C₈₉	—	4	24	no	—
165	#41×C₈₉	—	4	48	no	—
166	C₂×Q₈×C₈₉	—	2	4	no	—
167	C₂×Q₈×C₈₉	C₂⁴	1	3	yes	—
168	C₂×Q₈×C₈₉	C₂⁴	1	6	yes	—
169	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
170	C₂×Q₈×C₈₉	C₂⁴	1	1	yes	characteristic
171	C₂×D₄×C₈₉	—	2	12	no	—
172	C₂×D₄×C₈₉	—	4	12	no	—
173	C₂×D₄×C₈₉	—	4	24	no	—
174	C₂×D₄×C₈₉	—	4	48	no	—
175	C₂×D₄×C₈₉	—	4	48	no	—
176	C₂×D₄×C₈₉	—	4	48	no	—
177	C₂×D₄×C₈₉	—	8	6	no	—
178	C₂×D₄×C₈₉	—	8	24	no	—
179	C₂×D₄×C₈₉	C₂⁴	1	12	yes	—
180	C₂×D₄×C₈₉	C₂⁴	1	12	yes	—
181	C₂×D₄×C₈₉	C₂⁴	1	12	yes	—
182	C₂×D₄×C₈₉	C₂⁴	1	6	yes	—
183	C₂×D₄×C₈₉	C₂⁴	1	12	yes	—
184	C₂×D₄×C₈₉	C₂⁴	1	12	yes	—
185	C₂×D₄×C₈₉	C₂⁴	1	24	yes	—
186	C₂×D₄×C₈₉	C₂⁴	1	24	yes	—
187	C₂×D₄×C₈₉	C₂⁴	1	12	yes	—
188	C₂×D₄×C₈₉	C₂⁴	1	12	yes	—
189	C₂×D₄×C₈₉	C₂⁴	1	48	yes	—
190	C₂×D₄×C₈₉	C₂⁴	1	24	yes	—
191	C₂²×C₄×C₈₉	—	4	12	no	—
192	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
193	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
194	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
195	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
196	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
197	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
198	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
200	C₂²×C₄×C₈₉	C₂⁴	1	24	yes	—
201	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
202	C₄⋊C₄×C₈₉	C₂⁴	1	12	yes	—
203	C₄⋊C₄×C₈₉	C₂⁴	1	12	yes	—
204	C₄⋊C₄×C₈₉	C₂⁴	1	6	yes	—
205	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
206	C₄⋊C₄×C₈₉	C₂⁴	1	6	yes	—
207	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
208	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
209	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
210	C₄⋊C₄×C₈₉	C₂⁴	1	12	yes	—
211	C₄⋊C₄×C₈₉	C₂⁴	1	12	yes	—
212	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
213	C₄⋊C₄×C₈₉	C₂⁴	1	12	yes	—
214	C₂²⋊C₄×C₈₉	C₂⁴	1	24	yes	—
215	C₂²⋊C₄×C₈₉	C₂⁴	1	24	yes	—
216	C₂²⋊C₄×C₈₉	C₂⁴	1	24	yes	—
217	C₂²⋊C₄×C₈₉	C₂⁴	1	4	yes	—
218	C₂²⋊C₄×C₈₉	C₂⁴	1	12	yes	—
219	C₂²⋊C₄×C₈₉	C₂⁴	1	24	yes	—
220	C₂²⋊C₄×C₈₉	C₂⁴	1	12	yes	—
221	C₂²⋊C₄×C₈₉	C₂⁴	1	24	yes	—
222	C₂²⋊C₄×C₈₉	C₂⁴	1	24	yes	—
223	C₄²×C₈₉	C₂⁴	1	4	yes	—
224	C₄²×C₈₉	C₂⁴	1	6	yes	—
225	C₄²×C₈₉	C₂⁴	1	12	yes	—
226	C₄²×C₈₉	C₂⁴	1	6	yes	—
227	C₄²×C₈₉	C₂⁴	1	12	yes	—
228	C₂³×C₈₉	—	2	12	no	—
229	C₂³×C₈₉	—	4	6	no	—
230	C₂³×C₈₉	—	4	6	no	—
231	C₂³×C₈₉	—	4	24	no	—
232	C₂³×C₈₉	—	4	24	no	—
233	C₂³×C₈₉	—	4	24	no	—
234	C₂³×C₈₉	C₂⁵	1	12	yes	—
235	C₂³×C₈₉	C₂⁵	1	12	yes	—
236	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
237	Q₈×C₈₉	—	2	6	no	—
238	Q₈×C₈₉	—	2	12	no	—
239	Q₈×C₈₉	—	4	4	no	—
240	Q₈×C₈₉	H(5,𝔽₂)	1	4	yes	complemented
241	D₄×C₈₉	—	2	24	no	—
242	D₄×C₈₉	—	2	24	no	—
243	D₄×C₈₉	—	2	24	no	—
244	D₄×C₈₉	—	2	12	no	—
245	D₄×C₈₉	—	2	24	no	—
246	D₄×C₈₉	—	4	12	no	—
247	D₄×C₈₉	—	4	24	no	—
248	D₄×C₈₉	—	4	48	no	—
249	D₄×C₈₉	—	4	12	no	—
250	D₄×C₈₉	—	4	12	no	—
251	D₄×C₈₉	—	4	24	no	—
252	D₄×C₈₉	—	4	24	no	—
253	C₂×C₄×C₈₉	—	2	6	no	—
254	C₂×C₄×C₈₉	—	2	24	no	—
255	C₂×C₄×C₈₉	—	2	24	no	—
256	C₂×C₄×C₈₉	—	2	24	no	—
257	C₂×C₄×C₈₉	—	4	2	no	—
258	C₂×C₄×C₈₉	—	4	12	no	—
259	C₂×C₄×C₈₉	—	4	24	no	—
260	C₂×C₄×C₈₉	—	4	3	no	—
261	C₂×C₄×C₈₉	—	4	12	no	—
262	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	complemented
263	C₂×C₄×C₈₉	C₂⁵	1	4	yes	—
264	C₂×C₄×C₈₉	C₂⁵	1	6	yes	—
265	C₂×C₄×C₈₉	C₂⁵	1	4	yes	—
266	C₂×C₄×C₈₉	C₂⁵	1	3	yes	—
267	C₂×C₄×C₈₉	C₂⁵	1	12	yes	—
268	C₂×C₄×C₈₉	C₂⁵	1	4	yes	—
269	C₂×C₄×C₈₉	C₂⁵	1	2	yes	—
270	C₂×C₄×C₈₉	C₂⁵	1	3	yes	—
271	C₂²×C₈₉	—	2	1	no	—
272	C₂²×C₈₉	—	2	1	no	—
273	C₂²×C₈₉	—	2	12	no	—
274	C₂²×C₈₉	—	2	12	no	—
275	C₂²×C₈₉	—	2	12	no	—
276	C₂²×C₈₉	—	2	12	no	—
277	C₂²×C₈₉	—	2	12	no	—
278	C₂²×C₈₉	—	2	12	no	—
279	C₂²×C₈₉	—	8	12	no	—
280	C₂²×C₈₉	—	16	24	no	—
281	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
282	C₂²×C₈₉	C₂⁶	1	1	yes	characteristic
283	C₄×C₈₉	—	2	4	no	—
284	C₄×C₈₉	—	2	4	no	—
285	C₄×C₈₉	—	2	12	no	—
286	C₄×C₈₉	—	2	3	no	—
287	C₄×C₈₉	—	2	6	no	—
288	C₄×C₈₉	—	2	4	no	—
289	C₄×C₈₉	C₂×H(5,𝔽₂)	1	6	yes	—
290	C₄×C₈₉	C₂×H(5,𝔽₂)	1	4	yes	—
291	#62916	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂×C₈₉	—	2	2	no	—
293	C₂×C₈₉	—	4	12	no	—
294	C₂×C₈₉	—	4	12	no	—
295	C₂×C₈₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
296	C₂×C₈₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
297	C₂×C₈₉	H(7,𝔽₂)	1	1	yes	characteristic
298	#3546	C₂×C₈₉	1	12	yes	complemented
299	#3543	C₂×C₈₉	1	12	yes	complemented
300	#3537	C₂×C₈₉	1	12	yes	complemented
301	#3517	C₂×C₈₉	1	4	yes	complemented
302	#3498	C₂×C₈₉	1	6	yes	complemented
303	#3498	C₂×C₈₉	1	4	yes	complemented
304	#3496	C₂×C₈₉	1	3	yes	complemented
305	#3488	C₂×C₈₉	1	4	yes	complemented
306	#3480	C₂×C₈₉	1	2	yes	complemented
307	C₂×#550	C₂×C₈₉	1	1	yes	characteristic, complemented
308	#3460	C₂×C₈₉	1	3	yes	complemented
309	C₈₉	#62916	1	1	yes	characteristic, direct product factor, Sylow
310	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	4	yes	complemented
311	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	6	yes	complemented
312	C₂²×#41	C₂²×C₈₉	1	2	yes	complemented
313	#560	C₂²×C₈₉	1	48	yes	complemented
314	#559	C₂²×C₈₉	1	6	yes	complemented
315	#557	C₂²×C₈₉	1	2	yes	complemented
316	#556	C₂²×C₈₉	1	6	yes	complemented
317	#553	C₂²×C₈₉	1	12	yes	complemented
318	#553	C₂²×C₈₉	1	24	yes	complemented
319	#550	C₂²×C₈₉	1	4	yes	complemented
320	#550	C₂²×C₈₉	1	12	yes	complemented
321	#550	C₂²×C₈₉	1	2	yes	complemented
322	#550	C₂²×C₈₉	1	8	yes	complemented
323	#550	C₂²×C₈₉	1	12	yes	complemented
324	#550	C₂²×C₈₉	1	12	yes	complemented
325	#548	C₂²×C₈₉	1	4	yes	complemented
326	#547	C₂²×C₈₉	1	12	yes	complemented
327	#546	C₂²×C₈₉	1	24	yes	complemented
328	#546	C₂²×C₈₉	1	24	yes	complemented
329	D₄²	C₂²×C₈₉	1	24	yes	complemented
330	D₄²	C₂²×C₈₉	1	12	yes	complemented
331	D₄²	C₂²×C₈₉	1	12	yes	complemented
332	D₄²	C₂²×C₈₉	1	24	yes	complemented
333	D₄²	C₂²×C₈₉	1	24	yes	complemented
334	D₄²	C₂²×C₈₉	1	12	yes	complemented
335	#544	C₂²×C₈₉	1	1	yes	characteristic, complemented
336	#542	C₂²×C₈₉	1	12	yes	complemented
337	#540	C₂²×C₈₉	1	12	yes	complemented
338	#540	C₂²×C₈₉	1	24	yes	complemented
339	#538	C₂²×C₈₉	1	12	yes	complemented
340	#538	C₂²×C₈₉	1	24	yes	complemented
341	#537	C₂²×C₈₉	1	3	yes	complemented
342	#537	C₂²×C₈₉	1	12	yes	complemented
343	#535	C₂²×C₈₉	1	6	yes	complemented
344	#535	C₂²×C₈₉	1	12	yes	complemented
345	#535	C₂²×C₈₉	1	24	yes	complemented
346	#535	C₂²×C₈₉	1	24	yes	complemented
347	#534	C₂²×C₈₉	1	12	yes	complemented
348	#534	C₂²×C₈₉	1	24	yes	complemented
349	#532	C₂²×C₈₉	1	12	yes	complemented
350	#532	C₂²×C₈₉	1	6	yes	complemented
351	#532	C₂²×C₈₉	1	12	yes	complemented
352	C₂×#127	C₂²×C₈₉	1	3	yes	complemented
353	C₂×#127	C₂²×C₈₉	1	12	yes	complemented
354	#529	C₂²×C₈₉	1	12	yes	complemented
355	#529	C₂²×C₈₉	1	12	yes	complemented
356	#525	C₂²×C₈₉	1	12	yes	complemented
357	C₂×#121	C₂²×C₈₉	1	6	yes	complemented
358	C₂×#121	C₂²×C₈₉	1	12	yes	complemented
359	#520	C₂²×C₈₉	1	4	yes	complemented
360	C₄×#41	C₂²×C₈₉	1	6	yes	complemented
361	C₄×#41	C₂²×C₈₉	1	4	yes	complemented
362	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
363	C₂×C₄×D₄	C₂²×C₈₉	1	3	yes	complemented
364	H(5,𝔽₂)	—	4	24	no	—
365	H(5,𝔽₂)	—	4	16	no	—
366	C₂×#41	—	4	8	no	—
367	C₂×#41	C₂³×C₈₉	1	6	yes	—
368	C₂×#41	C₂³×C₈₉	1	4	yes	—
369	C₂×#41	C₂³×C₈₉	1	4	yes	—
370	C₂×#41	C₂³×C₈₉	1	12	yes	—
371	C₂×#41	C₂³×C₈₉	1	24	yes	—
372	C₂×#41	C₂³×C₈₉	1	12	yes	—
373	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
374	C₂²×D₄	C₂³×C₈₉	1	3	yes	—
375	C₂²×D₄	C₂³×C₈₉	1	6	yes	—
376	C₂²×D₄	C₂³×C₈₉	1	24	yes	—
377	C₂²×D₄	C₂³×C₈₉	1	12	yes	—
378	C₂²×D₄	C₂³×C₈₉	1	24	yes	—
379	C₂²×D₄	C₂³×C₈₉	1	24	yes	—
380	C₂²×D₄	C₂³×C₈₉	1	6	yes	—
381	C₂²×D₄	C₂³×C₈₉	1	24	yes	—
382	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
383	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
384	C₄⋊Q₈	C₂³×C₈₉	1	6	yes	—
385	#127	C₂³×C₈₉	1	12	yes	—
386	#127	C₂³×C₈₉	1	6	yes	—
387	#127	C₂³×C₈₉	1	12	yes	—
388	#127	C₂³×C₈₉	1	12	yes	—
389	#127	C₂³×C₈₉	1	24	yes	—
390	#127	C₂³×C₈₉	1	24	yes	—
391	#126	C₂³×C₈₉	1	24	yes	—
392	#126	C₂³×C₈₉	1	24	yes	—
393	#125	C₂³×C₈₉	1	4	yes	—
394	#125	C₂³×C₈₉	1	12	yes	—
395	#124	C₂³×C₈₉	1	12	yes	—
396	#124	C₂³×C₈₉	1	12	yes	—
397	#124	C₂³×C₈₉	1	24	yes	—
398	#124	C₂³×C₈₉	1	12	yes	—
399	#123	C₂³×C₈₉	1	24	yes	—
400	#123	C₂³×C₈₉	1	24	yes	—
401	#123	C₂³×C₈₉	1	24	yes	—
402	#123	C₂³×C₈₉	1	48	yes	—
403	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
404	C₂²⋊Q₈	C₂³×C₈₉	1	12	yes	—
405	C₂²⋊Q₈	C₂³×C₈₉	1	12	yes	—
406	C₂²⋊Q₈	C₂³×C₈₉	1	12	yes	—
407	#121	C₂³×C₈₉	1	12	yes	—
408	#121	C₂³×C₈₉	1	12	yes	—
409	#121	C₂³×C₈₉	1	24	yes	—
410	#121	C₂³×C₈₉	1	24	yes	—
411	#121	C₂³×C₈₉	1	24	yes	—
412	#121	C₂³×C₈₉	1	24	yes	—
413	#121	C₂³×C₈₉	1	12	yes	—
414	#121	C₂³×C₈₉	1	12	yes	—
415	#121	C₂³×C₈₉	1	12	yes	—
416	#121	C₂³×C₈₉	1	12	yes	—
417	#121	C₂³×C₈₉	1	24	yes	—
418	#121	C₂³×C₈₉	1	12	yes	—
419	#121	C₂³×C₈₉	1	12	yes	—
420	#121	C₂³×C₈₉	1	24	yes	—
421	#121	C₂³×C₈₉	1	24	yes	—
422	#121	C₂³×C₈₉	1	48	yes	—
423	#121	C₂³×C₈₉	1	48	yes	—
424	#120	C₂³×C₈₉	1	24	yes	—
425	#120	C₂³×C₈₉	1	48	yes	—
426	#120	C₂³×C₈₉	1	48	yes	—
427	#120	C₂³×C₈₉	1	48	yes	—
428	C₄×Q₈	C₂³×C₈₉	1	2	yes	—
429	C₄×Q₈	C₂³×C₈₉	1	6	yes	—
430	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
431	C₄×Q₈	C₂³×C₈₉	1	6	yes	—
432	C₄×Q₈	C₂³×C₈₉	1	6	yes	—
433	C₄×D₄	C₂³×C₈₉	1	12	yes	—
434	C₄×D₄	C₂³×C₈₉	1	12	yes	—
435	C₄×D₄	C₂³×C₈₉	1	12	yes	—
436	C₄×D₄	C₂³×C₈₉	1	12	yes	—
437	C₄×D₄	C₂³×C₈₉	1	6	yes	—
438	C₄×D₄	C₂³×C₈₉	1	12	yes	—
439	C₄×D₄	C₂³×C₈₉	1	12	yes	—
440	C₄×D₄	C₂³×C₈₉	1	6	yes	—
441	C₄×D₄	C₂³×C₈₉	1	24	yes	—
442	C₄×D₄	C₂³×C₈₉	1	24	yes	—
443	C₄×D₄	C₂³×C₈₉	1	12	yes	—
444	C₄×D₄	C₂³×C₈₉	1	24	yes	—
445	#117	C₂³×C₈₉	1	2	yes	—
446	#117	C₂³×C₈₉	1	12	yes	—
447	#117	C₂³×C₈₉	1	12	yes	—
448	#117	C₂³×C₈₉	1	12	yes	—
449	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
451	C₂×C₄⋊C₄	C₂³×C₈₉	1	12	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₈₉	1	6	yes	—
453	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	—
454	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
455	C₂×C₄²	C₂³×C₈₉	1	12	yes	—
456	C₂⁴	C₂⁴×C₈₉	1	6	yes	—
457	C₂⁴	C₂⁴×C₈₉	1	24	yes	—
458	#41	—	2	16	no	—
459	#41	—	2	16	no	—
460	#41	—	2	24	no	—
461	#41	—	4	24	no	—
462	#41	—	4	24	no	—
463	#41	—	4	48	no	—
464	C₂×Q₈	—	2	4	no	—
465	C₂×Q₈	C₂⁴×C₈₉	1	3	yes	—
466	C₂×Q₈	C₂⁴×C₈₉	1	6	yes	—
467	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
468	C₂×Q₈	C₂⁴×C₈₉	1	1	yes	characteristic
469	C₂×D₄	—	2	12	no	—
470	C₂×D₄	—	4	12	no	—
471	C₂×D₄	—	4	24	no	—
472	C₂×D₄	—	4	48	no	—
473	C₂×D₄	—	4	48	no	—
474	C₂×D₄	—	4	48	no	—
475	C₂×D₄	—	8	6	no	—
476	C₂×D₄	—	8	24	no	—
477	C₂×D₄	C₂⁴×C₈₉	1	12	yes	—
478	C₂×D₄	C₂⁴×C₈₉	1	12	yes	—
479	C₂×D₄	C₂⁴×C₈₉	1	12	yes	—
480	C₂×D₄	C₂⁴×C₈₉	1	6	yes	—
481	C₂×D₄	C₂⁴×C₈₉	1	12	yes	—
482	C₂×D₄	C₂⁴×C₈₉	1	12	yes	—
483	C₂×D₄	C₂⁴×C₈₉	1	24	yes	—
484	C₂×D₄	C₂⁴×C₈₉	1	24	yes	—
485	C₂×D₄	C₂⁴×C₈₉	1	12	yes	—
486	C₂×D₄	C₂⁴×C₈₉	1	12	yes	—
487	C₂×D₄	C₂⁴×C₈₉	1	48	yes	—
488	C₂×D₄	C₂⁴×C₈₉	1	24	yes	—
489	C₂²×C₄	—	4	12	no	—
490	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
491	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
492	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
493	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
494	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
495	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
496	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
498	C₂²×C₄	C₂⁴×C₈₉	1	24	yes	—
499	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
500	C₄⋊C₄	C₂⁴×C₈₉	1	12	yes	—
501	C₄⋊C₄	C₂⁴×C₈₉	1	12	yes	—
502	C₄⋊C₄	C₂⁴×C₈₉	1	6	yes	—
503	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
504	C₄⋊C₄	C₂⁴×C₈₉	1	6	yes	—
505	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
506	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
507	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
508	C₄⋊C₄	C₂⁴×C₈₉	1	12	yes	—
509	C₄⋊C₄	C₂⁴×C₈₉	1	12	yes	—
510	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
511	C₄⋊C₄	C₂⁴×C₈₉	1	12	yes	—
512	C₂²⋊C₄	C₂⁴×C₈₉	1	24	yes	—
513	C₂²⋊C₄	C₂⁴×C₈₉	1	24	yes	—
514	C₂²⋊C₄	C₂⁴×C₈₉	1	24	yes	—
515	C₂²⋊C₄	C₂⁴×C₈₉	1	4	yes	—
516	C₂²⋊C₄	C₂⁴×C₈₉	1	12	yes	—
517	C₂²⋊C₄	C₂⁴×C₈₉	1	24	yes	—
518	C₂²⋊C₄	C₂⁴×C₈₉	1	12	yes	—
519	C₂²⋊C₄	C₂⁴×C₈₉	1	24	yes	—
520	C₂²⋊C₄	C₂⁴×C₈₉	1	24	yes	—
521	C₄²	C₂⁴×C₈₉	1	4	yes	—
522	C₄²	C₂⁴×C₈₉	1	6	yes	—
523	C₄²	C₂⁴×C₈₉	1	12	yes	—
524	C₄²	C₂⁴×C₈₉	1	6	yes	—
525	C₄²	C₂⁴×C₈₉	1	12	yes	—
526	C₂³	—	2	12	no	—
527	C₂³	—	4	6	no	—
528	C₂³	—	4	6	no	—
529	C₂³	—	4	24	no	—
530	C₂³	—	4	24	no	—
531	C₂³	—	4	24	no	—
532	C₂³	C₂⁵×C₈₉	1	12	yes	—
533	C₂³	C₂⁵×C₈₉	1	12	yes	—
534	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
535	Q₈	—	2	6	no	—
536	Q₈	—	2	12	no	—
537	Q₈	—	4	4	no	—
538	Q₈	H(5,𝔽₂)×C₈₉	1	4	yes	complemented
539	D₄	—	2	24	no	—
540	D₄	—	2	24	no	—
541	D₄	—	2	24	no	—
542	D₄	—	2	12	no	—
543	D₄	—	2	24	no	—
544	D₄	—	4	12	no	—
545	D₄	—	4	24	no	—
546	D₄	—	4	48	no	—
547	D₄	—	4	12	no	—
548	D₄	—	4	12	no	—
549	D₄	—	4	24	no	—
550	D₄	—	4	24	no	—
551	C₂×C₄	—	2	6	no	—
552	C₂×C₄	—	2	24	no	—
553	C₂×C₄	—	2	24	no	—
554	C₂×C₄	—	2	24	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	12	no	—
557	C₂×C₄	—	4	24	no	—
558	C₂×C₄	—	4	3	no	—
559	C₂×C₄	—	4	12	no	—
560	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	complemented
561	C₂×C₄	C₂⁵×C₈₉	1	4	yes	—
562	C₂×C₄	C₂⁵×C₈₉	1	6	yes	—
563	C₂×C₄	C₂⁵×C₈₉	1	4	yes	—
564	C₂×C₄	C₂⁵×C₈₉	1	3	yes	—
565	C₂×C₄	C₂⁵×C₈₉	1	12	yes	—
566	C₂×C₄	C₂⁵×C₈₉	1	4	yes	—
567	C₂×C₄	C₂⁵×C₈₉	1	2	yes	—
568	C₂×C₄	C₂⁵×C₈₉	1	3	yes	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	12	no	—
572	C₂²	—	2	12	no	—
573	C₂²	—	2	12	no	—
574	C₂²	—	2	12	no	—
575	C₂²	—	2	12	no	—
576	C₂²	—	2	12	no	—
577	C₂²	—	8	12	no	—
578	C₂²	—	16	24	no	—
579	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
580	C₂²	C₂⁶×C₈₉	1	1	yes	characteristic
581	C₄	—	2	4	no	—
582	C₄	—	2	4	no	—
583	C₄	—	2	12	no	—
584	C₄	—	2	3	no	—
585	C₄	—	2	6	no	—
586	C₄	—	2	4	no	—
587	C₄	C₂×H(5,𝔽₂)×C₈₉	1	6	yes	—
588	C₄	C₂×H(5,𝔽₂)×C₈₉	1	4	yes	—
589	C₂	—	2	2	no	—
590	C₂	—	4	12	no	—
591	C₂	—	4	12	no	—
592	C₂	C₂²×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
593	C₂	C₂²×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
594	C₂	C₈₉×H(7,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁶×C₈₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₈₉×#3546	C₂	semidirect
2. C₈₉×#3543	C₂	semidirect
3. C₈₉×#3537	C₂	semidirect
4. C₈₉×#3517	C₂	semidirect
5. C₈₉×#3498	C₂	semidirect
6. C₈₉×#3498	C₂	semidirect
7. C₈₉×#3496	C₂	semidirect
8. C₈₉×#3488	C₂	semidirect
9. C₈₉×#3480	C₂	semidirect
10. C₂×#550×C₈₉	C₂	semidirect
11. C₈₉×#3460	C₂	semidirect
12. C₂×H(5,𝔽₂)×C₈₉	C₂²	semidirect
13. C₂×H(5,𝔽₂)×C₈₉	C₂²	semidirect
14. C₂²×#41×C₈₉	C₂²	semidirect
15. #560×C₈₉	C₂²	semidirect
16. #559×C₈₉	C₂²	semidirect
17. #557×C₈₉	C₂²	semidirect
18. #556×C₈₉	C₂²	semidirect
19. #553×C₈₉	C₂²	semidirect
20. #553×C₈₉	C₂²	semidirect
21. #550×C₈₉	C₂²	semidirect
22. #550×C₈₉	C₂²	semidirect
23. #550×C₈₉	C₂²	semidirect
24. #550×C₈₉	C₂²	semidirect
25. #550×C₈₉	C₂²	semidirect
26. #550×C₈₉	C₂²	semidirect
27. #548×C₈₉	C₂²	semidirect
28. #547×C₈₉	C₂²	semidirect
29. #546×C₈₉	C₂²	semidirect
30. #546×C₈₉	C₂²	semidirect
31. D₄²×C₈₉	C₂²	semidirect
32. D₄²×C₈₉	C₂²	semidirect
33. D₄²×C₈₉	C₂²	semidirect
34. D₄²×C₈₉	C₂²	semidirect
35. D₄²×C₈₉	C₂²	semidirect
36. D₄²×C₈₉	C₂²	semidirect
37. #544×C₈₉	C₂²	semidirect
38. #542×C₈₉	C₂²	semidirect
39. #540×C₈₉	C₂²	semidirect
40. #540×C₈₉	C₂²	semidirect
41. #538×C₈₉	C₂²	semidirect
42. #538×C₈₉	C₂²	semidirect
43. #537×C₈₉	C₂²	semidirect
44. #537×C₈₉	C₂²	semidirect
45. #535×C₈₉	C₂²	semidirect
46. #535×C₈₉	C₂²	semidirect
47. #535×C₈₉	C₂²	semidirect
48. #535×C₈₉	C₂²	semidirect
49. #534×C₈₉	C₂²	semidirect
50. #534×C₈₉	C₂²	semidirect
51. #532×C₈₉	C₂²	semidirect
52. #532×C₈₉	C₂²	semidirect
53. #532×C₈₉	C₂²	semidirect
54. C₂×#127×C₈₉	C₂²	semidirect
55. C₂×#127×C₈₉	C₂²	semidirect
56. #529×C₈₉	C₂²	semidirect
57. #529×C₈₉	C₂²	semidirect
58. #525×C₈₉	C₂²	semidirect
59. C₂×#121×C₈₉	C₂²	semidirect
60. C₂×#121×C₈₉	C₂²	semidirect
61. #520×C₈₉	C₂²	semidirect
62. C₄×#41×C₈₉	C₂²	semidirect
63. C₄×#41×C₈₉	C₂²	semidirect
64. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
65. C₂×C₄×D₄×C₈₉	C₂²	semidirect
240. Q₈×C₈₉	H(5,𝔽₂)	semidirect
262. C₂×C₄×C₈₉	H(5,𝔽₂)	semidirect
291. #62916	C₈₉	direct (N × H)
298. #3546	C₂×C₈₉	semidirect
299. #3543	C₂×C₈₉	semidirect
300. #3537	C₂×C₈₉	semidirect
301. #3517	C₂×C₈₉	semidirect
302. #3498	C₂×C₈₉	semidirect
303. #3498	C₂×C₈₉	semidirect
304. #3496	C₂×C₈₉	semidirect
305. #3488	C₂×C₈₉	semidirect
306. #3480	C₂×C₈₉	semidirect
307. C₂×#550	C₂×C₈₉	semidirect
308. #3460	C₂×C₈₉	semidirect
309. C₈₉	#62916	direct (N × H)
310. C₂×H(5,𝔽₂)	C₂²×C₈₉	semidirect
311. C₂×H(5,𝔽₂)	C₂²×C₈₉	semidirect
312. C₂²×#41	C₂²×C₈₉	semidirect
313. #560	C₂²×C₈₉	semidirect
314. #559	C₂²×C₈₉	semidirect
315. #557	C₂²×C₈₉	semidirect
316. #556	C₂²×C₈₉	semidirect
317. #553	C₂²×C₈₉	semidirect
318. #553	C₂²×C₈₉	semidirect
319. #550	C₂²×C₈₉	semidirect
320. #550	C₂²×C₈₉	semidirect
321. #550	C₂²×C₈₉	semidirect
322. #550	C₂²×C₈₉	semidirect
323. #550	C₂²×C₈₉	semidirect
324. #550	C₂²×C₈₉	semidirect
325. #548	C₂²×C₈₉	semidirect
326. #547	C₂²×C₈₉	semidirect
327. #546	C₂²×C₈₉	semidirect
328. #546	C₂²×C₈₉	semidirect
329. D₄²	C₂²×C₈₉	semidirect
330. D₄²	C₂²×C₈₉	semidirect
331. D₄²	C₂²×C₈₉	semidirect
332. D₄²	C₂²×C₈₉	semidirect
333. D₄²	C₂²×C₈₉	semidirect
334. D₄²	C₂²×C₈₉	semidirect
335. #544	C₂²×C₈₉	semidirect
336. #542	C₂²×C₈₉	semidirect
337. #540	C₂²×C₈₉	semidirect
338. #540	C₂²×C₈₉	semidirect
339. #538	C₂²×C₈₉	semidirect
340. #538	C₂²×C₈₉	semidirect
341. #537	C₂²×C₈₉	semidirect
342. #537	C₂²×C₈₉	semidirect
343. #535	C₂²×C₈₉	semidirect
344. #535	C₂²×C₈₉	semidirect
345. #535	C₂²×C₈₉	semidirect
346. #535	C₂²×C₈₉	semidirect
347. #534	C₂²×C₈₉	semidirect
348. #534	C₂²×C₈₉	semidirect
349. #532	C₂²×C₈₉	semidirect
350. #532	C₂²×C₈₉	semidirect
351. #532	C₂²×C₈₉	semidirect
352. C₂×#127	C₂²×C₈₉	semidirect
353. C₂×#127	C₂²×C₈₉	semidirect
354. #529	C₂²×C₈₉	semidirect
355. #529	C₂²×C₈₉	semidirect
356. #525	C₂²×C₈₉	semidirect
357. C₂×#121	C₂²×C₈₉	semidirect
358. C₂×#121	C₂²×C₈₉	semidirect
359. #520	C₂²×C₈₉	semidirect
360. C₄×#41	C₂²×C₈₉	semidirect
361. C₄×#41	C₂²×C₈₉	semidirect
362. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
363. C₂×C₄×D₄	C₂²×C₈₉	semidirect
538. Q₈	H(5,𝔽₂)×C₈₉	semidirect
560. C₂×C₄	H(5,𝔽₂)×C₈₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₈₉×#3546	2
C₈₉×#3543	2
C₈₉×#3537	2
C₈₉×#3517	2
C₈₉×#3498	2
C₈₉×#3496	2
C₈₉×#3488	2
C₈₉×#3480	2
C₂×#550×C₈₉	2
C₈₉×#3460	2
#62916	89

Subgroups — C89×#62916 #36440764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#62916 #36440764