Subgroups — C₈₉×#60496 #36438344

Order 22784.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3084	C₂	1	3	yes	maximal, complemented
4	C₈₉×#3083	C₂	1	3	yes	maximal, complemented
5	C₈₉×#3081	C₂	1	3	yes	maximal, complemented
6	C₈₉×#3081	C₂	1	3	yes	maximal, complemented
7	C₈₉×#3080	C₂	1	3	yes	maximal, complemented
8	C₈₉×#3077	C₂	1	3	yes	maximal, complemented
9	C₈₉×#3076	C₂	1	3	yes	maximal, complemented
10	C₂×#478×C₈₉	C₂	1	3	yes	maximal, complemented
11	C₈₉×#2913	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#2912	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#2906	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#2897	C₂	1	1	yes	maximal, characteristic
15	C₈₉×#2897	C₂	1	1	yes	maximal, characteristic
16	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
19	#557×C₈₉	—	2	1	no	—
20	#557×C₈₉	—	2	1	no	—
21	#554×C₈₉	—	2	3	no	—
22	#554×C₈₉	—	2	3	no	—
23	#544×C₈₉	C₂²	1	3	yes	complemented
24	#544×C₈₉	C₂²	1	3	yes	complemented
25	#533×C₈₉	C₂²	1	3	yes	complemented
26	C₂×#125×C₈₉	C₂²	1	3	yes	complemented
27	#520×C₈₉	—	2	1	no	—
28	#520×C₈₉	C₂²	1	3	yes	—
29	#520×C₈₉	C₂²	1	1	yes	characteristic
30	#520×C₈₉	C₂²	1	1	yes	characteristic
31	#520×C₈₉	C₂²	1	1	yes	characteristic
32	#518×C₈₉	—	2	3	no	—
33	#518×C₈₉	C₂²	1	3	yes	—
34	C₄×#41×C₈₉	—	2	1	no	—
35	C₄×#41×C₈₉	C₂²	1	3	yes	—
36	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₈₉	—	2	3	no	—
38	C₂×C₄×D₄×C₈₉	C₂²	1	3	yes	—
39	#479×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	#479×C₈₉	C₂²	1	3	yes	complemented
41	#479×C₈₉	C₂²	1	6	yes	complemented
42	#478×C₈₉	C₂²	1	3	yes	complemented
43	#478×C₈₉	C₂²	1	6	yes	complemented
44	#478×C₈₉	C₂²	1	6	yes	complemented
45	#478×C₈₉	C₂²	1	6	yes	complemented
46	#478×C₈₉	C₂²	1	3	yes	complemented
47	#478×C₈₉	C₂²	1	6	yes	complemented
48	#477×C₈₉	C₂²	1	3	yes	complemented
49	#476×C₈₉	C₂²	1	3	yes	complemented
50	#476×C₈₉	C₂²	1	6	yes	complemented
51	#475×C₈₉	C₂²	1	3	yes	complemented
52	#474×C₈₉	C₂²	1	3	yes	complemented
53	#474×C₈₉	C₂²	1	6	yes	complemented
54	#428×C₈₉	C₂²	1	3	yes	complemented
55	#428×C₈₉	C₂²	1	3	yes	complemented
56	#428×C₈₉	C₂²	1	3	yes	complemented
57	#428×C₈₉	C₂²	1	3	yes	complemented
58	#427×C₈₉	C₂²	1	3	yes	complemented
59	#427×C₈₉	C₂²	1	3	yes	complemented
60	C₂×#107×C₈₉	C₂²	1	3	yes	complemented
61	C₂×#106×C₈₉	C₂²	1	3	yes	complemented
62	#424×C₈₉	C₂²	1	3	yes	complemented
63	#424×C₈₉	C₂²	1	3	yes	complemented
64	#423×C₈₉	C₂²	1	3	yes	complemented
65	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	complemented
66	#418×C₈₉	C₂²	1	3	yes	—
67	#418×C₈₉	C₂²	1	3	yes	—
68	#417×C₈₉	C₂²	1	3	yes	—
69	#417×C₈₉	C₂²	1	3	yes	—
70	#416×C₈₉	C₂²	1	3	yes	—
71	#416×C₈₉	C₂²	1	3	yes	—
72	C₂×#102×C₈₉	C₂²	1	3	yes	—
73	C₂×#102×C₈₉	C₂²	1	3	yes	—
74	H(5,𝔽₂)×C₈₉	—	2	4	no	—
75	H(5,𝔽₂)×C₈₉	Q₈	1	8	yes	complemented
76	C₂×#41×C₈₉	—	2	1	no	—
77	C₂×#41×C₈₉	C₂³	1	3	yes	—
78	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
79	C₂²×D₄×C₈₉	—	2	3	no	—
80	C₂²×D₄×C₈₉	C₂³	1	3	yes	—
81	C₂³×C₄×C₈₉	—	2	3	no	—
82	#131×C₈₉	—	2	4	no	—
83	#131×C₈₉	—	2	4	no	—
84	C₂×M₁₆×C₈₉	C₂³	1	3	yes	—
85	C₂×M₁₆×C₈₉	C₂³	1	3	yes	—
86	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
87	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
88	C₄⋊Q₈×C₈₉	—	2	3	no	—
89	C₄⋊Q₈×C₈₉	—	2	3	no	—
90	C₄⋊Q₈×C₈₉	C₂³	1	3	yes	—
91	C₄⋊Q₈×C₈₉	C₂³	1	3	yes	—
92	#125×C₈₉	—	2	3	no	—
93	#125×C₈₉	—	2	3	no	—
94	#125×C₈₉	—	2	3	no	—
95	#125×C₈₉	—	2	3	no	—
96	#125×C₈₉	C₂³	1	3	yes	—
97	#125×C₈₉	C₂³	1	6	yes	—
98	#125×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²⋊Q₈×C₈₉	—	2	3	no	—
100	C₂²⋊Q₈×C₈₉	—	2	3	no	—
101	C₂²⋊Q₈×C₈₉	—	2	6	no	—
102	C₂²⋊Q₈×C₈₉	—	2	6	no	—
103	C₄×Q₈×C₈₉	—	2	1	no	—
104	C₄×Q₈×C₈₉	—	2	1	no	—
105	C₄×Q₈×C₈₉	—	2	1	no	—
106	C₄×Q₈×C₈₉	—	2	3	no	—
107	C₄×Q₈×C₈₉	—	2	3	no	—
108	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
109	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
110	C₄×Q₈×C₈₉	C₂³	1	3	yes	—
111	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₄×D₄×C₈₉	—	2	3	no	—
113	C₄×D₄×C₈₉	—	2	3	no	—
114	C₄×D₄×C₈₉	—	2	3	no	—
115	C₄×D₄×C₈₉	—	2	3	no	—
116	C₄×D₄×C₈₉	—	2	3	no	—
117	C₄×D₄×C₈₉	—	2	3	no	—
118	C₄×D₄×C₈₉	—	2	3	no	—
119	C₄×D₄×C₈₉	—	2	6	no	—
120	C₄×D₄×C₈₉	—	2	3	no	—
121	C₄×D₄×C₈₉	—	2	3	no	—
122	C₄×D₄×C₈₉	D₄	1	3	yes	—
123	C₄×D₄×C₈₉	D₄	1	3	yes	—
124	C₄×D₄×C₈₉	C₂³	1	6	yes	—
125	C₄×D₄×C₈₉	C₂³	1	3	yes	—
126	C₄×D₄×C₈₉	C₂³	1	3	yes	—
127	#117×C₈₉	—	2	3	no	—
128	#117×C₈₉	—	2	3	no	—
129	#117×C₈₉	C₂³	1	3	yes	—
130	#117×C₈₉	C₂³	1	3	yes	—
131	#117×C₈₉	C₂³	1	3	yes	—
132	#117×C₈₉	C₂³	1	3	yes	—
133	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
134	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
135	C₂×C₄⋊C₄×C₈₉	—	2	3	no	—
136	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
137	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
138	C₂×C₄⋊C₄×C₈₉	C₂³	1	3	yes	—
139	C₂×C₂²⋊C₄×C₈₉	—	2	3	no	—
140	C₂×C₂²⋊C₄×C₈₉	—	2	6	no	—
141	C₂×C₄²×C₈₉	—	2	3	no	—
142	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
143	#107×C₈₉	C₂³	1	3	yes	—
144	#107×C₈₉	C₂³	1	6	yes	—
145	#107×C₈₉	C₂³	1	1	yes	characteristic
146	#107×C₈₉	C₂³	1	3	yes	—
147	#107×C₈₉	C₂³	1	3	yes	—
148	#106×C₈₉	C₂³	1	3	yes	—
149	#106×C₈₉	C₂³	1	6	yes	—
150	#106×C₈₉	C₂³	1	1	yes	characteristic
151	#106×C₈₉	C₂³	1	3	yes	—
152	#106×C₈₉	C₂³	1	3	yes	—
153	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
154	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
155	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
156	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
157	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
158	#103×C₈₉	C₂³	1	1	yes	characteristic
159	#103×C₈₉	C₂³	1	1	yes	characteristic
160	#103×C₈₉	C₂³	1	3	yes	—
161	#103×C₈₉	C₂³	1	3	yes	—
162	#102×C₈₉	C₂³	1	3	yes	—
163	#102×C₈₉	C₂³	1	3	yes	—
164	#102×C₈₉	C₂³	1	6	yes	—
165	#102×C₈₉	C₂³	1	6	yes	—
166	#102×C₈₉	C₂³	1	3	yes	—
167	#102×C₈₉	C₂³	1	3	yes	—
168	C₂⁴×C₈₉	—	2	3	no	—
169	#41×C₈₉	—	2	8	no	—
170	#41×C₈₉	—	2	6	no	—
171	#41×C₈₉	C₂×Q₈	1	12	yes	—
172	#41×C₈₉	C₂×Q₈	1	4	yes	—
173	#41×C₈₉	#41	1	4	yes	—
174	C₂×Q₈×C₈₉	—	2	3	no	—
175	C₂×Q₈×C₈₉	—	2	3	no	—
176	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
177	C₂×Q₈×C₈₉	C₂⁴	1	1	yes	characteristic
178	C₂×D₄×C₈₉	—	2	3	no	—
179	C₂×D₄×C₈₉	—	2	24	no	—
180	C₂×D₄×C₈₉	—	2	3	no	—
181	C₂×D₄×C₈₉	—	2	6	no	—
182	C₂×D₄×C₈₉	C₂×D₄	1	3	yes	—
183	C₂×D₄×C₈₉	C₂×Q₈	1	12	yes	—
184	C₂×D₄×C₈₉	C₂⁴	1	3	yes	—
185	C₂²×C₄×C₈₉	—	2	3	no	—
186	C₂²×C₄×C₈₉	—	2	3	no	—
187	C₂²×C₄×C₈₉	—	2	3	no	—
188	C₂²×C₄×C₈₉	—	2	3	no	—
189	C₂²×C₄×C₈₉	—	2	3	no	—
190	C₂²×C₄×C₈₉	—	2	3	no	—
191	C₂²×C₄×C₈₉	—	2	6	no	—
192	C₂²×C₄×C₈₉	—	8	6	no	—
193	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
194	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
195	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
196	M₁₆×C₈₉	—	2	6	no	—
197	M₁₆×C₈₉	—	2	6	no	—
198	C₂×C₈×C₈₉	—	2	6	no	—
199	C₂×C₈×C₈₉	—	2	6	no	—
200	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
201	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
202	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
203	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
204	C₄⋊C₄×C₈₉	—	2	3	no	—
205	C₄⋊C₄×C₈₉	—	2	3	no	—
206	C₄⋊C₄×C₈₉	—	2	3	no	—
207	C₄⋊C₄×C₈₉	—	2	3	no	—
208	C₄⋊C₄×C₈₉	—	2	6	no	—
209	C₄⋊C₄×C₈₉	—	2	6	no	—
210	C₄⋊C₄×C₈₉	—	2	3	no	—
211	C₄⋊C₄×C₈₉	—	2	3	no	—
212	C₄⋊C₄×C₈₉	—	2	1	no	—
213	C₄⋊C₄×C₈₉	—	2	1	no	—
214	C₄⋊C₄×C₈₉	—	2	1	no	—
215	C₄⋊C₄×C₈₉	—	2	3	no	—
216	C₄⋊C₄×C₈₉	—	2	3	no	—
217	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
218	C₄⋊C₄×C₈₉	C₂×D₄	1	3	yes	—
219	C₄⋊C₄×C₈₉	C₂⁴	1	3	yes	—
220	C₄⋊C₄×C₈₉	C₂⁴	1	1	yes	characteristic
221	C₄⋊C₄×C₈₉	C₂⁴	1	3	yes	—
222	C₄⋊C₄×C₈₉	C₂⁴	1	1	yes	characteristic
223	C₄⋊C₄×C₈₉	C₂⁴	1	3	yes	—
224	C₄⋊C₄×C₈₉	C₂⁴	1	1	yes	characteristic
225	C₂²⋊C₄×C₈₉	—	2	3	no	—
226	C₂²⋊C₄×C₈₉	—	2	3	no	—
227	C₂²⋊C₄×C₈₉	—	2	3	no	—
228	C₂²⋊C₄×C₈₉	—	2	3	no	—
229	C₂²⋊C₄×C₈₉	—	2	6	no	—
230	C₂²⋊C₄×C₈₉	—	2	3	no	—
231	C₂²⋊C₄×C₈₉	C₂×D₄	1	6	yes	—
232	C₄²×C₈₉	—	2	1	no	—
233	C₄²×C₈₉	—	2	3	no	—
234	C₄²×C₈₉	—	2	3	no	—
235	C₄²×C₈₉	—	2	3	no	—
236	C₄²×C₈₉	—	2	3	no	—
237	C₄²×C₈₉	C₂×D₄	1	3	yes	—
238	C₄²×C₈₉	C₂×D₄	1	3	yes	—
239	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
240	C₄²×C₈₉	C₂⁴	1	3	yes	—
241	C₂³×C₈₉	—	2	3	no	—
242	C₂³×C₈₉	—	2	12	no	—
243	C₂³×C₈₉	—	8	3	no	—
244	C₂³×C₈₉	—	8	3	no	—
245	C₂³×C₈₉	C₂²×D₄	1	3	yes	—
246	Q₈×C₈₉	—	2	1	no	—
247	Q₈×C₈₉	—	8	3	no	—
248	Q₈×C₈₉	—	8	3	no	—
249	Q₈×C₈₉	C₂²⋊Q₈	1	4	yes	—
250	Q₈×C₈₉	C₂²×Q₈	1	2	yes	—
251	D₄×C₈₉	—	2	12	no	—
252	D₄×C₈₉	—	2	12	no	—
253	D₄×C₈₉	—	2	3	no	—
254	D₄×C₈₉	C₂²⋊Q₈	1	12	yes	—
255	D₄×C₈₉	C₂²×Q₈	1	6	yes	—
256	C₂×C₄×C₈₉	—	2	1	no	—
257	C₂×C₄×C₈₉	—	2	12	no	—
258	C₂×C₄×C₈₉	—	2	1	no	—
259	C₂×C₄×C₈₉	—	2	1	no	—
260	C₂×C₄×C₈₉	—	2	1	no	—
261	C₂×C₄×C₈₉	—	2	3	no	—
262	C₂×C₄×C₈₉	—	2	3	no	—
263	C₂×C₄×C₈₉	—	2	3	no	—
264	C₂×C₄×C₈₉	—	4	3	no	—
265	C₂×C₄×C₈₉	—	4	3	no	—
266	C₂×C₄×C₈₉	—	4	6	no	—
267	C₂×C₄×C₈₉	—	8	3	no	—
268	C₂×C₄×C₈₉	—	8	6	no	—
269	C₂×C₄×C₈₉	—	8	3	no	—
270	C₂×C₄×C₈₉	—	8	3	no	—
271	C₂×C₄×C₈₉	—	8	3	no	—
272	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
274	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
276	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
277	C₂×C₄×C₈₉	C₂²×Q₈	1	6	yes	—
278	C₂×C₄×C₈₉	C₂×#41	1	6	yes	—
279	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
280	C₈×C₈₉	—	2	1	no	—
281	C₈×C₈₉	—	2	1	no	—
282	C₈×C₈₉	—	2	3	no	—
283	C₈×C₈₉	—	2	3	no	—
284	C₂²×C₈₉	—	2	3	no	—
285	C₂²×C₈₉	—	2	3	no	—
286	C₂²×C₈₉	—	2	6	no	—
287	C₂²×C₈₉	—	4	3	no	—
288	C₂²×C₈₉	—	4	3	no	—
289	C₂²×C₈₉	—	8	12	no	—
290	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	6	yes	—
291	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
292	C₄×C₈₉	—	2	1	no	—
293	C₄×C₈₉	—	2	3	no	—
294	C₄×C₈₉	—	2	1	no	—
295	C₄×C₈₉	—	2	3	no	—
296	C₄×C₈₉	—	2	2	no	—
297	C₄×C₈₉	—	4	1	no	—
298	C₄×C₈₉	—	4	3	no	—
299	C₄×C₈₉	—	4	1	no	—
300	C₄×C₈₉	—	4	3	no	—
301	C₄×C₈₉	—	4	3	no	—
302	C₄×C₈₉	—	4	1	no	—
303	C₄×C₈₉	C₂×C₂²⋊Q₈	1	6	yes	—
304	C₄×C₈₉	C₂³×Q₈	1	1	yes	characteristic
305	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
306	#60496	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
307	C₂×C₈₉	—	2	6	no	—
308	C₂×C₈₉	—	4	6	no	—
309	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
310	C₂×C₈₉	#3584	1	1	yes	characteristic
311	C₂×C₈₉	#3585	1	1	yes	characteristic
312	#3504	C₂×C₈₉	1	1	yes	characteristic, complemented
313	C₄×H(5,𝔽₂)	C₂×C₈₉	1	1	yes	characteristic
314	#3084	C₂×C₈₉	1	3	yes	complemented
315	#3083	C₂×C₈₉	1	3	yes	complemented
316	#3081	C₂×C₈₉	1	3	yes	complemented
317	#3081	C₂×C₈₉	1	3	yes	complemented
318	#3080	C₂×C₈₉	1	3	yes	complemented
319	#3077	C₂×C₈₉	1	3	yes	complemented
320	#3076	C₂×C₈₉	1	3	yes	complemented
321	C₂×#478	C₂×C₈₉	1	3	yes	complemented
322	#2913	C₂×C₈₉	1	1	yes	characteristic, complemented
323	#2912	C₂×C₈₉	1	1	yes	characteristic, complemented
324	#2906	C₂×C₈₉	1	1	yes	characteristic, complemented
325	#2897	C₂×C₈₉	1	1	yes	characteristic
326	#2897	C₂×C₈₉	1	1	yes	characteristic
327	C₈₉	#60496	1	1	yes	characteristic, direct product factor, Sylow
328	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	1	yes	characteristic
329	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
330	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
331	#557	—	2	1	no	—
332	#557	—	2	1	no	—
333	#554	—	2	3	no	—
334	#554	—	2	3	no	—
335	#544	C₂²×C₈₉	1	3	yes	complemented
336	#544	C₂²×C₈₉	1	3	yes	complemented
337	#533	C₂²×C₈₉	1	3	yes	complemented
338	C₂×#125	C₂²×C₈₉	1	3	yes	complemented
339	#520	—	2	1	no	—
340	#520	C₂²×C₈₉	1	3	yes	—
341	#520	C₂²×C₈₉	1	1	yes	characteristic
342	#520	C₂²×C₈₉	1	1	yes	characteristic
343	#520	C₂²×C₈₉	1	1	yes	characteristic
344	#518	—	2	3	no	—
345	#518	C₂²×C₈₉	1	3	yes	—
346	C₄×#41	—	2	1	no	—
347	C₄×#41	C₂²×C₈₉	1	3	yes	—
348	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
349	C₂×C₄×D₄	—	2	3	no	—
350	C₂×C₄×D₄	C₂²×C₈₉	1	3	yes	—
351	#479	C₂²×C₈₉	1	1	yes	characteristic, complemented
352	#479	C₂²×C₈₉	1	3	yes	complemented
353	#479	C₂²×C₈₉	1	6	yes	complemented
354	#478	C₂²×C₈₉	1	3	yes	complemented
355	#478	C₂²×C₈₉	1	6	yes	complemented
356	#478	C₂²×C₈₉	1	6	yes	complemented
357	#478	C₂²×C₈₉	1	6	yes	complemented
358	#478	C₂²×C₈₉	1	3	yes	complemented
359	#478	C₂²×C₈₉	1	6	yes	complemented
360	#477	C₂²×C₈₉	1	3	yes	complemented
361	#476	C₂²×C₈₉	1	3	yes	complemented
362	#476	C₂²×C₈₉	1	6	yes	complemented
363	#475	C₂²×C₈₉	1	3	yes	complemented
364	#474	C₂²×C₈₉	1	3	yes	complemented
365	#474	C₂²×C₈₉	1	6	yes	complemented
366	#428	C₂²×C₈₉	1	3	yes	complemented
367	#428	C₂²×C₈₉	1	3	yes	complemented
368	#428	C₂²×C₈₉	1	3	yes	complemented
369	#428	C₂²×C₈₉	1	3	yes	complemented
370	#427	C₂²×C₈₉	1	3	yes	complemented
371	#427	C₂²×C₈₉	1	3	yes	complemented
372	C₂×#107	C₂²×C₈₉	1	3	yes	complemented
373	C₂×#106	C₂²×C₈₉	1	3	yes	complemented
374	#424	C₂²×C₈₉	1	3	yes	complemented
375	#424	C₂²×C₈₉	1	3	yes	complemented
376	#423	C₂²×C₈₉	1	3	yes	complemented
377	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	complemented
378	#418	C₂²×C₈₉	1	3	yes	—
379	#418	C₂²×C₈₉	1	3	yes	—
380	#417	C₂²×C₈₉	1	3	yes	—
381	#417	C₂²×C₈₉	1	3	yes	—
382	#416	C₂²×C₈₉	1	3	yes	—
383	#416	C₂²×C₈₉	1	3	yes	—
384	C₂×#102	C₂²×C₈₉	1	3	yes	—
385	C₂×#102	C₂²×C₈₉	1	3	yes	—
386	H(5,𝔽₂)	—	2	4	no	—
387	H(5,𝔽₂)	Q₈×C₈₉	1	8	yes	complemented
388	C₂×#41	—	2	1	no	—
389	C₂×#41	C₂³×C₈₉	1	3	yes	—
390	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
391	C₂²×D₄	—	2	3	no	—
392	C₂²×D₄	C₂³×C₈₉	1	3	yes	—
393	C₂³×C₄	—	2	3	no	—
394	#131	—	2	4	no	—
395	#131	—	2	4	no	—
396	C₂×M₁₆	C₂³×C₈₉	1	3	yes	—
397	C₂×M₁₆	C₂³×C₈₉	1	3	yes	—
398	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
399	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
400	C₄⋊Q₈	—	2	3	no	—
401	C₄⋊Q₈	—	2	3	no	—
402	C₄⋊Q₈	C₂³×C₈₉	1	3	yes	—
403	C₄⋊Q₈	C₂³×C₈₉	1	3	yes	—
404	#125	—	2	3	no	—
405	#125	—	2	3	no	—
406	#125	—	2	3	no	—
407	#125	—	2	3	no	—
408	#125	C₂³×C₈₉	1	3	yes	—
409	#125	C₂³×C₈₉	1	6	yes	—
410	#125	C₂³×C₈₉	1	1	yes	characteristic
411	C₂²⋊Q₈	—	2	3	no	—
412	C₂²⋊Q₈	—	2	3	no	—
413	C₂²⋊Q₈	—	2	6	no	—
414	C₂²⋊Q₈	—	2	6	no	—
415	C₄×Q₈	—	2	1	no	—
416	C₄×Q₈	—	2	1	no	—
417	C₄×Q₈	—	2	1	no	—
418	C₄×Q₈	—	2	3	no	—
419	C₄×Q₈	—	2	3	no	—
420	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
421	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
422	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
423	C₄×Q₈	C₂³×C₈₉	1	3	yes	—
424	C₄×D₄	—	2	3	no	—
425	C₄×D₄	—	2	3	no	—
426	C₄×D₄	—	2	3	no	—
427	C₄×D₄	—	2	3	no	—
428	C₄×D₄	—	2	3	no	—
429	C₄×D₄	—	2	3	no	—
430	C₄×D₄	—	2	3	no	—
431	C₄×D₄	—	2	6	no	—
432	C₄×D₄	—	2	3	no	—
433	C₄×D₄	—	2	3	no	—
434	C₄×D₄	D₄×C₈₉	1	3	yes	—
435	C₄×D₄	D₄×C₈₉	1	3	yes	—
436	C₄×D₄	C₂³×C₈₉	1	3	yes	—
437	C₄×D₄	C₂³×C₈₉	1	6	yes	—
438	C₄×D₄	C₂³×C₈₉	1	3	yes	—
439	#117	—	2	3	no	—
440	#117	—	2	3	no	—
441	#117	C₂³×C₈₉	1	3	yes	—
442	#117	C₂³×C₈₉	1	3	yes	—
443	#117	C₂³×C₈₉	1	3	yes	—
444	#117	C₂³×C₈₉	1	3	yes	—
445	C₂×C₄⋊C₄	—	2	3	no	—
446	C₂×C₄⋊C₄	—	2	3	no	—
447	C₂×C₄⋊C₄	—	2	3	no	—
448	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
449	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
450	C₂×C₄⋊C₄	C₂³×C₈₉	1	3	yes	—
451	C₂×C₂²⋊C₄	—	2	3	no	—
452	C₂×C₂²⋊C₄	—	2	6	no	—
453	C₂×C₄²	—	2	3	no	—
454	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
455	#107	C₂³×C₈₉	1	3	yes	—
456	#107	C₂³×C₈₉	1	6	yes	—
457	#107	C₂³×C₈₉	1	1	yes	characteristic
458	#107	C₂³×C₈₉	1	3	yes	—
459	#107	C₂³×C₈₉	1	3	yes	—
460	#106	C₂³×C₈₉	1	3	yes	—
461	#106	C₂³×C₈₉	1	6	yes	—
462	#106	C₂³×C₈₉	1	1	yes	characteristic
463	#106	C₂³×C₈₉	1	3	yes	—
464	#106	C₂³×C₈₉	1	3	yes	—
465	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
466	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
467	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
468	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
469	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
470	#103	C₂³×C₈₉	1	1	yes	characteristic
471	#103	C₂³×C₈₉	1	1	yes	characteristic
472	#103	C₂³×C₈₉	1	3	yes	—
473	#103	C₂³×C₈₉	1	3	yes	—
474	#102	C₂³×C₈₉	1	3	yes	—
475	#102	C₂³×C₈₉	1	3	yes	—
476	#102	C₂³×C₈₉	1	6	yes	—
477	#102	C₂³×C₈₉	1	6	yes	—
478	#102	C₂³×C₈₉	1	3	yes	—
479	#102	C₂³×C₈₉	1	3	yes	—
480	C₂⁴	—	2	3	no	—
481	#41	—	2	8	no	—
482	#41	—	2	6	no	—
483	#41	C₂×Q₈×C₈₉	1	12	yes	—
484	#41	C₂×Q₈×C₈₉	1	4	yes	—
485	#41	#41×C₈₉	1	4	yes	—
486	C₂×Q₈	—	2	3	no	—
487	C₂×Q₈	—	2	3	no	—
488	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
489	C₂×Q₈	C₂⁴×C₈₉	1	1	yes	characteristic
490	C₂×D₄	—	2	3	no	—
491	C₂×D₄	—	2	24	no	—
492	C₂×D₄	—	2	3	no	—
493	C₂×D₄	—	2	6	no	—
494	C₂×D₄	C₂×D₄×C₈₉	1	3	yes	—
495	C₂×D₄	C₂×Q₈×C₈₉	1	12	yes	—
496	C₂×D₄	C₂⁴×C₈₉	1	3	yes	—
497	C₂²×C₄	—	2	3	no	—
498	C₂²×C₄	—	2	3	no	—
499	C₂²×C₄	—	2	3	no	—
500	C₂²×C₄	—	2	3	no	—
501	C₂²×C₄	—	2	3	no	—
502	C₂²×C₄	—	2	3	no	—
503	C₂²×C₄	—	2	6	no	—
504	C₂²×C₄	—	8	6	no	—
505	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
506	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
507	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
508	M₁₆	—	2	6	no	—
509	M₁₆	—	2	6	no	—
510	C₂×C₈	—	2	6	no	—
511	C₂×C₈	—	2	6	no	—
512	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
513	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
514	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
515	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
516	C₄⋊C₄	—	2	3	no	—
517	C₄⋊C₄	—	2	3	no	—
518	C₄⋊C₄	—	2	3	no	—
519	C₄⋊C₄	—	2	3	no	—
520	C₄⋊C₄	—	2	6	no	—
521	C₄⋊C₄	—	2	6	no	—
522	C₄⋊C₄	—	2	3	no	—
523	C₄⋊C₄	—	2	3	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	3	no	—
528	C₄⋊C₄	—	2	3	no	—
529	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
530	C₄⋊C₄	C₂×D₄×C₈₉	1	3	yes	—
531	C₄⋊C₄	C₂⁴×C₈₉	1	3	yes	—
532	C₄⋊C₄	C₂⁴×C₈₉	1	1	yes	characteristic
533	C₄⋊C₄	C₂⁴×C₈₉	1	3	yes	—
534	C₄⋊C₄	C₂⁴×C₈₉	1	1	yes	characteristic
535	C₄⋊C₄	C₂⁴×C₈₉	1	3	yes	—
536	C₄⋊C₄	C₂⁴×C₈₉	1	1	yes	characteristic
537	C₂²⋊C₄	—	2	3	no	—
538	C₂²⋊C₄	—	2	3	no	—
539	C₂²⋊C₄	—	2	3	no	—
540	C₂²⋊C₄	—	2	3	no	—
541	C₂²⋊C₄	—	2	6	no	—
542	C₂²⋊C₄	—	2	3	no	—
543	C₂²⋊C₄	C₂×D₄×C₈₉	1	6	yes	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	3	no	—
546	C₄²	—	2	3	no	—
547	C₄²	—	2	3	no	—
548	C₄²	—	2	3	no	—
549	C₄²	C₂×D₄×C₈₉	1	3	yes	—
550	C₄²	C₂×D₄×C₈₉	1	3	yes	—
551	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
552	C₄²	C₂⁴×C₈₉	1	3	yes	—
553	C₂³	—	2	3	no	—
554	C₂³	—	2	12	no	—
555	C₂³	—	8	3	no	—
556	C₂³	—	8	3	no	—
557	C₂³	C₂²×D₄×C₈₉	1	3	yes	—
558	Q₈	—	2	1	no	—
559	Q₈	—	8	3	no	—
560	Q₈	—	8	3	no	—
561	Q₈	C₂²⋊Q₈×C₈₉	1	4	yes	—
562	Q₈	C₂²×Q₈×C₈₉	1	2	yes	—
563	D₄	—	2	12	no	—
564	D₄	—	2	12	no	—
565	D₄	—	2	3	no	—
566	D₄	C₂²⋊Q₈×C₈₉	1	12	yes	—
567	D₄	C₂²×Q₈×C₈₉	1	6	yes	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	12	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	3	no	—
574	C₂×C₄	—	2	3	no	—
575	C₂×C₄	—	2	3	no	—
576	C₂×C₄	—	4	3	no	—
577	C₂×C₄	—	4	3	no	—
578	C₂×C₄	—	4	6	no	—
579	C₂×C₄	—	8	3	no	—
580	C₂×C₄	—	8	6	no	—
581	C₂×C₄	—	8	3	no	—
582	C₂×C₄	—	8	3	no	—
583	C₂×C₄	—	8	3	no	—
584	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
586	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
587	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
588	C₂×C₄	C₂²×D₄×C₈₉	1	3	yes	—
589	C₂×C₄	C₂²×Q₈×C₈₉	1	6	yes	—
590	C₂×C₄	C₂×#41×C₈₉	1	6	yes	—
591	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
592	C₈	—	2	1	no	—
593	C₈	—	2	1	no	—
594	C₈	—	2	3	no	—
595	C₈	—	2	3	no	—
596	C₂²	—	2	3	no	—
597	C₂²	—	2	3	no	—
598	C₂²	—	2	6	no	—
599	C₂²	—	4	3	no	—
600	C₂²	—	4	3	no	—
601	C₂²	—	8	12	no	—
602	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	6	yes	—
603	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
604	C₄	—	2	1	no	—
605	C₄	—	2	3	no	—
606	C₄	—	2	1	no	—
607	C₄	—	2	3	no	—
608	C₄	—	2	2	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	3	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	3	no	—
613	C₄	—	4	3	no	—
614	C₄	—	4	1	no	—
615	C₄	C₂×C₂²⋊Q₈×C₈₉	1	6	yes	—
616	C₄	C₂³×Q₈×C₈₉	1	1	yes	characteristic
617	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
618	C₂	—	2	6	no	—
619	C₂	—	4	6	no	—
620	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
621	C₂	C₈₉×#3584	1	1	yes	characteristic
622	C₂	C₈₉×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂×H(5,𝔽₂)×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3504	C₂	semidirect
3. C₈₉×#3084	C₂	semidirect
4. C₈₉×#3083	C₂	semidirect
5. C₈₉×#3081	C₂	semidirect
6. C₈₉×#3081	C₂	semidirect
7. C₈₉×#3080	C₂	semidirect
8. C₈₉×#3077	C₂	semidirect
9. C₈₉×#3076	C₂	semidirect
10. C₂×#478×C₈₉	C₂	semidirect
11. C₈₉×#2913	C₂	semidirect
12. C₈₉×#2912	C₂	semidirect
13. C₈₉×#2906	C₂	semidirect
23. #544×C₈₉	C₂²	semidirect
24. #544×C₈₉	C₂²	semidirect
25. #533×C₈₉	C₂²	semidirect
26. C₂×#125×C₈₉	C₂²	semidirect
39. #479×C₈₉	C₂²	semidirect
40. #479×C₈₉	C₂²	semidirect
41. #479×C₈₉	C₂²	semidirect
42. #478×C₈₉	C₂²	semidirect
43. #478×C₈₉	C₂²	semidirect
44. #478×C₈₉	C₂²	semidirect
45. #478×C₈₉	C₂²	semidirect
46. #478×C₈₉	C₂²	semidirect
47. #478×C₈₉	C₂²	semidirect
48. #477×C₈₉	C₂²	semidirect
49. #476×C₈₉	C₂²	semidirect
50. #476×C₈₉	C₂²	semidirect
51. #475×C₈₉	C₂²	semidirect
52. #474×C₈₉	C₂²	semidirect
53. #474×C₈₉	C₂²	semidirect
54. #428×C₈₉	C₂²	semidirect
55. #428×C₈₉	C₂²	semidirect
56. #428×C₈₉	C₂²	semidirect
57. #428×C₈₉	C₂²	semidirect
58. #427×C₈₉	C₂²	semidirect
59. #427×C₈₉	C₂²	semidirect
60. C₂×#107×C₈₉	C₂²	semidirect
61. C₂×#106×C₈₉	C₂²	semidirect
62. #424×C₈₉	C₂²	semidirect
63. #424×C₈₉	C₂²	semidirect
64. #423×C₈₉	C₂²	semidirect
65. C₂×C₄⋊C₈×C₈₉	C₂²	semidirect
75. H(5,𝔽₂)×C₈₉	Q₈	semidirect
306. #60496	C₈₉	direct (N × H)
312. #3504	C₂×C₈₉	semidirect
314. #3084	C₂×C₈₉	semidirect
315. #3083	C₂×C₈₉	semidirect
316. #3081	C₂×C₈₉	semidirect
317. #3081	C₂×C₈₉	semidirect
318. #3080	C₂×C₈₉	semidirect
319. #3077	C₂×C₈₉	semidirect
320. #3076	C₂×C₈₉	semidirect
321. C₂×#478	C₂×C₈₉	semidirect
322. #2913	C₂×C₈₉	semidirect
323. #2912	C₂×C₈₉	semidirect
324. #2906	C₂×C₈₉	semidirect
327. C₈₉	#60496	direct (N × H)
335. #544	C₂²×C₈₉	semidirect
336. #544	C₂²×C₈₉	semidirect
337. #533	C₂²×C₈₉	semidirect
338. C₂×#125	C₂²×C₈₉	semidirect
351. #479	C₂²×C₈₉	semidirect
352. #479	C₂²×C₈₉	semidirect
353. #479	C₂²×C₈₉	semidirect
354. #478	C₂²×C₈₉	semidirect
355. #478	C₂²×C₈₉	semidirect
356. #478	C₂²×C₈₉	semidirect
357. #478	C₂²×C₈₉	semidirect
358. #478	C₂²×C₈₉	semidirect
359. #478	C₂²×C₈₉	semidirect
360. #477	C₂²×C₈₉	semidirect
361. #476	C₂²×C₈₉	semidirect
362. #476	C₂²×C₈₉	semidirect
363. #475	C₂²×C₈₉	semidirect
364. #474	C₂²×C₈₉	semidirect
365. #474	C₂²×C₈₉	semidirect
366. #428	C₂²×C₈₉	semidirect
367. #428	C₂²×C₈₉	semidirect
368. #428	C₂²×C₈₉	semidirect
369. #428	C₂²×C₈₉	semidirect
370. #427	C₂²×C₈₉	semidirect
371. #427	C₂²×C₈₉	semidirect
372. C₂×#107	C₂²×C₈₉	semidirect
373. C₂×#106	C₂²×C₈₉	semidirect
374. #424	C₂²×C₈₉	semidirect
375. #424	C₂²×C₈₉	semidirect
376. #423	C₂²×C₈₉	semidirect
377. C₂×C₄⋊C₈	C₂²×C₈₉	semidirect
387. H(5,𝔽₂)	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3504	2
C₄×H(5,𝔽₂)×C₈₉	2
C₈₉×#3084	2
C₈₉×#3083	2
C₈₉×#3081	2
C₈₉×#3080	2
C₈₉×#3077	2
C₈₉×#3076	2
C₂×#478×C₈₉	2
C₈₉×#2913	2
C₈₉×#2912	2
C₈₉×#2906	2
C₈₉×#2897	2
#60496	89

Subgroups — C89×#60496 #36438344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60496 #36438344