Subgroups — C₈₉×#35556 #36413404

Order 22784.

2670 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#535×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#533×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2592	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2548	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2491	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2490	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2472	C₂	1	8	yes	maximal, complemented
8	C₈₉×#2311	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2309	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#2297	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#390×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#2274	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	#535×C₈₉	—	2	8	no	—
16	#533×C₈₉	—	2	8	no	—
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	complemented
18	C₂×#127×C₈₉	C₂²	1	2	yes	—
19	C₂×#125×C₈₉	C₂²	1	2	yes	complemented
20	C₂×#124×C₈₉	C₂²	1	2	yes	—
21	C₂×#124×C₈₉	C₂²	1	8	yes	complemented
22	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
23	C₂×#121×C₈₉	C₂²	1	4	yes	—
24	C₂×#120×C₈₉	C₂²	1	4	yes	—
25	C₂×C₄×Q₈×C₈₉	C₂²	1	4	yes	complemented
26	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
31	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
32	#397×C₈₉	C₂²	1	8	yes	complemented
33	#394×C₈₉	C₂²	1	8	yes	complemented
34	#394×C₈₉	C₂²	1	8	yes	complemented
35	#393×C₈₉	C₂²	1	8	yes	complemented
36	#391×C₈₉	C₂²	1	2	yes	complemented
37	#391×C₈₉	C₂²	1	2	yes	complemented
38	#390×C₈₉	C₂²	1	4	yes	—
39	#390×C₈₉	C₂²	1	4	yes	—
40	#390×C₈₉	C₂²	1	4	yes	—
41	#390×C₈₉	C₂²	1	2	yes	—
42	#390×C₈₉	C₂²	1	2	yes	—
43	#390×C₈₉	C₂²	1	2	yes	complemented
44	#390×C₈₉	C₂²	1	2	yes	complemented
45	#390×C₈₉	C₂²	1	8	yes	complemented
46	#388×C₈₉	C₂²	1	8	yes	complemented
47	#388×C₈₉	C₂²	1	8	yes	complemented
48	#386×C₈₉	C₂²	1	4	yes	—
49	#386×C₈₉	C₂²	1	4	yes	—
50	#380×C₈₉	C₂²	1	4	yes	complemented
51	#379×C₈₉	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	complemented
53	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	complemented
54	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
55	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	complemented
56	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	complemented
57	C₂⁵×C₈₉	C₂³	1	2	yes	—
58	C₂×#41×C₈₉	—	2	4	no	—
59	C₂×#41×C₈₉	—	2	4	no	—
60	C₂×#41×C₈₉	—	2	4	no	—
61	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
63	C₂²×D₄×C₈₉	—	2	4	no	—
64	C₂²×D₄×C₈₉	—	2	4	no	—
65	C₂²×D₄×C₈₉	—	2	4	no	—
66	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
67	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
68	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
69	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
71	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
72	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
73	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
74	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₈₉	—	2	8	no	—
76	#127×C₈₉	—	2	8	no	—
77	#125×C₈₉	—	2	8	no	—
78	#124×C₈₉	—	2	8	no	—
79	#124×C₈₉	—	4	16	no	—
80	C₂²⋊Q₈×C₈₉	—	2	16	no	—
81	#121×C₈₉	—	2	16	no	—
82	#120×C₈₉	—	2	16	no	—
83	C₄×Q₈×C₈₉	—	2	16	no	—
84	#117×C₈₉	—	2	4	no	—
85	#117×C₈₉	—	2	4	no	—
86	#117×C₈₉	D₄	1	8	yes	—
87	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₈₉	—	4	8	no	—
97	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
107	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
108	C₂×C₄²×C₈₉	—	2	4	no	—
109	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
110	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
113	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
114	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
115	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
116	#95×C₈₉	C₂³	1	8	yes	—
117	#95×C₈₉	C₂³	1	4	yes	—
118	#95×C₈₉	C₂³	1	4	yes	—
119	#95×C₈₉	C₂³	1	8	yes	—
120	C₂⁴×C₈₉	—	2	4	no	—
121	C₂⁴×C₈₉	—	2	4	no	—
122	C₂⁴×C₈₉	—	2	4	no	—
123	C₂⁴×C₈₉	—	4	4	no	—
124	C₂⁴×C₈₉	—	4	4	no	—
125	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
126	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
127	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
128	#41×C₈₉	—	4	16	no	—
129	C₂×Q₈×C₈₉	—	2	8	no	—
130	C₂×Q₈×C₈₉	—	2	2	no	—
131	C₂×Q₈×C₈₉	—	2	2	no	—
132	C₂×Q₈×C₈₉	—	2	2	no	—
133	C₂×Q₈×C₈₉	—	2	8	no	—
134	C₂×Q₈×C₈₉	—	4	4	no	—
135	C₂×D₄×C₈₉	—	2	4	no	—
136	C₂×D₄×C₈₉	—	2	4	no	—
137	C₂×D₄×C₈₉	—	2	4	no	—
138	C₂×D₄×C₈₉	—	2	4	no	—
139	C₂×D₄×C₈₉	—	2	4	no	—
140	C₂×D₄×C₈₉	—	2	2	no	—
141	C₂×D₄×C₈₉	—	2	16	no	—
142	C₂×D₄×C₈₉	—	2	4	no	—
143	C₂×D₄×C₈₉	—	2	4	no	—
144	C₂×D₄×C₈₉	—	2	4	no	—
145	C₂×D₄×C₈₉	—	2	4	no	—
146	C₂×D₄×C₈₉	—	2	2	no	—
147	C₂×D₄×C₈₉	—	2	2	no	—
148	C₂×D₄×C₈₉	—	4	8	no	—
149	C₂×D₄×C₈₉	—	4	8	no	—
150	C₂×D₄×C₈₉	—	4	16	no	—
151	C₂²×C₄×C₈₉	—	2	4	no	—
152	C₂²×C₄×C₈₉	—	2	4	no	—
153	C₂²×C₄×C₈₉	—	2	4	no	—
154	C₂²×C₄×C₈₉	—	2	4	no	—
155	C₂²×C₄×C₈₉	—	2	4	no	—
156	C₂²×C₄×C₈₉	—	2	4	no	—
157	C₂²×C₄×C₈₉	—	4	2	no	—
158	C₂²×C₄×C₈₉	—	4	4	no	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₈₉	#41	1	4	yes	—
161	C₂²×C₄×C₈₉	#41	1	4	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
171	C₄⋊C₄×C₈₉	—	2	4	no	—
172	C₄⋊C₄×C₈₉	—	2	8	no	—
173	C₄⋊C₄×C₈₉	—	2	8	no	—
174	C₄⋊C₄×C₈₉	—	2	8	no	—
175	C₄⋊C₄×C₈₉	—	2	4	no	—
176	C₄⋊C₄×C₈₉	—	2	4	no	—
177	C₄⋊C₄×C₈₉	—	2	4	no	—
178	C₄⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
179	C₂²⋊C₄×C₈₉	—	2	4	no	—
180	C₂²⋊C₄×C₈₉	—	2	4	no	—
181	C₂²⋊C₄×C₈₉	—	2	16	no	—
182	C₂²⋊C₄×C₈₉	—	2	16	no	—
183	C₂²⋊C₄×C₈₉	—	2	8	no	—
184	C₂²⋊C₄×C₈₉	—	4	8	no	—
185	C₂²⋊C₄×C₈₉	—	4	4	no	—
186	C₂²⋊C₄×C₈₉	—	4	8	no	—
187	C₂²⋊C₄×C₈₉	—	4	8	no	—
188	C₂²⋊C₄×C₈₉	—	4	8	no	—
189	C₂²⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
190	C₄²×C₈₉	—	2	4	no	—
191	C₄²×C₈₉	—	2	2	no	—
192	C₄²×C₈₉	—	2	2	no	—
193	C₄²×C₈₉	—	2	4	no	—
194	C₄²×C₈₉	—	2	4	no	—
195	C₄²×C₈₉	—	4	8	no	—
196	C₄²×C₈₉	C₂×D₄	1	8	yes	—
197	C₂³×C₈₉	—	2	2	no	—
198	C₂³×C₈₉	—	2	2	no	—
199	C₂³×C₈₉	—	2	4	no	—
200	C₂³×C₈₉	—	2	2	no	—
201	C₂³×C₈₉	—	2	2	no	—
202	C₂³×C₈₉	—	2	2	no	—
203	C₂³×C₈₉	—	2	4	no	—
204	C₂³×C₈₉	—	2	4	no	—
205	C₂³×C₈₉	—	2	4	no	—
206	C₂³×C₈₉	—	2	8	no	—
207	C₂³×C₈₉	—	2	4	no	—
208	C₂³×C₈₉	—	2	8	no	—
209	C₂³×C₈₉	—	4	16	no	—
210	C₂³×C₈₉	—	4	8	no	—
211	C₂³×C₈₉	—	8	8	no	—
212	C₂³×C₈₉	—	8	4	no	—
213	C₂³×C₈₉	—	8	4	no	—
214	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
215	C₂³×C₈₉	C₂×#41	1	2	yes	—
216	C₂³×C₈₉	C₂×#41	1	2	yes	—
217	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
218	Q₈×C₈₉	—	4	16	no	—
219	Q₈×C₈₉	—	4	4	no	—
220	D₄×C₈₉	—	4	8	no	—
221	D₄×C₈₉	—	4	8	no	—
222	D₄×C₈₉	—	4	8	no	—
223	D₄×C₈₉	—	4	4	no	—
224	D₄×C₈₉	—	8	16	no	—
225	C₂×C₄×C₈₉	—	2	2	no	—
226	C₂×C₄×C₈₉	—	2	4	no	—
227	C₂×C₄×C₈₉	—	2	2	no	—
228	C₂×C₄×C₈₉	—	2	4	no	—
229	C₂×C₄×C₈₉	—	2	2	no	—
230	C₂×C₄×C₈₉	—	2	4	no	—
231	C₂×C₄×C₈₉	—	2	2	no	—
232	C₂×C₄×C₈₉	—	2	4	no	—
233	C₂×C₄×C₈₉	—	2	4	no	—
234	C₂×C₄×C₈₉	—	2	2	no	—
235	C₂×C₄×C₈₉	—	2	2	no	—
236	C₂×C₄×C₈₉	—	2	4	no	—
237	C₂×C₄×C₈₉	—	2	8	no	—
238	C₂×C₄×C₈₉	—	2	2	no	—
239	C₂×C₄×C₈₉	—	2	4	no	—
240	C₂×C₄×C₈₉	—	2	4	no	—
241	C₂×C₄×C₈₉	—	2	2	no	—
242	C₂×C₄×C₈₉	—	2	4	no	—
243	C₂×C₄×C₈₉	—	4	8	no	—
244	C₂×C₄×C₈₉	—	4	8	no	—
245	C₂×C₄×C₈₉	—	8	4	no	—
246	C₂×C₄×C₈₉	#124	1	16	yes	complemented
247	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
248	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
251	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
252	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
253	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
254	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
255	C₂²×C₈₉	—	2	4	no	—
256	C₂²×C₈₉	—	2	4	no	—
257	C₂²×C₈₉	—	2	2	no	—
258	C₂²×C₈₉	—	2	2	no	—
259	C₂²×C₈₉	—	4	2	no	—
260	C₂²×C₈₉	—	4	4	no	—
261	C₂²×C₈₉	—	4	4	no	—
262	C₂²×C₈₉	—	4	8	no	—
263	C₂²×C₈₉	—	4	4	no	—
264	C₂²×C₈₉	—	4	8	no	—
265	C₂²×C₈₉	—	4	2	no	—
266	C₂²×C₈₉	—	4	4	no	—
267	C₂²×C₈₉	—	4	2	no	—
268	C₂²×C₈₉	—	4	4	no	—
269	C₂²×C₈₉	—	8	16	no	—
270	C₂²×C₈₉	C₂×#124	1	4	yes	—
271	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
272	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
273	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
274	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
275	C₂²×C₈₉	#585	1	2	yes	—
276	C₄×C₈₉	—	2	4	no	—
277	C₄×C₈₉	—	2	8	no	—
278	C₄×C₈₉	—	4	2	no	—
279	C₄×C₈₉	—	4	2	no	—
280	C₄×C₈₉	—	4	4	no	—
281	C₄×C₈₉	—	4	4	no	—
282	C₄×C₈₉	—	4	4	no	—
283	C₄×C₈₉	C₂×#124	1	4	yes	—
284	C₄×C₈₉	C₂×#124	1	4	yes	—
285	#35556	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
286	C₂×C₈₉	—	2	4	no	—
287	C₂×C₈₉	—	8	2	no	—
288	C₂×C₈₉	—	8	4	no	—
289	C₂×C₈₉	C₂²×#124	1	1	yes	characteristic
290	C₂×C₈₉	#3488	1	2	yes	—
291	C₂×C₈₉	#3506	1	2	yes	—
292	C₂×C₈₉	#3508	1	1	yes	characteristic
293	C₂×C₈₉	#3510	1	1	yes	characteristic
294	C₂×#535	C₂×C₈₉	1	1	yes	characteristic
295	C₂×#533	C₂×C₈₉	1	1	yes	characteristic, complemented
296	#2592	C₂×C₈₉	1	4	yes	complemented
297	#2548	C₂×C₈₉	1	4	yes	complemented
298	#2491	C₂×C₈₉	1	4	yes	complemented
299	#2490	C₂×C₈₉	1	4	yes	complemented
300	#2472	C₂×C₈₉	1	8	yes	complemented
301	#2311	C₂×C₈₉	1	1	yes	characteristic, complemented
302	#2309	C₂×C₈₉	1	1	yes	characteristic, complemented
303	#2297	C₂×C₈₉	1	1	yes	characteristic, complemented
304	C₂×#390	C₂×C₈₉	1	1	yes	characteristic, complemented
305	#2274	C₂×C₈₉	1	1	yes	characteristic, complemented
306	C₈₉	#35556	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
308	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
309	#535	—	2	8	no	—
310	#533	—	2	8	no	—
311	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	complemented
312	C₂×#127	C₂²×C₈₉	1	2	yes	—
313	C₂×#125	C₂²×C₈₉	1	2	yes	complemented
314	C₂×#124	C₂²×C₈₉	1	2	yes	—
315	C₂×#124	C₂²×C₈₉	1	8	yes	complemented
316	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
317	C₂×#121	C₂²×C₈₉	1	4	yes	—
318	C₂×#120	C₂²×C₈₉	1	4	yes	—
319	C₂×C₄×Q₈	C₂²×C₈₉	1	4	yes	complemented
320	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
321	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
322	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
323	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
324	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
325	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
326	#397	C₂²×C₈₉	1	8	yes	complemented
327	#394	C₂²×C₈₉	1	8	yes	complemented
328	#394	C₂²×C₈₉	1	8	yes	complemented
329	#393	C₂²×C₈₉	1	8	yes	complemented
330	#391	C₂²×C₈₉	1	2	yes	complemented
331	#391	C₂²×C₈₉	1	2	yes	complemented
332	#390	C₂²×C₈₉	1	4	yes	—
333	#390	C₂²×C₈₉	1	4	yes	—
334	#390	C₂²×C₈₉	1	4	yes	—
335	#390	C₂²×C₈₉	1	2	yes	—
336	#390	C₂²×C₈₉	1	2	yes	—
337	#390	C₂²×C₈₉	1	2	yes	complemented
338	#390	C₂²×C₈₉	1	8	yes	complemented
339	#390	C₂²×C₈₉	1	2	yes	complemented
340	#388	C₂²×C₈₉	1	8	yes	complemented
341	#388	C₂²×C₈₉	1	8	yes	complemented
342	#386	C₂²×C₈₉	1	4	yes	—
343	#386	C₂²×C₈₉	1	4	yes	—
344	#380	C₂²×C₈₉	1	4	yes	complemented
345	#379	C₂²×C₈₉	1	4	yes	—
346	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	complemented
347	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	complemented
348	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
349	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	complemented
350	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	complemented
351	C₂⁵	C₂³×C₈₉	1	2	yes	—
352	C₂×#41	—	2	4	no	—
353	C₂×#41	—	2	4	no	—
354	C₂×#41	—	2	4	no	—
355	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
356	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
357	C₂²×D₄	—	2	4	no	—
358	C₂²×D₄	—	2	4	no	—
359	C₂²×D₄	—	2	4	no	—
360	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
361	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
362	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
363	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
364	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
365	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
366	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
367	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
368	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
369	C₄⋊Q₈	—	2	8	no	—
370	#127	—	2	8	no	—
371	#125	—	2	8	no	—
372	#124	—	2	8	no	—
373	#124	—	4	16	no	—
374	C₂²⋊Q₈	—	2	16	no	—
375	#121	—	2	16	no	—
376	#120	—	2	16	no	—
377	C₄×Q₈	—	2	16	no	—
378	#117	—	2	4	no	—
379	#117	—	2	4	no	—
380	#117	D₄×C₈₉	1	8	yes	—
381	C₂×C₄⋊C₄	—	2	4	no	—
382	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
387	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
388	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
390	C₂×C₂²⋊C₄	—	4	8	no	—
391	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
402	C₂×C₄²	—	2	4	no	—
403	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
404	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
405	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
406	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
408	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
409	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
410	#95	C₂³×C₈₉	1	8	yes	—
411	#95	C₂³×C₈₉	1	4	yes	—
412	#95	C₂³×C₈₉	1	4	yes	—
413	#95	C₂³×C₈₉	1	8	yes	—
414	C₂⁴	—	2	4	no	—
415	C₂⁴	—	2	4	no	—
416	C₂⁴	—	2	4	no	—
417	C₂⁴	—	4	4	no	—
418	C₂⁴	—	4	4	no	—
419	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
420	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
421	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
422	#41	—	4	16	no	—
423	C₂×Q₈	—	2	8	no	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	2	8	no	—
428	C₂×Q₈	—	4	4	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	16	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	4	8	no	—
443	C₂×D₄	—	4	8	no	—
444	C₂×D₄	—	4	16	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	4	no	—
453	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
454	C₂²×C₄	#41×C₈₉	1	4	yes	—
455	C₂²×C₄	#41×C₈₉	1	4	yes	—
456	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
458	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
461	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
462	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	—	2	8	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	2	16	no	—
476	C₂²⋊C₄	—	2	16	no	—
477	C₂²⋊C₄	—	2	8	no	—
478	C₂²⋊C₄	—	4	8	no	—
479	C₂²⋊C₄	—	4	8	no	—
480	C₂²⋊C₄	—	4	8	no	—
481	C₂²⋊C₄	—	4	4	no	—
482	C₂²⋊C₄	—	4	8	no	—
483	C₂²⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
484	C₄²	—	2	4	no	—
485	C₄²	—	2	2	no	—
486	C₄²	—	2	2	no	—
487	C₄²	—	2	4	no	—
488	C₄²	—	2	4	no	—
489	C₄²	—	4	8	no	—
490	C₄²	C₂×D₄×C₈₉	1	8	yes	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	4	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	4	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	8	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	2	8	no	—
503	C₂³	—	4	16	no	—
504	C₂³	—	4	8	no	—
505	C₂³	—	8	8	no	—
506	C₂³	—	8	4	no	—
507	C₂³	—	8	4	no	—
508	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
509	C₂³	C₂×#41×C₈₉	1	2	yes	—
510	C₂³	C₂×#41×C₈₉	1	2	yes	—
511	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
512	Q₈	—	4	16	no	—
513	Q₈	—	4	4	no	—
514	D₄	—	4	8	no	—
515	D₄	—	4	8	no	—
516	D₄	—	4	8	no	—
517	D₄	—	4	4	no	—
518	D₄	—	8	16	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	8	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	4	8	no	—
538	C₂×C₄	—	4	8	no	—
539	C₂×C₄	—	8	4	no	—
540	C₂×C₄	#124×C₈₉	1	16	yes	complemented
541	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
542	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
543	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
544	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
545	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
546	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
547	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
548	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	2	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	4	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	4	8	no	—
557	C₂²	—	4	8	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	4	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	8	16	no	—
564	C₂²	C₂×#124×C₈₉	1	4	yes	—
565	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
566	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
567	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
568	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
569	C₂²	#585×C₈₉	1	2	yes	—
570	C₄	—	2	4	no	—
571	C₄	—	2	8	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	2	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	4	no	—
576	C₄	—	4	4	no	—
577	C₄	C₂×#124×C₈₉	1	4	yes	—
578	C₄	C₂×#124×C₈₉	1	4	yes	—
579	C₂	—	2	4	no	—
580	C₂	—	8	2	no	—
581	C₂	—	8	4	no	—
582	C₂	C₂²×#124×C₈₉	1	1	yes	characteristic
583	C₂	C₈₉×#3488	1	2	yes	—
584	C₂	C₈₉×#3506	1	2	yes	—
585	C₂	C₈₉×#3508	1	1	yes	characteristic
586	C₂	C₈₉×#3510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂×#535×C₈₉	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#533×C₈₉	C₂	semidirect
3. C₈₉×#2592	C₂	semidirect
4. C₈₉×#2548	C₂	semidirect
5. C₈₉×#2491	C₂	semidirect
6. C₈₉×#2490	C₂	semidirect
7. C₈₉×#2472	C₂	semidirect
8. C₈₉×#2311	C₂	semidirect
9. C₈₉×#2309	C₂	semidirect
10. C₈₉×#2297	C₂	semidirect
11. C₂×#390×C₈₉	C₂	semidirect
12. C₈₉×#2274	C₂	semidirect
17. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
19. C₂×#125×C₈₉	C₂²	semidirect
21. C₂×#124×C₈₉	C₂²	semidirect
25. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
32. #397×C₈₉	C₂²	semidirect
33. #394×C₈₉	C₂²	semidirect
34. #394×C₈₉	C₂²	semidirect
35. #393×C₈₉	C₂²	semidirect
36. #391×C₈₉	C₂²	semidirect
37. #391×C₈₉	C₂²	semidirect
43. #390×C₈₉	C₂²	semidirect
44. #390×C₈₉	C₂²	semidirect
45. #390×C₈₉	C₂²	semidirect
46. #388×C₈₉	C₂²	semidirect
47. #388×C₈₉	C₂²	semidirect
50. #380×C₈₉	C₂²	semidirect
52. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
53. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
55. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
56. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
246. C₂×C₄×C₈₉	#124	semidirect
285. #35556	C₈₉	direct (N × H)
295. C₂×#533	C₂×C₈₉	semidirect
296. #2592	C₂×C₈₉	semidirect
297. #2548	C₂×C₈₉	semidirect
298. #2491	C₂×C₈₉	semidirect
299. #2490	C₂×C₈₉	semidirect
300. #2472	C₂×C₈₉	semidirect
301. #2311	C₂×C₈₉	semidirect
302. #2309	C₂×C₈₉	semidirect
303. #2297	C₂×C₈₉	semidirect
304. C₂×#390	C₂×C₈₉	semidirect
305. #2274	C₂×C₈₉	semidirect
306. C₈₉	#35556	direct (N × H)
311. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
313. C₂×#125	C₂²×C₈₉	semidirect
315. C₂×#124	C₂²×C₈₉	semidirect
319. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
326. #397	C₂²×C₈₉	semidirect
327. #394	C₂²×C₈₉	semidirect
328. #394	C₂²×C₈₉	semidirect
329. #393	C₂²×C₈₉	semidirect
330. #391	C₂²×C₈₉	semidirect
331. #391	C₂²×C₈₉	semidirect
337. #390	C₂²×C₈₉	semidirect
338. #390	C₂²×C₈₉	semidirect
339. #390	C₂²×C₈₉	semidirect
340. #388	C₂²×C₈₉	semidirect
341. #388	C₂²×C₈₉	semidirect
344. #380	C₂²×C₈₉	semidirect
346. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
347. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
349. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
350. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
540. C₂×C₄	#124×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#535×C₈₉	2
C₂×#533×C₈₉	2
C₈₉×#2592	2
C₈₉×#2548	2
C₈₉×#2491	2
C₈₉×#2490	2
C₈₉×#2472	2
C₈₉×#2311	2
C₈₉×#2309	2
C₈₉×#2297	2
C₂×#390×C₈₉	2
C₈₉×#2274	2
#35556	89

Subgroups — C89×#35556 #36413404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35556 #36413404