Subgroups — #36712063

Order 22784.

482 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032674	C₂	1	1	yes	maximal, characteristic
2	#26032644	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022462	C₂	1	2	yes	maximal, complemented
4	#26022434	C₂	1	2	yes	maximal, complemented
5	#26022401	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022396	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022366	C₂	1	2	yes	maximal, complemented
8	C₄×#15667957	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022085	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022037	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152314	—	2	1	no	—
13	#19152309	—	2	1	no	—
14	C₂×#15667957	C₂²	1	1	yes	characteristic
15	C₂×#15667951	C₂²	1	2	yes	—
16	#19151935	—	2	2	no	—
17	#19151925	—	2	2	no	—
18	#19151890	—	2	1	no	—
19	#19151885	—	2	1	no	—
20	#19151840	C₂²	1	2	yes	—
21	#19151835	C₂²	1	2	yes	—
22	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
25	#19151635	C₂²	1	2	yes	—
26	#107×D₈₉	C₂²	1	2	yes	—
27	#19151426	C₂²	1	1	yes	characteristic
28	#19151425	C₂²	1	2	yes	—
29	#19151423	C₂²	1	2	yes	—
30	C₄⋊#13083549	C₂²	1	1	yes	characteristic
31	#19151403	C₂²	1	1	yes	characteristic
32	#19151401	C₂²	1	2	yes	—
33	#19151400	C₂²	1	1	yes	characteristic
34	#19151398	C₂²	1	2	yes	—
35	#19151377	C₂²	1	2	yes	—
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	#500×C₈₉	C₂²	1	1	yes	characteristic
38	#366×C₈₉	C₂²	1	2	yes	—
39	#366×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₈₉	C₂³	1	2	yes	—
44	#15668003	—	4	1	no	—
45	#15668001	—	4	2	no	—
46	#15667992	—	2	2	no	—
47	#15667988	—	4	2	no	—
48	#15667987	—	2	2	no	—
49	#15667987	—	4	1	no	—
50	C₄×Dic₁₇₈	—	2	1	no	—
51	C₄×D₃₅₆	—	2	1	no	—
52	C₄×D₃₅₆	D₄	1	1	yes	characteristic
53	#15667974	—	4	1	no	—
54	#15667970	—	2	2	no	—
55	#15667970	—	4	1	no	—
56	C₄⋊C₄×D₈₉	—	2	2	no	—
57	#15667957	—	2	2	no	—
58	#15667951	—	2	4	no	—
59	C₂×#13083549	C₂³	1	2	yes	—
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667945	—	2	2	no	—
63	#15667943	—	2	2	no	—
64	C₄⋊#10623494	D₄	1	1	yes	characteristic
65	#15667937	—	2	1	no	—
66	#15667935	—	2	2	no	—
67	#15667934	—	2	1	no	—
68	#15667932	—	2	2	no	—
69	#15667924	D₄	1	2	yes	—
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₄⋊Q₈×C₈₉	—	2	1	no	—
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	#107×C₈₉	—	2	1	no	—
75	#107×C₈₉	—	2	1	no	—
76	#107×C₈₉	C₂³	1	2	yes	—
77	#107×C₈₉	C₂³	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×D₃₅₆	—	2	1	no	—
81	C₂×D₃₅₆	—	4	2	no	—
82	C₂×D₃₅₆	D₈	1	1	yes	characteristic
83	C₂×C₄×D₈₉	—	4	1	no	—
84	C₂×C₄×D₈₉	—	4	2	no	—
85	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
86	Dic₃₅₆	—	2	1	no	—
87	Dic₃₅₆	—	2	1	no	—
88	D₇₁₂	—	2	1	no	—
89	D₇₁₂	—	2	1	no	—
90	C₂×#10623494	C₂×D₄	1	2	yes	—
91	C₈×D₈₉	—	2	2	no	—
92	C₈×D₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	2	1	no	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	—	4	2	no	—
96	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
97	#13083553	—	4	1	no	—
98	#13083553	—	4	2	no	—
99	#13083551	—	4	2	no	—
100	#13083551	—	4	2	no	—
101	#13083551	—	4	1	no	—
102	#13083551	—	4	2	no	—
103	#13083551	D₈	1	2	yes	—
104	C₄×Dic₈₉	—	4	1	no	—
105	C₄×Dic₈₉	—	4	2	no	—
106	#13083549	—	2	2	no	—
107	C₂×Q₈×C₈₉	—	4	1	no	—
108	C₂×Q₈×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	2	no	—
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	4	1	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₈₉	—	2	1	no	—
117	C₁₆×C₈₉	#41	1	2	yes	—
118	C₂²×D₈₉	—	4	1	no	—
119	C₂²×D₈₉	C₂×D₈	1	2	yes	—
120	Q₈×C₈₉	—	16	1	no	—
121	Q₈×C₈₉	—	16	1	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₈₉	—	4	1	no	—
128	C₂×Dic₈₉	—	4	1	no	—
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
131	D₃₅₆	—	2	2	no	—
132	D₃₅₆	—	4	1	no	—
133	D₃₅₆	—	4	1	no	—
134	D₃₅₆	—	16	2	no	—
135	C₄×D₈₉	—	2	2	no	—
136	C₄×D₈₉	—	2	2	no	—
137	C₄×D₈₉	—	8	2	no	—
138	C₄×D₈₉	—	16	2	no	—
139	Dic₁₇₈	—	4	1	no	—
140	Dic₁₇₈	—	4	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
144	#10623494	—	2	2	no	—
145	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₈₉	—	2	2	no	—
147	C₂×D₈₉	—	2	2	no	—
148	C₂×D₈₉	—	2	2	no	—
149	C₂×D₈₉	—	8	1	no	—
150	C₂×D₈₉	—	8	1	no	—
151	C₂×D₈₉	—	8	1	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	8	2	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	#510	1	2	yes	—
156	C₄×C₈₉	#547	1	1	yes	characteristic
157	C₄×C₈₉	#578	1	1	yes	characteristic
158	Dic₈₉	—	2	2	no	—
159	Dic₈₉	—	8	1	no	—
160	Dic₈₉	—	8	1	no	—
161	Dic₈₉	—	8	1	no	—
162	#33924	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	#3334	1	1	yes	characteristic
164	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
165	C₂×C₈₉	#3415	1	1	yes	characteristic
166	D₈₉	—	4	2	no	—
167	D₈₉	—	16	1	no	—
168	#3341	—	89	1	no	—
169	#3338	—	89	1	no	—
170	#2252	D₈₉	1	1	yes	characteristic, complemented
171	#2237	—	89	2	no	—
172	#2232	—	89	2	no	—
173	#2227	—	89	1	no	—
174	#2226	—	89	1	no	—
175	#2220	—	89	2	no	—
176	C₄×SD₃₂	—	89	1	no	—
177	#2169	—	89	1	no	—
178	#2159	—	89	2	no	—
179	C₈₉	#33924	1	1	yes	characteristic, complemented, Sylow
180	#555	—	178	1	no	—
181	#554	—	178	1	no	—
182	C₂×SD₃₂	—	89	1	no	—
183	C₂×M₃₂	—	89	2	no	—
184	#500	C₂×D₈₉	1	1	yes	characteristic
185	#500	C₂×D₈₉	1	1	yes	characteristic
186	#484	—	178	2	no	—
187	#482	—	178	2	no	—
188	#477	—	178	1	no	—
189	#476	—	178	1	no	—
190	#467	—	89	2	no	—
191	#466	—	89	2	no	—
192	C₄×Q₁₆	—	89	1	no	—
193	C₄×D₈	—	89	1	no	—
194	C₈×D₄	—	89	1	no	—
195	#427	—	89	2	no	—
196	C₂×#107	—	89	2	no	—
197	#367	—	89	1	no	—
198	#367	—	89	2	no	—
199	#366	—	89	2	no	—
200	#366	C₂×D₈₉	1	2	yes	—
201	#366	C₂×D₈₉	1	2	yes	—
202	C₄⋊C₁₆	—	89	1	no	—
203	#358	—	89	1	no	—
204	#358	—	89	2	no	—
205	#357	—	89	1	no	—
206	#357	—	89	2	no	—
207	C₂²⋊C₁₆	—	89	2	no	—
208	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
209	C₂×Q₁₆	—	89	1	no	—
210	C₂×D₈	—	89	1	no	—
211	C₂²×C₈	—	89	2	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	#125	—	356	1	no	—
215	#124	—	356	2	no	—
216	C₂²⋊Q₈	—	356	2	no	—
217	C₂²⋊Q₈	—	178	2	no	—
218	#121	—	356	1	no	—
219	#121	—	178	2	no	—
220	C₄×Q₈	—	178	1	no	—
221	C₄×D₄	—	89	1	no	—
222	C₄×D₄	—	356	1	no	—
223	C₄×D₄	—	178	1	no	—
224	#117	—	356	1	no	—
225	#117	—	178	2	no	—
226	C₂×C₄⋊C₄	—	178	2	no	—
227	SD₃₂	—	178	2	no	—
228	M₃₂	—	178	4	no	—
229	C₂×C₁₆	—	89	2	no	—
230	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
232	#107	—	2	1	no	—
233	#107	—	2	1	no	—
234	#107	—	89	1	no	—
235	#107	—	89	1	no	—
236	#107	—	178	2	no	—
237	#107	C₂²×D₈₉	1	2	yes	—
238	#107	C₂²×D₈₉	1	2	yes	—
239	#106	—	178	2	no	—
240	C₄⋊C₈	—	89	1	no	—
241	#103	—	178	2	no	—
242	#103	—	178	1	no	—
243	#102	—	178	2	no	—
244	#102	—	178	1	no	—
245	C₂²⋊C₈	—	89	2	no	—
246	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	178	1	no	—
250	C₂×D₄	—	89	1	no	—
251	C₂×D₄	—	356	2	no	—
252	C₂×D₄	—	178	1	no	—
253	C₂²×C₄	—	89	2	no	—
254	C₂²×C₄	—	356	2	no	—
255	C₂²×C₄	—	356	1	no	—
256	Q₁₆	—	178	1	no	—
257	Q₁₆	—	178	1	no	—
258	D₈	—	178	1	no	—
259	D₈	—	178	1	no	—
260	C₂×C₈	—	89	2	no	—
261	C₂×C₈	—	178	2	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	89	1	no	—
270	C₄⋊C₄	—	356	2	no	—
271	C₄⋊C₄	—	356	2	no	—
272	C₄⋊C₄	—	356	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₂²⋊C₄	—	89	2	no	—
276	C₂²⋊C₄	—	356	2	no	—
277	C₂²⋊C₄	—	356	2	no	—
278	C₂²⋊C₄	—	356	2	no	—
279	C₂²⋊C₄	—	356	1	no	—
280	C₄²	—	356	2	no	—
281	C₄²	—	356	1	no	—
282	C₄²	D₄×D₈₉	1	1	yes	characteristic
283	C₁₆	—	2	1	no	—
284	C₁₆	—	178	2	no	—
285	C₁₆	#13083575	1	2	yes	—
286	C₂³	—	89	2	no	—
287	C₂³	—	356	1	no	—
288	Q₈	—	16	1	no	—
289	Q₈	—	16	1	no	—
290	Q₈	—	356	1	no	—
291	Q₈	—	356	1	no	—
292	D₄	—	356	1	no	—
293	D₄	—	356	1	no	—
294	D₄	—	1424	2	no	—
295	D₄	—	178	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	89	2	no	—
299	C₂×C₄	—	356	1	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	1424	2	no	—
303	C₂×C₄	—	178	2	no	—
304	C₂×C₄	—	178	2	no	—
305	C₂×C₄	—	712	2	no	—
306	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
307	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	178	2	no	—
311	C₈	C₂×#13083575	1	1	yes	characteristic
312	C₈	#15668060	1	1	yes	characteristic
313	C₂²	—	178	2	no	—
314	C₂²	—	178	2	no	—
315	C₂²	—	178	2	no	—
316	C₂²	—	712	1	no	—
317	C₂²	—	712	1	no	—
318	C₂²	—	712	1	no	—
319	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	178	2	no	—
324	C₄	—	712	1	no	—
325	C₄	—	712	1	no	—
326	C₄	—	712	1	no	—
327	C₄	#19152040	1	2	yes	—
328	C₄	#19152251	1	1	yes	characteristic
329	C₄	#19152411	1	1	yes	characteristic
330	C₂	—	356	2	no	—
331	C₂	—	1424	1	no	—
332	C₂	#26032599	1	1	yes	characteristic
333	C₂	C₂×#19152040	1	1	yes	characteristic
334	C₂	#26033389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2252	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26032674	C₂
hypercenter	#2252	D₈₉
hypocenter	C₈₉	#33924
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #26032644	C₂	semidirect
3. #26022462	C₂	semidirect
4. #26022434	C₂	semidirect
5. #26022401	C₂	semidirect
6. #26022396	C₂	semidirect
7. #26022366	C₂	semidirect
8. C₄×#15667957	C₂	semidirect
9. #26022085	C₂	semidirect
10. #26022037	C₂	semidirect
11. C₈₉×#2252	C₂	semidirect
170. #2252	D₈₉	semidirect
179. C₈₉	#33924	semidirect

Maximal subgroups

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

subgroup	index
#26032674	2
#26032644	2
#26022462	2
#26022434	2
#26022401	2
#26022396	2
#26022366	2
C₄×#15667957	2
#26022085	2
#26022037	2
C₈₉×#2252	2
#33924	89