Subgroups — #36711784

Order 22784.

548 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032601	C₂	1	1	yes	maximal, characteristic, complemented
2	#26032599	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022525	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022434	C₂	1	2	yes	maximal, complemented
5	#26022431	C₂	1	2	yes	maximal, complemented
6	#26022390	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022337	C₂	1	2	yes	maximal, complemented
8	C₄×#15667953	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022071	C₂	1	1	yes	maximal, characteristic, complemented
10	#366×D₈₉	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2224	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152253	—	2	1	no	—
13	#19152251	—	2	1	no	—
14	C₂×#15667953	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151913	—	2	2	no	—
18	#19151910	—	2	2	no	—
19	#19151878	—	2	1	no	—
20	#19151875	—	2	1	no	—
21	#19151835	C₂²	1	2	yes	—
22	#19151831	C₂²	1	2	yes	—
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
25	#107×D₈₉	C₂²	1	2	yes	—
26	#107×D₈₉	C₂²	1	2	yes	—
27	#19151424	C₂²	1	2	yes	—
28	#19151423	C₂²	1	2	yes	—
29	#19151423	C₂²	1	2	yes	—
30	#19151423	C₂²	1	1	yes	characteristic
31	#19151400	C₂²	1	2	yes	—
32	#19151399	C₂²	1	1	yes	characteristic
33	#19151398	C₂²	1	1	yes	characteristic
34	#19151377	C₂²	1	2	yes	—
35	C₄×#13083549	C₂²	1	1	yes	characteristic
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
38	#366×C₈₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#357×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083575	—	4	1	no	—
42	C₂×#13083574	—	4	1	no	—
43	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₈₉	C₂³	1	2	yes	—
45	#15668012	—	2	1	no	—
46	#15667997	—	4	2	no	—
47	#15667994	—	4	2	no	—
48	#15667991	—	4	1	no	—
49	#15667988	—	4	1	no	—
50	#15667987	—	2	2	no	—
51	#15667983	—	2	2	no	—
52	C₄×D₃₅₆	—	2	1	no	—
53	C₄×D₃₅₆	D₄	1	1	yes	characteristic
54	C₄⋊C₄×D₈₉	—	2	2	no	—
55	C₄⋊C₄×D₈₉	—	2	2	no	—
56	#15667953	—	2	2	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₁₆×D₈₉	—	2	4	no	—
60	#15667946	—	2	1	no	—
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667946	C₂³	1	2	yes	—
63	#15667945	—	2	2	no	—
64	#15667945	—	2	2	no	—
65	C₄⋊#10623494	D₄	1	1	yes	characteristic
66	#15667934	—	2	1	no	—
67	#15667933	—	2	2	no	—
68	#15667932	—	2	2	no	—
69	#15667924	D₄	1	2	yes	—
70	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₈₉	—	2	1	no	—
72	C₄×D₄×C₈₉	—	2	1	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
74	#107×C₈₉	—	2	1	no	—
75	#107×C₈₉	C₂³	1	1	yes	characteristic
76	#107×C₈₉	C₂³	1	2	yes	—
77	#102×C₈₉	—	2	1	no	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	#13083575	—	8	4	no	—
80	#13083574	—	8	4	no	—
81	C₂×Dic₁₇₈	—	4	1	no	—
82	C₂×D₃₅₆	—	2	1	no	—
83	C₂×D₃₅₆	—	4	2	no	—
84	C₂×D₃₅₆	D₈	1	1	yes	characteristic
85	C₂×C₂²⋊D₈₉	—	4	2	no	—
86	C₂×C₄×D₈₉	—	4	2	no	—
87	C₂×C₄×D₈₉	—	4	2	no	—
88	C₂×C₄×D₈₉	—	4	1	no	—
89	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
90	D₇₁₂	—	2	1	no	—
91	D₇₁₂	—	2	1	no	—
92	C₂×#10623494	C₂×D₄	1	2	yes	—
93	C₈×D₈₉	—	2	2	no	—
94	C₈×D₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
99	#13083553	—	4	2	no	—
100	#13083553	—	4	2	no	—
101	C₂²⋊Dic₈₉	—	4	2	no	—
102	#13083551	—	4	1	no	—
103	#13083551	—	4	2	no	—
104	#13083551	D₈	1	2	yes	—
105	#13083549	—	2	1	no	—
106	#13083549	#41	1	2	yes	complemented
107	C₂×Q₈×C₈₉	—	4	1	no	—
108	C₂×D₄×C₈₉	—	2	1	no	—
109	C₂²×C₄×C₈₉	—	4	1	no	—
110	D₈×C₈₉	—	2	1	no	—
111	D₈×C₈₉	—	2	1	no	—
112	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	—	2	2	no	—
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₂²⋊C₄×C₈₉	—	4	1	no	—
118	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₈₉	—	2	2	no	—
120	C₂³×C₈₉	—	4	1	no	—
121	C₂²×D₈₉	—	4	1	no	—
122	C₂²×D₈₉	C₂×D₈	1	2	yes	—
123	Q₈×C₈₉	—	8	2	no	—
124	D₄×C₈₉	—	4	1	no	—
125	D₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	8	2	no	—
129	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₈₉	—	8	4	no	—
133	C₂×Dic₈₉	—	4	1	no	—
134	C₂×Dic₈₉	—	4	2	no	—
135	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
136	D₃₅₆	—	4	1	no	—
137	D₃₅₆	—	4	1	no	—
138	D₃₅₆	—	8	4	no	—
139	D₃₅₆	D₁₆	1	4	yes	complemented
140	C₄×D₈₉	—	2	2	no	—
141	C₄×D₈₉	—	2	2	no	—
142	C₄×D₈₉	—	8	4	no	—
143	C₄×D₈₉	—	8	4	no	—
144	C₄×D₈₉	—	8	2	no	—
145	Dic₁₇₈	—	8	2	no	—
146	C₈×C₈₉	—	2	1	no	—
147	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
148	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
149	#10623494	—	2	2	no	—
150	C₂²×C₈₉	—	8	1	no	—
151	C₂²×C₈₉	—	8	1	no	—
152	C₂²×C₈₉	—	8	1	no	—
153	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₈₉	—	2	2	no	—
155	C₂×D₈₉	—	2	2	no	—
156	C₂×D₈₉	—	8	1	no	—
157	C₂×D₈₉	—	8	1	no	—
158	C₂×D₈₉	—	8	1	no	—
159	C₂×D₈₉	C₂×D₁₆	1	4	yes	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	8	1	no	—
162	C₄×C₈₉	—	8	2	no	—
163	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
164	C₄×C₈₉	#547	1	1	yes	characteristic
165	C₄×C₈₉	#578	1	1	yes	characteristic
166	Dic₈₉	—	2	2	no	—
167	Dic₈₉	—	8	2	no	—
168	Dic₈₉	—	8	1	no	—
169	#33901	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	—	16	1	no	—
171	C₂×C₈₉	#3334	1	1	yes	characteristic
172	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
173	C₂×C₈₉	#3417	1	1	yes	characteristic
174	D₈₉	—	2	4	no	—
175	D₈₉	—	16	1	no	—
176	#3334	—	89	1	no	—
177	#3334	—	89	1	no	—
178	#2252	—	89	1	no	—
179	#2232	—	89	2	no	—
180	#2232	—	89	2	no	—
181	#2224	—	89	1	no	—
182	#2224	D₈₉	1	1	yes	characteristic, complemented
183	#2215	—	89	2	no	—
184	C₄×D₁₆	—	89	1	no	—
185	D₄×C₁₆	—	89	1	no	—
186	C₂×#366	—	89	2	no	—
187	C₈₉	#33901	1	1	yes	characteristic, complemented, Sylow
188	#547	—	178	1	no	—
189	#547	—	178	1	no	—
190	C₂×D₁₆	—	89	1	no	—
191	C₂²×C₁₆	—	89	2	no	—
192	#500	—	89	1	no	—
193	#500	C₂×D₈₉	1	1	yes	characteristic
194	#480	—	178	2	no	—
195	#480	—	178	2	no	—
196	#474	—	178	1	no	—
197	#474	—	178	1	no	—
198	#466	—	89	2	no	—
199	#466	—	89	2	no	—
200	C₄×D₈	—	89	1	no	—
201	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
202	C₈×D₄	—	89	1	no	—
203	C₂×#107	—	89	2	no	—
204	C₂×#107	—	89	2	no	—
205	#366	—	89	2	no	—
206	#366	—	89	1	no	—
207	#366	—	89	2	no	—
208	#366	—	89	2	no	—
209	#366	C₂×D₈₉	1	2	yes	—
210	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
211	#357	—	89	1	no	—
212	#357	—	89	2	no	—
213	#357	—	89	1	no	—
214	#357	C₂×D₈₉	1	2	yes	—
215	C₂²⋊C₁₆	—	89	2	no	—
216	C₄×C₁₆	—	89	1	no	—
217	C₂×#41	—	356	1	no	—
218	C₂×#41	—	356	1	no	—
219	C₂×D₈	—	89	1	no	—
220	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
221	C₂²×C₈	—	89	2	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	178	1	no	—
224	#123	—	356	2	no	—
225	#123	—	356	2	no	—
226	C₂²⋊Q₈	—	356	1	no	—
227	C₂²⋊Q₈	—	356	1	no	—
228	#121	—	178	2	no	—
229	#121	—	178	2	no	—
230	C₄×D₄	—	2	1	no	—
231	C₄×D₄	—	89	1	no	—
232	C₄×D₄	—	178	1	no	—
233	C₂×C₄⋊C₄	—	178	2	no	—
234	C₂×C₄⋊C₄	—	178	2	no	—
235	D₁₆	—	178	2	no	—
236	C₂×C₁₆	—	89	1	no	—
237	C₂×C₁₆	—	89	1	no	—
238	C₂×C₁₆	—	178	4	no	—
239	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
240	#107	—	2	1	no	—
241	#107	—	89	1	no	—
242	#107	—	89	2	no	—
243	#107	—	178	2	no	—
244	#107	—	178	2	no	—
245	#107	—	178	1	no	—
246	#107	C₂²×D₈₉	1	1	yes	characteristic
247	#107	C₂²×D₈₉	1	2	yes	—
248	C₄⋊C₈	—	89	1	no	—
249	#102	—	2	1	no	—
250	#102	—	178	2	no	—
251	#102	—	178	2	no	—
252	#102	—	178	1	no	—
253	C₂²⋊C₈	—	89	2	no	—
254	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
255	#41	—	712	4	no	—
256	#41	—	712	4	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	356	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	89	1	no	—
261	C₂×D₄	—	356	2	no	—
262	C₂×D₄	—	356	2	no	—
263	C₂×D₄	—	178	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	—	89	2	no	—
266	C₂²×C₄	—	356	2	no	—
267	C₂²×C₄	—	356	2	no	—
268	C₂²×C₄	—	356	1	no	—
269	D₈	—	2	1	no	—
270	D₈	—	2	1	no	—
271	D₈	—	178	1	no	—
272	D₈	—	178	1	no	—
273	C₂×C₈	—	89	2	no	—
274	C₂×C₈	—	178	2	no	—
275	C₂×C₈	—	178	2	no	—
276	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
277	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	89	1	no	—
282	C₄⋊C₄	—	356	2	no	—
283	C₄⋊C₄	—	356	2	no	—
284	C₄⋊C₄	—	356	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₄⋊C₄	—	178	2	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	89	2	no	—
289	C₂²⋊C₄	—	356	2	no	—
290	C₂²⋊C₄	—	356	2	no	—
291	C₂²⋊C₄	—	356	1	no	—
292	C₄²	D₄×D₈₉	1	1	yes	characteristic
293	C₁₆	—	2	2	no	—
294	C₁₆	—	89	2	no	—
295	C₁₆	—	178	1	no	—
296	C₂³	—	4	1	no	—
297	C₂³	—	89	2	no	—
298	C₂³	—	356	1	no	—
299	Q₈	—	8	2	no	—
300	Q₈	—	712	2	no	—
301	D₄	—	4	1	no	—
302	D₄	—	4	1	no	—
303	D₄	—	89	4	no	—
304	D₄	—	356	1	no	—
305	D₄	—	356	1	no	—
306	D₄	—	712	4	no	—
307	D₄	—	712	4	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	89	2	no	—
312	C₂×C₄	—	356	1	no	—
313	C₂×C₄	—	356	2	no	—
314	C₂×C₄	—	178	2	no	—
315	C₂×C₄	—	178	2	no	—
316	C₂×C₄	—	712	4	no	—
317	C₂×C₄	—	712	4	no	—
318	C₂×C₄	—	712	2	no	—
319	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
320	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
321	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	178	2	no	—
324	C₈	#41×D₈₉	1	1	yes	characteristic
325	C₈	#15668060	1	1	yes	characteristic
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	89	4	no	—
330	C₂²	—	178	2	no	—
331	C₂²	—	178	2	no	—
332	C₂²	—	712	1	no	—
333	C₂²	—	712	1	no	—
334	C₂²	—	712	1	no	—
335	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	2	no	—
339	C₄	—	178	2	no	—
340	C₄	—	712	1	no	—
341	C₄	—	712	2	no	—
342	C₄	D₁₆×D₈₉	1	2	yes	—
343	C₄	#19152250	1	1	yes	characteristic
344	C₄	#19152414	1	1	yes	characteristic
345	C₂	—	16	1	no	—
346	C₂	—	1424	1	no	—
347	C₂	—	178	4	no	—
348	C₂	#26032598	1	1	yes	characteristic
349	C₂	C₂×D₁₆×D₈₉	1	1	yes	characteristic
350	C₂	#26033404	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×#2224	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#2224	D₈₉
hypocenter	C₈₉	#33901
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2224

Complements

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

N	H ≅ G/N	type
1. #26032601	C₂	semidirect
2. #26032599	C₂	semidirect
3. #26022525	C₂	semidirect
4. #26022434	C₂	semidirect
5. #26022431	C₂	semidirect
6. #26022390	C₂	semidirect
7. #26022337	C₂	semidirect
8. C₄×#15667953	C₂	semidirect
9. #26022071	C₂	semidirect
10. #366×D₈₉	C₂	semidirect
11. C₈₉×#2224	C₂	semidirect
106. #13083549	#41	semidirect
139. D₃₅₆	D₁₆	semidirect
182. #2224	D₈₉	semidirect
187. C₈₉	#33901	semidirect

Maximal subgroups

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

subgroup	index
#26032601	2
#26032599	2
#26022525	2
#26022434	2
#26022431	2
#26022390	2
#26022337	2
C₄×#15667953	2
#26022071	2
#366×D₈₉	2
C₈₉×#2224	2
#33901	89