Subgroups — #36712100

Order 22784.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033265	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033120	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022484	C₂	1	2	yes	maximal, complemented
4	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022410	C₂	1	2	yes	maximal, complemented
6	#26022405	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022308	C₂	1	2	yes	maximal, complemented
8	#26022288	C₂	1	1	yes	maximal, characteristic
9	C₄×#15667957	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₄₂₄	C₂	1	2	yes	maximal, complemented
11	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152333	—	2	1	no	—
13	#19152308	—	2	1	no	—
14	C₂×#15667957	C₂²	1	1	yes	characteristic
15	C₂×D₁₄₂₄	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19152000	C₂²	1	2	yes	—
18	#19151982	C₂²	1	2	yes	—
19	#19151975	C₂²	1	1	yes	characteristic
20	#19151899	—	2	2	no	—
21	#19151889	—	2	1	no	—
22	#19151814	—	2	2	no	—
23	#19151799	—	2	1	no	—
24	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
25	C₄×D₇₁₂	C₂²	1	2	yes	—
26	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
27	C₄×D₇₁₂	C₂²	1	2	yes	—
28	#19151426	C₂²	1	1	yes	characteristic
29	#19151424	C₂²	1	2	yes	—
30	#19151403	C₂²	1	2	yes	—
31	#19151403	C₂²	1	1	yes	characteristic
32	#19151400	C₂²	1	2	yes	—
33	#19151400	C₂²	1	2	yes	—
34	#19151400	C₂²	1	2	yes	—
35	#19151400	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
39	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×#13083563	—	2	2	no	—
43	C₂×D₇₁₂	—	2	1	no	—
44	C₂×D₇₁₂	C₂³	1	2	yes	—
45	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
46	#15668012	—	2	1	no	—
47	#15668009	—	2	1	no	—
48	#15668004	—	2	2	no	—
49	#15668001	—	4	1	no	—
50	#15668000	—	2	2	no	—
51	#15667997	—	4	2	no	—
52	#15667990	—	4	2	no	—
53	#15667988	—	4	1	no	—
54	C₄×Dic₁₇₈	—	2	1	no	—
55	C₄×D₃₅₆	—	2	1	no	—
56	C₄×D₃₅₆	—	2	2	no	—
57	C₄×D₃₅₆	—	2	2	no	—
58	#15667974	—	4	1	no	—
59	C₄⋊C₄×D₈₉	—	4	1	no	—
60	#15667957	—	2	2	no	—
61	D₁₄₂₄	—	2	4	no	—
62	#15667946	—	2	1	no	—
63	#15667946	C₂³	1	2	yes	—
64	#15667946	C₂³	1	1	yes	characteristic
65	#15667946	C₂³	1	1	yes	characteristic
66	#15667944	—	2	2	no	—
67	#15667937	—	2	1	no	—
68	#15667934	—	2	2	no	—
69	#15667934	—	2	1	no	—
70	#15667934	—	2	2	no	—
71	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
77	C₄×C₈×C₈₉	C₂³	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×D₃₅₆	—	2	2	no	—
83	C₂×D₃₅₆	—	4	1	no	—
84	C₂×D₃₅₆	—	4	1	no	—
85	C₂×C₄×D₈₉	—	4	1	no	—
86	C₂×C₄×D₈₉	—	4	2	no	—
87	C₂×C₄×D₈₉	—	4	2	no	—
88	Dic₃₅₆	—	2	1	no	—
89	Dic₃₅₆	—	2	1	no	—
90	#13083563	—	8	2	no	—
91	D₇₁₂	—	2	2	no	—
92	D₇₁₂	—	2	1	no	—
93	D₇₁₂	—	2	2	no	—
94	D₇₁₂	—	8	1	no	—
95	D₇₁₂	#41	1	2	yes	—
96	C₄⋊Dic₈₉	—	2	1	no	—
97	C₄⋊Dic₈₉	—	2	1	no	—
98	C₄⋊Dic₈₉	—	2	2	no	—
99	C₄⋊Dic₈₉	—	4	1	no	—
100	#13083553	—	4	1	no	—
101	#13083553	—	4	2	no	—
102	#13083553	—	4	2	no	—
103	#13083551	—	4	2	no	—
104	#13083551	—	4	2	no	—
105	#13083551	—	4	1	no	—
106	#13083551	—	4	2	no	—
107	#13083551	—	4	2	no	—
108	C₄×Dic₈₉	—	4	1	no	—
109	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
115	C₄²×C₈₉	C₂×D₄	1	2	yes	—
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	2	no	—
118	C₁₆×C₈₉	—	2	1	no	—
119	C₁₆×C₈₉	—	2	1	no	—
120	C₂²×D₈₉	—	4	1	no	—
121	C₂²×D₈₉	—	4	2	no	—
122	Q₈×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₈₉	—	4	1	no	—
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	—	4	1	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	D₃₅₆	—	2	2	no	—
133	D₃₅₆	—	4	2	no	—
134	D₃₅₆	—	4	1	no	—
135	D₃₅₆	—	4	2	no	—
136	D₃₅₆	—	8	2	no	—
137	D₃₅₆	—	16	1	no	—
138	C₄×D₈₉	—	8	2	no	—
139	C₄×D₈₉	—	16	2	no	—
140	C₄×D₈₉	—	16	2	no	—
141	Dic₁₇₈	—	4	1	no	—
142	Dic₁₇₈	—	4	1	no	—
143	Dic₁₇₈	—	16	1	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	—	2	2	no	—
146	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₈₉	—	4	2	no	—
150	C₂×D₈₉	—	8	1	no	—
151	C₂×D₈₉	—	8	1	no	—
152	C₂×D₈₉	—	8	2	no	—
153	C₂×D₈₉	—	8	2	no	—
154	C₂×D₈₉	—	8	2	no	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	#508	1	2	yes	—
159	C₄×C₈₉	#548	1	1	yes	characteristic
160	C₄×C₈₉	#577	1	1	yes	characteristic
161	Dic₈₉	—	8	1	no	—
162	Dic₈₉	—	8	1	no	—
163	Dic₈₉	—	8	2	no	—
164	Dic₈₉	—	8	1	no	—
165	#33928	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	#3358	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
168	C₂×C₈₉	#3415	1	1	yes	characteristic
169	D₈₉	—	8	2	no	—
170	D₈₉	—	16	2	no	—
171	#3396	—	89	1	no	—
172	#3380	—	89	1	no	—
173	#2242	—	89	2	no	—
174	#2240	—	89	1	no	—
175	#2228	—	89	2	no	—
176	#2227	—	89	1	no	—
177	#2210	—	89	2	no	—
178	#2206	—	89	1	no	—
179	Q₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
180	C₄×SD₃₂	—	89	1	no	—
181	C₄×D₁₆	—	89	2	no	—
182	C₈₉	#33928	1	1	yes	characteristic, complemented, Sylow
183	#559	—	178	1	no	—
184	#554	—	178	1	no	—
185	C₂×SD₃₂	—	89	1	no	—
186	C₂×D₁₆	—	89	2	no	—
187	#500	—	89	1	no	—
188	#499	—	89	2	no	—
189	#495	—	89	2	no	—
190	#493	—	89	1	no	—
191	#478	—	178	2	no	—
192	#476	—	178	1	no	—
193	#463	—	178	2	no	—
194	#460	—	178	1	no	—
195	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
196	C₄×Q₁₆	—	89	1	no	—
197	C₄×D₈	—	89	2	no	—
198	C₄×D₈	—	89	1	no	—
199	C₄×D₈	—	89	2	no	—
200	#367	—	89	1	no	—
201	#366	—	89	2	no	—
202	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
204	#358	—	89	1	no	—
205	#358	—	89	2	no	—
206	#357	—	89	1	no	—
207	#357	—	89	2	no	—
208	#357	—	89	2	no	—
209	#357	—	89	2	no	—
210	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
211	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
212	C₂×Q₁₆	—	89	1	no	—
213	C₂×SD₁₆	—	178	2	no	—
214	C₂×D₈	—	89	1	no	—
215	C₂×D₈	—	89	2	no	—
216	C₂×D₈	—	178	1	no	—
217	C₄⋊Q₈	—	178	1	no	—
218	#127	—	178	1	no	—
219	#125	—	178	2	no	—
220	#124	—	356	1	no	—
221	#124	—	178	2	no	—
222	#123	—	356	2	no	—
223	C₂²⋊Q₈	—	356	1	no	—
224	C₂²⋊Q₈	—	356	2	no	—
225	C₄×Q₈	—	178	1	no	—
226	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
227	C₄×D₄	—	356	1	no	—
228	C₄×D₄	—	178	1	no	—
229	C₄×D₄	—	178	2	no	—
230	C₄×D₄	—	178	2	no	—
231	C₂×C₄⋊C₄	—	356	1	no	—
232	SD₃₂	—	178	2	no	—
233	D₁₆	—	178	4	no	—
234	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
237	#107	—	89	2	no	—
238	#107	—	89	1	no	—
239	#107	—	89	1	no	—
240	#107	—	178	1	no	—
241	#106	—	178	2	no	—
242	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
243	C₄⋊C₈	D₃₅₆	1	2	yes	—
244	#103	—	178	1	no	—
245	#102	—	178	2	no	—
246	#102	—	178	2	no	—
247	#102	—	178	1	no	—
248	C₄×C₈	C₂²×D₈₉	1	2	yes	—
249	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×Q₈	—	356	1	no	—
251	C₂×Q₈	—	178	1	no	—
252	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
253	C₂×D₄	—	356	1	no	—
254	C₂×D₄	—	356	1	no	—
255	C₂×D₄	—	178	2	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂²×C₄	—	356	2	no	—
258	C₂²×C₄	—	356	2	no	—
259	C₂²×C₄	—	356	1	no	—
260	Q₁₆	—	178	1	no	—
261	Q₁₆	—	178	1	no	—
262	SD₁₆	—	712	2	no	—
263	D₈	—	89	2	no	—
264	D₈	—	178	2	no	—
265	D₈	—	178	2	no	—
266	D₈	—	178	1	no	—
267	D₈	—	712	1	no	—
268	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
269	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	356	2	no	—
272	C₄⋊C₄	—	356	1	no	—
273	C₄⋊C₄	—	356	1	no	—
274	C₄⋊C₄	—	356	2	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
279	C₄⋊C₄	D₇₁₂	1	2	yes	—
280	C₂²⋊C₄	—	356	2	no	—
281	C₂²⋊C₄	—	356	1	no	—
282	C₂²⋊C₄	—	356	2	no	—
283	C₂²⋊C₄	—	356	2	no	—
284	C₂²⋊C₄	—	356	2	no	—
285	C₄²	—	356	1	no	—
286	C₄²	C₂×D₃₅₆	1	2	yes	—
287	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
288	C₁₆	—	2	2	no	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	1	no	—
291	C₂³	—	356	1	no	—
292	C₂³	—	356	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	356	1	no	—
295	Q₈	—	356	1	no	—
296	Q₈	—	1424	1	no	—
297	D₄	—	356	1	no	—
298	D₄	—	356	2	no	—
299	D₄	—	356	2	no	—
300	D₄	—	1424	1	no	—
301	D₄	—	178	2	no	—
302	D₄	—	712	2	no	—
303	C₂×C₄	—	356	1	no	—
304	C₂×C₄	—	356	1	no	—
305	C₂×C₄	—	356	1	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	1424	2	no	—
308	C₂×C₄	—	1424	2	no	—
309	C₂×C₄	—	712	2	no	—
310	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
311	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
312	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
313	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	#41×D₈₉	1	1	yes	characteristic
318	C₈	#15668060	1	1	yes	characteristic
319	C₂²	—	356	2	no	—
320	C₂²	—	712	2	no	—
321	C₂²	—	712	2	no	—
322	C₂²	—	712	2	no	—
323	C₂²	—	712	1	no	—
324	C₂²	—	712	1	no	—
325	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	712	1	no	—
330	C₄	—	712	1	no	—
331	C₄	—	712	2	no	—
332	C₄	—	712	1	no	—
333	C₄	#19152030	1	2	yes	—
334	C₄	#19152260	1	1	yes	characteristic
335	C₄	#19152410	1	1	yes	characteristic
336	C₂	—	1424	2	no	—
337	C₂	—	712	2	no	—
338	C₂	#26032888	1	1	yes	characteristic
339	C₂	C₂×#19152030	1	1	yes	characteristic
340	C₂	#26033393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	Q₈×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26022288	C₂
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#33928
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #26033265	C₂	semidirect
2. #26033120	C₂	semidirect
3. #26022484	C₂	semidirect
4. #26022476	C₂	semidirect
5. #26022410	C₂	semidirect
6. #26022405	C₂	semidirect
7. #26022308	C₂	semidirect
9. C₄×#15667957	C₂	semidirect
10. C₄×D₁₄₂₄	C₂	semidirect
11. Q₈×C₁₆×C₈₉	C₂	semidirect
179. Q₈×C₁₆	D₈₉	semidirect
182. C₈₉	#33928	semidirect

Maximal subgroups

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

subgroup	index
#26033265	2
#26033120	2
#26022484	2
#26022476	2
#26022410	2
#26022405	2
#26022308	2
#26022288	2
C₄×#15667957	2
C₄×D₁₄₂₄	2
Q₈×C₁₆×C₈₉	2
#33928	89