Subgroups — #36712304

Order 22784.

476 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033121	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033117	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022530	C₂	1	2	yes	maximal, complemented
4	#26022525	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022411	C₂	1	2	yes	maximal, complemented
6	#26022408	C₂	1	2	yes	maximal, complemented
7	#26022400	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022154	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#15667953	C₂	1	1	yes	maximal, characteristic
10	#26022096	C₂	1	2	yes	maximal
11	C₈₉×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152309	—	2	1	no	—
13	#19152305	—	2	1	no	—
14	C₂×#15667953	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151999	C₂²	1	2	yes	—
17	#19151999	C₂²	1	2	yes	—
18	#19151901	—	2	2	no	—
19	#19151896	—	2	2	no	—
20	#19151888	—	2	1	no	—
21	#19151885	—	2	1	no	—
22	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	D₈×Dic₈₉	C₂²	1	2	yes	—
25	#19151680	C₂²	1	2	yes	—
26	#19151426	C₂²	1	2	yes	—
27	#19151423	C₂²	1	2	yes	—
28	#19151423	C₂²	1	2	yes	—
29	#19151423	C₂²	1	1	yes	characteristic
30	#19151416	C₂²	1	2	yes	—
31	#19151400	C₂²	1	2	yes	—
32	#19151399	C₂²	1	1	yes	characteristic
33	#19151398	C₂²	1	1	yes	characteristic
34	C₄×#13083549	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
38	#367×C₈₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#357×C₈₉	C₂²	1	2	yes	—
41	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15668003	—	2	2	no	—
44	#15668003	—	2	2	no	—
45	#15667992	—	4	1	no	—
46	#15667991	—	4	1	no	—
47	#15667989	—	4	2	no	—
48	#15667988	—	4	2	no	—
49	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
50	C₄×D₃₅₆	—	2	1	no	—
51	D₄×Dic₈₉	—	2	2	no	—
52	#15667974	—	2	2	no	—
53	C₂×C₄⋊Dic₈₉	—	4	1	no	—
54	C₄⋊C₄×D₈₉	—	4	1	no	—
55	#15667953	Q₈	1	4	yes	complemented
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	#15667946	—	2	1	no	—
59	#15667946	C₂³	1	1	yes	characteristic
60	#15667946	C₂³	1	2	yes	—
61	#15667943	—	2	2	no	—
62	#15667943	—	2	2	no	—
63	C₄⋊#10623494	D₄	1	1	yes	characteristic
64	#15667941	D₄	1	2	yes	—
65	#15667934	—	2	1	no	—
66	#15667933	—	2	2	no	—
67	#15667932	—	2	2	no	—
68	C₈×Dic₈₉	C₂³	1	2	yes	—
69	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₄×D₄×C₈₉	—	2	1	no	—
72	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
73	#107×C₈₉	—	2	1	no	—
74	#107×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#102×C₈₉	—	2	1	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₇₈	—	4	1	no	—
79	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
80	C₂×D₃₅₆	—	2	1	no	—
81	C₂²×Dic₈₉	—	4	2	no	—
82	C₂×C₄×D₈₉	—	4	1	no	—
83	C₂×C₄×D₈₉	—	4	2	no	—
84	D₇₁₂	—	2	1	no	—
85	D₇₁₂	C₂×Q₈	1	2	yes	—
86	C₂×#10623494	C₂×D₄	1	2	yes	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₄⋊Dic₈₉	—	4	2	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
93	#13083553	—	4	2	no	—
94	#13083553	—	4	2	no	—
95	#13083553	D₈	1	2	yes	—
96	C₂²⋊Dic₈₉	—	4	2	no	—
97	#13083551	—	4	1	no	—
98	#13083551	—	4	2	no	—
99	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
100	#13083549	—	2	1	no	—
101	#13083549	C₂×Q₈	1	2	yes	—
102	C₂×Q₈×C₈₉	—	4	1	no	—
103	C₂×D₄×C₈₉	—	2	1	no	—
104	C₂²×C₄×C₈₉	—	4	1	no	—
105	D₈×C₈₉	—	2	1	no	—
106	D₈×C₈₉	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₄⋊C₄×C₈₉	—	4	1	no	—
112	C₂²⋊C₄×C₈₉	—	4	1	no	—
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₁₆×C₈₉	—	2	2	no	—
115	C₂³×C₈₉	—	4	1	no	—
116	C₂²×D₈₉	—	4	1	no	—
117	Q₈×C₈₉	—	16	1	no	—
118	D₄×C₈₉	—	2	2	no	—
119	D₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	8	2	no	—
123	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₈₉	—	4	1	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	—	16	2	no	—
129	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
130	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
131	D₃₅₆	—	2	2	no	—
132	D₃₅₆	—	4	1	no	—
133	C₄×D₈₉	—	8	2	no	—
134	C₄×D₈₉	—	16	2	no	—
135	Dic₁₇₈	—	2	2	no	—
136	Dic₁₇₈	—	16	1	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
139	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
140	#10623494	—	2	2	no	—
141	C₂²×C₈₉	—	4	2	no	—
142	C₂²×C₈₉	—	8	1	no	—
143	C₂²×C₈₉	—	8	1	no	—
144	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₈₉	—	4	2	no	—
146	C₂×D₈₉	—	8	1	no	—
147	C₂×D₈₉	—	8	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	8	1	no	—
150	C₄×C₈₉	—	8	2	no	—
151	C₄×C₈₉	#508	1	2	yes	—
152	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
153	C₄×C₈₉	#578	1	1	yes	characteristic
154	Dic₈₉	—	2	2	no	—
155	Dic₈₉	—	2	2	no	—
156	Dic₈₉	—	8	1	no	—
157	Dic₈₉	—	8	2	no	—
158	#33947	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	—	8	2	no	—
160	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
162	C₂×C₈₉	#3415	1	1	yes	characteristic
163	D₈₉	—	8	2	no	—
164	#3380	—	89	1	no	—
165	#3380	—	89	1	no	—
166	#2253	—	89	2	no	—
167	#2252	—	89	1	no	—
168	#2228	—	89	2	no	—
169	#2228	—	89	2	no	—
170	#2226	—	89	1	no	—
171	#2226	D₈₉	1	1	yes	characteristic, complemented
172	Q₈×C₁₆	—	89	1	no	—
173	C₄×D₁₆	—	89	1	no	—
174	C₄×D₁₆	—	89	2	no	—
175	C₈₉	#33947	1	1	yes	characteristic, complemented, Sylow
176	#554	—	178	1	no	—
177	#554	—	178	1	no	—
178	C₂×D₁₆	—	89	1	no	—
179	#500	—	89	1	no	—
180	#500	C₂×D₈₉	1	1	yes	characteristic
181	#499	—	89	2	no	—
182	#499	—	89	2	no	—
183	#478	—	178	2	no	—
184	#478	—	178	2	no	—
185	#476	—	178	1	no	—
186	#476	—	178	1	no	—
187	C₈×Q₈	—	89	1	no	—
188	C₄×D₈	—	89	2	no	—
189	C₄×D₈	—	89	2	no	—
190	C₄×D₈	—	89	1	no	—
191	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
192	#367	—	89	2	no	—
193	#367	C₂×D₈₉	1	2	yes	—
194	#366	—	89	2	no	—
195	#366	—	89	1	no	—
196	#366	—	89	2	no	—
197	C₄⋊C₁₆	—	89	2	no	—
198	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
199	#357	—	89	1	no	—
200	#357	—	89	2	no	—
201	#357	—	89	1	no	—
202	#357	C₂×D₈₉	1	2	yes	—
203	C₄×C₁₆	—	89	2	no	—
204	C₄×C₁₆	—	89	1	no	—
205	C₂×D₈	—	89	1	no	—
206	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	178	1	no	—
209	#125	—	178	2	no	—
210	#125	—	178	2	no	—
211	C₂²⋊Q₈	—	356	2	no	—
212	C₂²⋊Q₈	—	356	1	no	—
213	C₂²⋊Q₈	—	356	2	no	—
214	C₂²⋊Q₈	—	356	1	no	—
215	C₄×Q₈	—	89	1	no	—
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	—	178	2	no	—
218	C₄×D₄	—	178	2	no	—
219	C₄×D₄	—	178	1	no	—
220	C₂×C₄⋊C₄	—	356	1	no	—
221	C₂×C₄⋊C₄	—	356	1	no	—
222	D₁₆	—	89	4	no	—
223	C₂×C₁₆	—	89	1	no	—
224	C₂×C₁₆	—	89	1	no	—
225	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
226	#107	—	2	1	no	—
227	#107	—	89	2	no	—
228	#107	—	89	1	no	—
229	#107	—	178	1	no	—
230	#107	C₂²×D₈₉	1	1	yes	characteristic
231	#107	C₂²×D₈₉	1	2	yes	—
232	#106	—	178	2	no	—
233	#106	—	178	2	no	—
234	C₄⋊C₈	—	89	2	no	—
235	C₄⋊C₈	—	89	1	no	—
236	#102	—	2	1	no	—
237	#102	—	178	2	no	—
238	#102	—	178	2	no	—
239	#102	—	178	1	no	—
240	C₄×C₈	—	89	2	no	—
241	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	89	1	no	—
244	C₂×Q₈	—	356	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	178	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	356	2	no	—
249	C₂²×C₄	—	356	2	no	—
250	C₂²×C₄	—	356	1	no	—
251	D₈	—	2	1	no	—
252	D₈	—	89	2	no	—
253	D₈	—	178	1	no	—
254	D₈	Q₈×D₈₉	1	2	yes	—
255	C₂×C₈	—	89	2	no	—
256	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	89	1	no	—
262	C₄⋊C₄	—	89	2	no	—
263	C₄⋊C₄	—	356	2	no	—
264	C₄⋊C₄	—	356	2	no	—
265	C₄⋊C₄	—	356	2	no	—
266	C₄⋊C₄	—	356	2	no	—
267	C₄⋊C₄	—	356	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₄⋊C₄	—	178	2	no	—
270	C₂²⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	356	2	no	—
272	C₂²⋊C₄	—	356	2	no	—
273	C₂²⋊C₄	—	356	1	no	—
274	C₄²	—	89	2	no	—
275	C₄²	D₄×D₈₉	1	1	yes	characteristic
276	C₁₆	—	2	2	no	—
277	C₁₆	—	89	2	no	—
278	C₁₆	—	178	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	356	1	no	—
281	Q₈	—	16	1	no	—
282	Q₈	—	1424	1	no	—
283	Q₈	—	178	2	no	—
284	D₄	—	2	2	no	—
285	D₄	—	4	1	no	—
286	D₄	—	356	1	no	—
287	D₄	—	178	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	89	2	no	—
292	C₂×C₄	—	89	2	no	—
293	C₂×C₄	—	356	1	no	—
294	C₂×C₄	—	356	2	no	—
295	C₂×C₄	—	1424	2	no	—
296	C₂×C₄	—	1424	2	no	—
297	C₂×C₄	—	712	2	no	—
298	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
299	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	178	2	no	—
303	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
304	C₈	#15668059	1	1	yes	characteristic
305	C₂²	—	4	2	no	—
306	C₂²	—	8	1	no	—
307	C₂²	—	8	1	no	—
308	C₂²	—	356	2	no	—
309	C₂²	—	712	1	no	—
310	C₂²	—	712	1	no	—
311	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	178	2	no	—
316	C₄	—	178	2	no	—
317	C₄	—	712	1	no	—
318	C₄	—	712	2	no	—
319	C₄	#19152029	1	2	yes	—
320	C₄	#19152261	1	1	yes	characteristic
321	C₄	#19152414	1	1	yes	characteristic
322	C₂	—	8	2	no	—
323	C₂	—	712	2	no	—
324	C₂	#26033102	1	1	yes	characteristic
325	C₂	C₂×#19152029	1	1	yes	characteristic
326	C₂	#26033392	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×#2226	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15667953	C₂²
hypercenter	#2226	D₈₉
hypocenter	C₈₉	#33947
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
1. #26033121	C₂	semidirect
2. #26033117	C₂	semidirect
3. #26022530	C₂	semidirect
4. #26022525	C₂	semidirect
5. #26022411	C₂	semidirect
6. #26022408	C₂	semidirect
7. #26022400	C₂	semidirect
8. #26022154	C₂	semidirect
11. C₈₉×#2226	C₂	semidirect
55. #15667953	Q₈	semidirect
171. #2226	D₈₉	semidirect
175. C₈₉	#33947	semidirect

Maximal subgroups

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

subgroup	index
#26033121	2
#26033117	2
#26022530	2
#26022525	2
#26022411	2
#26022408	2
#26022400	2
#26022154	2
C₄×#15667953	2
#26022096	2
C₈₉×#2226	2
#33947	89