Subgroups — #36712288

Order 22784.

528 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	D₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033104	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022526	C₂	1	2	yes	maximal, complemented
4	#26022525	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022390	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022388	C₂	1	2	yes	maximal, complemented
7	#26022387	C₂	1	2	yes	maximal, 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×#2224	C₂	1	1	yes	maximal, characteristic, complemented
12	D₄×Dic₁₇₈	—	2	1	no	—
13	#19152263	—	2	1	no	—
14	C₂×#15667953	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	C₂²	1	2	yes	—
17	#19152002	C₂²	1	2	yes	—
18	#19151878	—	2	1	no	—
19	#19151877	—	2	2	no	—
20	#19151876	—	2	2	no	—
21	#19151875	—	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	#19151424	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	#366×C₈₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#357×C₈₉	C₂²	1	2	yes	—
41	C₂²×Dic₁₇₈	—	4	1	no	—
42	C₂×Q₈×D₈₉	—	4	1	no	—
43	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
44	#15668012	—	2	1	no	—
45	#15668011	—	2	2	no	—
46	#15668011	—	2	2	no	—
47	#15667991	—	4	1	no	—
48	#15667990	—	4	2	no	—
49	#15667989	—	4	2	no	—
50	#15667988	—	4	1	no	—
51	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
52	C₄×D₃₅₆	—	2	1	no	—
53	D₄×Dic₈₉	—	2	2	no	—
54	#15667974	—	2	2	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	#15667945	—	2	2	no	—
62	#15667945	—	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	—	4	2	no	—
80	C₂×Dic₁₇₈	—	4	2	no	—
81	C₂×Dic₁₇₈	—	8	4	no	—
82	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
83	Q₈×D₈₉	—	8	4	no	—
84	C₂×D₃₅₆	—	2	1	no	—
85	C₂²×Dic₈₉	—	4	2	no	—
86	C₂×C₄×D₈₉	—	4	1	no	—
87	C₂×C₄×D₈₉	—	4	2	no	—
88	D₇₁₂	—	2	1	no	—
89	D₇₁₂	C₂×Q₈	1	2	yes	complemented
90	C₂×#10623494	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₄⋊Dic₈₉	—	4	1	no	—
94	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
95	#13083553	—	4	2	no	—
96	#13083553	—	4	2	no	—
97	#13083553	D₈	1	2	yes	—
98	C₂²⋊Dic₈₉	—	4	2	no	—
99	#13083551	—	4	1	no	—
100	#13083551	—	4	2	no	—
101	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
102	#13083549	—	2	1	no	—
103	#13083549	C₂×Q₈	1	2	yes	complemented
104	C₂×Q₈×C₈₉	—	4	1	no	—
105	C₂×D₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	—	4	1	no	—
107	D₈×C₈₉	—	2	1	no	—
108	D₈×C₈₉	C₂×Q₈	1	2	yes	complemented
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	1	no	—
113	C₄⋊C₄×C₈₉	—	4	1	no	—
114	C₂²⋊C₄×C₈₉	—	4	1	no	—
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₈₉	—	2	2	no	—
117	C₂³×C₈₉	—	4	1	no	—
118	C₂²×D₈₉	—	4	1	no	—
119	Q₈×C₈₉	—	8	2	no	—
120	D₄×C₈₉	—	2	2	no	—
121	D₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	8	2	no	—
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
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	2	no	—
130	C₂×Dic₈₉	—	8	4	no	—
131	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
132	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
133	D₃₅₆	—	2	2	no	—
134	D₃₅₆	—	4	1	no	—
135	C₄×D₈₉	—	8	2	no	—
136	C₄×D₈₉	—	8	4	no	—
137	Dic₁₇₈	—	8	2	no	—
138	Dic₁₇₈	—	8	4	no	—
139	Dic₁₇₈	—	8	4	no	—
140	Dic₁₇₈	D₁₆	1	4	yes	complemented
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
143	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
144	#10623494	—	2	2	no	—
145	C₂²×C₈₉	—	4	2	no	—
146	C₂²×C₈₉	—	8	1	no	—
147	C₂²×C₈₉	—	8	1	no	—
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	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	8	1	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
156	C₄×C₈₉	D₄×Q₈	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	2	no	—
161	Dic₈₉	C₂×D₁₆	1	4	yes	—
162	Q₈×D₁₆	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	—	8	2	no	—
164	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
165	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
166	C₂×C₈₉	#3417	1	1	yes	characteristic
167	D₈₉	—	8	2	no	—
168	Q₈×D₈	—	89	1	no	—
169	Q₈×D₈	—	89	1	no	—
170	#2252	—	89	1	no	—
171	#2252	—	89	2	no	—
172	#2224	—	89	2	no	—
173	#2224	—	89	1	no	—
174	#2224	—	89	2	no	—
175	#2224	D₈₉	1	1	yes	characteristic, complemented
176	Q₈×C₁₆	—	89	1	no	—
177	C₄×D₁₆	—	89	1	no	—
178	C₄×D₁₆	—	89	2	no	—
179	C₈₉	Q₈×D₁₆	1	1	yes	characteristic, complemented, Sylow
180	D₄×Q₈	—	178	1	no	—
181	D₄×Q₈	—	178	1	no	—
182	C₂×D₁₆	—	89	1	no	—
183	#500	—	89	2	no	—
184	#500	—	89	2	no	—
185	#500	—	89	1	no	—
186	#500	C₂×D₈₉	1	1	yes	characteristic
187	#474	—	178	2	no	—
188	#474	—	178	2	no	—
189	#474	—	178	1	no	—
190	#474	—	178	1	no	—
191	C₈×Q₈	—	89	1	no	—
192	C₄×D₈	—	89	2	no	—
193	C₄×D₈	—	89	2	no	—
194	C₄×D₈	—	89	1	no	—
195	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
196	#366	—	89	2	no	—
197	#366	—	89	1	no	—
198	#366	—	89	2	no	—
199	#366	—	89	2	no	—
200	#366	C₂×D₈₉	1	2	yes	—
201	C₄⋊C₁₆	—	89	2	no	—
202	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
203	#357	—	89	1	no	—
204	#357	—	89	2	no	—
205	#357	—	89	1	no	—
206	#357	C₂×D₈₉	1	2	yes	—
207	C₄×C₁₆	—	89	2	no	—
208	C₄×C₁₆	—	89	1	no	—
209	C₂²×Q₈	—	356	1	no	—
210	C₂²×Q₈	—	356	1	no	—
211	C₂×D₈	—	89	1	no	—
212	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	178	2	no	—
215	C₄⋊Q₈	—	178	2	no	—
216	C₄⋊Q₈	—	178	1	no	—
217	C₂²⋊Q₈	—	356	2	no	—
218	C₂²⋊Q₈	—	356	2	no	—
219	C₂²⋊Q₈	—	356	1	no	—
220	C₂²⋊Q₈	—	356	1	no	—
221	C₄×Q₈	—	89	1	no	—
222	C₄×D₄	—	2	1	no	—
223	C₄×D₄	—	178	2	no	—
224	C₄×D₄	—	178	2	no	—
225	C₄×D₄	—	178	1	no	—
226	D₁₆	—	89	4	no	—
227	C₂×C₁₆	—	89	1	no	—
228	C₂×C₁₆	—	89	1	no	—
229	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
230	#107	—	2	1	no	—
231	#107	—	89	1	no	—
232	#107	—	89	2	no	—
233	#107	—	178	2	no	—
234	#107	—	178	2	no	—
235	#107	—	178	1	no	—
236	#107	C₂²×D₈₉	1	1	yes	characteristic
237	#107	C₂²×D₈₉	1	2	yes	—
238	C₄⋊C₈	—	89	2	no	—
239	C₄⋊C₈	—	89	1	no	—
240	#102	—	2	1	no	—
241	#102	—	178	2	no	—
242	#102	—	178	2	no	—
243	#102	—	178	1	no	—
244	C₄×C₈	—	89	2	no	—
245	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	89	1	no	—
248	C₂×Q₈	—	356	2	no	—
249	C₂×Q₈	—	356	2	no	—
250	C₂×Q₈	—	356	1	no	—
251	C₂×Q₈	—	712	4	no	—
252	C₂×Q₈	—	712	4	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	178	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	356	2	no	—
257	C₂²×C₄	—	356	2	no	—
258	C₂²×C₄	—	356	1	no	—
259	D₈	—	2	1	no	—
260	D₈	—	89	2	no	—
261	D₈	—	178	1	no	—
262	D₈	Q₈×D₈₉	1	2	yes	complemented
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	89	2	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	356	2	no	—
272	C₄⋊C₄	—	356	2	no	—
273	C₄⋊C₄	—	356	1	no	—
274	C₄⋊C₄	—	178	2	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₂²⋊C₄	—	356	2	no	—
278	C₂²⋊C₄	—	356	2	no	—
279	C₂²⋊C₄	—	356	1	no	—
280	C₄²	—	89	2	no	—
281	C₄²	D₄×D₈₉	1	1	yes	characteristic
282	C₁₆	—	2	2	no	—
283	C₁₆	—	89	2	no	—
284	C₁₆	—	178	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	356	1	no	—
287	Q₈	—	8	2	no	—
288	Q₈	—	89	4	no	—
289	Q₈	—	712	4	no	—
290	Q₈	—	712	4	no	—
291	Q₈	—	712	2	no	—
292	D₄	—	2	2	no	—
293	D₄	—	4	1	no	—
294	D₄	—	356	1	no	—
295	D₄	—	178	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	89	2	no	—
300	C₂×C₄	—	89	2	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	712	4	no	—
304	C₂×C₄	—	712	4	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×Q₈×D₈₉	1	1	yes	characteristic
312	C₈	#15668059	1	1	yes	characteristic
313	C₂²	—	4	2	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	356	2	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	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	89	4	no	—
324	C₄	—	178	2	no	—
325	C₄	—	712	2	no	—
326	C₄	—	712	1	no	—
327	C₄	D₁₆×D₈₉	1	2	yes	—
328	C₄	#19152261	1	1	yes	characteristic
329	C₄	#19152414	1	1	yes	characteristic
330	C₂	—	8	2	no	—
331	C₂	—	712	2	no	—
332	C₂	#26033102	1	1	yes	characteristic
333	C₂	C₂×D₁₆×D₈₉	1	1	yes	characteristic
334	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
prime radical	C₂×#15667953	C₂²
hypercenter	#2224	D₈₉
hypocenter	C₈₉	Q₈×D₁₆
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. D₈×Dic₁₇₈	C₂	semidirect
2. #26033104	C₂	semidirect
3. #26022526	C₂	semidirect
4. #26022525	C₂	semidirect
5. #26022390	C₂	semidirect
6. #26022388	C₂	semidirect
7. #26022387	C₂	semidirect
8. #26022154	C₂	semidirect
11. C₈₉×#2224	C₂	semidirect
55. #15667953	Q₈	semidirect
89. D₇₁₂	C₂×Q₈	semidirect
103. #13083549	C₂×Q₈	semidirect
108. D₈×C₈₉	C₂×Q₈	semidirect
140. Dic₁₇₈	D₁₆	semidirect
175. #2224	D₈₉	semidirect
179. C₈₉	Q₈×D₁₆	semidirect
262. D₈	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
D₈×Dic₁₇₈	2
#26033104	2
#26022526	2
#26022525	2
#26022390	2
#26022388	2
#26022387	2
#26022154	2
C₄×#15667953	2
#26022096	2
C₈₉×#2224	2
Q₈×D₁₆	89