Subgroups — #36712226

Order 22784.

520 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022488	C₂	1	2	yes	maximal, complemented
3	#26022475	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022399	C₂	1	2	yes	maximal, complemented
5	#26022394	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022304	C₂	1	2	yes	maximal, complemented
7	#26022284	C₂	1	1	yes	maximal, characteristic
8	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022126	C₂	1	2	yes	maximal, complemented
10	C₄×#15667955	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3396	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×D₃₅₆	—	2	1	no	—
13	C₂×#15667955	C₂²	1	1	yes	characteristic
14	C₂×#15667953	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152003	C₂²	1	2	yes	—
17	#19151879	—	2	1	no	—
18	#19151879	—	2	2	no	—
19	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₄×D₇₁₂	C₂²	1	2	yes	—
21	#19151425	C₂²	1	1	yes	characteristic
22	#19151425	C₂²	1	2	yes	—
23	C₄⋊#13083549	C₂²	1	1	yes	characteristic
24	C₄⋊#13083549	C₂²	1	2	yes	—
25	#19151402	C₂²	1	1	yes	characteristic
26	#19151402	C₂²	1	2	yes	—
27	#19151399	C₂²	1	2	yes	—
28	#19151399	C₂²	1	2	yes	—
29	#19151399	C₂²	1	2	yes	—
30	#19151398	C₂²	1	1	yes	characteristic
31	#19151374	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#559×C₈₉	—	2	1	no	—
34	#495×C₈₉	C₂²	1	2	yes	—
35	#493×C₈₉	C₂²	1	1	yes	characteristic
36	#463×C₈₉	—	2	2	no	—
37	#460×C₈₉	—	2	1	no	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₈₉	C₂²	1	2	yes	—
41	C₂×Q₈×D₈₉	—	4	1	no	—
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	#15668012	—	2	2	no	—
44	#15668012	—	2	1	no	—
45	#15667992	—	4	1	no	—
46	#15667992	—	4	2	no	—
47	C₄×D₃₅₆	—	2	2	no	—
48	C₄×D₃₅₆	—	2	1	no	—
49	#15667955	—	2	2	no	—
50	#15667953	—	2	4	no	—
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	C₂×#13083549	C₂³	1	2	yes	—
54	#15667946	—	2	2	no	—
55	#15667946	—	2	1	no	—
56	#15667946	C₂³	1	2	yes	—
57	#15667946	C₂³	1	1	yes	characteristic
58	#15667934	—	2	2	no	—
59	#15667934	—	2	1	no	—
60	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
61	C₂×SD₁₆×C₈₉	—	2	2	no	—
62	C₂×D₈×C₈₉	—	2	1	no	—
63	C₂×D₈×C₈₉	C₂³	1	2	yes	—
64	#127×C₈₉	—	2	1	no	—
65	#124×C₈₉	—	2	2	no	—
66	#124×C₈₉	—	4	1	no	—
67	#123×C₈₉	—	4	2	no	—
68	C₄×Q₈×C₈₉	—	2	1	no	—
69	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
70	C₄×D₄×C₈₉	—	2	2	no	—
71	C₄×D₄×C₈₉	—	4	1	no	—
72	#107×C₈₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
75	#103×C₈₉	—	2	1	no	—
76	#102×C₈₉	—	2	2	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	2	yes	—
79	C₂×Dic₁₇₈	—	4	2	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	Q₈×D₈₉	—	8	4	no	—
82	C₂×D₃₅₆	—	2	1	no	—
83	C₂×C₄×D₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	—	4	1	no	—
85	D₇₁₂	—	2	1	no	—
86	D₇₁₂	#41	1	2	yes	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	4	2	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	#13083551	—	4	2	no	—
92	#13083551	—	4	1	no	—
93	#13083549	—	2	2	no	—
94	#13083549	—	2	1	no	—
95	#13083549	—	2	1	no	—
96	C₂×Q₈×C₈₉	—	2	1	no	—
97	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
98	C₂×D₄×C₈₉	—	2	2	no	—
99	C₂×D₄×C₈₉	—	4	1	no	—
100	C₂×D₄×C₈₉	—	4	1	no	—
101	C₂²×C₄×C₈₉	—	4	2	no	—
102	Q₁₆×C₈₉	—	2	1	no	—
103	Q₁₆×C₈₉	—	2	1	no	—
104	SD₁₆×C₈₉	—	8	2	no	—
105	D₈×C₈₉	—	2	2	no	—
106	D₈×C₈₉	—	2	2	no	—
107	D₈×C₈₉	—	8	1	no	—
108	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	—	4	1	no	—
113	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
115	C₂²⋊C₄×C₈₉	—	4	2	no	—
116	C₂²⋊C₄×C₈₉	—	4	2	no	—
117	C₂²⋊C₄×C₈₉	—	4	2	no	—
118	C₄²×C₈₉	—	4	1	no	—
119	C₄²×C₈₉	C₂×D₄	1	2	yes	—
120	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₈₉	—	4	2	no	—
122	C₂²×D₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	2	2	no	—
124	Q₈×C₈₉	—	4	1	no	—
125	Q₈×C₈₉	—	4	1	no	—
126	D₄×C₈₉	—	4	2	no	—
127	D₄×C₈₉	—	4	2	no	—
128	D₄×C₈₉	—	8	2	no	—
129	D₄×C₈₉	—	16	1	no	—
130	C₂×C₄×C₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	16	2	no	—
133	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₈₉	—	4	1	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	D₃₅₆	—	2	2	no	—
141	D₃₅₆	—	4	1	no	—
142	C₄×D₈₉	—	8	4	no	—
143	C₄×D₈₉	—	8	2	no	—
144	Dic₁₇₈	—	8	4	no	—
145	Dic₁₇₈	—	8	2	no	—
146	C₈×C₈₉	—	2	1	no	—
147	C₈×C₈₉	—	2	2	no	—
148	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
149	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂²×C₈₉	—	8	2	no	—
151	C₂²×C₈₉	—	8	2	no	—
152	C₂²×C₈₉	—	8	2	no	—
153	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₈₉	—	4	2	no	—
155	C₂×D₈₉	—	8	1	no	—
156	C₂×D₈₉	—	8	1	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	8	1	no	—
161	C₄×C₈₉	—	8	1	no	—
162	C₄×C₈₉	#509	1	2	yes	—
163	C₄×C₈₉	#548	1	1	yes	characteristic
164	C₄×C₈₉	#577	1	1	yes	characteristic
165	Dic₈₉	—	8	1	no	—
166	Dic₈₉	—	8	2	no	—
167	#33940	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	16	2	no	—
169	C₂×C₈₉	#3358	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
171	C₂×C₈₉	#3416	1	1	yes	characteristic
172	D₈₉	—	8	2	no	—
173	#3396	D₈₉	1	1	yes	characteristic, complemented
174	Q₈×D₈	—	89	1	no	—
175	#2243	—	89	2	no	—
176	#2240	—	89	1	no	—
177	#2226	—	89	2	no	—
178	#2225	—	89	1	no	—
179	#2210	—	89	2	no	—
180	#2206	—	89	1	no	—
181	#2183	—	89	1	no	—
182	#2177	—	89	2	no	—
183	C₄×SD₃₂	—	89	1	no	—
184	C₈₉	#33940	1	1	yes	characteristic, complemented, Sylow
185	#559	—	2	1	no	—
186	D₄×Q₈	—	178	1	no	—
187	C₂×SD₃₂	—	89	1	no	—
188	C₂×D₁₆	—	89	2	no	—
189	#500	—	89	1	no	—
190	#500	—	89	2	no	—
191	#495	C₂×D₈₉	1	2	yes	—
192	#493	C₂×D₈₉	1	1	yes	characteristic
193	#474	—	178	1	no	—
194	#474	—	178	2	no	—
195	#463	—	2	2	no	—
196	#460	—	2	1	no	—
197	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
198	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
199	C₄×D₈	—	89	1	no	—
200	C₄×D₈	—	89	2	no	—
201	C₄×D₈	C₂×D₈₉	1	2	yes	—
202	#367	—	89	1	no	—
203	#367	—	89	2	no	—
204	C₄⋊C₁₆	—	89	1	no	—
205	C₄⋊C₁₆	—	89	2	no	—
206	#358	—	89	1	no	—
207	#358	—	89	2	no	—
208	#357	—	89	2	no	—
209	#357	—	89	2	no	—
210	#357	—	89	2	no	—
211	#357	—	89	1	no	—
212	#346	—	89	2	no	—
213	C₄×C₁₆	—	89	1	no	—
214	C₂²×Q₈	—	356	1	no	—
215	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	—	89	1	no	—
219	C₂×D₈	C₂²×D₈₉	1	2	yes	—
220	C₄⋊Q₈	—	178	2	no	—
221	C₄⋊Q₈	—	178	1	no	—
222	#127	—	2	1	no	—
223	#124	—	2	2	no	—
224	#124	—	4	1	no	—
225	#123	—	4	2	no	—
226	C₂²⋊Q₈	—	356	1	no	—
227	C₂²⋊Q₈	—	356	2	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	—	4	1	no	—
232	C₄×D₄	—	178	2	no	—
233	C₄×D₄	—	178	1	no	—
234	SD₃₂	—	178	2	no	—
235	D₁₆	—	178	4	no	—
236	C₂×C₁₆	—	89	1	no	—
237	C₂×C₁₆	—	89	1	no	—
238	C₂×C₁₆	—	89	2	no	—
239	#107	—	89	2	no	—
240	#107	—	89	1	no	—
241	#107	—	178	2	no	—
242	#107	—	178	1	no	—
243	#107	C₂²×D₈₉	1	1	yes	characteristic
244	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
245	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
246	#103	—	2	1	no	—
247	#102	—	2	2	no	—
248	#102	—	178	2	no	—
249	#102	—	178	1	no	—
250	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₈₉	1	2	yes	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	356	2	no	—
254	C₂×Q₈	—	356	1	no	—
255	C₂×Q₈	—	712	4	no	—
256	C₂×Q₈	#13083559	1	1	yes	characteristic
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	356	2	no	—
263	C₂²×C₄	—	356	1	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	—	2	1	no	—
266	SD₁₆	—	8	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	2	2	no	—
269	D₈	—	8	1	no	—
270	D₈	—	89	2	no	—
271	D₈	—	178	1	no	—
272	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	356	2	no	—
278	C₄⋊C₄	—	356	1	no	—
279	C₄⋊C₄	—	178	2	no	—
280	C₄⋊C₄	—	178	1	no	—
281	C₄⋊C₄	#13083559	1	1	yes	characteristic
282	C₄⋊C₄	#13083559	1	2	yes	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	356	2	no	—
287	C₂²⋊C₄	—	356	1	no	—
288	C₄²	—	4	1	no	—
289	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
290	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
291	C₁₆	—	178	2	no	—
292	C₁₆	—	178	1	no	—
293	C₁₆	—	178	1	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	356	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	712	4	no	—
300	Q₈	—	712	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	16	1	no	—
305	D₄	—	356	1	no	—
306	D₄	—	178	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	—	356	2	no	—
311	C₂×C₄	—	356	1	no	—
312	C₂×C₄	—	712	4	no	—
313	C₂×C₄	—	712	2	no	—
314	C₂×C₄	C₂×#13083559	1	2	yes	—
315	C₂×C₄	C₂×#13083559	1	2	yes	—
316	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
317	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	2	no	—
321	C₈	#41×D₈₉	1	1	yes	characteristic
322	C₈	#15668059	1	1	yes	characteristic
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	356	2	no	—
327	C₂²	—	712	1	no	—
328	C₂²	—	712	1	no	—
329	C₂²	C₂²×#13083559	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	712	2	no	—
336	C₄	—	712	1	no	—
337	C₄	#19152033	1	2	yes	—
338	C₄	#19152255	1	1	yes	characteristic
339	C₄	#19152409	1	1	yes	characteristic
340	C₂	—	16	2	no	—
341	C₂	—	712	2	no	—
342	C₂	#26032883	1	1	yes	characteristic
343	C₂	C₂×#19152033	1	1	yes	characteristic
344	C₂	#26033399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3396	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26022284	C₂
hypercenter	#3396	D₈₉
hypocenter	C₈₉	#33940
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3396

Complements

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

N	H ≅ G/N	type
1. Q₈×D₇₁₂	C₂	semidirect
2. #26022488	C₂	semidirect
3. #26022475	C₂	semidirect
4. #26022399	C₂	semidirect
5. #26022394	C₂	semidirect
6. #26022304	C₂	semidirect
8. #26022161	C₂	semidirect
9. #26022126	C₂	semidirect
10. C₄×#15667955	C₂	semidirect
11. C₈₉×#3396	C₂	semidirect
173. #3396	D₈₉	semidirect
184. C₈₉	#33940	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₇₁₂	2
#26022488	2
#26022475	2
#26022399	2
#26022394	2
#26022304	2
#26022284	2
#26022161	2
#26022126	2
C₄×#15667955	2
C₈₉×#3396	2
#33940	89