Subgroups — #36708939

Order 22784.

682 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	C₂²×#15667957	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029907	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029900	C₂	1	1	yes	maximal, characteristic
4	#26022365	C₂	1	4	yes	maximal, complemented
5	#26022348	C₂	1	4	yes	maximal, complemented
6	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021973	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
11	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
12	#19152169	—	2	1	no	—
13	#19152162	—	2	1	no	—
14	C₂×#15667957	—	2	4	no	—
15	C₂×#15667957	C₂²	1	2	yes	—
16	C₂×#15667957	C₂²	1	2	yes	—
17	#19151856	—	2	2	no	—
18	#19151851	—	2	2	no	—
19	#19151839	C₂²	1	4	yes	complemented
20	#19151834	C₂²	1	4	yes	—
21	#19151636	C₂²	1	1	yes	characteristic
22	#19151636	C₂²	1	1	yes	characteristic
23	#19151608	—	2	1	no	—
24	#19151604	—	2	1	no	—
25	#19151426	C₂²	1	2	yes	complemented
26	#19151424	C₂²	1	2	yes	complemented
27	#19151403	C₂²	1	4	yes	complemented
28	#19151400	C₂²	1	4	yes	—
29	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
30	#504×C₈₉	—	2	4	no	—
31	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₈₉	C₂²	1	2	yes	complemented
33	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
34	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	complemented
35	C₂²×Dic₁₇₈	—	2	1	no	—
36	C₂²×D₃₅₆	—	2	1	no	—
37	C₂×Dic₃₅₆	—	2	2	no	—
38	C₂×Dic₃₅₆	—	2	1	no	—
39	C₂×Dic₃₅₆	—	2	2	no	—
40	C₂×Dic₃₅₆	C₂³	1	2	yes	—
41	C₂×D₇₁₂	—	2	2	no	—
42	C₂×D₇₁₂	—	2	1	no	—
43	C₂×D₇₁₂	—	2	2	no	—
44	C₂×D₇₁₂	C₂³	1	2	yes	—
45	#15668010	—	4	1	no	—
46	#15668008	—	4	1	no	—
47	#15668001	—	4	1	no	—
48	#15667999	—	4	1	no	—
49	#15667993	—	4	2	no	—
50	#15667991	—	2	4	no	—
51	#15667986	—	2	4	no	—
52	#15667980	—	4	2	no	—
53	#15667971	—	2	1	no	—
54	#15667971	—	2	1	no	—
55	#15667957	—	2	4	no	—
56	#15667957	—	2	4	no	—
57	#15667946	C₂³	1	2	yes	—
58	#15667946	C₂³	1	2	yes	—
59	#15667944	—	2	1	no	—
60	#15667944	—	2	1	no	—
61	#15667937	—	2	4	no	—
62	#15667934	—	2	4	no	—
63	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
64	#131×C₈₉	—	2	2	no	—
65	#131×C₈₉	—	2	2	no	—
66	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
67	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	M₃₂×C₈₉	—	2	2	no	—
71	M₃₂×C₈₉	—	2	4	no	—
72	C₂×C₁₆×C₈₉	—	2	4	no	—
73	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	C₂³×D₈₉	—	4	1	no	—
78	C₂×Dic₁₇₈	—	2	2	no	—
79	C₂×Dic₁₇₈	—	4	2	no	—
80	C₂×Dic₁₇₈	—	4	2	no	—
81	C₂×Dic₁₇₈	—	4	1	no	—
82	C₂×D₃₅₆	—	2	2	no	—
83	C₂×D₃₅₆	—	4	2	no	—
84	C₂×D₃₅₆	—	4	2	no	—
85	C₂×D₃₅₆	—	4	1	no	—
86	C₂×C₂²⋊D₈₉	—	4	4	no	—
87	C₂²×Dic₈₉	—	4	1	no	—
88	Dic₃₅₆	—	2	2	no	—
89	Dic₃₅₆	—	2	2	no	—
90	Dic₃₅₆	—	4	2	no	—
91	D₇₁₂	—	2	2	no	—
92	D₇₁₂	—	2	2	no	—
93	D₇₁₂	—	4	2	no	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	#13083553	—	4	4	no	—
97	C₂²⋊Dic₈₉	—	4	1	no	—
98	C₂²⋊Dic₈₉	—	4	1	no	—
99	C₂²⋊Dic₈₉	—	4	4	no	—
100	#13083551	—	4	4	no	—
101	C₄×Dic₈₉	—	4	1	no	—
102	C₄×Dic₈₉	—	4	1	no	—
103	#41×C₈₉	—	2	2	no	—
104	#41×C₈₉	—	2	2	no	—
105	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
106	C₂×D₄×C₈₉	D₈	1	2	yes	—
107	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	M₁₆×C₈₉	—	2	2	no	—
111	M₁₆×C₈₉	—	2	4	no	—
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₈₉	#41	1	2	yes	—
118	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₁₆×C₈₉	—	2	2	no	—
120	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
121	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
122	C₂³×C₈₉	C₂×D₈	1	2	yes	—
123	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₂²×D₈₉	—	4	2	no	—
125	C₂²×D₈₉	—	8	2	no	—
126	C₂²×D₈₉	—	16	2	no	—
127	Q₈×C₈₉	—	2	2	no	—
128	D₄×C₈₉	—	2	4	no	—
129	D₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	#121	1	2	yes	—
133	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#135	1	2	yes	—
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₈₉	—	16	4	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	8	2	no	—
142	D₃₅₆	—	4	2	no	—
143	D₃₅₆	—	4	2	no	—
144	D₃₅₆	—	8	2	no	—
145	Dic₁₇₈	—	4	2	no	—
146	Dic₁₇₈	—	4	2	no	—
147	Dic₁₇₈	—	8	2	no	—
148	C₈×C₈₉	—	2	2	no	—
149	C₈×C₈₉	#121	1	2	yes	—
150	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₂²×C₈₉	—	2	2	no	—
153	C₂²×C₈₉	—	2	4	no	—
154	C₂²×C₈₉	—	2	2	no	—
155	C₂²×C₈₉	#466	1	2	yes	—
156	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₈₉	—	8	4	no	—
158	C₂×D₈₉	—	8	2	no	—
159	C₂×D₈₉	—	16	4	no	—
160	C₄×C₈₉	—	2	2	no	—
161	C₄×C₈₉	#466	1	2	yes	—
162	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
163	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
164	Dic₈₉	—	8	2	no	—
165	Dic₈₉	—	8	2	no	—
166	Dic₈₉	—	8	2	no	—
167	#33609	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	2	2	no	—
169	C₂×C₈₉	—	4	2	no	—
170	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
171	C₂×C₈₉	#3416	1	2	yes	—
172	D₈₉	—	16	2	no	—
173	C₂²×SD₃₂	—	89	1	no	—
174	C₂×#504	D₈₉	1	1	yes	characteristic, complemented
175	#3059	—	89	1	no	—
176	#3058	—	89	1	no	—
177	#2220	—	89	4	no	—
178	#2217	—	89	4	no	—
179	#2158	—	89	1	no	—
180	#2141	—	89	1	no	—
181	#2140	—	89	1	no	—
182	C₈₉	#33609	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₁₆	—	89	1	no	—
184	C₂²×D₈	—	89	1	no	—
185	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
186	#536	—	178	1	no	—
187	#535	—	178	1	no	—
188	C₂×SD₃₂	—	89	2	no	—
189	C₂×SD₃₂	—	89	2	no	—
190	C₂×SD₃₂	—	178	4	no	—
191	#504	—	2	4	no	—
192	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
193	C₂×M₃₂	C₂×D₈₉	1	2	yes	complemented
194	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
195	C₂²×C₁₆	C₂×D₈₉	1	2	yes	complemented
196	#470	—	178	2	no	—
197	#469	—	178	2	no	—
198	#467	—	89	4	no	—
199	#466	—	89	4	no	—
200	#427	—	89	1	no	—
201	#427	—	89	1	no	—
202	#419	—	178	1	no	—
203	#418	—	178	1	no	—
204	#367	—	89	2	no	—
205	#366	—	89	2	no	—
206	#358	—	89	4	no	—
207	#357	—	89	4	no	—
208	C₂×#41	D₃₅₆	1	1	yes	characteristic
209	C₂²×Q₈	—	178	1	no	—
210	C₂²×D₄	—	178	1	no	—
211	C₂×Q₁₆	—	89	2	no	—
212	C₂×Q₁₆	—	178	2	no	—
213	C₂×Q₁₆	—	178	1	no	—
214	C₂×Q₁₆	—	178	2	no	—
215	C₂×D₈	—	89	2	no	—
216	C₂×D₈	—	178	2	no	—
217	C₂×D₈	—	178	1	no	—
218	C₂×D₈	—	178	2	no	—
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
222	C₂×M₁₆	D₃₅₆	1	2	yes	—
223	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
224	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
225	C₄⋊Q₈	—	356	1	no	—
226	#127	—	356	1	no	—
227	#124	—	356	1	no	—
228	#124	—	356	1	no	—
229	#123	—	356	2	no	—
230	C₂²⋊Q₈	—	178	4	no	—
231	#121	—	178	4	no	—
232	#120	—	356	2	no	—
233	#117	—	178	1	no	—
234	#117	—	178	1	no	—
235	SD₃₂	—	178	4	no	—
236	SD₃₂	—	178	4	no	—
237	M₃₂	—	2	2	no	—
238	M₃₂	—	2	4	no	—
239	C₂×C₁₆	—	2	4	no	—
240	C₂×C₁₆	C₂²⋊D₈₉	1	4	yes	complemented
241	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
242	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
244	#107	—	89	2	no	—
245	#107	—	89	2	no	—
246	#106	—	178	1	no	—
247	#106	—	178	1	no	—
248	#103	—	178	4	no	—
249	#102	—	178	4	no	—
250	C₂⁴	—	356	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	—	356	1	no	—
254	C₂×Q₈	—	356	2	no	—
255	C₂×Q₈	—	356	2	no	—
256	C₂×Q₈	—	178	2	no	—
257	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
258	C₂×D₄	—	356	4	no	—
259	C₂×D₄	—	356	2	no	—
260	C₂×D₄	—	356	1	no	—
261	C₂×D₄	—	356	2	no	—
262	C₂×D₄	—	178	2	no	—
263	C₂×D₄	D₇₁₂	1	2	yes	—
264	C₂×D₄	D₇₁₂	1	1	yes	characteristic
265	C₂²×C₄	—	356	1	no	—
266	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
267	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
268	Q₁₆	—	356	2	no	—
269	Q₁₆	—	178	2	no	—
270	Q₁₆	—	178	2	no	—
271	D₈	—	356	2	no	—
272	D₈	—	178	2	no	—
273	D₈	—	178	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
279	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
280	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
281	C₂×C₈	#13083574	1	2	yes	—
282	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
283	C₄⋊C₄	—	356	4	no	—
284	C₄⋊C₄	—	178	2	no	—
285	C₄⋊C₄	—	178	2	no	—
286	C₂²⋊C₄	—	356	1	no	—
287	C₂²⋊C₄	—	356	1	no	—
288	C₂²⋊C₄	—	356	4	no	—
289	C₂²⋊C₄	—	356	4	no	—
290	C₄²	—	356	1	no	—
291	C₄²	—	356	1	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
294	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
295	C₂³	—	356	2	no	—
296	C₂³	—	1424	2	no	—
297	C₂³	—	712	2	no	—
298	C₂³	C₂×D₇₁₂	1	2	yes	—
299	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	356	2	no	—
302	Q₈	—	356	2	no	—
303	Q₈	—	712	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	356	2	no	—
307	D₄	—	356	2	no	—
308	D₄	—	1424	4	no	—
309	D₄	—	712	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	356	2	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	—	712	2	no	—
316	C₂×C₄	#15667986	1	2	yes	—
317	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
318	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
319	C₂×C₄	#15668034	1	2	yes	—
320	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#15667986	1	2	yes	—
323	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
324	C₈	C₂×#13083574	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	1424	4	no	—
329	C₂²	—	712	4	no	—
330	C₂²	—	712	2	no	—
331	C₂²	#19151834	1	2	yes	—
332	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	712	2	no	—
335	C₄	—	712	2	no	—
336	C₄	—	712	2	no	—
337	C₄	#19151834	1	2	yes	—
338	C₄	C₂×#15667986	1	1	yes	characteristic
339	C₄	C₂×#15668034	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	1424	2	no	—
343	C₂	C₂×#19151834	1	1	yes	characteristic
344	C₂	#26033400	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#504×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26029900	C₂
hypercenter	C₂×#504	D₈₉
hypocenter	C₈₉	#33609
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×#15667957	C₂	semidirect
2. #26029907	C₂	semidirect
4. #26022365	C₂	semidirect
5. #26022348	C₂	semidirect
6. #26022036	C₂	semidirect
7. #26021973	C₂	semidirect
8. #26021969	C₂	semidirect
9. C₂×#504×C₈₉	C₂	semidirect
19. #19151839	C₂²	semidirect
25. #19151426	C₂²	semidirect
26. #19151424	C₂²	semidirect
27. #19151403	C₂²	semidirect
32. C₂×M₃₂×C₈₉	C₂²	semidirect
34. C₂²×C₁₆×C₈₉	C₂²	semidirect
73. C₂×C₁₆×C₈₉	D₄	semidirect
174. C₂×#504	D₈₉	semidirect
182. C₈₉	#33609	semidirect
193. C₂×M₃₂	C₂×D₈₉	semidirect
195. C₂²×C₁₆	C₂×D₈₉	semidirect
240. C₂×C₁₆	C₂²⋊D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667957	2
#26029907	2
#26029900	2
#26022365	2
#26022348	2
#26022036	2
#26021973	2
#26021969	2
C₂×#504×C₈₉	2
#33609	89