Subgroups — #36709858

Order 22784.

918 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152036	C₂	1	4	yes	maximal, complemented
2	#26030680	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151975	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022518	C₂	1	4	yes	maximal, complemented
5	#26022508	C₂	1	2	yes	maximal, complemented
6	#26022502	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	—	2	1	no	—
9	C₂×#15668034	C₂²	1	2	yes	—
10	C₂²×D₇₁₂	—	2	1	no	—
11	C₂²×D₇₁₂	C₂²	1	2	yes	—
12	#19152148	—	2	1	no	—
13	C₂×#15668009	—	2	1	no	—
14	#19152036	—	2	16	no	—
15	C₂×#15667957	C₂²	1	8	yes	complemented
16	C₂×D₁₄₂₄	C₂²	1	8	yes	complemented
17	#19151982	—	2	1	no	—
18	#19151982	C₂²	1	2	yes	complemented
19	#19151978	C₂²	1	1	yes	characteristic, complemented
20	#19151975	—	2	1	no	—
21	#19151975	—	2	1	no	—
22	#19151975	C₂²	1	2	yes	—
23	#19151975	C₂²	1	2	yes	—
24	#19151975	C₂²	1	1	yes	characteristic, complemented
25	#19151971	—	2	1	no	—
26	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	#346×C₈₉	C₂²	1	2	yes	complemented
29	C₄×C₁₆×C₈₉	C₂²	1	2	yes	complemented
30	C₂×#13083574	—	2	2	no	—
31	C₂²×D₃₅₆	—	2	2	no	—
32	C₂²×D₃₅₆	—	4	2	no	—
33	#15668034	—	2	8	no	—
34	#15668034	—	4	4	no	—
35	C₂×Dic₃₅₆	—	2	1	no	—
36	C₂×Dic₃₅₆	C₂³	1	2	yes	—
37	C₂×#13083563	—	2	2	no	—
38	C₂×#13083563	—	2	2	no	—
39	C₂×D₇₁₂	—	2	8	no	—
40	C₂×D₇₁₂	—	2	2	no	—
41	C₂×D₇₁₂	—	2	2	no	—
42	C₂×D₇₁₂	—	2	2	no	—
43	C₂×D₇₁₂	—	2	1	no	—
44	C₂×D₇₁₂	—	4	4	no	—
45	C₂×D₇₁₂	C₂³	1	4	yes	—
46	C₂×D₇₁₂	C₂³	1	2	yes	—
47	#15668012	—	2	1	no	—
48	#15668009	—	2	2	no	—
49	#15668009	—	2	2	no	—
50	#15668009	—	2	1	no	—
51	#15668009	—	4	2	no	—
52	#15668000	—	2	2	no	—
53	#15667986	—	4	2	no	—
54	C₄×D₃₅₆	—	4	2	no	—
55	#15667957	—	2	16	no	—
56	D₁₄₂₄	—	2	16	no	—
57	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
60	M₃₂×C₈₉	D₄	1	8	yes	complemented
61	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
62	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
64	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂³×D₈₉	—	4	2	no	—
67	#13083574	—	4	4	no	—
68	#13083574	—	4	4	no	—
69	C₂×Dic₁₇₈	—	2	2	no	—
70	C₂×D₃₅₆	—	2	2	no	—
71	C₂×D₃₅₆	—	2	4	no	—
72	C₂×D₃₅₆	—	4	4	no	—
73	C₂×D₃₅₆	—	4	4	no	—
74	C₂×D₃₅₆	—	4	2	no	—
75	C₂×D₃₅₆	—	4	4	no	—
76	C₂×D₃₅₆	—	4	4	no	—
77	C₂×D₃₅₆	—	4	2	no	—
78	C₂×D₃₅₆	—	8	8	no	—
79	C₂×C₂²⋊D₈₉	—	4	2	no	—
80	C₂×C₄×D₈₉	—	4	2	no	—
81	Dic₃₅₆	—	2	4	no	—
82	Dic₃₅₆	—	4	2	no	—
83	#13083563	—	4	4	no	—
84	#13083563	—	4	4	no	—
85	D₇₁₂	—	2	8	no	—
86	D₇₁₂	—	2	4	no	—
87	D₇₁₂	—	4	4	no	—
88	D₇₁₂	—	4	4	no	—
89	D₇₁₂	—	4	4	no	—
90	D₇₁₂	—	4	2	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	#13083551	—	4	4	no	—
93	C₂²×C₄×C₈₉	D₈	1	2	yes	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	D₈	1	4	yes	—
96	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
101	C₄²×C₈₉	D₈	1	2	yes	—
102	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₁₆×C₈₉	C₂×D₄	1	8	yes	complemented
105	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
106	C₂²×D₈₉	—	4	2	no	—
107	C₂²×D₈₉	—	4	4	no	—
108	C₂²×D₈₉	—	8	8	no	—
109	C₂²×D₈₉	—	8	4	no	—
110	C₂×C₄×C₈₉	—	2	4	no	—
111	C₂×C₄×C₈₉	#127	1	2	yes	—
112	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
117	C₂²⋊D₈₉	—	8	4	no	—
118	C₂×Dic₈₉	—	4	2	no	—
119	D₃₅₆	—	4	4	no	—
120	D₃₅₆	—	4	2	no	—
121	D₃₅₆	—	4	4	no	—
122	D₃₅₆	—	4	2	no	—
123	D₃₅₆	—	8	8	no	—
124	D₃₅₆	—	8	4	no	—
125	D₃₅₆	—	8	4	no	—
126	D₃₅₆	—	8	8	no	—
127	C₄×D₈₉	—	8	4	no	—
128	Dic₁₇₈	—	4	2	no	—
129	Dic₁₇₈	—	4	2	no	—
130	C₈×C₈₉	#127	1	2	yes	—
131	C₈×C₈₉	C₂×D₈	1	2	yes	—
132	C₈×C₈₉	C₂×D₈	1	2	yes	—
133	C₈×C₈₉	C₂²×D₄	1	2	yes	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	#493	1	2	yes	—
136	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
137	C₂×D₈₉	—	8	2	no	—
138	C₂×D₈₉	—	8	4	no	—
139	C₂×D₈₉	—	8	8	no	—
140	C₂×D₈₉	—	8	4	no	—
141	C₂×D₈₉	—	16	8	no	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	#493	1	2	yes	—
144	C₄×C₈₉	#509	1	4	yes	complemented
145	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
146	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
147	Dic₈₉	—	8	2	no	—
148	#33717	—	89	1	no	maximal, Sylow
149	C₂×C₈₉	—	2	2	no	—
150	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
151	C₂×C₈₉	C₂×#509	1	2	yes	—
152	D₈₉	—	16	2	no	—
153	D₈₉	—	16	4	no	—
154	C₂×#509	—	89	4	no	—
155	#3147	—	89	1	no	—
156	C₂×#493	—	89	1	no	—
157	#2250	—	89	4	no	—
158	#2248	—	89	2	no	—
159	#2246	—	89	2	no	—
160	C₄×M₃₂	D₈₉	1	1	yes	characteristic, complemented
161	C₈₉	#33717	1	1	yes	characteristic, complemented, Sylow
162	C₂×#135	—	89	2	no	—
163	C₂×#135	—	178	1	no	—
164	C₂²×D₈	—	89	2	no	—
165	C₂²×D₈	—	178	1	no	—
166	#532	—	178	1	no	—
167	C₂×#127	—	178	1	no	—
168	#509	—	178	16	no	—
169	C₂×SD₃₂	—	89	8	no	—
170	C₂×D₁₆	—	89	8	no	—
171	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
172	#495	—	89	2	no	—
173	#495	—	178	1	no	—
174	#494	—	89	1	no	—
175	#493	—	89	2	no	—
176	#493	—	89	2	no	—
177	#493	—	89	1	no	—
178	#493	—	178	1	no	—
179	#493	—	178	1	no	—
180	#492	—	178	1	no	—
181	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
182	#346	C₂×D₈₉	1	2	yes	complemented
183	C₄×C₁₆	C₂×D₈₉	1	2	yes	complemented
184	C₂×#41	—	178	2	no	—
185	C₂²×D₄	—	356	2	no	—
186	C₂²×D₄	—	178	2	no	—
187	#135	—	356	4	no	—
188	#135	—	178	8	no	—
189	C₂×Q₁₆	—	89	2	no	—
190	C₂×Q₁₆	—	178	1	no	—
191	C₂×SD₁₆	—	178	2	no	—
192	C₂×SD₁₆	—	178	2	no	—
193	C₂×D₈	—	89	4	no	—
194	C₂×D₈	—	89	2	no	—
195	C₂×D₈	—	356	4	no	—
196	C₂×D₈	—	178	2	no	—
197	C₂×D₈	—	178	2	no	—
198	C₂×D₈	—	178	2	no	—
199	C₂×D₈	—	178	1	no	—
200	C₂×D₈	—	178	8	no	—
201	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	178	1	no	—
204	#127	—	356	2	no	—
205	#127	—	178	2	no	—
206	#127	—	178	2	no	—
207	#127	—	178	1	no	—
208	#124	—	178	2	no	—
209	#121	—	356	2	no	—
210	C₄×D₄	—	356	2	no	—
211	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
212	SD₃₂	—	178	16	no	—
213	D₁₆	—	178	16	no	—
214	M₃₂	D₃₅₆	1	8	yes	complemented
215	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
216	C₄×C₈	D₃₅₆	1	1	yes	characteristic
217	C₄×C₈	D₃₅₆	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
220	C₂⁴	—	356	2	no	—
221	#41	—	356	4	no	—
222	#41	—	356	4	no	—
223	C₂×Q₈	—	178	2	no	—
224	C₂×D₄	—	356	2	no	—
225	C₂×D₄	—	356	4	no	—
226	C₂×D₄	—	356	4	no	—
227	C₂×D₄	—	356	2	no	—
228	C₂×D₄	—	356	2	no	—
229	C₂×D₄	—	356	4	no	—
230	C₂×D₄	—	356	4	no	—
231	C₂×D₄	—	178	2	no	—
232	C₂×D₄	—	178	4	no	—
233	C₂×D₄	—	712	8	no	—
234	C₂²×C₄	—	356	2	no	—
235	C₂²×C₄	D₇₁₂	1	2	yes	—
236	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
237	Q₁₆	—	356	2	no	—
238	Q₁₆	—	178	4	no	—
239	SD₁₆	—	356	4	no	—
240	SD₁₆	—	356	4	no	—
241	D₈	—	356	4	no	—
242	D₈	—	356	4	no	—
243	D₈	—	356	4	no	—
244	D₈	—	356	2	no	—
245	D₈	—	178	4	no	—
246	D₈	—	178	8	no	—
247	C₂×C₈	D₇₁₂	1	4	yes	—
248	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
249	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
250	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
251	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	—	356	2	no	—
254	C₂²⋊C₄	—	356	4	no	—
255	C₄²	D₇₁₂	1	2	yes	—
256	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
257	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
258	C₁₆	C₂×D₃₅₆	1	8	yes	complemented
259	C₂³	—	356	2	no	—
260	C₂³	—	356	4	no	—
261	C₂³	—	712	8	no	—
262	C₂³	—	712	4	no	—
263	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
264	Q₈	—	356	2	no	—
265	Q₈	—	356	2	no	—
266	D₄	—	356	4	no	—
267	D₄	—	356	2	no	—
268	D₄	—	356	4	no	—
269	D₄	—	356	2	no	—
270	D₄	—	712	8	no	—
271	D₄	—	712	4	no	—
272	D₄	—	712	4	no	—
273	D₄	—	712	4	no	—
274	D₄	—	712	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	712	4	no	—
278	C₂×C₄	#15668009	1	2	yes	—
279	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
280	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
281	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
282	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
283	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
284	C₈	#15668009	1	2	yes	—
285	C₈	C₂×D₇₁₂	1	2	yes	—
286	C₈	C₂×D₇₁₂	1	2	yes	—
287	C₈	C₂²×D₃₅₆	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	1424	8	no	—
290	C₂²	—	712	2	no	—
291	C₂²	—	712	4	no	—
292	C₂²	—	712	8	no	—
293	C₂²	—	712	4	no	—
294	C₂²	#19151975	1	2	yes	—
295	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	712	2	no	—
298	C₄	#19151975	1	2	yes	—
299	C₄	#19152036	1	4	yes	complemented
300	C₄	C₂×#15668009	1	1	yes	characteristic
301	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	1424	4	no	—
304	C₂	—	1424	2	no	—
305	C₂	C₂×#19151975	1	1	yes	characteristic
306	C₂	C₂×#19152036	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#33717
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#19152036	C₂	semidirect
2. #26030680	C₂	semidirect
3. C₂×#19151975	C₂	semidirect
4. #26022518	C₂	semidirect
5. #26022508	C₂	semidirect
6. #26022502	C₂	semidirect
7. C₄×M₃₂×C₈₉	C₂	semidirect
15. C₂×#15667957	C₂²	semidirect
16. C₂×D₁₄₂₄	C₂²	semidirect
18. #19151982	C₂²	semidirect
19. #19151978	C₂²	semidirect
24. #19151975	C₂²	semidirect
28. #346×C₈₉	C₂²	semidirect
29. C₄×C₁₆×C₈₉	C₂²	semidirect
60. M₃₂×C₈₉	D₄	semidirect
104. C₁₆×C₈₉	C₂×D₄	semidirect
144. C₄×C₈₉	#509	semidirect
160. C₄×M₃₂	D₈₉	semidirect
161. C₈₉	#33717	semidirect
182. #346	C₂×D₈₉	semidirect
183. C₄×C₁₆	C₂×D₈₉	semidirect
214. M₃₂	D₃₅₆	semidirect
258. C₁₆	C₂×D₃₅₆	semidirect
299. C₄	#19152036	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152036	2
#26030680	2
C₂×#19151975	2
#26022518	2
#26022508	2
#26022502	2
C₄×M₃₂×C₈₉	2
#33717	89