Subgroups — #36708720

Order 22784.

806 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152033	C₂	1	2	yes	maximal, complemented
2	C₂×#19151975	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022292	C₂	1	4	yes	maximal, complemented
4	#26022287	C₂	1	4	yes	maximal, complemented
5	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021967	C₂	1	2	yes	maximal, complemented
7	C₄×#135×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₁₂	—	2	1	no	—
9	C₂²×D₇₁₂	C₂²	1	2	yes	—
10	C₂×#15668009	—	2	1	no	—
11	#19152033	—	2	8	no	—
12	C₂×#15667955	C₂²	1	4	yes	complemented
13	C₂×#15667953	C₂²	1	4	yes	complemented
14	C₂×#15667952	C₂²	1	2	yes	—
15	#19151975	—	2	2	no	—
16	#19151975	C₂²	1	4	yes	—
17	C₄⋊#13083549	C₂²	1	4	yes	complemented
18	#19151398	C₂²	1	8	yes	complemented
19	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×#41×C₈₉	—	2	1	no	—
21	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	complemented
22	C₄×SD₁₆×C₈₉	—	2	2	no	—
23	C₄×D₈×C₈₉	C₂²	1	2	yes	complemented
24	#427×C₈₉	C₂²	1	1	yes	characteristic
25	#416×C₈₉	—	2	1	no	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×D₃₅₆	—	2	2	no	—
28	C₂²×D₃₅₆	—	4	2	no	—
29	C₂×D₇₁₂	—	2	4	no	—
30	C₂×D₇₁₂	—	2	2	no	—
31	C₂×D₇₁₂	—	2	4	no	—
32	C₂×D₇₁₂	—	2	4	no	—
33	C₂×D₇₁₂	—	4	4	no	—
34	C₂×D₇₁₂	C₂³	1	4	yes	—
35	#15668009	—	2	4	no	—
36	#15668009	—	4	1	no	—
37	#15668009	—	4	1	no	—
38	#15667955	—	2	8	no	—
39	#15667953	—	2	8	no	—
40	#15667952	—	2	4	no	—
41	C₂×#13083549	C₂³	1	4	yes	—
42	C₂×#41×C₈₉	—	2	1	no	—
43	#135×C₈₉	—	2	2	no	—
44	#135×C₈₉	D₄	1	4	yes	complemented
45	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆×C₈₉	—	2	1	no	—
47	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₈₉	—	2	2	no	—
51	C₄×D₄×C₈₉	—	2	2	no	—
52	C₄×D₄×C₈₉	—	4	2	no	—
53	#117×C₈₉	—	2	1	no	—
54	#117×C₈₉	—	4	1	no	—
55	C₂×C₄²×C₈₉	—	4	1	no	—
56	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
57	#107×C₈₉	C₂³	1	2	yes	—
58	#106×C₈₉	—	2	1	no	—
59	#103×C₈₉	—	2	2	no	—
60	#102×C₈₉	—	2	2	no	—
61	C₄×C₈×C₈₉	D₄	1	2	yes	—
62	C₄×C₈×C₈₉	C₂³	1	2	yes	—
63	C₂³×D₈₉	—	4	2	no	—
64	C₂×D₃₅₆	—	2	4	no	—
65	C₂×D₃₅₆	—	4	4	no	—
66	C₂×D₃₅₆	—	4	4	no	—
67	C₂×D₃₅₆	—	4	4	no	—
68	C₂×D₃₅₆	—	4	4	no	—
69	C₂×D₃₅₆	—	4	2	no	—
70	C₂×D₃₅₆	—	8	8	no	—
71	D₇₁₂	—	2	4	no	—
72	D₇₁₂	—	2	4	no	—
73	D₇₁₂	—	4	8	no	—
74	D₇₁₂	—	4	4	no	—
75	#13083549	—	2	4	no	—
76	#41×C₈₉	—	2	4	no	—
77	#41×C₈₉	—	4	2	no	—
78	C₂×Q₈×C₈₉	—	2	1	no	—
79	C₂×D₄×C₈₉	—	2	1	no	—
80	C₂×D₄×C₈₉	—	4	1	no	—
81	C₂²×C₄×C₈₉	—	4	1	no	—
82	C₂²×C₄×C₈₉	—	4	2	no	—
83	C₂²×C₄×C₈₉	D₈	1	2	yes	—
84	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
85	Q₁₆×C₈₉	—	2	1	no	—
86	Q₁₆×C₈₉	C₂×D₄	1	2	yes	complemented
87	SD₁₆×C₈₉	—	2	4	no	—
88	D₈×C₈₉	—	2	1	no	—
89	D₈×C₈₉	C₂×D₄	1	2	yes	complemented
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₈₉	#41	1	2	yes	—
95	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	—	2	2	no	—
97	C₄⋊C₄×C₈₉	—	4	2	no	—
98	C₂²⋊C₄×C₈₉	—	4	1	no	—
99	C₂²⋊C₄×C₈₉	—	4	2	no	—
100	C₄²×C₈₉	—	4	1	no	—
101	C₄²×C₈₉	—	4	2	no	—
102	C₄²×C₈₉	D₈	1	1	yes	characteristic
103	C₄²×C₈₉	D₈	1	1	yes	characteristic
104	C₄²×C₈₉	C₂×D₄	1	2	yes	—
105	C₂³×C₈₉	—	4	1	no	—
106	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₈₉	—	4	4	no	—
108	C₂²×D₈₉	—	8	8	no	—
109	C₂²×D₈₉	—	8	4	no	—
110	Q₈×C₈₉	—	2	2	no	—
111	Q₈×C₈₉	—	4	1	no	—
112	D₄×C₈₉	—	2	2	no	—
113	D₄×C₈₉	—	4	4	no	—
114	D₄×C₈₉	—	4	1	no	—
115	C₂×C₄×C₈₉	—	2	4	no	—
116	C₂×C₄×C₈₉	—	4	1	no	—
117	C₂×C₄×C₈₉	—	4	4	no	—
118	C₂×C₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	8	4	no	—
120	C₂×C₄×C₈₉	#121	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	D₃₅₆	—	4	4	no	—
127	D₃₅₆	—	4	4	no	—
128	D₃₅₆	—	8	8	no	—
129	D₃₅₆	—	8	4	no	—
130	D₃₅₆	—	8	8	no	—
131	C₈×C₈₉	—	2	2	no	—
132	C₈×C₈₉	#121	1	2	yes	—
133	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
135	C₂²×C₈₉	—	2	2	no	—
136	C₂²×C₈₉	—	4	2	no	—
137	C₂²×C₈₉	—	8	2	no	—
138	C₂²×C₈₉	#459	1	2	yes	—
139	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₈₉	—	8	8	no	—
141	C₂×D₈₉	—	8	4	no	—
142	C₂×D₈₉	—	16	8	no	—
143	C₄×C₈₉	—	2	2	no	—
144	C₄×C₈₉	—	4	2	no	—
145	C₄×C₈₉	—	8	2	no	—
146	C₄×C₈₉	#459	1	2	yes	—
147	C₄×C₈₉	#509	1	4	yes	complemented
148	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
149	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	#33582	—	89	1	no	maximal, Sylow
151	C₂×C₈₉	—	2	2	no	—
152	C₂×C₈₉	—	8	2	no	—
153	C₂×C₈₉	C₂×#459	1	1	yes	characteristic
154	C₂×C₈₉	C₂×#509	1	2	yes	—
155	D₈₉	—	16	4	no	—
156	C₂×#509	—	89	2	no	—
157	C₂×#493	—	89	1	no	—
158	C₄×#135	D₈₉	1	1	yes	characteristic, complemented
159	#2207	—	89	4	no	—
160	#2206	—	89	4	no	—
161	#2150	—	89	1	no	—
162	#2140	—	89	2	no	—
163	C₈₉	#33582	1	1	yes	characteristic, complemented, Sylow
164	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
165	C₂²×D₈	—	89	2	no	—
166	C₂²×D₈	—	178	1	no	—
167	C₂×#127	—	178	1	no	—
168	C₄×#41	—	2	1	no	—
169	#509	—	178	8	no	—
170	C₂×SD₃₂	—	89	4	no	—
171	C₂×D₁₆	—	89	4	no	—
172	C₂×M₃₂	—	89	2	no	—
173	#493	—	89	4	no	—
174	#493	—	178	2	no	—
175	C₄×Q₁₆	C₂×D₈₉	1	2	yes	complemented
176	C₄×SD₁₆	—	2	2	no	—
177	C₄×D₈	C₂×D₈₉	1	2	yes	complemented
178	#427	C₂×D₈₉	1	1	yes	characteristic
179	#416	—	2	1	no	—
180	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
181	C₄⋊C₁₆	—	89	4	no	—
182	#357	—	89	8	no	—
183	C₂×#41	—	2	1	no	—
184	C₂²×D₄	—	356	2	no	—
185	C₂²×D₄	—	178	2	no	—
186	#135	—	2	2	no	—
187	#135	D₃₅₆	1	4	yes	complemented
188	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
189	C₂×SD₁₆	—	2	1	no	—
190	C₂×D₈	—	89	4	no	—
191	C₂×D₈	—	356	4	no	—
192	C₂×D₈	—	178	2	no	—
193	C₂×D₈	—	178	4	no	—
194	C₂×D₈	—	178	4	no	—
195	C₂×D₈	—	178	4	no	—
196	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
199	#127	—	356	1	no	—
200	#127	—	356	1	no	—
201	#127	—	178	4	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×D₄	—	2	2	no	—
204	C₄×D₄	—	4	2	no	—
205	#117	—	2	1	no	—
206	#117	—	4	1	no	—
207	C₂×C₄²	—	4	1	no	—
208	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
209	SD₃₂	—	178	8	no	—
210	D₁₆	—	178	8	no	—
211	M₃₂	—	178	4	no	—
212	C₂×C₁₆	—	89	4	no	—
213	#107	C₂²×D₈₉	1	2	yes	—
214	#106	—	2	1	no	—
215	#103	—	2	2	no	—
216	#102	—	2	2	no	—
217	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
218	C₄×C₈	C₂²×D₈₉	1	2	yes	—
219	C₂⁴	—	356	2	no	—
220	#41	—	2	4	no	—
221	#41	—	4	2	no	—
222	C₂×Q₈	—	2	1	no	—
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	356	4	no	—
226	C₂×D₄	—	356	2	no	—
227	C₂×D₄	—	356	4	no	—
228	C₂×D₄	—	356	4	no	—
229	C₂×D₄	—	356	4	no	—
230	C₂×D₄	—	178	4	no	—
231	C₂×D₄	—	712	8	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	#13083559	1	2	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
236	Q₁₆	—	2	1	no	—
237	Q₁₆	C₂×D₃₅₆	1	2	yes	complemented
238	SD₁₆	—	2	4	no	—
239	D₈	—	2	1	no	—
240	D₈	—	356	8	no	—
241	D₈	—	356	4	no	—
242	D₈	—	178	4	no	—
243	D₈	—	178	4	no	—
244	D₈	C₂×D₃₅₆	1	2	yes	complemented
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
247	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
248	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
249	C₂×C₈	#13083574	1	2	yes	—
250	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	4	2	no	—
257	C₄²	#13083559	1	1	yes	characteristic
258	C₄²	#13083559	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
260	C₁₆	—	178	4	no	—
261	C₂³	—	4	1	no	—
262	C₂³	—	356	4	no	—
263	C₂³	—	712	8	no	—
264	C₂³	—	712	4	no	—
265	C₂³	C₂×#13083559	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	—	4	1	no	—
268	D₄	—	2	2	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	1	no	—
271	D₄	—	356	4	no	—
272	D₄	—	356	4	no	—
273	D₄	—	712	8	no	—
274	D₄	—	712	8	no	—
275	D₄	—	712	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	#15667986	1	2	yes	—
282	C₂×C₄	C₂×#13083559	1	2	yes	—
283	C₂×C₄	C₂×#13083559	1	2	yes	—
284	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
285	C₂×C₄	#15668037	1	2	yes	—
286	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	#15667986	1	2	yes	—
289	C₈	C₂²×D₃₅₆	1	1	yes	characteristic
290	C₈	C₂×#13083574	1	1	yes	characteristic
291	C₂²	—	2	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	1424	8	no	—
295	C₂²	—	712	4	no	—
296	C₂²	—	712	8	no	—
297	C₂²	#19151793	1	2	yes	—
298	C₂²	C₂²×#13083559	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	2	no	—
302	C₄	#19151793	1	2	yes	—
303	C₄	#19152033	1	4	yes	complemented
304	C₄	C₂×#15667986	1	1	yes	characteristic
305	C₄	C₂×#15668037	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	8	2	no	—
308	C₂	—	1424	4	no	—
309	C₂	C₂×#19151793	1	1	yes	characteristic
310	C₂	C₂×#19152033	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×#135×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	C₄×#135	D₈₉
hypocenter	C₈₉	#33582
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#19152033	C₂	semidirect
2. C₂×#19151975	C₂	semidirect
3. #26022292	C₂	semidirect
4. #26022287	C₂	semidirect
5. #26022008	C₂	semidirect
6. #26021967	C₂	semidirect
7. C₄×#135×C₈₉	C₂	semidirect
12. C₂×#15667955	C₂²	semidirect
13. C₂×#15667953	C₂²	semidirect
17. C₄⋊#13083549	C₂²	semidirect
18. #19151398	C₂²	semidirect
21. C₄×Q₁₆×C₈₉	C₂²	semidirect
23. C₄×D₈×C₈₉	C₂²	semidirect
44. #135×C₈₉	D₄	semidirect
86. Q₁₆×C₈₉	C₂×D₄	semidirect
89. D₈×C₈₉	C₂×D₄	semidirect
147. C₄×C₈₉	#509	semidirect
158. C₄×#135	D₈₉	semidirect
163. C₈₉	#33582	semidirect
175. C₄×Q₁₆	C₂×D₈₉	semidirect
177. C₄×D₈	C₂×D₈₉	semidirect
187. #135	D₃₅₆	semidirect
237. Q₁₆	C₂×D₃₅₆	semidirect
244. D₈	C₂×D₃₅₆	semidirect
303. C₄	#19152033	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152033	2
C₂×#19151975	2
#26022292	2
#26022287	2
#26022008	2
#26021967	2
C₄×#135×C₈₉	2
#33582	89