Subgroups — #36708948

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152030	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029922	C₂	1	2	yes	maximal, complemented
3	#26022365	C₂	1	4	yes	maximal, complemented
4	#26022343	C₂	1	2	yes	maximal, complemented
5	#26022338	C₂	1	2	yes	maximal, complemented
6	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021965	C₂	1	2	yes	maximal, complemented
8	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	2	yes	—
10	#19152176	—	2	2	no	—
11	#19152030	—	2	4	no	—
12	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
13	C₂×#15667957	C₂²	1	2	yes	—
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
15	#19151844	—	2	4	no	—
16	#19151839	C₂²	1	4	yes	—
17	#19151834	C₂²	1	4	yes	—
18	C₂×#15667946	C₂²	1	2	yes	—
19	#19151600	—	2	2	no	—
20	#19151424	C₂²	1	2	yes	—
21	#19151424	C₂²	1	2	yes	—
22	#19151403	C₂²	1	4	yes	—
23	#19151400	C₂²	1	4	yes	—
24	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
25	#504×C₈₉	—	2	4	no	—
26	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
28	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
29	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#13083574	—	2	2	no	—
31	#15668034	—	2	4	no	—
32	#15668034	—	2	4	no	—
33	C₂×Dic₃₅₆	C₂³	1	2	yes	—
34	C₂×#13083563	—	2	2	no	—
35	C₂×D₇₁₂	C₂³	1	2	yes	—
36	#15667991	—	2	4	no	—
37	#15667988	—	4	2	no	—
38	#15667986	—	2	4	no	—
39	#15667984	—	4	4	no	—
40	#15667983	—	4	2	no	—
41	C₂×C₄⋊Dic₈₉	—	2	2	no	—
42	Dic₇₁₂	—	2	2	no	—
43	#15667957	—	2	4	no	—
44	D₁₄₂₄	—	2	2	no	—
45	#15667946	—	2	2	no	—
46	#15667946	C₂³	1	4	yes	—
47	#15667937	—	2	4	no	—
48	#15667934	—	2	4	no	—
49	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
50	#131×C₈₉	—	2	2	no	—
51	#131×C₈₉	—	2	2	no	—
52	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
53	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	M₃₂×C₈₉	—	2	2	no	—
57	M₃₂×C₈₉	—	2	4	no	—
58	C₂×C₁₆×C₈₉	—	2	4	no	—
59	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
60	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
61	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
63	#13083574	—	4	4	no	—
64	#13083574	—	4	4	no	—
65	C₂×Dic₁₇₈	—	2	2	no	—
66	C₂×D₃₅₆	—	2	2	no	—
67	C₂×C₂²⋊D₈₉	—	4	2	no	—
68	C₂×C₂²⋊D₈₉	—	4	4	no	—
69	C₂²×Dic₈₉	—	4	2	no	—
70	C₂×C₄×D₈₉	—	4	2	no	—
71	Dic₃₅₆	—	2	2	no	—
72	Dic₃₅₆	—	2	2	no	—
73	#13083563	—	4	4	no	—
74	D₇₁₂	—	2	2	no	—
75	D₇₁₂	—	2	2	no	—
76	C₄⋊Dic₈₉	—	2	4	no	—
77	C₄⋊Dic₈₉	—	4	2	no	—
78	#13083553	—	4	4	no	—
79	C₂²⋊Dic₈₉	—	4	4	no	—
80	#13083551	—	4	4	no	—
81	#41×C₈₉	—	2	2	no	—
82	#41×C₈₉	—	2	2	no	—
83	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
84	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
85	C₂×D₄×C₈₉	D₈	1	2	yes	—
86	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	M₁₆×C₈₉	—	2	2	no	—
89	M₁₆×C₈₉	—	2	4	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₈₉	#41	1	2	yes	—
96	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
97	C₁₆×C₈₉	—	2	2	no	—
98	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
99	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
100	C₂³×C₈₉	C₂×D₈	1	2	yes	—
101	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₈₉	—	4	2	no	—
103	Q₈×C₈₉	—	2	2	no	—
104	D₄×C₈₉	—	2	4	no	—
105	D₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	2	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	#121	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#135	1	2	yes	—
112	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₈₉	—	8	4	no	—
114	C₂²⋊D₈₉	—	16	4	no	—
115	C₂×Dic₈₉	—	4	4	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	C₂×Dic₈₉	—	16	2	no	—
118	D₃₅₆	—	4	2	no	—
119	D₃₅₆	—	4	2	no	—
120	C₄×D₈₉	—	8	4	no	—
121	Dic₁₇₈	—	4	2	no	—
122	Dic₁₇₈	—	4	2	no	—
123	C₈×C₈₉	—	2	2	no	—
124	C₈×C₈₉	#121	1	2	yes	—
125	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	—	2	2	no	—
132	C₂²×C₈₉	#466	1	2	yes	—
133	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
134	C₂×D₈₉	—	8	2	no	—
135	C₂×D₈₉	—	8	2	no	—
136	C₂×D₈₉	—	8	2	no	—
137	C₄×C₈₉	—	2	2	no	—
138	C₄×C₈₉	#466	1	2	yes	—
139	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
140	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
141	Dic₈₉	—	8	4	no	—
142	Dic₈₉	—	8	2	no	—
143	#33610	—	89	1	no	maximal, Sylow
144	C₂×C₈₉	—	2	2	no	—
145	C₂×C₈₉	—	4	2	no	—
146	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
147	C₂×C₈₉	#3415	1	1	yes	characteristic
148	C₂×C₈₉	#3417	1	1	yes	characteristic
149	D₈₉	—	16	2	no	—
150	C₂×#508	—	89	1	no	—
151	C₂×#504	D₈₉	1	1	yes	characteristic, complemented
152	#3061	—	89	2	no	—
153	#2220	—	89	4	no	—
154	#2216	—	89	2	no	—
155	#2215	—	89	2	no	—
156	C₂×#366	—	89	1	no	—
157	#2139	—	89	2	no	—
158	C₈₉	#33610	1	1	yes	characteristic, complemented, Sylow
159	C₂×#135	—	89	2	no	—
160	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
161	#537	—	178	2	no	—
162	#508	—	178	4	no	—
163	C₂×Q₃₂	—	89	1	no	—
164	C₂×SD₃₂	—	89	2	no	—
165	C₂×D₁₆	—	89	1	no	—
166	#504	—	2	4	no	—
167	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
168	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
169	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
170	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
171	#468	—	178	4	no	—
172	#467	—	89	4	no	—
173	#466	—	89	4	no	—
174	C₂×#107	—	89	2	no	—
175	#417	—	178	2	no	—
176	#366	—	89	2	no	—
177	#366	—	89	2	no	—
178	#358	—	89	4	no	—
179	#357	—	89	4	no	—
180	C₂×#41	—	178	2	no	—
181	C₂×#41	D₃₅₆	1	1	yes	characteristic
182	#135	—	178	4	no	—
183	#135	—	178	4	no	—
184	C₂×Q₁₆	—	89	2	no	—
185	C₂×SD₁₆	—	178	2	no	—
186	C₂×D₈	—	89	2	no	—
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
190	C₂×M₁₆	D₃₅₆	1	2	yes	—
191	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
192	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
193	C₂²⋊Q₈	—	356	2	no	—
194	C₂²⋊Q₈	—	178	4	no	—
195	#121	—	356	2	no	—
196	#121	—	356	4	no	—
197	#121	—	178	4	no	—
198	C₂×C₄⋊C₄	—	178	2	no	—
199	Q₃₂	—	178	2	no	—
200	SD₃₂	—	178	4	no	—
201	D₁₆	—	178	2	no	—
202	M₃₂	—	2	2	no	—
203	M₃₂	—	2	4	no	—
204	C₂×C₁₆	—	2	4	no	—
205	C₂×C₁₆	C₂²⋊D₈₉	1	4	yes	complemented
206	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
207	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
208	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
209	#107	—	89	4	no	—
210	#107	—	178	2	no	—
211	#103	—	178	4	no	—
212	#102	—	178	4	no	—
213	#41	—	2	2	no	—
214	#41	—	2	2	no	—
215	#41	—	356	4	no	—
216	#41	—	356	4	no	—
217	C₂×Q₈	—	178	2	no	—
218	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
219	C₂×D₄	—	356	4	no	—
220	C₂×D₄	—	356	2	no	—
221	C₂×D₄	—	178	2	no	—
222	C₂×D₄	D₇₁₂	1	2	yes	—
223	C₂×D₄	D₇₁₂	1	1	yes	characteristic
224	C₂²×C₄	—	356	2	no	—
225	C₂²×C₄	—	356	2	no	—
226	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
227	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
228	Q₁₆	—	178	2	no	—
229	Q₁₆	—	178	2	no	—
230	SD₁₆	—	356	4	no	—
231	D₈	—	178	2	no	—
232	D₈	—	178	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
238	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
239	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
240	C₂×C₈	#13083574	1	2	yes	—
241	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
242	C₄⋊C₄	—	356	2	no	—
243	C₄⋊C₄	—	356	4	no	—
244	C₄⋊C₄	—	178	4	no	—
245	C₂²⋊C₄	—	356	4	no	—
246	C₂²⋊C₄	—	356	4	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
249	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
250	C₂³	—	356	2	no	—
251	C₂³	C₂×D₇₁₂	1	2	yes	—
252	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
253	Q₈	—	2	2	no	—
254	Q₈	—	356	2	no	—
255	Q₈	—	356	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	2	no	—
258	D₄	—	356	2	no	—
259	D₄	—	356	2	no	—
260	D₄	—	1424	4	no	—
261	D₄	—	712	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	356	4	no	—
265	C₂×C₄	—	356	2	no	—
266	C₂×C₄	—	1424	2	no	—
267	C₂×C₄	—	712	4	no	—
268	C₂×C₄	#15667986	1	2	yes	—
269	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
270	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
271	C₂×C₄	#15668034	1	2	yes	—
272	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	#15667986	1	2	yes	—
275	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
276	C₈	C₂×#13083574	1	1	yes	characteristic
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	712	2	no	—
283	C₂²	—	712	2	no	—
284	C₂²	—	712	2	no	—
285	C₂²	#19151834	1	2	yes	—
286	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	712	4	no	—
289	C₄	—	712	2	no	—
290	C₄	#19151834	1	2	yes	—
291	C₄	C₂×#15667986	1	1	yes	characteristic
292	C₄	C₂×#15668034	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	1424	2	no	—
296	C₂	C₂×#19151834	1	1	yes	characteristic
297	C₂	#26033393	1	1	yes	characteristic
298	C₂	#26033405	1	1	yes	characteristic

	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
hypercenter	C₂×#504	D₈₉
hypocenter	C₈₉	#33610
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. C₂×#19152030	C₂	semidirect
2. #26029922	C₂	semidirect
3. #26022365	C₂	semidirect
4. #26022343	C₂	semidirect
5. #26022338	C₂	semidirect
6. C₂×#19151424	C₂	semidirect
7. #26021965	C₂	semidirect
8. C₂×#504×C₈₉	C₂	semidirect
59. C₂×C₁₆×C₈₉	D₄	semidirect
151. C₂×#504	D₈₉	semidirect
158. C₈₉	#33610	semidirect
205. 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
C₂×#19152030	2
#26029922	2
#26022365	2
#26022343	2
#26022338	2
C₂×#19151424	2
#26021965	2
C₂×#504×C₈₉	2
#33610	89