Subgroups — #36709952

Order 22784.

514 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	1	yes	maximal, characteristic
2	C₂×#19152003	C₂	1	1	yes	maximal, characteristic
3	#26022538	C₂	1	4	yes	maximal, complemented
4	#26022534	C₂	1	2	yes	maximal, complemented
5	#26022527	C₂	1	2	yes	maximal, complemented
6	#26022040	C₂	1	4	yes	maximal
7	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#19152155	—	2	1	no	—
9	C₂×#15668012	—	2	1	no	—
10	#19152003	—	2	1	no	—
11	#19152003	—	2	1	no	—
12	#19152003	C₂²	1	1	yes	characteristic
13	#19152003	C₂²	1	2	yes	—
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	2	yes	—
16	#19152000	—	2	1	no	—
17	#19152000	C₂²	1	2	yes	—
18	#19151996	—	2	1	no	—
19	#19151636	—	2	1	no	—
20	#19151636	C₂²	1	2	yes	—
21	C₂×#15667946	—	2	1	no	—
22	C₂×#15667946	C₂²	1	2	yes	—
23	#19151426	C₂²	1	8	yes	—
24	#19151424	C₂²	1	8	yes	—
25	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	#346×C₈₉	C₂²	1	2	yes	—
28	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
29	C₂²×Dic₁₇₈	—	4	1	no	—
30	#15668012	—	2	2	no	—
31	#15668012	—	2	2	no	—
32	#15668012	—	2	1	no	—
33	#15668012	—	2	1	no	—
34	#15668012	—	4	1	no	—
35	#15668012	—	4	1	no	—
36	#15668004	—	2	2	no	—
37	#15667991	—	4	1	no	—
38	#15667991	—	4	1	no	—
39	C₄×Dic₁₇₈	—	4	1	no	—
40	C₄×Dic₁₇₈	—	4	1	no	—
41	#15667971	—	2	2	no	—
42	C₂×C₄⋊Dic₈₉	—	2	2	no	—
43	C₂×C₄⋊Dic₈₉	—	4	1	no	—
44	#15667946	—	2	2	no	—
45	#15667946	—	2	2	no	—
46	#15667946	—	2	2	no	—
47	#15667946	—	2	1	no	—
48	#15667946	—	2	1	no	—
49	#15667946	C₂³	1	4	yes	—
50	#15667946	C₂³	1	2	yes	—
51	#15667946	C₂³	1	2	yes	—
52	#15667944	—	2	2	no	—
53	#15667944	—	2	2	no	—
54	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
57	M₃₂×C₈₉	Q₈	1	8	yes	complemented
58	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₇₈	—	4	1	no	—
64	C₂×Dic₁₇₈	—	4	1	no	—
65	C₂×Dic₁₇₈	—	4	2	no	—
66	C₂×Dic₁₇₈	—	4	2	no	—
67	C₂×Dic₁₇₈	—	8	4	no	—
68	C₂²×Dic₈₉	—	4	2	no	—
69	C₄⋊Dic₈₉	—	2	2	no	—
70	C₄⋊Dic₈₉	—	2	2	no	—
71	C₄⋊Dic₈₉	—	2	4	no	—
72	C₄⋊Dic₈₉	—	4	1	no	—
73	C₄⋊Dic₈₉	—	4	1	no	—
74	C₄⋊Dic₈₉	—	4	2	no	—
75	C₄⋊Dic₈₉	—	4	2	no	—
76	C₄⋊Dic₈₉	—	4	2	no	—
77	#13083553	—	4	2	no	—
78	#13083553	—	4	2	no	—
79	C₂²⋊Dic₈₉	—	4	2	no	—
80	C₄×Dic₈₉	—	4	2	no	—
81	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
85	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
89	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₄²×C₈₉	D₈	1	1	yes	characteristic
91	C₄²×C₈₉	D₈	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₁₆×C₈₉	C₂×Q₈	1	8	yes	complemented
95	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
99	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×Dic₈₉	—	4	2	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	C₂×Dic₈₉	—	4	4	no	—
109	C₂×Dic₈₉	—	8	4	no	—
110	Dic₁₇₈	—	8	4	no	—
111	Dic₁₇₈	—	8	2	no	—
112	Dic₁₇₈	—	8	4	no	—
113	Dic₁₇₈	—	8	2	no	—
114	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
115	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
116	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
117	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
118	C₂²×C₈₉	—	2	1	no	—
119	C₂²×C₈₉	—	2	1	no	—
120	C₂²×C₈₉	#500	1	2	yes	—
121	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₈₉	—	2	1	no	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	#500	1	2	yes	—
125	C₄×C₈₉	#509	1	2	yes	—
126	C₄×C₈₉	#510	1	2	yes	—
127	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
129	Dic₈₉	—	8	4	no	—
130	Dic₈₉	—	8	2	no	—
131	Dic₈₉	—	8	2	no	—
132	#33731	—	89	1	no	maximal, Sylow
133	C₂×C₈₉	—	2	2	no	—
134	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
135	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
136	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
137	#3160	—	89	1	no	—
138	C₂×#500	—	89	1	no	—
139	#2255	—	89	4	no	—
140	#2254	—	89	2	no	—
141	#2252	—	89	2	no	—
142	#2159	—	89	4	no	—
143	C₄×M₃₂	D₈₉	1	1	yes	characteristic
144	C₈₉	#33731	1	1	yes	characteristic, complemented, Sylow
145	#533	—	178	1	no	—
146	C₂×C₄⋊Q₈	—	178	1	no	—
147	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
148	#500	—	89	2	no	—
149	#500	—	89	2	no	—
150	#500	—	89	1	no	—
151	#500	—	89	1	no	—
152	#500	—	178	1	no	—
153	#500	—	178	1	no	—
154	#499	—	89	2	no	—
155	#499	—	178	1	no	—
156	#498	—	178	1	no	—
157	#427	—	89	2	no	—
158	#427	—	178	1	no	—
159	C₂×#107	—	89	2	no	—
160	C₂×#107	—	178	1	no	—
161	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
162	#367	—	89	8	no	—
163	#366	—	89	8	no	—
164	#346	C₂×D₈₉	1	2	yes	—
165	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
166	C₂²×Q₈	—	356	1	no	—
167	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
169	C₄⋊Q₈	—	356	1	no	—
170	C₄⋊Q₈	—	356	1	no	—
171	C₄⋊Q₈	—	178	2	no	—
172	C₄⋊Q₈	—	178	2	no	—
173	C₄⋊Q₈	—	178	1	no	—
174	C₄⋊Q₈	—	178	1	no	—
175	#125	—	178	2	no	—
176	C₂²⋊Q₈	—	356	1	no	—
177	C₂²⋊Q₈	—	356	1	no	—
178	C₄×Q₈	—	356	1	no	—
179	C₄×Q₈	—	356	1	no	—
180	#117	—	178	2	no	—
181	C₂×C₄⋊C₄	—	356	1	no	—
182	C₂×C₄⋊C₄	—	178	2	no	—
183	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
184	M₃₂	Dic₁₇₈	1	8	yes	complemented
185	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
186	#107	—	89	2	no	—
187	#107	—	89	4	no	—
188	#107	—	89	2	no	—
189	#107	—	178	2	no	—
190	#107	—	178	2	no	—
191	#107	—	178	2	no	—
192	#107	—	178	1	no	—
193	#107	—	178	1	no	—
194	#106	—	178	2	no	—
195	#106	—	178	2	no	—
196	C₄×C₈	D₃₅₆	1	1	yes	characteristic
197	C₄×C₈	D₃₅₆	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
200	C₂×Q₈	—	356	1	no	—
201	C₂×Q₈	—	356	1	no	—
202	C₂×Q₈	—	356	2	no	—
203	C₂×Q₈	—	356	2	no	—
204	C₂×Q₈	—	712	4	no	—
205	C₂²×C₄	—	356	2	no	—
206	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
207	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
209	C₂×C₈	Dic₃₅₆	1	4	yes	—
210	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
211	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
212	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
213	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
214	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
215	C₄⋊C₄	—	356	2	no	—
216	C₄⋊C₄	—	356	2	no	—
217	C₄⋊C₄	—	356	2	no	—
218	C₄⋊C₄	—	356	2	no	—
219	C₄⋊C₄	—	356	2	no	—
220	C₄⋊C₄	—	356	1	no	—
221	C₄⋊C₄	—	356	1	no	—
222	C₄⋊C₄	—	178	2	no	—
223	C₄⋊C₄	—	178	2	no	—
224	C₄⋊C₄	—	178	4	no	—
225	C₂²⋊C₄	—	356	2	no	—
226	C₄²	—	356	2	no	—
227	C₄²	D₇₁₂	1	1	yes	characteristic
228	C₄²	D₇₁₂	1	1	yes	characteristic
229	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
230	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
231	C₁₆	C₂×Dic₁₇₈	1	8	yes	complemented
232	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
233	Q₈	—	712	4	no	—
234	Q₈	—	712	2	no	—
235	Q₈	—	712	4	no	—
236	Q₈	—	712	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	356	2	no	—
240	C₂×C₄	—	356	2	no	—
241	C₂×C₄	—	356	4	no	—
242	C₂×C₄	—	712	4	no	—
243	C₂×C₄	#15668012	1	2	yes	—
244	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
245	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
246	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
247	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
248	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
249	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
250	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
251	C₈	#15668012	1	2	yes	—
252	C₈	C₂×Dic₃₅₆	1	2	yes	—
253	C₈	C₂×Dic₃₅₆	1	2	yes	—
254	C₈	C₂²×Dic₁₇₈	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	#19152003	1	2	yes	—
258	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	712	4	no	—
262	C₄	—	712	2	no	—
263	C₄	—	712	2	no	—
264	C₄	#19152003	1	2	yes	—
265	C₄	#19152036	1	2	yes	—
266	C₄	#19152041	1	2	yes	—
267	C₄	C₂×#15668012	1	1	yes	characteristic
268	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	C₂×#19152003	1	1	yes	characteristic
271	C₂	C₂×#19152036	1	1	yes	characteristic
272	C₂	C₂×#19152041	1	1	yes	characteristic

	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
prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#33731
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
3. #26022538	C₂	semidirect
4. #26022534	C₂	semidirect
5. #26022527	C₂	semidirect
57. M₃₂×C₈₉	Q₈	semidirect
94. C₁₆×C₈₉	C₂×Q₈	semidirect
144. C₈₉	#33731	semidirect
184. M₃₂	Dic₁₇₈	semidirect
231. C₁₆	C₂×Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
C₂×#19152003	2
#26022538	2
#26022534	2
#26022527	2
#26022040	2
C₄×M₃₂×C₈₉	2
#33731	89