Subgroups — #36709781

Order 22784.

682 conjugacy classes of subgroups.

The subgroup lattice has 248 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, complemented
2	C₂×#19151975	C₂	1	1	yes	maximal, characteristic
3	#26022494	C₂	1	4	yes	maximal, complemented
4	#26022476	C₂	1	4	yes	maximal, complemented
5	#26021969	C₂	1	4	yes	maximal, complemented
6	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₇₁₂	—	2	1	no	—
8	C₂²×D₇₁₂	C₂²	1	2	yes	—
9	#19152155	—	2	1	no	—
10	C₂×#15668009	—	2	1	no	—
11	#19152003	C₂²	1	2	yes	complemented
12	#19152000	—	2	1	no	—
13	#19152000	C₂²	1	2	yes	complemented
14	#19151996	—	2	1	no	—
15	#19151975	—	2	1	no	—
16	#19151975	—	2	1	no	—
17	#19151975	C₂²	1	2	yes	—
18	#19151975	C₂²	1	2	yes	—
19	#19151636	—	2	1	no	—
20	#19151636	C₂²	1	2	yes	—
21	#19151400	C₂²	1	16	yes	complemented
22	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#346×C₈₉	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₈₉	C₂²	1	2	yes	complemented
26	C₂²×D₃₅₆	—	2	2	no	—
27	C₂²×D₃₅₆	—	4	2	no	—
28	C₂×D₇₁₂	—	2	2	no	—
29	C₂×D₇₁₂	—	2	2	no	—
30	C₂×D₇₁₂	—	2	2	no	—
31	C₂×D₇₁₂	—	2	8	no	—
32	C₂×D₇₁₂	—	4	4	no	—
33	C₂×D₇₁₂	C₂³	1	4	yes	—
34	#15668012	—	2	2	no	—
35	#15668009	—	2	2	no	—
36	#15668009	—	2	2	no	—
37	#15668009	—	4	2	no	—
38	#15668004	—	2	2	no	—
39	#15667991	—	4	2	no	—
40	C₄×Dic₁₇₈	—	4	2	no	—
41	#15667971	—	2	2	no	—
42	#15667946	—	2	2	no	—
43	#15667946	C₂³	1	4	yes	—
44	#15667944	—	2	2	no	—
45	#15667944	—	2	2	no	—
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
49	M₃₂×C₈₉	—	2	4	no	—
50	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₂³×D₈₉	—	4	2	no	—
56	C₂×Dic₁₇₈	—	4	2	no	—
57	C₂×D₃₅₆	—	2	4	no	—
58	C₂×D₃₅₆	—	4	4	no	—
59	C₂×D₃₅₆	—	4	4	no	—
60	C₂×D₃₅₆	—	4	4	no	—
61	C₂×D₃₅₆	—	4	4	no	—
62	C₂×D₃₅₆	—	4	2	no	—
63	C₂×D₃₅₆	—	8	8	no	—
64	D₇₁₂	—	2	8	no	—
65	D₇₁₂	—	4	4	no	—
66	D₇₁₂	—	4	4	no	—
67	D₇₁₂	—	4	4	no	—
68	C₄⋊Dic₈₉	—	2	4	no	—
69	C₄⋊Dic₈₉	—	4	2	no	—
70	#13083553	—	4	4	no	—
71	C₂²⋊Dic₈₉	—	4	2	no	—
72	C₄×Dic₈₉	—	4	2	no	—
73	C₂²×C₄×C₈₉	D₈	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
76	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	#41	1	4	yes	—
80	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
81	C₄²×C₈₉	D₈	1	2	yes	—
82	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₁₆×C₈₉	—	2	4	no	—
85	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
86	C₂²×D₈₉	—	4	4	no	—
87	C₂²×D₈₉	—	8	8	no	—
88	C₂²×D₈₉	—	8	4	no	—
89	C₂×C₄×C₈₉	—	2	4	no	—
90	C₂×C₄×C₈₉	#124	1	2	yes	—
91	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×Dic₈₉	—	4	4	no	—
97	D₃₅₆	—	4	4	no	—
98	D₃₅₆	—	4	4	no	—
99	D₃₅₆	—	8	8	no	—
100	D₃₅₆	—	8	4	no	—
101	D₃₅₆	—	8	8	no	—
102	Dic₁₇₈	—	8	4	no	—
103	C₈×C₈₉	#124	1	2	yes	—
104	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
105	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
106	C₈×C₈₉	C₂×#41	1	2	yes	—
107	C₂²×C₈₉	—	2	2	no	—
108	C₂²×C₈₉	#486	1	2	yes	—
109	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
110	C₂×D₈₉	—	8	4	no	—
111	C₂×D₈₉	—	8	8	no	—
112	C₂×D₈₉	—	16	8	no	—
113	C₄×C₈₉	—	2	2	no	—
114	C₄×C₈₉	#486	1	2	yes	—
115	C₄×C₈₉	#509	1	4	yes	—
116	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
117	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
118	Dic₈₉	—	8	4	no	—
119	#33704	—	89	1	no	maximal, Sylow
120	C₂×C₈₉	—	2	2	no	—
121	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
122	C₂×C₈₉	C₂×#509	1	2	yes	—
123	D₈₉	—	16	4	no	—
124	#3160	—	89	1	no	—
125	C₂×#493	—	89	1	no	—
126	#2244	—	89	4	no	—
127	#2240	—	89	4	no	—
128	#2140	—	89	4	no	—
129	C₄×M₃₂	D₈₉	1	1	yes	characteristic, complemented
130	C₈₉	#33704	1	1	yes	characteristic, complemented, Sylow
131	C₂²×D₈	—	89	2	no	—
132	C₂²×D₈	—	178	1	no	—
133	#533	—	178	1	no	—
134	C₂×#127	—	178	1	no	—
135	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
136	#500	—	89	2	no	—
137	#499	—	89	2	no	—
138	#499	—	178	1	no	—
139	#498	—	178	1	no	—
140	#493	—	89	2	no	—
141	#493	—	89	2	no	—
142	#493	—	178	1	no	—
143	#493	—	178	1	no	—
144	#427	—	89	2	no	—
145	#427	—	178	1	no	—
146	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
147	#357	—	89	16	no	—
148	#346	C₂×D₈₉	1	2	yes	complemented
149	C₄×C₁₆	C₂×D₈₉	1	2	yes	complemented
150	C₂²×D₄	—	356	2	no	—
151	C₂²×D₄	—	178	2	no	—
152	C₂×D₈	—	89	4	no	—
153	C₂×D₈	—	356	4	no	—
154	C₂×D₈	—	178	2	no	—
155	C₂×D₈	—	178	2	no	—
156	C₂×D₈	—	178	2	no	—
157	C₂×D₈	—	178	8	no	—
158	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
160	C₄⋊Q₈	—	178	2	no	—
161	#127	—	356	2	no	—
162	#127	—	178	2	no	—
163	#127	—	178	2	no	—
164	#125	—	178	2	no	—
165	C₂²⋊Q₈	—	356	2	no	—
166	C₄×Q₈	—	356	2	no	—
167	#117	—	178	2	no	—
168	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
169	M₃₂	—	2	4	no	—
170	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
171	#107	—	89	4	no	—
172	#107	—	178	2	no	—
173	#106	—	178	2	no	—
174	#106	—	178	2	no	—
175	C₄×C₈	D₃₅₆	1	1	yes	characteristic
176	C₄×C₈	D₃₅₆	1	1	yes	characteristic
177	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
178	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
179	C₂⁴	—	356	2	no	—
180	C₂×Q₈	—	356	2	no	—
181	C₂×D₄	—	356	4	no	—
182	C₂×D₄	—	356	2	no	—
183	C₂×D₄	—	356	4	no	—
184	C₂×D₄	—	356	4	no	—
185	C₂×D₄	—	356	4	no	—
186	C₂×D₄	—	178	4	no	—
187	C₂×D₄	—	712	8	no	—
188	C₂²×C₄	D₇₁₂	1	2	yes	—
189	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
190	D₈	—	356	4	no	—
191	D₈	—	356	4	no	—
192	D₈	—	356	4	no	—
193	D₈	—	178	8	no	—
194	C₂×C₈	#13083563	1	4	yes	—
195	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
196	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
197	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
198	C₂×C₈	#13083574	1	4	yes	—
199	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
200	C₄⋊C₄	—	356	4	no	—
201	C₄⋊C₄	—	356	2	no	—
202	C₄⋊C₄	—	178	4	no	—
203	C₂²⋊C₄	—	356	2	no	—
204	C₄²	—	356	2	no	—
205	C₄²	D₇₁₂	1	2	yes	—
206	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
207	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
208	C₁₆	—	2	4	no	—
209	C₂³	—	356	4	no	—
210	C₂³	—	712	8	no	—
211	C₂³	—	712	4	no	—
212	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
213	Q₈	—	712	4	no	—
214	D₄	—	356	4	no	—
215	D₄	—	356	4	no	—
216	D₄	—	712	8	no	—
217	D₄	—	712	4	no	—
218	D₄	—	712	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	356	4	no	—
221	C₂×C₄	#15668000	1	2	yes	—
222	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
223	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
224	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
225	C₂×C₄	C₂×#13083563	1	2	yes	—
226	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
227	C₈	#15668000	1	2	yes	—
228	C₈	C₂×#13083563	1	2	yes	—
229	C₈	C₂×#13083563	1	2	yes	—
230	C₈	C₂×#13083574	1	2	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	1424	8	no	—
233	C₂²	—	712	4	no	—
234	C₂²	—	712	8	no	—
235	C₂²	#19151945	1	2	yes	—
236	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	712	4	no	—
239	C₄	#19151945	1	2	yes	—
240	C₄	#19152036	1	4	yes	—
241	C₄	C₂×#15668000	1	1	yes	characteristic
242	C₄	C₂²×#13083563	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	1424	4	no	—
245	C₂	C₂×#19151945	1	1	yes	characteristic
246	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
prime radical	C₂×#19151975	C₂
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#33704
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. #26030766	C₂	semidirect
3. #26022494	C₂	semidirect
4. #26022476	C₂	semidirect
5. #26021969	C₂	semidirect
6. C₄×M₃₂×C₈₉	C₂	semidirect
11. #19152003	C₂²	semidirect
13. #19152000	C₂²	semidirect
21. #19151400	C₂²	semidirect
24. #346×C₈₉	C₂²	semidirect
25. C₄×C₁₆×C₈₉	C₂²	semidirect
129. C₄×M₃₂	D₈₉	semidirect
130. C₈₉	#33704	semidirect
148. #346	C₂×D₈₉	semidirect
149. C₄×C₁₆	C₂×D₈₉	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×#19151975	2
#26022494	2
#26022476	2
#26021969	2
C₄×M₃₂×C₈₉	2
#33704	89