Subgroups — #36711786

Order 22784.

548 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032600	C₂	1	2	yes	maximal, complemented
2	#26022527	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022433	C₂	1	4	yes	maximal, complemented
4	#26022390	C₂	1	2	yes	maximal, complemented
5	#26022338	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151424	C₂	1	2	yes	maximal, complemented
8	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#19152252	—	2	2	no	—
10	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
11	#19152003	C₂²	1	2	yes	—
12	#19151912	—	2	4	no	—
13	#19151879	—	2	2	no	—
14	#19151834	C₂²	1	4	yes	—
15	C₄×D₇₁₂	C₂²	1	2	yes	—
16	C₂×#15667946	C₂²	1	4	yes	—
17	#19151424	C₂²	1	2	yes	—
18	#19151424	C₂²	1	4	yes	—
19	#19151424	C₂²	1	2	yes	—
20	#19151424	C₂²	1	1	yes	characteristic
21	#19151400	C₂²	1	2	yes	—
22	#19151400	C₂²	1	4	yes	—
23	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
24	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
27	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#13083573	—	4	2	no	—
29	C₂×D₇₁₂	C₂³	1	2	yes	—
30	#15668012	—	2	2	no	—
31	#15667996	—	4	4	no	—
32	#15667992	—	4	2	no	—
33	#15667986	—	2	4	no	—
34	C₄×D₃₅₆	—	2	2	no	—
35	C₂×C₄⋊Dic₈₉	—	2	4	no	—
36	D₁₄₂₄	—	2	2	no	—
37	#15667946	—	2	4	no	—
38	#15667946	—	2	2	no	—
39	#15667946	C₂³	1	2	yes	—
40	#15667946	C₂³	1	4	yes	—
41	#15667934	—	2	4	no	—
42	#15667934	—	2	2	no	—
43	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
44	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₈₉	—	2	4	no	—
46	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
47	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
51	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	#13083573	—	8	8	no	—
53	C₂×Dic₁₇₈	—	4	2	no	—
54	C₂×D₃₅₆	—	2	2	no	—
55	C₂×C₂²⋊D₈₉	—	4	4	no	—
56	C₂²×Dic₈₉	—	4	4	no	—
57	C₂×C₄×D₈₉	—	4	2	no	—
58	D₇₁₂	—	2	2	no	—
59	D₇₁₂	—	2	2	no	—
60	C₄⋊Dic₈₉	—	2	2	no	—
61	C₄⋊Dic₈₉	—	2	4	no	—
62	C₄⋊Dic₈₉	—	4	4	no	—
63	C₄⋊Dic₈₉	—	4	2	no	—
64	#13083551	—	4	2	no	—
65	#13083551	—	4	4	no	—
66	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
74	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
75	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₈₉	—	2	2	no	—
77	C₁₆×C₈₉	—	2	1	no	—
78	C₁₆×C₈₉	#41	1	2	yes	complemented
79	C₂³×C₈₉	C₂×D₈	1	2	yes	—
80	C₂²×D₈₉	—	4	2	no	—
81	D₄×C₈₉	D₁₆	1	4	yes	complemented
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
88	C₂²⋊D₈₉	—	8	8	no	—
89	C₂×Dic₈₉	—	4	2	no	—
90	C₂×Dic₈₉	—	4	4	no	—
91	C₂×Dic₈₉	—	8	8	no	—
92	D₃₅₆	—	4	2	no	—
93	D₃₅₆	—	4	2	no	—
94	C₄×D₈₉	—	8	4	no	—
95	Dic₁₇₈	—	8	4	no	—
96	C₈×C₈₉	—	2	2	no	—
97	C₈×C₈₉	—	2	1	no	—
98	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
99	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
100	C₂²×C₈₉	—	2	2	no	—
101	C₂²×C₈₉	—	2	2	no	—
102	C₂²×C₈₉	C₂×D₁₆	1	4	yes	—
103	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
104	C₂×D₈₉	—	8	2	no	—
105	C₂×D₈₉	—	8	2	no	—
106	C₂×D₈₉	—	8	2	no	—
107	C₄×C₈₉	—	2	2	no	—
108	C₄×C₈₉	—	2	1	no	—
109	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
110	C₄×C₈₉	#547	1	1	yes	characteristic
111	C₄×C₈₉	#578	1	1	yes	characteristic
112	Dic₈₉	—	8	4	no	—
113	Dic₈₉	—	8	2	no	—
114	#33901	—	89	1	no	maximal, Sylow
115	C₂×C₈₉	—	2	4	no	—
116	C₂×C₈₉	#3334	1	1	yes	characteristic
117	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
118	C₂×C₈₉	#3417	1	1	yes	characteristic
119	D₈₉	—	16	2	no	—
120	#3334	—	89	2	no	—
121	#2252	—	89	1	no	—
122	#2232	—	89	4	no	—
123	#2224	—	89	2	no	—
124	#2215	—	89	2	no	—
125	C₄×D₁₆	—	89	1	no	—
126	D₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
127	C₂×#366	—	89	2	no	—
128	C₈₉	#33901	1	1	yes	characteristic, complemented, Sylow
129	#547	—	178	2	no	—
130	C₂×D₁₆	—	89	1	no	—
131	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
132	#500	—	89	2	no	—
133	#480	—	178	4	no	—
134	#474	—	178	2	no	—
135	#466	—	89	4	no	—
136	C₄×D₈	—	89	2	no	—
137	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
138	C₂×#107	—	89	4	no	—
139	#366	—	89	1	no	—
140	#366	—	89	2	no	—
141	#366	—	89	4	no	—
142	#366	—	89	2	no	—
143	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
144	#357	—	89	2	no	—
145	#357	—	89	4	no	—
146	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
147	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
148	C₂×#41	—	356	2	no	—
149	C₂×D₈	—	89	2	no	—
150	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
151	C₄⋊Q₈	—	178	2	no	—
152	#123	—	356	4	no	—
153	C₂²⋊Q₈	—	356	2	no	—
154	#121	—	178	4	no	—
155	C₄×D₄	—	178	2	no	—
156	C₄×D₄	D₃₅₆	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	178	4	no	—
158	D₁₆	—	178	2	no	—
159	C₂×C₁₆	—	2	4	no	—
160	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
161	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
162	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
163	#107	—	89	2	no	—
164	#107	—	89	4	no	—
165	#107	—	178	4	no	—
166	#107	—	178	2	no	—
167	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
168	#102	—	178	2	no	—
169	#102	—	178	4	no	—
170	C₂²⋊C₈	D₃₅₆	1	2	yes	—
171	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
172	#41	—	712	8	no	—
173	C₂×Q₈	—	356	2	no	—
174	C₂×D₄	—	356	4	no	—
175	C₂×D₄	—	178	2	no	—
176	C₂×D₄	D₇₁₂	1	1	yes	characteristic
177	C₂²×C₄	—	356	4	no	—
178	C₂²×C₄	—	356	2	no	—
179	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
180	D₈	—	178	2	no	—
181	D₈	—	178	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
185	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
186	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
187	C₄⋊C₄	—	356	4	no	—
188	C₄⋊C₄	—	356	2	no	—
189	C₄⋊C₄	—	178	2	no	—
190	C₄⋊C₄	—	178	4	no	—
191	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
192	C₂²⋊C₄	—	356	2	no	—
193	C₂²⋊C₄	—	356	4	no	—
194	C₂²⋊C₄	D₇₁₂	1	2	yes	—
195	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
196	C₁₆	—	2	2	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#13083573	1	2	yes	complemented
199	C₂³	—	356	2	no	—
200	C₂³	C₂×D₇₁₂	1	2	yes	—
201	Q₈	—	712	4	no	—
202	D₄	—	356	2	no	—
203	D₄	—	356	2	no	—
204	D₄	—	712	8	no	—
205	D₄	D₁₄₂₄	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	356	2	no	—
209	C₂×C₄	—	356	4	no	—
210	C₂×C₄	—	712	8	no	—
211	C₂×C₄	—	712	4	no	—
212	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
213	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
214	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	C₂×#13083573	1	1	yes	characteristic
219	C₈	#15668059	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	712	2	no	—
223	C₂²	—	712	2	no	—
224	C₂²	—	712	2	no	—
225	C₂²	C₂×D₁₄₂₄	1	4	yes	—
226	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	712	2	no	—
230	C₄	—	712	4	no	—
231	C₄	C₂×D₁₄₂₄	1	2	yes	—
232	C₄	#19152252	1	1	yes	characteristic
233	C₄	#19152415	1	1	yes	characteristic
234	C₂	—	2	4	no	—
235	C₂	—	1424	2	no	—
236	C₂	#26032600	1	1	yes	characteristic
237	C₂	C₂²×D₁₄₂₄	1	1	yes	characteristic
238	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	D₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33901
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #26032600	C₂	semidirect
2. #26022527	C₂	semidirect
3. #26022433	C₂	semidirect
4. #26022390	C₂	semidirect
5. #26022338	C₂	semidirect
6. C₄×D₁₄₂₄	C₂	semidirect
7. C₂×#19151424	C₂	semidirect
8. D₄×C₁₆×C₈₉	C₂	semidirect
78. C₁₆×C₈₉	#41	semidirect
81. D₄×C₈₉	D₁₆	semidirect
126. D₄×C₁₆	D₈₉	semidirect
128. C₈₉	#33901	semidirect
198. C₁₆	#13083573	semidirect
205. D₄	D₁₄₂₄	semidirect

Maximal subgroups

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

subgroup	index
#26032600	2
#26022527	2
#26022433	2
#26022390	2
#26022338	2
C₄×D₁₄₂₄	2
C₂×#19151424	2
D₄×C₁₆×C₈₉	2
#33901	89