Subgroups — #36711831

Order 22784.

484 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032645	C₂	1	2	yes	maximal
2	#26022527	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022455	C₂	1	4	yes	maximal, complemented
4	#26022405	C₂	1	2	yes	maximal, complemented
5	#26022343	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₇₁₂	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
9	#19152305	—	2	2	no	—
10	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
11	#19152003	C₂²	1	2	yes	—
12	#19151934	—	2	4	no	—
13	#19151894	—	2	2	no	—
14	#19151839	C₂²	1	4	yes	—
15	C₄×Dic₃₅₆	C₂²	1	2	yes	—
16	C₂×#15667946	C₂²	1	4	yes	—
17	#19151424	C₂²	1	4	yes	—
18	#19151424	C₂²	1	2	yes	—
19	#19151424	C₂²	1	2	yes	—
20	#19151424	C₂²	1	1	yes	characteristic
21	#19151403	C₂²	1	4	yes	—
22	#19151403	C₂²	1	2	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×Dic₃₅₆	C₂³	1	2	yes	—
29	#15668012	—	2	2	no	—
30	#15668003	—	4	2	no	—
31	#15667991	—	2	4	no	—
32	#15667989	—	4	4	no	—
33	C₄×Dic₁₇₈	—	2	2	no	—
34	D₄×Dic₈₉	—	4	2	no	—
35	C₂×C₄⋊Dic₈₉	—	2	4	no	—
36	Dic₇₁₂	—	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	#15667937	—	2	4	no	—
42	#15667937	—	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	C₂×Dic₁₇₈	—	2	2	no	—
53	C₂×Dic₁₇₈	—	4	2	no	—
54	C₂²×Dic₈₉	—	4	4	no	—
55	Dic₃₅₆	—	2	2	no	—
56	Dic₃₅₆	—	2	2	no	—
57	C₄⋊Dic₈₉	—	2	2	no	—
58	C₄⋊Dic₈₉	—	2	4	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	C₄⋊Dic₈₉	—	4	4	no	—
61	#13083553	—	4	2	no	—
62	#13083553	—	4	4	no	—
63	C₂²⋊Dic₈₉	—	4	4	no	—
64	C₄×Dic₈₉	—	4	2	no	—
65	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
73	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
74	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₈₉	—	2	2	no	—
76	C₁₆×C₈₉	—	2	1	no	—
77	C₁₆×C₈₉	#41	1	2	yes	—
78	C₂³×C₈₉	C₂×D₈	1	2	yes	—
79	D₄×C₈₉	Q₃₂	1	4	yes	complemented
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₈₉	—	4	2	no	—
87	C₂×Dic₈₉	—	4	2	no	—
88	C₂×Dic₈₉	—	4	4	no	—
89	C₂×Dic₈₉	—	16	4	no	—
90	Dic₁₇₈	—	4	2	no	—
91	Dic₁₇₈	—	4	2	no	—
92	Dic₁₇₈	—	16	2	no	—
93	C₈×C₈₉	—	2	2	no	—
94	C₈×C₈₉	—	2	1	no	—
95	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
96	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
97	C₂²×C₈₉	—	2	2	no	—
98	C₂²×C₈₉	—	2	2	no	—
99	C₂²×C₈₉	C₂×Q₃₂	1	4	yes	—
100	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
101	C₄×C₈₉	—	2	2	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	C₂×Q₃₂	1	2	yes	—
104	C₄×C₈₉	#547	1	1	yes	characteristic
105	C₄×C₈₉	#578	1	1	yes	characteristic
106	Dic₈₉	—	8	2	no	—
107	Dic₈₉	—	8	4	no	—
108	Dic₈₉	—	8	2	no	—
109	#33905	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	—	2	4	no	—
111	C₂×C₈₉	#3334	1	1	yes	characteristic
112	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
113	C₂×C₈₉	#3415	1	1	yes	characteristic
114	#3338	—	89	2	no	—
115	#2252	—	89	1	no	—
116	#2236	—	89	4	no	—
117	#2227	—	89	2	no	—
118	#2216	—	89	2	no	—
119	C₄×Q₃₂	—	89	1	no	—
120	D₄×C₁₆	D₈₉	1	1	yes	characteristic
121	C₂×#366	—	89	2	no	—
122	C₈₉	#33905	1	1	yes	characteristic, complemented, Sylow
123	#554	—	178	2	no	—
124	C₂×Q₃₂	—	89	1	no	—
125	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
126	#500	—	89	2	no	—
127	#484	—	178	4	no	—
128	#477	—	178	2	no	—
129	#467	—	89	4	no	—
130	C₄×Q₁₆	—	89	2	no	—
131	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
132	C₂×#107	—	89	4	no	—
133	#366	—	89	1	no	—
134	#366	—	89	2	no	—
135	#366	—	89	4	no	—
136	#366	—	89	2	no	—
137	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
138	#358	—	89	2	no	—
139	#358	—	89	4	no	—
140	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
141	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
142	C₂×Q₁₆	—	89	2	no	—
143	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
144	C₄⋊Q₈	—	178	2	no	—
145	#125	—	356	2	no	—
146	C₂²⋊Q₈	—	356	4	no	—
147	C₂²⋊Q₈	—	178	4	no	—
148	C₄×Q₈	—	178	2	no	—
149	C₄×D₄	—	356	2	no	—
150	C₄×D₄	D₃₅₆	1	1	yes	characteristic
151	C₂×C₄⋊C₄	—	178	4	no	—
152	Q₃₂	—	178	2	no	—
153	C₂×C₁₆	—	2	4	no	—
154	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
156	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
157	#107	—	89	2	no	—
158	#107	—	89	4	no	—
159	#107	—	178	4	no	—
160	#107	—	178	2	no	—
161	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
162	#103	—	178	2	no	—
163	#103	—	178	4	no	—
164	C₂²⋊C₈	D₃₅₆	1	2	yes	—
165	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
166	C₂×Q₈	—	356	2	no	—
167	C₂×Q₈	—	178	2	no	—
168	C₂×D₄	D₇₁₂	1	1	yes	characteristic
169	C₂²×C₄	—	356	4	no	—
170	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
171	Q₁₆	—	178	2	no	—
172	Q₁₆	—	178	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
176	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
177	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
178	C₄⋊C₄	—	356	4	no	—
179	C₄⋊C₄	—	356	2	no	—
180	C₄⋊C₄	—	356	2	no	—
181	C₄⋊C₄	—	356	4	no	—
182	C₄⋊C₄	—	178	2	no	—
183	C₄⋊C₄	—	178	4	no	—
184	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
185	C₂²⋊C₄	—	356	4	no	—
186	C₂²⋊C₄	D₇₁₂	1	2	yes	—
187	C₄²	—	356	2	no	—
188	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
189	C₁₆	—	2	2	no	—
190	C₁₆	—	2	1	no	—
191	C₁₆	#13083573	1	2	yes	—
192	C₂³	C₂×D₇₁₂	1	2	yes	—
193	Q₈	—	356	2	no	—
194	Q₈	—	356	2	no	—
195	Q₈	—	1424	2	no	—
196	D₄	Dic₇₁₂	1	4	yes	complemented
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	356	2	no	—
200	C₂×C₄	—	356	2	no	—
201	C₂×C₄	—	356	4	no	—
202	C₂×C₄	—	1424	4	no	—
203	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
204	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
205	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	2	no	—
209	C₈	C₂×#13083573	1	1	yes	characteristic
210	C₈	#15668059	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	C₂×Dic₇₁₂	1	4	yes	—
214	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	712	2	no	—
218	C₄	—	712	2	no	—
219	C₄	—	712	4	no	—
220	C₄	C₂×Dic₇₁₂	1	2	yes	—
221	C₄	#19152252	1	1	yes	characteristic
222	C₄	#19152415	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	#26032600	1	1	yes	characteristic
225	C₂	C₂²×Dic₇₁₂	1	1	yes	characteristic
226	C₂	#26033393	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
prime radical	C₂×D₄×C₈₉	D₈
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33905
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
2. #26022527	C₂	semidirect
3. #26022455	C₂	semidirect
4. #26022405	C₂	semidirect
5. #26022343	C₂	semidirect
6. C₄×Dic₇₁₂	C₂	semidirect
7. C₂×#19151424	C₂	semidirect
79. D₄×C₈₉	Q₃₂	semidirect
122. C₈₉	#33905	semidirect
196. D₄	Dic₇₁₂	semidirect

Maximal subgroups

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

subgroup	index
#26032645	2
#26022527	2
#26022455	2
#26022405	2
#26022343	2
C₄×Dic₇₁₂	2
C₂×#19151424	2
D₄×C₁₆×C₈₉	2
#33905	89