Subgroups — #27213301

Order 13504.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16809952	C₂	1	2	yes	maximal, complemented
3	#20269900	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	—	2	4	no	—
7	C₂×Dic₄₂₂	—	2	1	no	—
8	C₂×Dic₄₂₂	C₂²	1	2	yes	—
9	C₂²×Dic₂₁₁	—	2	1	no	—
10	#16809952	—	2	4	no	—
11	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
12	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₂₁₁	—	2	1	no	—
15	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
17	Dic₄₂₂	—	2	4	no	—
18	Dic₄₂₂	—	2	2	no	—
19	Dic₄₂₂	D₄	1	4	yes	—
20	C₂×Dic₂₁₁	—	2	4	no	—
21	C₂×Dic₂₁₁	—	2	2	no	—
22	C₂₁₁⋊C₈	—	2	2	no	—
23	C₂³×C₂₁₁	D₄	1	1	yes	characteristic
24	Q₈×C₂₁₁	—	2	2	no	—
25	C₂×C₄×C₂₁₁	—	2	2	no	—
26	C₂×C₄×C₂₁₁	—	2	1	no	—
27	C₂×C₄×C₂₁₁	—	2	1	no	—
28	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₂²×C₂₁₁	—	2	1	no	—
31	C₂²×C₂₁₁	—	2	1	no	—
32	C₂²×C₂₁₁	Q₁₆	1	2	yes	complemented
33	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₁₁	—	2	1	no	—
35	C₄×C₂₁₁	—	4	1	no	—
36	C₄×C₂₁₁	—	4	1	no	—
37	C₄×C₂₁₁	C₂×D₄	1	2	yes	—
38	Dic₂₁₁	—	2	4	no	—
39	C₂×C₂₁₁	—	2	2	no	—
40	C₂×C₂₁₁	#120	1	1	yes	characteristic
41	C₂×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₂₁₁	#137	1	1	yes	characteristic
43	C₂₁₁	#451	1	1	yes	characteristic, complemented, Sylow
44	#451	—	211	1	no	maximal, Sylow
45	C₂²×Q₈	—	211	1	no	—
46	C₂×Q₁₆	—	211	2	no	—
47	C₂²⋊Q₈	D₂₁₁	1	1	yes	characteristic
48	#103	—	211	2	no	—
49	C₂²⋊C₈	—	211	1	no	—
50	C₂×Q₈	—	422	4	no	—
51	C₂×Q₈	—	422	1	no	—
52	C₂×Q₈	—	211	2	no	—
53	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
54	C₂²×C₄	—	422	1	no	—
55	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
56	Q₁₆	—	422	4	no	—
57	C₂×C₈	—	211	2	no	—
58	C₄⋊C₄	—	2	1	no	—
59	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
60	C₂²⋊C₄	—	2	1	no	—
61	C₂³	C₂²⋊D₂₁₁	1	1	yes	characteristic
62	Q₈	—	2	2	no	—
63	Q₈	—	422	4	no	—
64	Q₈	—	422	2	no	—
65	Q₈	—	211	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	422	4	no	—
70	C₂×C₄	—	422	2	no	—
71	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
73	C₈	—	422	2	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	#16809952	1	2	yes	complemented
77	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	422	4	no	—
82	C₄	D₄×D₂₁₁	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#20269945	1	1	yes	characteristic
85	C₂	C₂×#16809952	1	1	yes	characteristic
86	C₂	#20270007	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₂²⋊Q₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂³×C₂₁₁	D₄
hypercenter	C₂²⋊Q₈	D₂₁₁
hypocenter	C₂₁₁	#451
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16809952	C₂	semidirect
3. #20269900	C₂	semidirect
32. C₂²×C₂₁₁	Q₁₆	semidirect
43. C₂₁₁	#451	semidirect
76. C₂²	#16809952	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₂₂	2
C₂×#16809952	2
#20269900	2
C₂²⋊(C₂₁₁⋊C₈)	2
C₂²⋊Q₈×C₂₁₁	2
#451	211