Subgroups — #27213067

Order 13504.

182 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×C₄⋊Dic₂₁₁	C₂	1	2	yes	maximal
3	C₂×Dic₂₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂₁₁⋊#95	C₂	1	2	yes	maximal
5	#95×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	—	2	4	no	—
7	C₂×Dic₄₂₂	—	2	2	no	—
8	C₂²×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₁₁	C₂²	1	2	yes	—
10	C₂²×Dic₂₁₁	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₁₁	—	2	4	no	—
12	Dic₂₁₁⋊C₄	—	2	2	no	—
13	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₂₁₁	C₂²	1	2	yes	—
15	Dic₄₂₂	—	2	8	no	—
16	C₂×Dic₂₁₁	—	2	1	no	—
17	C₂×Dic₂₁₁	—	2	2	no	—
18	C₂×Dic₂₁₁	—	2	2	no	—
19	C₂×Dic₂₁₁	—	2	1	no	—
20	C₂×Dic₂₁₁	—	2	1	no	—
21	C₂×Dic₂₁₁	—	2	2	no	—
22	C₂×Dic₂₁₁	D₄	1	4	yes	—
23	C₂×Dic₂₁₁	Q₈	1	2	yes	complemented
24	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₂₁₁	—	2	2	no	—
26	C₂×C₄×C₂₁₁	—	2	2	no	—
27	C₂×C₄×C₂₁₁	—	2	2	no	—
28	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
29	C₂×C₄×C₂₁₁	Q₈	1	4	yes	complemented
30	C₂²×C₂₁₁	C₂×D₄	1	2	yes	—
31	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₂₁₁	C₂×Q₈	1	2	yes	—
34	C₂²×C₂₁₁	#41	1	1	yes	characteristic
35	C₄×C₂₁₁	—	2	2	no	—
36	C₄×C₂₁₁	—	2	4	no	—
37	Dic₂₁₁	—	2	4	no	—
38	Dic₂₁₁	—	2	2	no	—
39	Dic₂₁₁	—	4	1	no	—
40	C₂×C₂₁₁	#120	1	1	yes	characteristic
41	C₂×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
42	C₂×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₂₁₁	C₄⋊Q₈	1	2	yes	—
44	C₂×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
45	C₂₁₁	#395	1	1	yes	characteristic, complemented, Sylow
46	#395	—	211	1	no	maximal, Sylow
47	C₂²×Q₈	—	211	1	no	—
48	C₂×C₄⋊C₄	—	211	1	no	—
49	C₂×C₄⋊C₄	—	211	2	no	—
50	#95	—	211	2	no	—
51	#95	D₂₁₁	1	1	yes	characteristic
52	C₂×Q₈	—	422	4	no	—
53	C₂×Q₈	—	422	2	no	—
54	C₂²×C₄	—	211	1	no	—
55	C₂²×C₄	—	211	1	no	—
56	C₂²×C₄	—	211	2	no	—
57	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
58	C₂²×C₄	C₂×D₂₁₁	1	2	yes	—
59	C₄⋊C₄	—	422	4	no	—
60	C₄⋊C₄	—	422	2	no	—
61	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
62	Q₈	—	422	8	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	422	2	no	—
67	C₂×C₄	—	422	1	no	—
68	C₂×C₄	—	422	2	no	—
69	C₂×C₄	—	422	1	no	—
70	C₂×C₄	—	422	2	no	—
71	C₂×C₄	—	422	1	no	—
72	C₂×C₄	—	211	4	no	—
73	C₂×C₄	—	211	2	no	—
74	C₂×C₄	D₈₄₄	1	2	yes	—
75	C₂×C₄	Dic₄₂₂	1	4	yes	complemented
76	C₂²	D₄×D₂₁₁	1	2	yes	—
77	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
78	C₂²	Q₈×D₂₁₁	1	1	yes	characteristic
79	C₂²	C₂×Dic₄₂₂	1	2	yes	—
80	C₂²	#16809961	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	2	4	no	—
83	C₄	—	844	1	no	—
84	C₄	—	422	4	no	—
85	C₄	—	422	2	no	—
86	C₂	#20269947	1	1	yes	characteristic
87	C₂	#20269954	1	2	yes	—
88	C₂	#20269957	1	1	yes	characteristic
89	C₂	#20269976	1	2	yes	—
90	C₂	#20269977	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₁₁
commutator, socle, prime radical, cosocle	C₂³×C₂₁₁	C₂³
Fitting	#95×C₂₁₁	C₂
hypercenter	#95	D₂₁₁
hypocenter	C₂₁₁	#395
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₂₁₁	Q₈	semidirect
29. C₂×C₄×C₂₁₁	Q₈	semidirect
45. C₂₁₁	#395	semidirect
75. C₂×C₄	Dic₄₂₂	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×C₄⋊Dic₂₁₁	2
C₂×Dic₂₁₁⋊C₄	2
C₂₁₁⋊#95	2
#95×C₂₁₁	2
#395	211