Subgroups — #27213010

Order 13504.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₁₁⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂₁₁⋊#95	C₂	1	2	yes	maximal
4	C₂₁₁⋊#95	C₂	1	1	yes	maximal, characteristic
5	#95×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₁	C₂²	1	2	yes	—
7	C₂²×Dic₂₁₁	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	Dic₂₁₁⋊C₄	—	2	4	no	—
10	C₄×Dic₂₁₁	C₄	1	4	yes	complemented
11	C₂²×C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₁₁	—	2	2	no	—
14	C₂×Dic₂₁₁	—	2	2	no	—
15	C₂×Dic₂₁₁	—	2	2	no	—
16	C₂×Dic₂₁₁	—	2	2	no	—
17	C₂×Dic₂₁₁	C₂×C₄	1	2	yes	—
18	C₂×Dic₂₁₁	C₂×C₄	1	2	yes	—
19	C₂×Dic₂₁₁	D₄	1	2	yes	—
20	C₂×Dic₂₁₁	Q₈	1	2	yes	—
21	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₂₁₁	—	2	2	no	—
23	C₂×C₄×C₂₁₁	—	2	2	no	—
24	C₂×C₄×C₂₁₁	—	2	2	no	—
25	C₂×C₄×C₂₁₁	—	2	2	no	—
26	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
27	C₂²×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₂₁₁	#41	1	2	yes	—
31	C₂²×C₂₁₁	#41	1	2	yes	—
32	C₄×C₂₁₁	—	2	2	no	—
33	C₄×C₂₁₁	—	4	2	no	—
34	Dic₂₁₁	—	2	2	no	—
35	Dic₂₁₁	—	4	2	no	—
36	Dic₂₁₁	C₄⋊C₄	1	4	yes	—
37	C₂×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₂₁₁	#117	1	1	yes	characteristic
39	C₂×C₂₁₁	#117	1	1	yes	characteristic
40	C₂×C₂₁₁	#124	1	2	yes	—
41	C₂×C₂₁₁	#125	1	2	yes	—
42	C₂₁₁	#382	1	1	yes	characteristic, complemented, Sylow
43	#382	—	211	1	no	maximal, Sylow
44	C₂×C₄⋊C₄	—	211	2	no	—
45	C₂×C₄²	—	211	1	no	—
46	#95	—	211	1	no	—
47	#95	—	211	2	no	—
48	#95	D₂₁₁	1	1	yes	characteristic
49	C₂²×C₄	—	211	2	no	—
50	C₂²×C₄	—	211	1	no	—
51	C₂²×C₄	—	211	1	no	—
52	C₂²×C₄	C₂×D₂₁₁	1	2	yes	—
53	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
54	C₄⋊C₄	—	422	4	no	—
55	C₄²	—	211	4	no	—
56	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	422	2	no	—
62	C₂×C₄	—	422	2	no	—
63	C₂×C₄	—	422	2	no	—
64	C₂×C₄	—	422	2	no	—
65	C₂×C₄	—	211	2	no	—
66	C₂×C₄	—	211	2	no	—
67	C₂×C₄	—	211	2	no	—
68	C₂×C₄	—	211	2	no	—
69	C₂×C₄	C₄×D₂₁₁	1	2	yes	—
70	C₂²	C₂×C₄×D₂₁₁	1	1	yes	characteristic
71	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
72	C₂²	Q₈×D₂₁₁	1	1	yes	characteristic
73	C₂²	#16809961	1	2	yes	—
74	C₂²	#16809962	1	2	yes	—
75	C₄	—	2	2	no	—
76	C₄	—	4	2	no	—
77	C₄	—	844	2	no	—
78	C₄	—	422	2	no	—
79	C₄	—	211	4	no	—
80	C₂	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
81	C₂	#20269933	1	1	yes	characteristic
82	C₂	#20269934	1	1	yes	characteristic
83	C₂	#20269963	1	2	yes	—
84	C₂	#20269967	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₁₁
commutator	C₂²×C₂₁₁	C₂²×C₄
Fitting	#95×C₂₁₁	C₂
socle, prime radical, cosocle	C₂³×C₂₁₁	C₂³
hypercenter	#95	D₂₁₁
hypocenter	C₂₁₁	#382
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₂₁₁	C₄	semidirect
42. C₂₁₁	#382	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₁₁⋊C₄	2
C₂×C₄×Dic₂₁₁	2
C₂₁₁⋊#95	2
#95×C₂₁₁	2
#382	211