Group #41013551 — subgroups

Subgroups — #41013551

Order 27008.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₁₁⋊C₈	C₂	1	4	yes	maximal
3	#27212992	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₂₁₁	C₂²	1	2	yes	—
7	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
8	Dic₂₁₁⋊C₈	—	2	8	no	—
9	C₂₁₁⋊#95	C₄	1	2	yes	—
10	C₂₁₁⋊#95	C₄	1	2	yes	—
11	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₂₁₁	—	2	2	no	—
14	C₂²×Dic₂₁₁	C₂×C₄	1	4	yes	—
15	C₂×C₂₁₁⋊C₈	—	2	4	no	—
16	C₂×C₂₁₁⋊C₈	—	2	2	no	—
17	C₄×Dic₂₁₁	D₄	1	4	yes	—
18	C₄×Dic₂₁₁	Q₈	1	4	yes	—
19	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₁₁	—	2	2	no	—
23	C₂×C₈×C₂₁₁	—	2	4	no	—
24	C₄²×C₂₁₁	D₄	1	2	yes	—
25	C₄²×C₂₁₁	Q₈	1	2	yes	—
26	C₂×Dic₂₁₁	—	2	4	no	—
27	C₂×Dic₂₁₁	—	2	4	no	—
28	C₂×Dic₂₁₁	C₄⋊C₄	1	8	yes	—
29	C₂₁₁⋊C₈	—	4	2	no	—
30	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₁₁	—	2	2	no	—
32	C₂×C₄×C₂₁₁	—	2	2	no	—
33	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₂₁₁	—	4	2	no	—
40	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₂₁₁	#131	1	2	yes	—
43	C₂²×C₂₁₁	#131	1	2	yes	—
44	C₄×C₂₁₁	—	2	2	no	—
45	C₄×C₂₁₁	—	2	2	no	—
46	C₄×C₂₁₁	#127	1	1	yes	characteristic
47	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₂₁₁	C₄⋊Q₈	1	2	yes	—
49	Dic₂₁₁	—	2	8	no	—
50	C₂×C₂₁₁	#384	1	1	yes	characteristic
51	C₂×C₂₁₁	#424	1	4	yes	—
52	C₂×C₂₁₁	#424	1	1	yes	characteristic
53	C₂×C₂₁₁	#424	1	1	yes	characteristic
54	C₂₁₁	#1843	1	1	yes	characteristic, complemented, Sylow
55	#1843	—	211	1	no	maximal, Sylow
56	C₂×C₄⋊C₈	—	211	4	no	—
57	C₂×C₄⋊C₈	—	211	1	no	—
58	C₂×C₄⋊C₈	D₂₁₁	1	1	yes	characteristic
59	#376	—	211	1	no	—
60	C₂²×C₈	—	211	2	no	—
61	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
62	C₂×C₄²	—	211	2	no	—
63	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	422	8	no	—
66	C₄⋊C₈	—	422	2	no	—
67	#95	—	211	2	no	—
68	#95	—	211	2	no	—
69	C₂²×C₄	—	211	4	no	—
70	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	422	4	no	—
76	C₂×C₈	—	422	2	no	—
77	C₄²	—	211	4	no	—
78	C₄²	—	211	4	no	—
79	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
80	C₄²	Dic₄₂₂	1	2	yes	—
81	C₂³	C₂×C₄×D₂₁₁	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	422	4	no	—
85	C₂×C₄	—	422	4	no	—
86	C₂×C₄	—	211	8	no	—
87	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
88	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
89	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₂₁₁	1	2	yes	—
92	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	844	2	no	—
95	C₂²	C₄⋊C₄×D₂₁₁	1	2	yes	—
96	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
97	C₂²	#20269983	1	2	yes	—
98	C₂²	#20269985	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	422	8	no	—
102	C₄	#20269973	1	1	yes	characteristic
103	C₄	#20269975	1	1	yes	characteristic
104	C₄	#20269976	1	2	yes	—
105	C₂	#27213015	1	1	yes	characteristic
106	C₂	#27213171	1	4	yes	—
107	C₂	#27213172	1	1	yes	characteristic
108	C₂	#27213174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₂₁₁⋊C₈

C₂

yes

maximal

#27212992

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₂₁₁

C₂²

yes

—

C₄⋊(C₂₁₁⋊C₈)

—

Dic₂₁₁⋊C₈

—

C₂₁₁⋊#95

C₄

yes

—

C₂₁₁⋊#95

C₄

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₂²×Dic₂₁₁

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₄×Dic₂₁₁

D₄

yes

—

C₄×Dic₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

Q₈

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

#131

yes

—

C₂²×C₂₁₁

#131

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#127

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

#384

yes

characteristic

C₂×C₂₁₁

#424

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1843

—

211

maximal, Sylow

C₂×C₄⋊C₈

—

211

—

C₂×C₄⋊C₈

—

211

—

C₂×C₄⋊C₈

D₂₁₁

yes

characteristic

#376

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

—

C₂×C₄²

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

422

—

C₄⋊C₈

—

422

—

#95

—

211

—

#95

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₄×D₂₁₁

yes

characteristic

C₂²×C₄

C₄×D₂₁₁

yes

characteristic

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₄²

—

211

—

C₄²

—

211

—

C₄²

C₂²⋊D₂₁₁

yes

—

C₄²

Dic₄₂₂

yes

—

C₂³

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

—

211

—

C₂×C₄

Dic₂₁₁⋊C₄

yes

—

C₂×C₄

Dic₂₁₁⋊C₄

yes

—

C₂×C₄

D₄×D₂₁₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

C₂×C₄

Q₈×D₂₁₁

yes

—

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

C₈

—

C₈

—

844

—

C₂²

C₄⋊C₄×D₂₁₁

yes

—

C₂²

C₂×Dic₂₁₁⋊C₄

yes

characteristic

C₂²

#20269983

yes

—

C₂²

#20269985

yes

—

C₄

—

100

C₄

—

101

C₄

—

422

—

102

C₄

#20269973

yes

characteristic

103

C₄

#20269975

yes

characteristic

104

C₄

#20269976

yes

—

105

C₂

#27213015

yes

characteristic

106

C₂

#27213171

yes

—

107

C₂

#27213172

yes

characteristic

108

C₂

#27213174

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator, socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₂₁₁	C₂
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	#1843
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₁₁

commutator, socle, prime radical

C₂³×C₂₁₁

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₂₁₁

C₂

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₁₁

C₂₁₁

G ▷ C₂³×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
54. C₂₁₁	#1843	semidirect

H ≅ G/N

type

54. C₂₁₁

#1843

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₁₁⋊C₈)	2
C₂×Dic₂₁₁⋊C₈	2
#27212992	2
C₂×C₄⋊C₈×C₂₁₁	2
#1843	211

subgroup

index

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂×Dic₂₁₁⋊C₈

#27212992

C₂×C₄⋊C₈×C₂₁₁

#1843

211

Subgroups — #41013551

Conjugacy classes of subgroups

Complements

Maximal subgroups