Group #41011682 — subgroups

Subgroups — #41011682

Order 27008.

212 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 133 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	#27213199	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#27212868	C₂	1	2	yes	maximal, complemented
4	C₈×C₂₁₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	#431×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₄	1	2	yes	—
7	C₄⋊(C₂₁₁⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₂₁₁⋊C₈)	C₄	1	2	yes	—
9	C₄×C₂₁₁⋊C₈	C₄	1	2	yes	—
10	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₂₁₁⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₁₁	C₄	1	2	yes	—
15	C₂²⋊C₈×C₂₁₁	C₄	1	2	yes	—
16	C₄×C₈×C₂₁₁	C₂²	1	2	yes	—
17	C₂×C₂₁₁⋊C₈	C₈	1	8	yes	complemented
18	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
25	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
29	C₂₁₁⋊C₈	—	2	4	no	—
30	C₂₁₁⋊C₈	C₂×C₈	1	4	yes	—
31	C₂₁₁⋊C₈	C₂×C₈	1	4	yes	—
32	C₂³×C₂₁₁	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	—	2	2	no	—
34	C₂×C₄×C₂₁₁	—	2	2	no	—
35	C₂×C₄×C₂₁₁	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₂₁₁	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	C₂×C₈	1	4	yes	—
39	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
42	C₈×C₂₁₁	—	2	4	no	—
43	C₂²×C₂₁₁	—	2	1	no	—
44	C₂²×C₂₁₁	—	2	1	no	—
45	C₂²×C₂₁₁	C₄×C₈	1	2	yes	—
46	C₂²×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
47	C₄×C₂₁₁	—	2	1	no	—
48	C₄×C₂₁₁	—	2	1	no	—
49	C₄×C₂₁₁	C₄×C₈	1	2	yes	—
50	C₄×C₂₁₁	C₂²×C₈	1	2	yes	—
51	C₄×C₂₁₁	#131	1	2	yes	—
52	C₄×C₂₁₁	#131	1	2	yes	—
53	C₂×C₂₁₁	—	2	2	no	—
54	C₂×C₂₁₁	C₂×C₄×C₈	1	1	yes	characteristic
55	C₂×C₂₁₁	#405	1	1	yes	characteristic
56	C₂×C₂₁₁	#405	1	1	yes	characteristic
57	C₂₁₁	#1450	1	1	yes	characteristic, complemented, Sylow
58	#1450	—	211	1	no	maximal, Sylow
59	#431	—	211	1	no	—
60	#431	D₂₁₁	1	1	yes	characteristic
61	C₂×C₄×C₈	—	211	1	no	—
62	#322	—	211	2	no	—
63	C₈²	—	211	2	no	—
64	C₂²×C₈	—	211	2	no	—
65	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
66	C₄⋊C₈	—	211	2	no	—
67	C₄⋊C₈	Dic₂₁₁	1	2	yes	—
68	C₂²⋊C₈	—	211	2	no	—
69	C₂²⋊C₈	Dic₂₁₁	1	2	yes	—
70	C₄×C₈	—	211	2	no	—
71	C₄×C₈	—	211	1	no	—
72	C₄×C₈	—	211	2	no	—
73	C₄×C₈	—	211	1	no	—
74	C₄×C₈	C₂×D₂₁₁	1	2	yes	—
75	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
77	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
78	C₂×C₈	—	211	4	no	—
79	C₂×C₈	—	211	8	no	—
80	C₂×C₈	—	211	2	no	—
81	C₂×C₈	—	211	2	no	—
82	C₂×C₈	C₂×Dic₂₁₁	1	4	yes	—
83	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
84	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
85	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
86	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
87	C₂³	C₄×Dic₂₁₁	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	C₄×Dic₂₁₁	1	1	yes	characteristic
91	C₂×C₄	C₄×Dic₂₁₁	1	1	yes	characteristic
92	C₂×C₄	C₄×Dic₂₁₁	1	1	yes	characteristic
93	C₂×C₄	C₈×D₂₁₁	1	4	yes	—
94	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
95	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
96	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
97	C₈	—	2	4	no	—
98	C₈	—	422	4	no	—
99	C₈	—	211	4	no	—
100	C₈	—	211	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₈×Dic₂₁₁	1	2	yes	—
104	C₂²	C₂×C₄×Dic₂₁₁	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	C₈×Dic₂₁₁	1	2	yes	—
108	C₄	C₂×C₈×D₂₁₁	1	2	yes	—
109	C₄	#20269984	1	2	yes	—
110	C₄	#20269985	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	C₂×C₈×Dic₂₁₁	1	1	yes	characteristic
113	C₂	#27213102	1	1	yes	characteristic
114	C₂	#27213104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27213199

C₂

yes

maximal, characteristic

C₂×C₄×C₂₁₁⋊C₈

C₂

yes

maximal, characteristic

#27212868

C₂

yes

maximal, complemented

C₈×C₂₁₁⋊C₈

C₂

yes

maximal, complemented

#431×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₄

yes

—

C₄⋊(C₂₁₁⋊C₈)

C₄

yes

—

C₂²⋊(C₂₁₁⋊C₈)

C₄

yes

—

C₄×C₂₁₁⋊C₈

C₄

yes

—

C₄×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₄×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

C₄

yes

—

C₂²⋊C₈×C₂₁₁

C₄

yes

—

C₄×C₈×C₂₁₁

C₂²

yes

—

C₂×C₂₁₁⋊C₈

C₈

yes

complemented

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

C₂×C₈

yes

—

C₂₁₁⋊C₈

C₂×C₈

yes

—

C₂³×C₂₁₁

C₄²

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄²

yes

characteristic

C₂×C₄×C₂₁₁

C₄²

yes

characteristic

C₂×C₄×C₂₁₁

C₄²

yes

characteristic

C₂×C₄×C₂₁₁

C₂×C₈

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₄×C₈

yes

—

C₂²×C₂₁₁

C₂×C₄²

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₄×C₈

yes

—

C₄×C₂₁₁

C₂²×C₈

yes

—

C₄×C₂₁₁

#131

yes

—

C₄×C₂₁₁

#131

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₂×C₄×C₈

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1450

—

211

maximal, Sylow

#431

—

211

—

#431

D₂₁₁

yes

characteristic

C₂×C₄×C₈

—

211

—

#322

—

211

—

C₈²

—

211

—

C₂²×C₈

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

211

—

C₄⋊C₈

Dic₂₁₁

yes

—

C₂²⋊C₈

—

211

—

C₂²⋊C₈

Dic₂₁₁

yes

—

C₄×C₈

—

211

—

C₄×C₈

—

211

—

C₄×C₈

—

211

—

C₄×C₈

—

211

—

C₄×C₈

C₂×D₂₁₁

yes

—

C₂²×C₄

C₄×D₂₁₁

yes

characteristic

C₂²×C₄

C₄×D₂₁₁

yes

characteristic

C₂²×C₄

C₂×Dic₂₁₁

yes

characteristic

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

C₂×Dic₂₁₁

yes

—

C₄²

C₄×D₂₁₁

yes

characteristic

C₄²

C₄×D₂₁₁

yes

characteristic

C₄²

C₂×Dic₂₁₁

yes

characteristic

C₄²

C₂²×D₂₁₁

yes

characteristic

C₂³

C₄×Dic₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₁₁

yes

characteristic

C₂×C₄

C₄×Dic₂₁₁

yes

characteristic

C₂×C₄

C₄×Dic₂₁₁

yes

characteristic

C₂×C₄

C₈×D₂₁₁

yes

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

C₈

—

C₈

—

422

—

C₈

—

211

—

100

C₈

—

211

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₈×Dic₂₁₁

yes

—

104

C₂²

C₂×C₄×Dic₂₁₁

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

C₈×Dic₂₁₁

yes

—

108

C₄

C₂×C₈×D₂₁₁

yes

—

109

C₄

#20269984

yes

—

110

C₄

#20269985

yes

—

111

C₂

—

112

C₂

C₂×C₈×Dic₂₁₁

yes

characteristic

113

C₂

#27213102

yes

characteristic

114

C₂

#27213104

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₁₁
commutator	C₂×C₂₁₁	C₂×C₄×C₈
Fitting	#431×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄²
prime radical	C₂³×C₂₁₁	C₄²
cosocle	C₄²×C₂₁₁	C₂³
hypercenter	#431	D₂₁₁
hypocenter	C₂₁₁	#1450
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₂₁₁

commutator

C₂×C₂₁₁

C₂×C₄×C₈

Fitting

#431×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₄²

prime radical

C₂³×C₂₁₁

C₄²

cosocle

C₄²×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #27212868	C₂	semidirect
4. C₈×C₂₁₁⋊C₈	C₂	semidirect
17. C₂×C₂₁₁⋊C₈	C₈	semidirect
57. C₂₁₁	#1450	semidirect

H ≅ G/N

type

3. #27212868

C₂

semidirect

4. C₈×C₂₁₁⋊C₈

C₂

semidirect

17. C₂×C₂₁₁⋊C₈

C₈

semidirect

57. C₂₁₁

#1450

semidirect

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

subgroup	index
#27213199	2
C₂×C₄×C₂₁₁⋊C₈	2
#27212868	2
C₈×C₂₁₁⋊C₈	2
#431×C₂₁₁	2
#1450	211

subgroup

index

#27213199

C₂×C₄×C₂₁₁⋊C₈

#27212868

C₈×C₂₁₁⋊C₈

#431×C₂₁₁

#1450

211

Subgroups — #41011682

Conjugacy classes of subgroups

Complements

Maximal subgroups