Group #41013527 — subgroups

Subgroups — #41013527

Order 27008.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#27213011	C₂	1	2	yes	maximal
3	#27212909	C₂	1	1	yes	maximal, characteristic
4	#27212908	C₂	1	2	yes	maximal
5	#340×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	C₈×Dic₂₁₁	C₄	1	4	yes	complemented
10	C₂₁₁⋊#95	—	2	2	no	—
11	C₂₁₁⋊#95	—	2	2	no	—
12	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂²×Dic₂₁₁	—	2	2	no	—
15	C₂²×Dic₂₁₁	D₄	1	1	yes	characteristic
16	C₂²×Dic₂₁₁	D₄	1	1	yes	characteristic
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₂₁₁	—	2	2	no	—
20	C₄⋊Dic₂₁₁	—	2	2	no	—
21	C₄×Dic₂₁₁	C₂×C₄	1	2	yes	—
22	C₄×Dic₂₁₁	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₁₁	—	2	2	no	—
24	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₂₁₁	—	2	2	no	—
26	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₂₁₁	—	2	2	no	—
28	C₄⋊C₄×C₂₁₁	—	2	2	no	—
29	C₂×Dic₂₁₁	—	2	1	no	—
30	C₂×Dic₂₁₁	—	2	2	no	—
31	C₂×Dic₂₁₁	—	2	1	no	—
32	C₂×Dic₂₁₁	—	4	2	no	—
33	C₂×Dic₂₁₁	—	4	2	no	—
34	C₂×Dic₂₁₁	—	4	2	no	—
35	C₂×Dic₂₁₁	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₂₁₁	C₄⋊C₄	1	2	yes	—
37	C₂₁₁⋊C₈	—	2	2	no	—
38	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₁₁	—	4	2	no	—
40	C₂×C₄×C₂₁₁	—	4	2	no	—
41	C₂×C₄×C₂₁₁	—	4	2	no	—
42	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	#41	1	2	yes	—
45	C₂×C₄×C₂₁₁	#41	1	2	yes	—
46	C₈×C₂₁₁	—	2	2	no	—
47	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₁₁	#124	1	2	yes	—
49	C₂²×C₂₁₁	#135	1	1	yes	characteristic
50	C₂²×C₂₁₁	#135	1	2	yes	—
51	C₂²×C₂₁₁	#135	1	1	yes	characteristic
52	C₄×C₂₁₁	—	8	2	no	—
53	C₄×C₂₁₁	#117	1	1	yes	characteristic
54	C₄×C₂₁₁	#117	1	1	yes	characteristic
55	C₄×C₂₁₁	#125	1	2	yes	—
56	Dic₂₁₁	—	2	2	no	—
57	Dic₂₁₁	—	2	2	no	—
58	Dic₂₁₁	—	8	2	no	—
59	C₂×C₂₁₁	#382	1	1	yes	characteristic
60	C₂×C₂₁₁	#427	1	1	yes	characteristic
61	C₂×C₂₁₁	#427	1	1	yes	characteristic
62	C₂×C₂₁₁	#488	1	2	yes	—
63	C₂×C₂₁₁	#488	1	2	yes	—
64	C₂₁₁	#1835	1	1	yes	characteristic, complemented, Sylow
65	#1835	—	211	1	no	maximal, Sylow
66	C₂×C₄×C₈	—	211	1	no	—
67	#382	—	211	2	no	—
68	#340	—	211	1	no	—
69	#340	—	211	2	no	—
70	#340	D₂₁₁	1	1	yes	characteristic
71	C₂²×C₈	—	211	1	no	—
72	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	211	2	no	—
74	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
75	C₂×C₄²	—	211	1	no	—
76	C₄×C₈	—	211	4	no	—
77	#95	—	422	2	no	—
78	#95	—	422	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	422	2	no	—
81	C₂²×C₄	—	211	1	no	—
82	C₂²×C₄	—	211	1	no	—
83	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	422	2	no	—
86	C₂×C₈	—	211	2	no	—
87	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	422	2	no	—
91	C₄⋊C₄	—	422	2	no	—
92	C₄²	—	211	2	no	—
93	C₄²	—	211	2	no	—
94	C₂³	D₄×D₂₁₁	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	844	2	no	—
99	C₂×C₄	—	844	2	no	—
100	C₂×C₄	—	844	2	no	—
101	C₂×C₄	—	422	1	no	—
102	C₂×C₄	—	422	2	no	—
103	C₂×C₄	—	422	1	no	—
104	C₂×C₄	—	211	2	no	—
105	C₂×C₄	—	211	2	no	—
106	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
108	C₂×C₄	#16809961	1	2	yes	—
109	C₂×C₄	#16809962	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	422	2	no	—
112	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
113	C₂²	#20269963	1	2	yes	—
114	C₂²	#20269996	1	1	yes	characteristic
115	C₂²	#20269998	1	2	yes	—
116	C₂²	#20270000	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	1688	2	no	—
119	C₄	—	422	2	no	—
120	C₄	—	422	2	no	—
121	C₄	#20269933	1	1	yes	characteristic
122	C₄	#20269934	1	1	yes	characteristic
123	C₄	#20269967	1	2	yes	—
124	C₂	#27213010	1	1	yes	characteristic
125	C₂	#27213181	1	1	yes	characteristic
126	C₂	#27213183	1	1	yes	characteristic
127	C₂	#27213499	1	2	yes	—
128	C₂	#27213502	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₂₁₁

C₂

yes

maximal, characteristic

#27213011

C₂

yes

maximal

#27212909

C₂

yes

maximal, characteristic

#27212908

C₂

yes

maximal

#340×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄×Dic₂₁₁

C₂²

yes

characteristic

C₈×Dic₂₁₁

C₄

yes

complemented

C₂₁₁⋊#95

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

D₄

yes

characteristic

C₂²×Dic₂₁₁

D₄

yes

characteristic

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄×Dic₂₁₁

C₂×C₄

yes

—

C₄×Dic₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₂×Dic₂₁₁

C₄⋊C₄

yes

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

#124

yes

—

C₂²×C₂₁₁

#135

yes

characteristic

C₂²×C₂₁₁

#135

yes

—

C₂²×C₂₁₁

#135

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂×C₂₁₁

#488

yes

—

C₂₁₁

#1835

yes

characteristic, complemented, Sylow

#1835

—

211

maximal, Sylow

C₂×C₄×C₈

—

211

—

#382

—

211

—

#340

—

211

—

#340

—

211

—

#340

D₂₁₁

yes

characteristic

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

—

211

—

C₄×C₈

—

211

—

#95

—

422

—

#95

—

422

—

C₂²×C₄

—

C₂²×C₄

—

422

—

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

C₄×D₂₁₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄²

—

211

—

C₄²

—

211

—

C₂³

D₄×D₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

844

—

C₂×C₄

—

844

—

100

C₂×C₄

—

844

—

101

C₂×C₄

—

422

—

102

C₂×C₄

—

422

—

103

C₂×C₄

—

422

—

104

C₂×C₄

—

211

—

105

C₂×C₄

—

211

—

106

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

107

C₂×C₄

Q₈×D₂₁₁

yes

characteristic

108

C₂×C₄

#16809961

yes

—

109

C₂×C₄

#16809962

yes

—

110

C₈

—

111

C₈

—

422

—

112

C₂²

C₄⋊C₄×D₂₁₁

yes

characteristic

113

C₂²

#20269963

yes

—

114

C₂²

#20269996

yes

characteristic

115

C₂²

#20269998

yes

—

116

C₂²

#20270000

yes

characteristic

117

C₄

—

118

C₄

—

1688

—

119

C₄

—

422

—

120

C₄

—

422

—

121

C₄

#20269933

yes

characteristic

122

C₄

#20269934

yes

characteristic

123

C₄

#20269967

yes

—

124

C₂

#27213010

yes

characteristic

125

C₂

#27213181

yes

characteristic

126

C₂

#27213183

yes

characteristic

127

C₂

#27213499

yes

—

128

C₂

#27213502

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#340×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#340	D₂₁₁
hypocenter	C₂₁₁	#1835
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#340×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
9. C₈×Dic₂₁₁	C₄	semidirect
64. C₂₁₁	#1835	semidirect

H ≅ G/N

type

9. C₈×Dic₂₁₁

C₄

semidirect

64. C₂₁₁

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₂₁₁	2
#27213011	2
#27212909	2
#27212908	2
#340×C₂₁₁	2
#1835	211

subgroup

index

C₂×C₈×Dic₂₁₁

#340×C₂₁₁

211

Subgroups — #41013527

Conjugacy classes of subgroups

Complements

Maximal subgroups