Group #41014347 — subgroups

Subgroups — #41014347

Order 27008.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₄×C₄⋊Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#27212908	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
7	C₂×C₄×Dic₂₁₁	—	2	1	no	—
8	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
9	C₂₁₁⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₂₁₁	—	2	1	no	—
11	C₄⋊Q₈×C₂₁₁	—	2	2	no	—
12	C₄⋊Q₈×C₂₁₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₁₁	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	—	2	4	no	—
19	C₄⋊Dic₂₁₁	—	2	2	no	—
20	C₄⋊Dic₂₁₁	Q₈	1	4	yes	complemented
21	C₄×Dic₂₁₁	—	4	2	no	—
22	C₂×Q₈×C₂₁₁	—	2	4	no	—
23	C₂×Q₈×C₂₁₁	—	4	4	no	—
24	C₂²×C₄×C₂₁₁	—	2	2	no	—
25	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₂₁₁	—	2	4	no	—
29	C₄⋊C₄×C₂₁₁	—	2	2	no	—
30	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
31	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
32	C₄²×C₂₁₁	D₄	1	2	yes	—
33	C₂×Dic₂₁₁	—	2	4	no	—
34	C₂×Dic₂₁₁	—	4	2	no	—
35	C₂×Dic₂₁₁	—	4	2	no	—
36	C₂₁₁⋊C₈	—	4	2	no	—
37	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₂₁₁	—	4	8	no	—
39	C₂×C₄×C₂₁₁	—	2	2	no	—
40	C₂×C₄×C₂₁₁	—	2	4	no	—
41	C₂×C₄×C₂₁₁	—	4	4	no	—
42	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
45	C₂×C₄×C₂₁₁	Q₁₆	1	2	yes	—
46	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₂₁₁	#41	1	2	yes	—
50	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₂₁₁	—	2	2	no	—
56	C₄×C₂₁₁	—	4	4	no	—
57	C₄×C₂₁₁	#103	1	4	yes	—
58	C₄×C₂₁₁	C₄×Q₈	1	2	yes	—
59	C₄×C₂₁₁	#124	1	1	yes	characteristic
60	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₂₁₁	—	4	4	no	—
62	C₂×C₂₁₁	#391	1	1	yes	characteristic
63	C₂×C₂₁₁	C₂×#103	1	1	yes	characteristic
64	C₂×C₂₁₁	#418	1	1	yes	characteristic
65	C₂×C₂₁₁	#476	1	2	yes	—
66	C₂×C₂₁₁	#477	1	2	yes	—
67	C₂₁₁	#1987	1	1	yes	characteristic, complemented, Sylow
68	#1987	—	211	1	no	maximal, Sylow
69	C₂×C₄⋊Q₈	D₂₁₁	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	211	1	no	—
71	C₄×C₄⋊C₄	—	211	1	no	—
72	#340	—	211	4	no	—
73	C₂²×Q₈	—	2	1	no	—
74	C₂²×C₈	—	211	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	Dic₂₁₁	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	211	2	no	—
79	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
80	C₂×C₄²	—	422	1	no	—
81	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
82	C₄⋊C₈	—	422	2	no	—
83	#95	—	422	1	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	4	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	422	2	no	—
88	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
91	C₂×C₈	—	422	2	no	—
92	C₂×C₈	—	422	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	422	2	no	—
96	C₄⋊C₄	—	211	4	no	—
97	C₄⋊C₄	C₂×Dic₂₁₁	1	4	yes	—
98	C₄²	—	844	2	no	—
99	C₄²	C₂×Dic₂₁₁	1	2	yes	—
100	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
102	Q₈	—	4	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	844	2	no	—
107	C₂×C₄	—	844	2	no	—
108	C₂×C₄	—	422	4	no	—
109	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
111	C₂×C₄	#16809949	1	2	yes	—
112	C₂×C₄	#16809952	1	2	yes	—
113	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₂₁₁	1	2	yes	—
116	C₂×C₄	#16809963	1	2	yes	—
117	C₈	—	844	2	no	—
118	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
119	C₂²	#20269953	1	2	yes	—
120	C₂²	C₂×#16809949	1	1	yes	characteristic
121	C₂²	C₂×#16809952	1	1	yes	characteristic
122	C₂²	#20270003	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	844	4	no	—
126	C₄	#20269901	1	4	yes	—
127	C₄	Q₈×Dic₂₁₁	1	2	yes	—
128	C₄	#20269966	1	1	yes	characteristic
129	C₄	#20269975	1	1	yes	characteristic
130	C₂	#27213053	1	1	yes	characteristic
131	C₂	C₂×#20269901	1	1	yes	characteristic
132	C₂	#27213151	1	1	yes	characteristic
133	C₂	#27213433	1	2	yes	—
134	C₂	#27213438	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₂₁₁

C₂

yes

maximal, characteristic

#27212908

C₂

yes

maximal

C₂×C₄⋊Q₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄×Dic₂₁₁

—

C₄⋊(C₂₁₁⋊C₈)

—

C₂₁₁⋊#95

—

C₂²×Q₈×C₂₁₁

—

C₄⋊Q₈×C₂₁₁

—

C₄⋊Q₈×C₂₁₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

Q₈

yes

complemented

C₄×Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

—

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×Q₁₆

yes

characteristic

C₂²×C₂₁₁

AGL(1,ℤ/8)

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#103

yes

—

C₄×C₂₁₁

C₄×Q₈

yes

—

C₄×C₂₁₁

#124

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂×C₂₁₁

#477

yes

—

C₂₁₁

#1987

yes

characteristic, complemented, Sylow

#1987

—

211

maximal, Sylow

C₂×C₄⋊Q₈

D₂₁₁

yes

characteristic

C₂×C₄⋊C₈

—

211

—

C₄×C₄⋊C₄

—

211

—

#340

—

211

—

C₂²×Q₈

—

C₂²×C₈

—

211

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₂₁₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

—

422

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

422

—

#95

—

422

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²⋊D₂₁₁

yes

characteristic

C₂²×C₄

C₂²⋊D₂₁₁

yes

characteristic

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

211

—

C₄⋊C₄

C₂×Dic₂₁₁

yes

—

C₄²

—

844

—

C₄²

C₂×Dic₂₁₁

yes

—

100

C₄²

C₂²⋊D₂₁₁

yes

—

101

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

844

—

107

C₂×C₄

—

844

—

108

C₂×C₄

—

422

—

109

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

110

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

111

C₂×C₄

#16809949

yes

—

112

C₂×C₄

#16809952

yes

—

113

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

115

C₂×C₄

Q₈×D₂₁₁

yes

—

116

C₂×C₄

#16809963

yes

—

117

C₈

—

844

—

118

C₂²

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

119

C₂²

#20269953

yes

—

120

C₂²

C₂×#16809949

yes

characteristic

121

C₂²

C₂×#16809952

yes

characteristic

122

C₂²

#20270003

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

844

—

126

C₄

#20269901

yes

—

127

C₄

Q₈×Dic₂₁₁

yes

—

128

C₄

#20269966

yes

characteristic

129

C₄

#20269975

yes

characteristic

130

C₂

#27213053

yes

characteristic

131

C₂

C₂×#20269901

yes

characteristic

132

C₂

#27213151

yes

characteristic

133

C₂

#27213433

yes

—

134

C₂

#27213438

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄⋊Q₈×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₁₁
hypocenter	C₂₁₁	#1987
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×C₄⋊Q₈×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₁₁

hypocenter

C₂₁₁

#1987

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₂₁₁	C₄	semidirect
20. C₄⋊Dic₂₁₁	Q₈	semidirect
67. C₂₁₁	#1987	semidirect
76. C₄⋊Q₈	Dic₂₁₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₂₁₁

C₄

semidirect

20. C₄⋊Dic₂₁₁

Q₈

semidirect

67. C₂₁₁

#1987

semidirect

76. C₄⋊Q₈

Dic₂₁₁

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
C₄×C₄⋊Dic₂₁₁	2
#27212908	2
C₂×C₄⋊Q₈×C₂₁₁	2
#1987	211

subgroup

index

C₂×C₄⋊(C₂₁₁⋊C₈)

C₄×C₄⋊Dic₂₁₁

#27212908

C₂×C₄⋊Q₈×C₂₁₁

#1987

211

Subgroups — #41014347

Conjugacy classes of subgroups

Complements

Maximal subgroups