Group #41014351 — subgroups

Subgroups — #41014351

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	2	no	—
24	C₂×Q₈×C₂₁₁	—	4	2	no	—
25	C₂²×C₄×C₂₁₁	—	2	2	no	—
26	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₁₁	—	2	4	no	—
30	C₄⋊C₄×C₂₁₁	—	2	2	no	—
31	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
32	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
33	C₄²×C₂₁₁	D₄	1	2	yes	—
34	C₂×Dic₂₁₁	—	2	4	no	—
35	C₂×Dic₂₁₁	—	4	2	no	—
36	C₂×Dic₂₁₁	—	4	2	no	—
37	C₂₁₁⋊C₈	—	4	2	no	—
38	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₂₁₁	—	4	8	no	—
40	C₂×C₄×C₂₁₁	—	2	2	no	—
41	C₂×C₄×C₂₁₁	—	2	4	no	—
42	C₂×C₄×C₂₁₁	—	4	2	no	—
43	C₂×C₄×C₂₁₁	—	4	2	no	—
44	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	D₈	1	2	yes	—
47	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₂₁₁	#41	1	2	yes	—
52	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₂₁₁	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₂₁₁	#137	1	2	yes	—
57	C₄×C₂₁₁	—	2	2	no	—
58	C₄×C₂₁₁	—	4	4	no	—
59	C₄×C₂₁₁	#102	1	4	yes	—
60	C₄×C₂₁₁	C₄×Q₈	1	2	yes	—
61	C₄×C₂₁₁	#124	1	1	yes	characteristic
62	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₂₁₁	—	4	4	no	—
64	C₂×C₂₁₁	#391	1	1	yes	characteristic
65	C₂×C₂₁₁	C₂×#102	1	1	yes	characteristic
66	C₂×C₂₁₁	#419	1	1	yes	characteristic
67	C₂×C₂₁₁	#474	1	2	yes	—
68	C₂×C₂₁₁	#475	1	2	yes	—
69	C₂₁₁	#1988	1	1	yes	characteristic, complemented, Sylow
70	#1988	—	211	1	no	maximal, Sylow
71	C₂×C₄⋊Q₈	D₂₁₁	1	1	yes	characteristic
72	C₂×C₄⋊C₈	—	211	1	no	—
73	C₄×C₄⋊C₄	—	211	1	no	—
74	#340	—	211	4	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×C₈	—	211	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	Dic₂₁₁	1	4	yes	complemented
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	211	2	no	—
81	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
82	C₂×C₄²	—	422	1	no	—
83	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
84	C₄⋊C₈	—	422	2	no	—
85	#95	—	422	1	no	—
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	—	4	2	no	—
88	C₂×Q₈	—	4	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	422	2	no	—
91	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
94	C₂×C₈	—	422	2	no	—
95	C₂×C₈	—	422	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	422	2	no	—
99	C₄⋊C₄	—	211	4	no	—
100	C₄⋊C₄	C₂×Dic₂₁₁	1	4	yes	—
101	C₄²	—	844	2	no	—
102	C₄²	C₂×Dic₂₁₁	1	2	yes	—
103	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
104	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
105	Q₈	—	4	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	844	2	no	—
111	C₂×C₄	—	844	2	no	—
112	C₂×C₄	—	422	4	no	—
113	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
114	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
115	C₂×C₄	#16809947	1	2	yes	—
116	C₂×C₄	#16809950	1	2	yes	—
117	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₂₁₁	1	2	yes	—
120	C₂×C₄	#16809963	1	2	yes	—
121	C₈	—	844	2	no	—
122	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
123	C₂²	#20269953	1	2	yes	—
124	C₂²	C₂×#16809947	1	1	yes	characteristic
125	C₂²	C₂×#16809950	1	1	yes	characteristic
126	C₂²	#20270008	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	844	4	no	—
130	C₄	#20269898	1	4	yes	—
131	C₄	Q₈×Dic₂₁₁	1	2	yes	—
132	C₄	#20269966	1	1	yes	characteristic
133	C₄	#20269975	1	1	yes	characteristic
134	C₂	#27213053	1	1	yes	characteristic
135	C₂	C₂×#20269898	1	1	yes	characteristic
136	C₂	#27213155	1	1	yes	characteristic
137	C₂	#27213423	1	2	yes	—
138	C₂	#27213428	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

—

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

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

D₈

yes

—

C₂×C₄×C₂₁₁

SD₁₆

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×D₈

yes

characteristic

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

#137

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#102

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

#475

yes

—

C₂₁₁

#1988

yes

characteristic, complemented, Sylow

#1988

—

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

—

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

—

100

C₄⋊C₄

C₂×Dic₂₁₁

yes

—

101

C₄²

—

844

—

102

C₄²

C₂×Dic₂₁₁

yes

—

103

C₄²

C₂²⋊D₂₁₁

yes

—

104

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

844

—

111

C₂×C₄

—

844

—

112

C₂×C₄

—

422

—

113

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

114

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

115

C₂×C₄

#16809947

yes

—

116

C₂×C₄

#16809950

yes

—

117

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

119

C₂×C₄

Q₈×D₂₁₁

yes

—

120

C₂×C₄

#16809963

yes

—

121

C₈

—

844

—

122

C₂²

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

123

C₂²

#20269953

yes

—

124

C₂²

C₂×#16809947

yes

characteristic

125

C₂²

C₂×#16809950

yes

characteristic

126

C₂²

#20270008

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

844

—

130

C₄

#20269898

yes

—

131

C₄

Q₈×Dic₂₁₁

yes

—

132

C₄

#20269966

yes

characteristic

133

C₄

#20269975

yes

characteristic

134

C₂

#27213053

yes

characteristic

135

C₂

C₂×#20269898

yes

characteristic

136

C₂

#27213155

yes

characteristic

137

C₂

#27213423

yes

—

138

C₂

#27213428

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	#1988
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

#1988

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
69. C₂₁₁	#1988	semidirect
78. C₄⋊Q₈	Dic₂₁₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₂₁₁

C₄

semidirect

20. C₄⋊Dic₂₁₁

Q₈

semidirect

69. C₂₁₁

#1988

semidirect

78. 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
#1988	211

subgroup

index

C₂×C₄⋊(C₂₁₁⋊C₈)

C₄×C₄⋊Dic₂₁₁

#27212908

C₂×C₄⋊Q₈×C₂₁₁

#1988

211

Subgroups — #41014351

Conjugacy classes of subgroups

Complements

Maximal subgroups