Group #41014121 — subgroups

Subgroups — #41014121

Order 27008.

248 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 113 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	2	yes	maximal
3	C₄×C₄⋊Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#27212900	C₂	1	2	yes	maximal
5	C₂×#97×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₁₁	C₄	1	2	yes	—
8	C₂×C₄×Dic₂₁₁	C₂²	1	2	yes	—
9	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
10	Dic₂₁₁⋊C₈	—	2	4	no	—
11	C₂₁₁⋊#95	C₄	1	1	yes	characteristic
12	C₂₁₁⋊#95	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	#97×C₂₁₁	—	2	2	no	—
17	C₂²×Dic₂₁₁	C₂×C₄	1	2	yes	—
18	C₂²×Dic₂₁₁	C₂×C₄	1	2	yes	—
19	C₂×C₂₁₁⋊C₈	—	2	2	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₂₁₁⋊C₈	—	2	2	no	—
22	C₄⋊Dic₂₁₁	—	2	4	no	—
23	C₄×Dic₂₁₁	—	2	2	no	—
24	C₄×Dic₂₁₁	—	2	2	no	—
25	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₂₁₁	—	2	1	no	—
29	C₂×C₈×C₂₁₁	—	2	1	no	—
30	C₂×C₈×C₂₁₁	—	2	1	no	—
31	C₂×C₈×C₂₁₁	—	2	1	no	—
32	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
33	C₂×C₈×C₂₁₁	Q₈	1	2	yes	—
34	C₄²×C₂₁₁	D₄	1	2	yes	—
35	C₄²×C₂₁₁	Q₈	1	2	yes	—
36	C₂×Dic₂₁₁	—	2	2	no	—
37	C₂×Dic₂₁₁	—	2	2	no	—
38	C₂×Dic₂₁₁	—	2	2	no	—
39	C₂×Dic₂₁₁	—	2	2	no	—
40	C₂×Dic₂₁₁	—	2	2	no	—
41	C₂×Dic₂₁₁	M₁₆	1	4	yes	—
42	C₂₁₁⋊C₈	—	4	2	no	—
43	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	—	2	2	no	—
45	C₂×C₄×C₂₁₁	—	2	2	no	—
46	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
53	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
54	C₈×C₂₁₁	—	2	2	no	—
55	C₈×C₂₁₁	—	2	2	no	—
56	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₂₁₁	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₂₁₁	C₂×M₁₆	1	2	yes	—
60	C₂²×C₂₁₁	#131	1	2	yes	—
61	C₄×C₂₁₁	—	2	2	no	—
62	C₄×C₂₁₁	—	2	2	no	—
63	C₄×C₂₁₁	#121	1	1	yes	characteristic
64	C₄×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
65	C₄×C₂₁₁	#125	1	1	yes	characteristic
66	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₂₁₁	—	2	4	no	—
68	Dic₂₁₁	—	4	2	no	—
69	C₂×C₂₁₁	#389	1	1	yes	characteristic
70	C₂×C₂₁₁	#423	1	1	yes	characteristic
71	C₂×C₂₁₁	#424	1	1	yes	characteristic
72	C₂×C₂₁₁	#435	1	2	yes	—
73	C₂×C₂₁₁	#446	1	2	yes	—
74	C₂₁₁	#1940	1	1	yes	characteristic, complemented, Sylow
75	#1940	—	211	1	no	maximal, Sylow
76	C₂×C₄⋊C₈	—	211	2	no	—
77	C₂×C₄⋊C₈	—	211	1	no	—
78	C₂×#97	D₂₁₁	1	1	yes	characteristic
79	C₄×C₄⋊C₄	—	211	1	no	—
80	#336	—	211	2	no	—
81	C₂²×C₈	—	211	2	no	—
82	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
84	C₂×C₄⋊C₄	—	211	2	no	—
85	C₂×C₄²	—	211	2	no	—
86	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
87	C₄⋊C₈	—	422	4	no	—
88	C₄⋊C₈	—	422	2	no	—
89	#97	—	2	2	no	—
90	#95	—	211	1	no	—
91	#95	—	211	1	no	—
92	C₂²×C₄	—	211	2	no	—
93	C₂²×C₄	—	211	2	no	—
94	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₂₁₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	422	2	no	—
102	C₂×C₈	—	422	2	no	—
103	C₂×C₈	—	422	2	no	—
104	C₂×C₈	D₈₄₄	1	2	yes	—
105	C₂×C₈	Dic₄₂₂	1	2	yes	—
106	C₄⋊C₄	—	422	4	no	—
107	C₄²	—	422	2	no	—
108	C₄²	—	422	2	no	—
109	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
110	C₄²	Dic₄₂₂	1	2	yes	—
111	C₂³	C₂×C₄×D₂₁₁	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	422	2	no	—
115	C₂×C₄	—	422	2	no	—
116	C₂×C₄	—	422	2	no	—
117	C₂×C₄	—	422	2	no	—
118	C₂×C₄	—	422	2	no	—
119	C₂×C₄	—	211	4	no	—
120	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
121	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
124	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
126	C₂×C₄	#16809962	1	1	yes	characteristic
127	C₂×C₄	#16809962	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	844	2	no	—
131	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
132	C₂²	C₄×D₈₄₄	1	1	yes	characteristic
133	C₂²	C₄×Dic₄₂₂	1	1	yes	characteristic
134	C₂²	M₁₆×D₂₁₁	1	2	yes	—
135	C₂²	#20269985	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	844	2	no	—
139	C₄	—	422	4	no	—
140	C₄	#20269951	1	1	yes	characteristic
141	C₄	#20269956	1	1	yes	characteristic
142	C₄	#20269969	1	1	yes	characteristic
143	C₄	#20269977	1	1	yes	characteristic
144	C₂	#27213044	1	1	yes	characteristic
145	C₂	#27213169	1	1	yes	characteristic
146	C₂	#27213174	1	1	yes	characteristic
147	C₂	#27213219	1	2	yes	—
148	C₂	#27213274	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₂₁₁⋊C₈

C₂

yes

maximal

C₄×C₄⋊Dic₂₁₁

C₂

yes

maximal, characteristic

#27212900

C₂

yes

maximal

C₂×#97×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₄

yes

—

C₂×C₄×Dic₂₁₁

C₂²

yes

—

C₄⋊(C₂₁₁⋊C₈)

—

Dic₂₁₁⋊C₈

—

C₂₁₁⋊#95

C₄

yes

characteristic

C₂₁₁⋊#95

C₄

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#97×C₂₁₁

—

C₂²×Dic₂₁₁

C₂×C₄

yes

—

C₂²×Dic₂₁₁

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

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

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

Q₈

yes

—

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

Q₈

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

M₁₆

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

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₄×D₄

yes

characteristic

C₂²×C₂₁₁

C₄×Q₈

yes

characteristic

C₂²×C₂₁₁

C₂×M₁₆

yes

—

C₂²×C₂₁₁

#131

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#121

yes

characteristic

C₄×C₂₁₁

C₂²⋊Q₈

yes

characteristic

C₄×C₂₁₁

#125

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂×C₂₁₁

#446

yes

—

C₂₁₁

#1940

yes

characteristic, complemented, Sylow

#1940

—

211

maximal, Sylow

C₂×C₄⋊C₈

—

211

—

C₂×C₄⋊C₈

—

211

—

C₂×#97

D₂₁₁

yes

characteristic

C₄×C₄⋊C₄

—

211

—

#336

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

422

—

C₄⋊C₈

—

422

—

#97

—

#95

—

211

—

#95

—

211

—

C₂²×C₄

—

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

—

100

C₂×C₈

—

101

C₂×C₈

—

422

—

102

C₂×C₈

—

422

—

103

C₂×C₈

—

422

—

104

C₂×C₈

D₈₄₄

yes

—

105

C₂×C₈

Dic₄₂₂

yes

—

106

C₄⋊C₄

—

422

—

107

C₄²

—

422

—

108

C₄²

—

422

—

109

C₄²

C₂²⋊D₂₁₁

yes

—

110

C₄²

Dic₄₂₂

yes

—

111

C₂³

C₂×C₄×D₂₁₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

422

—

115

C₂×C₄

—

422

—

116

C₂×C₄

—

422

—

117

C₂×C₄

—

422

—

118

C₂×C₄

—

422

—

119

C₂×C₄

—

211

—

120

C₂×C₄

Dic₂₁₁⋊C₄

yes

—

121

C₂×C₄

Dic₂₁₁⋊C₄

yes

—

122

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

123

C₂×C₄

C₂×D₈₄₄

yes

characteristic

124

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

125

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

126

C₂×C₄

#16809962

yes

characteristic

127

C₂×C₄

#16809962

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

844

—

131

C₂²

C₂×Dic₂₁₁⋊C₄

yes

characteristic

132

C₂²

C₄×D₈₄₄

yes

characteristic

133

C₂²

C₄×Dic₄₂₂

yes

characteristic

134

C₂²

M₁₆×D₂₁₁

yes

—

135

C₂²

#20269985

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

844

—

139

C₄

—

422

—

140

C₄

#20269951

yes

characteristic

141

C₄

#20269956

yes

characteristic

142

C₄

#20269969

yes

characteristic

143

C₄

#20269977

yes

characteristic

144

C₂

#27213044

yes

characteristic

145

C₂

#27213169

yes

characteristic

146

C₂

#27213174

yes

characteristic

147

C₂

#27213219

yes

—

148

C₂

#27213274

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator, socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
Fitting	C₂×#97×C₂₁₁	C₂
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×#97	D₂₁₁
hypocenter	C₂₁₁	#1940
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₁₁

commutator, socle, prime radical

C₂³×C₂₁₁

C₂²×C₄

Fitting

C₂×#97×C₂₁₁

C₂

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×#97

D₂₁₁

C₂₁₁

G ▷ C₂³×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
74. C₂₁₁	#1940	semidirect

H ≅ G/N

type

74. C₂₁₁

#1940

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
C₄×C₄⋊Dic₂₁₁	2
#27212900	2
C₂×#97×C₂₁₁	2
#1940	211

subgroup

index

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂×Dic₂₁₁⋊C₈

C₄×C₄⋊Dic₂₁₁

#27212900

C₂×#97×C₂₁₁

#1940

211

Subgroups — #41014121

Conjugacy classes of subgroups

Complements

Maximal subgroups