Group #41014259 — subgroups

Subgroups — #41014259

Order 27008.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×Dic₈₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269911	C₂	1	1	yes	maximal, characteristic
3	C₂×#20269902	C₂	1	2	yes	maximal
4	#27213052	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₂	C₂²	1	2	yes	—
7	C₂×Dic₈₄₄	—	2	2	no	—
8	C₂×Dic₈₄₄	—	2	2	no	—
9	C₂×Dic₈₄₄	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
11	#20269911	—	2	2	no	—
12	#20269902	—	2	4	no	—
13	C₂₁₁⋊#95	—	2	4	no	—
14	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₂₁₁	—	2	2	no	—
18	C₂×Dic₄₂₂	—	2	2	no	—
19	C₂×Dic₄₂₂	—	2	2	no	—
20	C₂×Dic₄₂₂	—	2	2	no	—
21	C₂×Dic₄₂₂	—	2	2	no	—
22	C₂×Dic₄₂₂	—	2	2	no	—
23	C₂×Dic₄₂₂	C₂×C₄	1	4	yes	—
24	C₂²×Dic₂₁₁	—	2	2	no	—
25	C₂²×Dic₂₁₁	—	2	2	no	—
26	Dic₈₄₄	—	2	8	no	—
27	C₄⋊Dic₂₁₁	—	2	2	no	—
28	C₄⋊Dic₂₁₁	—	2	2	no	—
29	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
30	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₂₁₁	—	2	1	no	—
32	C₂×C₈×C₂₁₁	—	2	1	no	—
33	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
35	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
36	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
37	C₄²×C₂₁₁	—	2	1	no	—
38	C₄²×C₂₁₁	—	2	1	no	—
39	Dic₄₂₂	—	2	4	no	—
40	Dic₄₂₂	—	2	4	no	—
41	Dic₄₂₂	—	4	2	no	—
42	Dic₄₂₂	—	4	2	no	—
43	C₂×Dic₂₁₁	—	2	4	no	—
44	C₂×Dic₂₁₁	—	4	2	no	—
45	C₂×Dic₂₁₁	—	4	2	no	—
46	C₂×Dic₂₁₁	—	4	2	no	—
47	C₂×Dic₂₁₁	—	4	4	no	—
48	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₁₁	—	2	2	no	—
50	C₂×C₄×C₂₁₁	—	2	2	no	—
51	C₂×C₄×C₂₁₁	Q₁₆	1	4	yes	complemented
52	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
57	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
58	C₈×C₂₁₁	—	2	2	no	—
59	C₈×C₂₁₁	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₂₁₁	#121	1	1	yes	characteristic
62	C₂²×C₂₁₁	#127	1	1	yes	characteristic
63	C₂²×C₂₁₁	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₂₁₁	#135	1	2	yes	—
65	C₄×C₂₁₁	—	2	4	no	—
66	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
68	C₄×C₂₁₁	#121	1	1	yes	characteristic
69	C₄×C₂₁₁	#124	1	1	yes	characteristic
70	Dic₂₁₁	—	4	4	no	—
71	Dic₂₁₁	—	8	2	no	—
72	C₂×C₂₁₁	#390	1	1	yes	characteristic
73	C₂×C₂₁₁	C₄×Q₁₆	1	2	yes	—
74	C₂×C₂₁₁	#467	1	2	yes	—
75	C₂×C₂₁₁	#494	1	1	yes	characteristic
76	C₂×C₂₁₁	#495	1	1	yes	characteristic
77	C₂₁₁	#1969	1	1	yes	characteristic, complemented, Sylow
78	#1969	—	211	1	no	maximal, Sylow
79	C₂²×Q₁₆	—	211	1	no	—
80	C₂×#107	—	211	1	no	—
81	C₂×#103	—	211	2	no	—
82	C₂×C₄×C₈	D₂₁₁	1	1	yes	characteristic
83	#391	—	211	2	no	—
84	C₂²×Q₈	—	211	2	no	—
85	C₂×Q₁₆	—	422	2	no	—
86	C₂×Q₁₆	—	422	2	no	—
87	C₂×Q₁₆	—	211	4	no	—
88	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	211	2	no	—
91	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
92	#107	—	422	2	no	—
93	#103	—	422	4	no	—
94	C₄×C₈	—	2	2	no	—
95	#95	—	422	4	no	—
96	C₂×Q₈	—	422	2	no	—
97	C₂×Q₈	—	422	2	no	—
98	C₂×Q₈	—	422	2	no	—
99	C₂×Q₈	—	422	2	no	—
100	C₂×Q₈	—	422	2	no	—
101	C₂×Q₈	—	211	4	no	—
102	C₂²×C₄	—	422	2	no	—
103	C₂²×C₄	—	422	2	no	—
104	C₂²×C₄	D₈₄₄	1	2	yes	—
105	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
106	Q₁₆	—	422	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
110	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
111	C₂×C₈	D₈₄₄	1	2	yes	—
112	C₂×C₈	D₈₄₄	1	2	yes	—
113	C₄⋊C₄	—	422	2	no	—
114	C₄⋊C₄	—	422	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
118	Q₈	—	844	2	no	—
119	Q₈	—	844	2	no	—
120	Q₈	—	422	4	no	—
121	Q₈	—	422	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	844	2	no	—
125	C₂×C₄	—	844	2	no	—
126	C₂×C₄	—	844	2	no	—
127	C₂×C₄	—	844	4	no	—
128	C₂×C₄	—	422	4	no	—
129	C₂×C₄	Dic₈₄₄	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
132	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
133	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
134	C₂×C₄	#16809962	1	1	yes	characteristic
135	C₂×C₄	#16809962	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	(C₂×D₂₁₁)⋊C₄	1	4	yes	—
138	C₂²	C₄×D₈₄₄	1	1	yes	characteristic
139	C₂²	#20269951	1	1	yes	characteristic
140	C₂²	#20269974	1	1	yes	characteristic
141	C₂²	C₂×Dic₈₄₄	1	2	yes	—
142	C₂²	#20269999	1	2	yes	—
143	C₄	—	2	4	no	—
144	C₄	—	844	4	no	—
145	C₄	—	1688	2	no	—
146	C₄	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
147	C₄	C₄×D₈₄₄	1	1	yes	characteristic
148	C₄	#20269951	1	1	yes	characteristic
149	C₄	#20269965	1	1	yes	characteristic
150	C₂	#27213048	1	1	yes	characteristic
151	C₂	C₄×Dic₈₄₄	1	2	yes	—
152	C₂	#27213387	1	2	yes	—
153	C₂	#27213526	1	1	yes	characteristic
154	C₂	#27213530	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₈₄₄

C₂

yes

maximal, characteristic

C₂×#20269911

C₂

yes

maximal, characteristic

C₂×#20269902

C₂

yes

maximal

#27213052

C₂

yes

maximal

C₂×C₄×C₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

—

C₂×Dic₈₄₄

—

C₂×Dic₈₄₄

—

C₂×Dic₈₄₄

C₄

yes

complemented

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269911

—

#20269902

—

C₂₁₁⋊#95

—

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄×C₈×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

C₂×C₄

yes

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

Dic₈₄₄

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

Q₁₆

yes

complemented

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₈×C₂₁₁

—

C₈×C₂₁₁

C₂²⋊C₄

yes

—

C₂²×C₂₁₁

C₄×D₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

C₂×Q₁₆

yes

—

C₂²×C₂₁₁

#135

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

C₄×D₄

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#390

yes

characteristic

C₂×C₂₁₁

C₄×Q₁₆

yes

—

C₂×C₂₁₁

#467

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1969

—

211

maximal, Sylow

C₂²×Q₁₆

—

211

—

C₂×#107

—

211

—

C₂×#103

—

211

—

C₂×C₄×C₈

D₂₁₁

yes

characteristic

#391

—

211

—

C₂²×Q₈

—

211

—

C₂×Q₁₆

—

422

—

C₂×Q₁₆

—

422

—

C₂×Q₁₆

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#107

—

422

—

#103

—

422

—

C₄×C₈

—

#95

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

100

C₂×Q₈

—

422

—

101

C₂×Q₈

—

211

—

102

C₂²×C₄

—

422

—

103

C₂²×C₄

—

422

—

104

C₂²×C₄

D₈₄₄

yes

—

105

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

106

Q₁₆

—

422

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₂₁₁

yes

—

110

C₂×C₈

C₂²⋊D₂₁₁

yes

—

111

C₂×C₈

D₈₄₄

yes

—

112

C₂×C₈

D₈₄₄

yes

—

113

C₄⋊C₄

—

422

—

114

C₄⋊C₄

—

422

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₈₄₄

yes

characteristic

118

Q₈

—

844

—

119

Q₈

—

844

—

120

Q₈

—

422

—

121

Q₈

—

422

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

844

—

125

C₂×C₄

—

844

—

126

C₂×C₄

—

844

—

127

C₂×C₄

—

844

—

128

C₂×C₄

—

422

—

129

C₂×C₄

Dic₈₄₄

yes

complemented

130

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

132

C₂×C₄

C₂×D₈₄₄

yes

characteristic

133

C₂×C₄

C₂×D₈₄₄

yes

characteristic

134

C₂×C₄

#16809962

yes

characteristic

135

C₂×C₄

#16809962

yes

characteristic

136

C₈

—

137

C₈

(C₂×D₂₁₁)⋊C₄

yes

—

138

C₂²

C₄×D₈₄₄

yes

characteristic

139

C₂²

#20269951

yes

characteristic

140

C₂²

#20269974

yes

characteristic

141

C₂²

C₂×Dic₈₄₄

yes

—

142

C₂²

#20269999

yes

—

143

C₄

—

144

C₄

—

844

—

145

C₄

—

1688

—

146

C₄

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

147

C₄

C₄×D₈₄₄

yes

characteristic

148

C₄

#20269951

yes

characteristic

149

C₄

#20269965

yes

characteristic

150

C₂

#27213048

yes

characteristic

151

C₂

C₄×Dic₈₄₄

yes

—

152

C₂

#27213387

yes

—

153

C₂

#27213526

yes

characteristic

154

C₂

#27213530

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₈₄₄
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄×C₈×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	#1969
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₈₄₄

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×C₄×C₈×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×C₄×C₈

D₂₁₁

hypocenter

C₂₁₁

#1969

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₂×Dic₈₄₄	C₄	semidirect
51. C₂×C₄×C₂₁₁	Q₁₆	semidirect
77. C₂₁₁	#1969	semidirect
129. C₂×C₄	Dic₈₄₄	semidirect

H ≅ G/N

type

9. C₂×Dic₈₄₄

C₄

semidirect

51. C₂×C₄×C₂₁₁

Q₁₆

semidirect

77. C₂₁₁

#1969

semidirect

129. C₂×C₄

Dic₈₄₄

semidirect

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

subgroup	index
C₂²×Dic₈₄₄	2
C₂×#20269911	2
C₂×#20269902	2
#27213052	2
C₂×C₄×C₈×C₂₁₁	2
#1969	211

subgroup

index

C₂²×Dic₈₄₄

C₂×C₄×C₈×C₂₁₁

211

Subgroups — #41014259

Conjugacy classes of subgroups

Complements

Maximal subgroups