Group #41014314 — subgroups

Subgroups — #41014314

Order 27008.

276 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 97 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×#20269913	C₂	1	1	yes	maximal, characteristic
2	#27213155	C₂	1	2	yes	maximal
3	C₂×#20269896	C₂	1	2	yes	maximal
4	#27213101	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂×#16809946	C₂²	1	1	yes	characteristic
8	#20269936	C₂²	1	1	yes	characteristic
9	#20269913	—	2	2	no	—
10	#20269901	—	2	4	no	—
11	#20269896	—	2	4	no	—
12	#20269887	—	2	1	no	—
13	C₈×Dic₂₁₁	—	2	1	no	—
14	C₂²×Q₈×C₂₁₁	C₂²	1	2	yes	—
15	C₂×Q₁₆×C₂₁₁	—	2	1	no	—
16	C₂×Q₁₆×C₂₁₁	—	2	1	no	—
17	C₂×Q₁₆×C₂₁₁	—	2	2	no	—
18	C₂×Q₁₆×C₂₁₁	C₄	1	4	yes	complemented
19	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
20	#16809946	—	2	2	no	—
21	#16809946	—	2	2	no	—
22	#16809946	—	2	1	no	—
23	#16809946	D₄	1	2	yes	—
24	C₂×C₂₁₁⋊C₈	—	2	1	no	—
25	C₂×C₂₁₁⋊C₈	—	2	2	no	—
26	C₄⋊Dic₂₁₁	D₄	1	2	yes	—
27	C₂²⋊Dic₂₁₁	—	2	1	no	—
28	C₄×Dic₂₁₁	—	2	1	no	—
29	C₂×Q₈×C₂₁₁	—	2	2	no	—
30	C₂×Q₈×C₂₁₁	—	2	2	no	—
31	C₂×Q₈×C₂₁₁	—	2	2	no	—
32	C₂×Q₈×C₂₁₁	—	2	2	no	—
33	C₂×Q₈×C₂₁₁	—	2	2	no	—
34	C₂×Q₈×C₂₁₁	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₂₁₁	—	2	2	no	—
36	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
37	Q₁₆×C₂₁₁	—	2	4	no	—
38	Q₁₆×C₂₁₁	—	2	4	no	—
39	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
41	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
42	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
43	C₂×Dic₂₁₁	—	2	2	no	—
44	C₂₁₁⋊C₈	—	2	2	no	—
45	C₂₁₁⋊C₈	—	4	2	no	—
46	C₂³×C₂₁₁	#41	1	1	yes	characteristic
47	Q₈×C₂₁₁	—	2	4	no	—
48	Q₈×C₂₁₁	—	2	4	no	—
49	Q₈×C₂₁₁	—	4	2	no	—
50	Q₈×C₂₁₁	—	4	2	no	—
51	C₂×C₄×C₂₁₁	—	2	4	no	—
52	C₂×C₄×C₂₁₁	—	4	2	no	—
53	C₂×C₄×C₂₁₁	—	4	2	no	—
54	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
60	C₈×C₂₁₁	C₂²⋊C₄	1	2	yes	—
61	C₈×C₂₁₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₂₁₁	—	2	2	no	—
63	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₂₁₁	#121	1	1	yes	characteristic
65	C₂²×C₂₁₁	#124	1	1	yes	characteristic
66	C₄×C₂₁₁	—	4	4	no	—
67	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
69	C₄×C₂₁₁	#121	1	1	yes	characteristic
70	C₄×C₂₁₁	#127	1	1	yes	characteristic
71	Dic₂₁₁	—	4	2	no	—
72	C₂×C₂₁₁	—	2	1	no	—
73	C₂×C₂₁₁	—	2	1	no	—
74	C₂×C₂₁₁	#390	1	1	yes	characteristic
75	C₂×C₂₁₁	#473	1	2	yes	—
76	C₂₁₁	#1979	1	1	yes	characteristic, complemented, Sylow
77	#1979	—	211	1	no	maximal, Sylow
78	C₂²×Q₁₆	D₂₁₁	1	1	yes	characteristic
79	C₂×#108	—	211	1	no	—
80	#419	—	211	2	no	—
81	C₂×#101	—	211	2	no	—
82	#405	—	211	1	no	—
83	C₂²×Q₈	C₂×D₂₁₁	1	2	yes	—
84	C₂×Q₁₆	—	2	1	no	—
85	C₂×Q₁₆	—	2	1	no	—
86	C₂×Q₁₆	—	2	2	no	—
87	C₂×Q₁₆	Dic₂₁₁	1	4	yes	complemented
88	C₂×M₁₆	—	211	2	no	—
89	C₂×M₁₆	—	211	1	no	—
90	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
91	#117	—	211	1	no	—
92	#108	—	422	2	no	—
93	#103	—	422	4	no	—
94	#101	—	422	4	no	—
95	#97	—	422	1	no	—
96	C₄×C₈	—	422	1	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	C₂×Dic₂₁₁	1	4	yes	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
105	Q₁₆	—	2	4	no	—
106	Q₁₆	—	2	4	no	—
107	M₁₆	—	422	2	no	—
108	M₁₆	—	422	2	no	—
109	M₁₆	—	422	1	no	—
110	M₁₆	—	211	2	no	—
111	C₂×C₈	—	422	2	no	—
112	C₂×C₈	—	422	1	no	—
113	C₂×C₈	C₂×Dic₂₁₁	1	2	yes	—
114	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
117	C₄⋊C₄	—	211	2	no	—
118	C₂²⋊C₄	—	422	1	no	—
119	C₄²	—	422	1	no	—
120	C₂³	#16809961	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	422	2	no	—
129	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
130	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
131	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
134	C₂×C₄	#16809961	1	1	yes	characteristic
135	C₈	—	844	2	no	—
136	C₈	—	422	2	no	—
137	C₈	C₂²⋊Dic₂₁₁	1	2	yes	—
138	C₈	C₂²⋊Dic₂₁₁	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	D₄×Dic₂₁₁	1	1	yes	characteristic
141	C₂²	#20269948	1	1	yes	characteristic
142	C₂²	#20269964	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	844	2	no	—
145	C₄	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
146	C₄	D₄×Dic₂₁₁	1	1	yes	characteristic
147	C₄	#20269948	1	1	yes	characteristic
148	C₄	#20269973	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#27213049	1	1	yes	characteristic
152	C₂	#27213418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269913

C₂

yes

maximal, characteristic

#27213155

C₂

yes

maximal

C₂×#20269896

C₂

yes

maximal

#27213101

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂×#16809946

C₂²

yes

characteristic

#20269936

C₂²

yes

—

—

—

—

C₈×Dic₂₁₁

—

C₂²×Q₈×C₂₁₁

C₂²

yes

—

C₂×Q₁₆×C₂₁₁

—

C₂×Q₁₆×C₂₁₁

—

C₂×Q₁₆×C₂₁₁

—

C₂×Q₁₆×C₂₁₁

C₄

yes

complemented

C₂²×C₈×C₂₁₁

C₂²

yes

—

—

—

D₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

D₄

yes

—

C₂²⋊Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

Q₁₆×C₂₁₁

—

Q₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

D₄

yes

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

#41

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

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

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

C₂²⋊C₄

yes

—

C₈×C₂₁₁

C₂²⋊C₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₄×D₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

C₄×D₄

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#390

yes

characteristic

C₂×C₂₁₁

#473

yes

—

C₂₁₁

#1979

yes

characteristic, complemented, Sylow

#1979

—

211

maximal, Sylow

C₂²×Q₁₆

D₂₁₁

yes

characteristic

C₂×#108

—

211

—

#419

—

211

—

C₂×#101

—

211

—

#405

—

211

—

C₂²×Q₈

C₂×D₂₁₁

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₂₁₁

yes

complemented

C₂×M₁₆

—

211

—

C₂×M₁₆

—

211

—

C₂²×C₈

C₂×D₂₁₁

yes

characteristic

#117

—

211

—

#108

—

422

—

#103

—

422

—

#101

—

422

—

#97

—

422

—

C₄×C₈

—

422

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂×Dic₂₁₁

yes

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

105

Q₁₆

—

106

Q₁₆

—

107

M₁₆

—

422

—

108

M₁₆

—

422

—

109

M₁₆

—

422

—

110

M₁₆

—

211

—

111

C₂×C₈

—

422

—

112

C₂×C₈

—

422

—

113

C₂×C₈

C₂×Dic₂₁₁

yes

—

114

C₂×C₈

C₂²⋊D₂₁₁

yes

characteristic

115

C₂×C₈

C₂²⋊D₂₁₁

yes

characteristic

116

C₂×C₈

C₂²⋊D₂₁₁

yes

—

117

C₄⋊C₄

—

211

—

118

C₂²⋊C₄

—

422

—

119

C₄²

—

422

—

120

C₂³

#16809961

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

422

—

129

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

130

C₂×C₄

D₄×D₂₁₁

yes

characteristic

131

C₂×C₄

D₄×D₂₁₁

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

134

C₂×C₄

#16809961

yes

characteristic

135

C₈

—

844

—

136

C₈

—

422

—

137

C₈

C₂²⋊Dic₂₁₁

yes

—

138

C₈

C₂²⋊Dic₂₁₁

yes

—

139

C₂²

—

140

C₂²

D₄×Dic₂₁₁

yes

characteristic

141

C₂²

#20269948

yes

characteristic

142

C₂²

#20269964

yes

characteristic

143

C₄

—

144

C₄

—

844

—

145

C₄

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

146

C₄

D₄×Dic₂₁₁

yes

characteristic

147

C₄

#20269948

yes

characteristic

148

C₄

#20269973

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#27213049

yes

characteristic

152

C₂

#27213418

yes

—

	subgroup	G/H
center	C₂²	#20269948
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂²×Q₁₆×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#121
prime radical	C₂³×C₂₁₁	#41
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂²×Q₁₆	D₂₁₁
hypocenter	C₂₁₁	#1979
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#20269948

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂²×Q₁₆×C₂₁₁

C₂

C₂²×C₂₁₁

C₂³×C₂₁₁

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂²×Q₁₆

D₂₁₁

hypocenter

C₂₁₁

#1979

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
18. C₂×Q₁₆×C₂₁₁	C₄	semidirect
76. C₂₁₁	#1979	semidirect
87. C₂×Q₁₆	Dic₂₁₁	semidirect

H ≅ G/N

type

18. C₂×Q₁₆×C₂₁₁

C₄

semidirect

76. C₂₁₁

#1979

semidirect

87. C₂×Q₁₆

Dic₂₁₁

semidirect

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

subgroup	index
C₂×#20269913	2
#27213155	2
C₂×#20269896	2
#27213101	2
C₂²×Q₁₆×C₂₁₁	2
#1979	211

subgroup

index

C₂²×Q₁₆×C₂₁₁

211

Subgroups — #41014314

Conjugacy classes of subgroups

Complements

Maximal subgroups