Group #26371780 — subgroups

Subgroups — #26371780

Order 12032.

482 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 219 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	#20253902	C₂	1	2	yes	maximal
2	#20245674	C₂	1	8	yes	maximal, complemented
3	#20245172	C₂	1	4	yes	maximal
4	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
5	#15689680	—	2	2	no	—
6	#15689528	C₂²	1	4	yes	—
7	#15689525	—	2	2	no	—
8	#15689525	C₂²	1	4	yes	—
9	#15689521	—	2	2	no	—
10	#15689161	—	2	2	no	—
11	#15689161	C₂²	1	4	yes	—
12	#15688951	C₂²	1	8	yes	—
13	#15688949	C₂²	1	8	yes	—
14	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#346	C₂²	1	4	yes	—
17	#13094024	—	2	4	no	—
18	#13094016	—	2	4	no	—
19	#13094003	—	4	4	no	—
20	C₄×Dic₉₄	—	4	4	no	—
21	#13093983	—	2	4	no	—
22	#13093958	—	2	4	no	—
23	#13093958	C₂³	1	8	yes	—
24	#13093956	—	2	4	no	—
25	#13093956	—	2	4	no	—
26	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₄₇	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
31	C₄×C₈×C₄₇	D₄	1	2	yes	—
32	C₄×C₈×C₄₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₄₇	D₈	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₉₄	—	4	4	no	—
36	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
37	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
40	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
41	C₄²×C₄₇	D₈	1	2	yes	—
42	C₄²×C₄₇	C₂×D₄	1	2	yes	—
43	C₂²⋊Dic₄₇	—	4	4	no	—
44	C₄⋊Dic₄₇	—	2	8	no	—
45	C₄⋊Dic₄₇	—	4	4	no	—
46	Dic₄₇⋊C₄	—	4	8	no	—
47	C₄×Dic₄₇	—	4	4	no	—
48	C₁₆×C₄₇	C₂×Q₈	1	8	yes	—
49	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₄₇	—	4	8	no	—
57	Dic₉₄	—	8	8	no	—
58	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
59	C₈×C₄₇	C₂×Q₁₆	1	4	yes	—
60	C₈×C₄₇	C₂²×Q₈	1	2	yes	—
61	#33728	—	47	1	no	maximal, Sylow
62	C₂²×C₄₇	—	2	2	no	—
63	C₂²×C₄₇	#500	1	2	yes	—
64	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
65	C₄×C₄₇	—	2	4	no	—
66	C₄×C₄₇	#500	1	2	yes	—
67	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
69	Dic₄₇	—	8	8	no	—
70	#3160	—	47	2	no	—
71	#2255	—	47	8	no	—
72	#2158	—	47	4	no	—
73	C₂×#346	D₄₇	1	1	yes	characteristic
74	C₂×C₄₇	—	2	2	no	—
75	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
76	C₂×C₄₇	#3414	1	2	yes	—
77	#533	—	94	2	no	—
78	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
79	#500	—	47	4	no	—
80	#499	—	47	4	no	—
81	#499	—	94	2	no	—
82	#498	—	94	2	no	—
83	#427	—	47	4	no	—
84	#427	—	94	2	no	—
85	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
86	#367	—	47	8	no	—
87	#366	—	47	8	no	—
88	#346	C₂×D₄₇	1	4	yes	—
89	C₄₇	#33728	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
92	C₄⋊Q₈	—	94	4	no	—
93	#125	—	94	4	no	—
94	C₂²⋊Q₈	—	188	4	no	—
95	C₄×Q₈	—	188	4	no	—
96	#117	—	94	4	no	—
97	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
98	C₂×C₁₆	Dic₉₄	1	8	yes	complemented
99	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
100	#107	—	47	8	no	—
101	#107	—	94	4	no	—
102	#106	—	94	4	no	—
103	#106	—	94	4	no	—
104	C₄×C₈	D₁₈₈	1	2	yes	—
105	C₄×C₈	C₂²×D₄₇	1	2	yes	—
106	C₂×Q₈	—	188	4	no	—
107	C₂²×C₄	D₃₇₆	1	2	yes	—
108	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
109	C₂×C₈	Dic₁₈₈	1	4	yes	—
110	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
111	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
112	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
113	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
114	C₄⋊C₄	—	94	8	no	—
115	C₄⋊C₄	—	188	8	no	—
116	C₄⋊C₄	—	188	4	no	—
117	C₂²⋊C₄	—	188	4	no	—
118	C₄²	—	188	4	no	—
119	C₄²	D₃₇₆	1	2	yes	—
120	C₄²	C₂×D₁₈₈	1	2	yes	—
121	C₁₆	C₂×Dic₉₄	1	8	yes	—
122	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
123	Q₈	—	376	8	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	188	8	no	—
126	C₂×C₄	#13094024	1	2	yes	—
127	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
128	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
130	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
131	C₈	#13094024	1	2	yes	—
132	C₈	C₂×Dic₁₈₈	1	4	yes	—
133	C₈	C₂²×Dic₉₄	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#15689528	1	2	yes	—
136	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	376	8	no	—
139	C₄	#15689528	1	2	yes	—
140	C₄	C₂×#13094024	1	1	yes	characteristic
141	C₄	C₂²×Dic₁₈₈	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#15689528	1	1	yes	characteristic
144	C₂	#20256524	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20253902

C₂

yes

maximal

#20245674

C₂

yes

maximal, complemented

#20245172

C₂

yes

maximal

C₂×C₄₇×#346

C₂

yes

maximal, characteristic

#15689680

—

#15689528

C₂²

yes

—

#15689525

—

#15689525

C₂²

yes

—

#15689521

—

#15689161

—

#15689161

C₂²

yes

—

#15688951

C₂²

yes

—

#15688949

C₂²

yes

—

C₂²×C₁₆×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#346

C₂²

yes

—

#13094024

—

#13094016

—

#13094003

—

C₄×Dic₉₄

—

#13093983

—

#13093958

—

#13093958

C₂³

yes

—

#13093956

—

#13093956

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

Q₈

yes

complemented

C₂×C₁₆×C₄₇

C₂³

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

D₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

C₂⁴

yes

characteristic

C₄²×C₄₇

D₈

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₁₆×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

C₄⋊Q₈

yes

—

C₈×C₄₇

C₂×Q₁₆

yes

—

C₈×C₄₇

C₂²×Q₈

yes

—

#33728

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#500

yes

—

C₂²×C₄₇

C₂²×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#500

yes

—

C₄×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₄₇

C₂²×Q₁₆

yes

characteristic

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

C₂×#500

yes

characteristic

C₂×C₄₇

#3414

yes

—

#533

—

C₂²×C₁₆

C₂×D₄₇

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

—

C₄₇

#33728

yes

characteristic, complemented, Sylow

C₂²×C₈

D₁₈₈

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

188

—

C₄×Q₈

—

188

—

#117

—

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₁₆

Dic₉₄

yes

complemented

C₂×C₁₆

C₂²×D₄₇

yes

—

100

#107

—

101

#107

—

102

#106

—

103

#106

—

104

C₄×C₈

D₁₈₈

yes

—

105

C₄×C₈

C₂²×D₄₇

yes

—

106

C₂×Q₈

—

188

—

107

C₂²×C₄

D₃₇₆

yes

—

108

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

109

C₂×C₈

Dic₁₈₈

yes

—

110

C₂×C₈

C₂×Dic₉₄

yes

—

111

C₂×C₈

C₂×D₁₈₈

yes

characteristic

112

C₂×C₈

C₂×D₁₈₈

yes

—

113

C₂×C₈

C₂³×D₄₇

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

188

—

116

C₄⋊C₄

—

188

—

117

C₂²⋊C₄

—

188

—

118

C₄²

—

188

—

119

C₄²

D₃₇₆

yes

—

120

C₄²

C₂×D₁₈₈

yes

—

121

C₁₆

C₂×Dic₉₄

yes

—

122

C₂³

C₂×D₃₇₆

yes

characteristic

123

Q₈

—

376

—

124

C₂×C₄

—

125

C₂×C₄

—

188

—

126

C₂×C₄

#13094024

yes

—

127

C₂×C₄

C₂×D₃₇₆

yes

—

128

C₂×C₄

C₂×D₃₇₆

yes

characteristic

129

C₂×C₄

C₂×Dic₁₈₈

yes

—

130

C₂×C₄

C₂²×D₁₈₈

yes

characteristic

131

C₈

#13094024

yes

—

132

C₈

C₂×Dic₁₈₈

yes

—

133

C₈

C₂²×Dic₉₄

yes

—

134

C₂²

—

135

C₂²

#15689528

yes

—

136

C₂²

C₂²×D₃₇₆

yes

characteristic

137

C₄

—

138

C₄

—

376

—

139

C₄

#15689528

yes

—

140

C₄

C₂×#13094024

yes

characteristic

141

C₄

C₂²×Dic₁₈₈

yes

characteristic

142

C₂

—

143

C₂

C₂×#15689528

yes

characteristic

144

C₂

#20256524

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#346	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	#33728
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₃₇₆

commutator, cosocle

C₂×C₈×C₄₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₄₇

Fitting

C₂×C₄₇×#346

C₂

socle

C₂²×C₄₇

C₂²×D₈

prime radical

C₂³×C₄₇

C₂×D₈

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #20245674	C₂	semidirect
29. C₂×C₁₆×C₄₇	Q₈	semidirect
89. C₄₇	#33728	semidirect
98. C₂×C₁₆	Dic₉₄	semidirect

H ≅ G/N

type

2. #20245674

C₂

semidirect

29. C₂×C₁₆×C₄₇

Q₈

semidirect

89. C₄₇

#33728

semidirect

98. C₂×C₁₆

Dic₉₄

semidirect

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

subgroup	index
#20253902	2
#20245674	2
#20245172	2
C₂×C₄₇×#346	2
#33728	47

subgroup

index

C₂×C₄₇×#346

Subgroups — #26371780

Conjugacy classes of subgroups

Complements

Maximal subgroups