Group #26103164 — subgroups

Subgroups — #26103164

Order 12032.

202 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 119 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×#15688944	C₂	1	2	yes	maximal
2	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093964	—	2	2	no	—
4	#15688944	—	2	2	no	—
5	#15688944	C₄	1	4	yes	—
6	C₂²×C₁₆×C₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₄₇×#346	—	2	2	no	—
10	#13093964	—	2	4	no	—
11	#13093964	—	4	2	no	—
12	C₂×C₄₇⋊C₁₆	—	4	2	no	—
13	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
15	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
16	C₂×C₁₆×C₄₇	—	2	1	no	—
17	C₂×C₁₆×C₄₇	—	2	1	no	—
18	C₂×C₁₆×C₄₇	—	2	1	no	—
19	C₂×C₁₆×C₄₇	—	2	1	no	—
20	C₂×C₁₆×C₄₇	C₈	1	2	yes	—
21	C₂×C₁₆×C₄₇	C₈	1	2	yes	—
22	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₄₇	D₄	1	2	yes	—
24	C₂²×C₄×C₄₇	—	2	1	no	—
25	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
29	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
34	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
35	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
36	C₄²×C₄₇	—	2	1	no	—
37	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₄₇	—	2	2	no	—
39	C₁₆×C₄₇	—	2	2	no	—
40	C₄₇⋊C₁₆	—	4	4	no	—
41	C₂³×C₄₇	#108	1	1	yes	characteristic
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	4	2	no	—
44	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
50	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
51	C₈×C₄₇	#97	1	1	yes	characteristic
52	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
53	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
54	C₈×C₄₇	#102	1	2	yes	—
55	C₈×C₄₇	#103	1	2	yes	—
56	#7161	—	47	1	no	maximal, Sylow
57	C₂²×C₄₇	—	2	2	no	—
58	C₂²×C₄₇	#334	1	1	yes	characteristic
59	C₂²×C₄₇	#335	1	1	yes	characteristic
60	C₂²×C₄₇	#341	1	1	yes	characteristic
61	C₄×C₄₇	—	4	2	no	—
62	C₄×C₄₇	#334	1	1	yes	characteristic
63	C₄×C₄₇	#335	1	1	yes	characteristic
64	C₄×C₄₇	#336	1	1	yes	characteristic
65	C₄×C₄₇	#340	1	1	yes	characteristic
66	C₂×#364	—	47	2	no	—
67	C₂×#346	D₄₇	1	1	yes	characteristic
68	C₂×C₄₇	—	2	1	no	—
69	C₂×C₄₇	—	2	1	no	—
70	C₂×C₄₇	#1277	1	1	yes	characteristic
71	C₂×C₄₇	#1369	1	2	yes	—
72	C₂×M₃₂	—	94	2	no	—
73	C₂²×C₁₆	Dic₄₇	1	1	yes	characteristic
74	C₂²×C₁₆	Dic₄₇	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
76	#364	—	47	4	no	—
77	#364	—	94	2	no	—
78	#346	—	2	2	no	—
79	C₄₇	#7161	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
81	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
82	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
83	M₃₂	—	94	4	no	—
84	M₃₂	—	188	2	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	188	2	no	—
90	C₂×C₁₆	C₄₇⋊C₈	1	2	yes	—
91	C₂×C₁₆	C₄₇⋊C₈	1	2	yes	—
92	C₄×C₈	C₄×D₄₇	1	2	yes	—
93	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
96	C₂×C₈	#10625325	1	2	yes	—
97	C₂×C₈	D₃₇₆	1	1	yes	characteristic
98	C₂×C₈	Dic₁₈₈	1	1	yes	characteristic
99	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
100	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
101	C₂×C₈	#10625329	1	1	yes	characteristic
102	C₂×C₈	#10625329	1	1	yes	characteristic
103	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
104	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
105	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	Dic₄₇⋊C₄	1	2	yes	—
108	C₁₆	—	2	2	no	—
109	C₁₆	—	2	2	no	—
110	C₁₆	—	188	4	no	—
111	C₂³	#13093960	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
115	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
116	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
117	C₂×C₄	#13093956	1	1	yes	characteristic
118	C₂×C₄	#13093958	1	1	yes	characteristic
119	C₂×C₄	#13093960	1	1	yes	characteristic
120	C₈	C₄×C₄₇⋊C₈	1	1	yes	characteristic
121	C₈	#13093935	1	1	yes	characteristic
122	C₈	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
123	C₈	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
124	C₈	#13093946	1	2	yes	—
125	C₈	#13093949	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	#15688874	1	1	yes	characteristic
128	C₂²	#15688876	1	1	yes	characteristic
129	C₂²	#15688888	1	1	yes	characteristic
130	C₄	—	4	2	no	—
131	C₄	#15688874	1	1	yes	characteristic
132	C₄	#15688876	1	1	yes	characteristic
133	C₄	#15688878	1	1	yes	characteristic
134	C₄	#15688886	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	#20241223	1	1	yes	characteristic
138	C₂	#20241456	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688944

C₂

yes

maximal

C₂×C₄₇×#346

C₂

yes

maximal, characteristic

C₂×#13093964

—

#15688944

—

#15688944

C₄

yes

—

C₂²×C₁₆×C₄₇

C₄

yes

characteristic

C₂²×C₁₆×C₄₇

C₄

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#346

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

Q₈

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₈

yes

—

C₂×C₁₆×C₄₇

C₈

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

M₁₆

yes

characteristic

C₂×C₈×C₄₇

M₁₆

yes

characteristic

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#108

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

C₄×C₈

yes

characteristic

C₈×C₄₇

#97

yes

characteristic

C₈×C₄₇

C₂²⋊C₈

yes

characteristic

C₈×C₄₇

C₂²⋊C₈

yes

characteristic

C₈×C₄₇

#102

yes

—

C₈×C₄₇

#103

yes

—

#7161

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1277

yes

characteristic

C₂×C₄₇

#1369

yes

—

C₂×M₃₂

—

C₂²×C₁₆

Dic₄₇

yes

characteristic

C₂²×C₁₆

Dic₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

C₄₇

#7161

yes

characteristic, complemented, Sylow

C₂²×C₈

D₁₈₈

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

Dic₉₄

yes

characteristic

M₃₂

—

M₃₂

—

188

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

188

—

C₂×C₁₆

C₄₇⋊C₈

yes

—

C₂×C₁₆

C₄₇⋊C₈

yes

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₈

#10625325

yes

—

C₂×C₈

D₃₇₆

yes

characteristic

C₂×C₈

Dic₁₈₈

yes

characteristic

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

100

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

101

C₂×C₈

#10625329

yes

characteristic

102

C₂×C₈

#10625329

yes

characteristic

103

C₂×C₈

C₄×Dic₄₇

yes

characteristic

104

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

105

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

106

C₄²

—

107

C₄²

Dic₄₇⋊C₄

yes

—

108

C₁₆

—

109

C₁₆

—

110

C₁₆

—

188

—

111

C₂³

#13093960

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₄₇⋊#95

yes

characteristic

115

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

characteristic

116

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

characteristic

117

C₂×C₄

#13093956

yes

characteristic

118

C₂×C₄

#13093958

yes

characteristic

119

C₂×C₄

#13093960

yes

characteristic

120

C₈

C₄×C₄₇⋊C₈

yes

characteristic

121

C₈

#13093935

yes

characteristic

122

C₈

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

123

C₈

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

124

C₈

#13093946

yes

—

125

C₈

#13093949

yes

—

126

C₂²

—

127

C₂²

#15688874

yes

characteristic

128

C₂²

#15688876

yes

characteristic

129

C₂²

#15688888

yes

characteristic

130

C₄

—

131

C₄

#15688874

yes

characteristic

132

C₄

#15688876

yes

characteristic

133

C₄

#15688878

yes

characteristic

134

C₄

#15688886

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#20241223

yes

characteristic

138

C₂

#20241456

yes

—

	subgroup	G/H
center	C₂×C₄	#13093956
commutator	C₈×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#346	C₂
socle	C₂²×C₄₇	#334
prime radical	C₂³×C₄₇	#108
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	#7161
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

#13093956

commutator

C₈×C₄₇

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₂×C₄₇×#346

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
79. C₄₇	#7161	semidirect

H ≅ G/N

type

79. C₄₇

#7161

semidirect

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

subgroup	index
C₂×#15688944	2
C₂×C₄₇×#346	2
#7161	47

subgroup

index

C₂×#15688944

C₂×C₄₇×#346

#7161

Subgroups — #26103164

Conjugacy classes of subgroups

Complements

Maximal subgroups