Group #26115777 — subgroups

Subgroups — #26115777

Order 12032.

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

normal subgroups only

This group has 454 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×#15689503	C₂	1	1	yes	maximal, characteristic
2	#20242061	C₂	1	4	yes	maximal
3	#20241680	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1837	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₈₈	—	2	2	no	—
6	C₂×#13094024	C₂²	1	2	yes	—
7	#15689503	—	2	2	no	—
8	#15689503	C₄	1	4	yes	—
9	#15689176	C₂²	1	2	yes	—
10	#15688870	—	2	8	no	—
11	#15688845	—	2	2	no	—
12	C₄₇×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#341	—	2	2	no	—
15	C₂²×Dic₉₄	—	2	2	no	—
16	C₂×Dic₁₈₈	—	2	8	no	—
17	C₂×Dic₁₈₈	—	4	8	no	—
18	#13094024	—	2	4	no	—
19	#13094024	—	2	2	no	—
20	#13094024	C₂×C₄	1	4	yes	—
21	C₂×C₄⋊Dic₄₇	—	2	4	no	—
22	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	D₄	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₄₇	—	2	4	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₄₇	D₄	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	#41	1	2	yes	—
33	C₂²×Dic₄₇	—	4	2	no	—
34	C₂×Dic₉₄	—	2	4	no	—
35	C₂×Dic₉₄	—	4	8	no	—
36	C₂×Dic₉₄	—	4	2	no	—
37	M₁₆×C₄₇	—	4	4	no	—
38	C₂×C₈×C₄₇	—	2	4	no	—
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
41	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₄₇	C₂×D₄	1	2	yes	—
44	C₄⋊Dic₄₇	—	2	8	no	—
45	C₄⋊Dic₄₇	—	4	4	no	—
46	C₂×C₄₇⋊C₈	—	2	4	no	—
47	Dic₁₈₈	—	4	16	no	—
48	C₂³×C₄₇	#124	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	2	4	no	—
50	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	#121	1	2	yes	—
53	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
54	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
55	C₂×Dic₄₇	—	4	4	no	—
56	C₂×Dic₄₇	—	8	4	no	—
57	Dic₉₄	—	4	8	no	—
58	Dic₉₄	—	8	4	no	—
59	C₈×C₄₇	—	2	4	no	—
60	C₈×C₄₇	—	8	2	no	—
61	C₄₇⋊C₈	—	4	4	no	—
62	#9779	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	2	2	no	—
64	C₂²×C₄₇	#390	1	1	yes	characteristic
65	C₂²×C₄₇	#490	1	2	yes	—
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	#418	1	2	yes	—
68	C₄×C₄₇	#473	1	4	yes	—
69	C₄×C₄₇	#496	1	2	yes	—
70	Dic₄₇	—	8	4	no	—
71	C₂×#494	—	47	1	no	—
72	#1837	D₄₇	1	1	yes	characteristic
73	#1546	—	47	4	no	—
74	#1452	—	47	1	no	—
75	C₂×C₄₇	—	2	2	no	—
76	C₂×C₄₇	#1961	1	1	yes	characteristic
77	C₂×C₄₇	#1979	1	2	yes	—
78	C₂²×Q₁₆	—	94	2	no	—
79	C₂×C₄⋊Q₈	—	47	2	no	—
80	#494	—	47	4	no	—
81	#494	—	94	2	no	—
82	#431	—	47	2	no	—
83	#423	C₂×D₄₇	1	2	yes	—
84	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
85	#341	—	2	2	no	—
86	#332	—	94	8	no	—
87	#322	—	94	2	no	—
88	C₄₇	#9779	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	94	2	no	—
90	C₂×Q₁₆	—	94	8	no	—
91	C₂×Q₁₆	—	188	8	no	—
92	C₂×M₁₆	—	2	2	no	—
93	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
94	C₄⋊Q₈	—	47	4	no	—
95	C₄⋊Q₈	—	94	2	no	—
96	C₄⋊Q₈	—	94	4	no	—
97	C₂×C₄⋊C₄	—	94	4	no	—
98	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	47	4	no	—
101	C₂²⋊C₈	—	94	2	no	—
102	C₄×C₈	—	94	2	no	—
103	C₄×C₈	C₄×D₄₇	1	2	yes	—
104	C₄×C₈	D₁₈₈	1	2	yes	—
105	C₂×Q₈	—	94	4	no	—
106	C₂×Q₈	—	188	8	no	—
107	C₂×Q₈	—	188	2	no	—
108	C₂²×C₄	—	188	2	no	—
109	C₂²×C₄	#10625354	1	2	yes	—
110	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
111	Q₁₆	—	188	16	no	—
112	M₁₆	—	4	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	94	4	no	—
116	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
117	C₄⋊C₄	—	94	8	no	—
118	C₄⋊C₄	—	188	4	no	—
119	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
120	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
121	C₄²	D₄×D₄₇	1	2	yes	—
122	C₂³	#13094013	1	1	yes	characteristic
123	Q₈	—	376	4	no	—
124	Q₈	—	188	8	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	376	4	no	—
127	C₂×C₄	—	188	4	no	—
128	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	#13093986	1	2	yes	—
130	C₂×C₄	#13093999	1	2	yes	—
131	C₂×C₄	#13094020	1	1	yes	characteristic
132	C₂×C₄	#13094049	1	4	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	8	2	no	—
135	C₈	—	188	4	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#15689024	1	1	yes	characteristic
138	C₂²	#15689489	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	376	4	no	—
141	C₄	#15689127	1	2	yes	—
142	C₄	#15689399	1	4	yes	—
143	C₄	#15689513	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#20244211	1	1	yes	characteristic
146	C₂	#20244301	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689503

C₂

yes

maximal, characteristic

#20242061

C₂

yes

maximal

#20241680

C₂

yes

maximal, characteristic

C₄₇×#1837

C₂

yes

maximal, characteristic

C₂²×Dic₁₈₈

—

C₂×#13094024

C₂²

yes

—

#15689503

—

#15689503

C₄

yes

—

#15689176

C₂²

yes

—

#15688870

—

#15688845

—

C₄₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#341

—

C₂²×Dic₉₄

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

#13094024

—

#13094024

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

C₂³×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#9779

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#390

yes

characteristic

C₂²×C₄₇

#490

yes

—

C₄×C₄₇

—

C₄×C₄₇

#418

yes

—

C₄×C₄₇

#473

yes

—

C₄×C₄₇

#496

yes

—

Dic₄₇

—

C₂×#494

—

#1837

D₄₇

yes

characteristic

#1546

—

#1452

—

C₂×C₄₇

—

C₂×C₄₇

#1961

yes

characteristic

C₂×C₄₇

#1979

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

C₄₇

#9779

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

188

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₄×D₄₇

yes

—

104

C₄×C₈

D₁₈₈

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

188

—

107

C₂×Q₈

—

188

—

108

C₂²×C₄

—

188

—

109

C₂²×C₄

#10625354

yes

—

110

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

111

Q₁₆

—

188

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

188

—

119

C₄²

C₂×C₄×D₄₇

yes

characteristic

120

C₄²

C₂×D₁₈₈

yes

characteristic

121

C₄²

D₄×D₄₇

yes

—

122

C₂³

#13094013

yes

characteristic

123

Q₈

—

376

—

124

Q₈

—

188

—

125

C₂×C₄

—

126

C₂×C₄

—

376

—

127

C₂×C₄

—

188

—

128

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

129

C₂×C₄

#13093986

yes

—

130

C₂×C₄

#13093999

yes

—

131

C₂×C₄

#13094020

yes

characteristic

132

C₂×C₄

#13094049

yes

—

133

C₈

—

134

C₈

—

135

C₈

—

188

—

136

C₂²

—

137

C₂²

#15689024

yes

characteristic

138

C₂²

#15689489

yes

—

139

C₄

—

140

C₄

—

376

—

141

C₄

#15689127

yes

—

142

C₄

#15689399

yes

—

143

C₄

#15689513

yes

—

144

C₂

—

145

C₂

#20244211

yes

characteristic

146

C₂

#20244301

yes

—

	subgroup	G/H
center	C₂²	#15689024
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1837	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	#124
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1837	D₄₇
hypocenter	C₄₇	#9779
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#15689024

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1837

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
88. C₄₇	#9779	semidirect

H ≅ G/N

type

88. C₄₇

#9779

semidirect

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

subgroup	index
C₂×#15689503	2
#20242061	2
#20241680	2
C₄₇×#1837	2
#9779	47

subgroup

index

C₄₇×#1837

Subgroups — #26115777

Conjugacy classes of subgroups

Complements

Maximal subgroups