Group #26110096 — subgroups

Subgroups — #26110096

Order 12032.

306 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20243526	C₂	1	2	yes	maximal
2	#20242147	C₂	1	2	yes	maximal
3	#20242144	C₂	1	2	yes	maximal
4	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
5	#15689147	C₂²	1	4	yes	—
6	#15688888	—	2	4	no	—
7	#15688876	—	2	4	no	—
8	#15688874	—	2	4	no	—
9	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#322	C₄	1	4	yes	—
13	C₂×#10625329	—	2	4	no	—
14	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
15	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
16	C₂²×C₈×C₄₇	D₄	1	2	yes	—
17	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
18	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₄₇	D₄	1	2	yes	—
24	C₄×C₈×C₄₇	D₄	1	2	yes	—
25	C₄×C₈×C₄₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₈×C₄₇	—	2	4	no	—
30	C₂×C₈×C₄₇	—	4	2	no	—
31	C₂×C₈×C₄₇	—	4	2	no	—
32	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
35	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	#10625329	—	4	8	no	—
40	C₂×C₄₇⋊C₈	—	4	4	no	—
41	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#135	1	4	yes	—
51	C₈×C₄₇	—	2	4	no	—
52	C₈×C₄₇	—	2	4	no	—
53	C₈×C₄₇	—	2	4	no	—
54	C₄₇⋊C₈	—	8	4	no	—
55	#8827	—	47	1	no	maximal, Sylow
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	#384	1	1	yes	characteristic
59	C₂²×C₄₇	#499	1	2	yes	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	#427	1	2	yes	—
63	C₄×C₄₇	#430	1	2	yes	—
64	C₄×C₄₇	#430	1	2	yes	—
65	C₄×C₄₇	#495	1	2	yes	—
66	#1837	—	47	2	no	—
67	#1567	—	47	2	no	—
68	#1566	—	47	2	no	—
69	C₂×#322	D₄₇	1	1	yes	characteristic
70	C₂×C₄₇	—	2	2	no	—
71	C₂×C₄₇	#1846	1	1	yes	characteristic
72	C₂×C₄₇	#1849	1	1	yes	characteristic
73	C₂×C₄₇	#1849	1	1	yes	characteristic
74	#423	—	47	4	no	—
75	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
78	#341	—	94	4	no	—
79	#335	—	94	4	no	—
80	#334	—	94	4	no	—
81	#322	Dic₄₇	1	4	yes	—
82	C₄₇	#8827	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₁₆	—	94	4	no	—
84	C₂²×C₈	Dic₉₄	1	2	yes	—
85	C₂²×C₈	Dic₉₄	1	2	yes	—
86	C₂²×C₈	D₁₈₈	1	2	yes	—
87	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
88	C₄⋊C₈	—	94	4	no	—
89	C₄⋊C₈	—	94	4	no	—
90	C₄×C₈	Dic₉₄	1	2	yes	—
91	C₄×C₈	D₁₈₈	1	2	yes	—
92	C₄×C₈	D₁₈₈	1	2	yes	—
93	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
94	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
95	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
96	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
99	M₁₆	—	188	8	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	188	4	no	—
104	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
105	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
106	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
107	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
108	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
109	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
110	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
111	C₂³	#13094024	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
117	C₂×C₄	#13094021	1	1	yes	characteristic
118	C₂×C₄	#13094024	1	1	yes	characteristic
119	C₂×C₄	#13094024	1	1	yes	characteristic
120	C₂×C₄	#13094046	1	4	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	2	4	no	—
123	C₈	—	2	4	no	—
124	C₈	—	376	4	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#15688993	1	1	yes	characteristic
128	C₂²	#15689525	1	2	yes	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	#15689161	1	2	yes	—
132	C₄	#15689172	1	2	yes	—
133	C₄	#15689172	1	2	yes	—
134	C₄	#15689507	1	2	yes	—
135	C₂	—	2	2	no	—
136	C₂	#20243552	1	1	yes	characteristic
137	C₂	#20243563	1	1	yes	characteristic
138	C₂	#20243563	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243526

C₂

yes

maximal

#20242147

C₂

yes

maximal

#20242144

C₂

yes

maximal

C₂×C₄₇×#322

C₂

yes

maximal, characteristic

#15689147

C₂²

yes

—

#15688888

—

#15688876

—

#15688874

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

C₄

yes

—

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

Q₈

yes

—

C₂²×C₈×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#8827

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#384

yes

characteristic

C₂²×C₄₇

#499

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#427

yes

—

C₄×C₄₇

#430

yes

—

C₄×C₄₇

#430

yes

—

C₄×C₄₇

#495

yes

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

Dic₄₇

yes

—

C₄₇

#8827

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

Dic₉₄

yes

—

C₂²×C₈

Dic₉₄

yes

—

C₂²×C₈

D₁₈₈

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

M₁₆

—

188

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

188

—

104

C₂×C₈

C₄⋊Dic₄₇

yes

—

105

C₂×C₈

C₄⋊Dic₄₇

yes

—

106

C₂×C₈

C₄⋊Dic₄₇

yes

—

107

C₄²

C₂×Dic₉₄

yes

characteristic

108

C₄²

C₂×D₁₈₈

yes

characteristic

109

C₄²

C₂×D₁₈₈

yes

characteristic

110

C₄²

C₂²×Dic₄₇

yes

characteristic

111

C₂³

#13094024

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

115

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

116

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

117

C₂×C₄

#13094021

yes

characteristic

118

C₂×C₄

#13094024

yes

characteristic

119

C₂×C₄

#13094024

yes

characteristic

120

C₂×C₄

#13094046

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

376

—

125

C₂²

—

126

C₂²

—

127

C₂²

#15688993

yes

characteristic

128

C₂²

#15689525

yes

—

129

C₄

—

130

C₄

—

131

C₄

#15689161

yes

—

132

C₄

#15689172

yes

—

133

C₄

#15689172

yes

—

134

C₄

#15689507

yes

—

135

C₂

—

136

C₂

#20243552

yes

characteristic

137

C₂

#20243563

yes

characteristic

138

C₂

#20243563

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	#384
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#8827
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#13094021

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#322

C₂

socle

C₂²×C₄₇

#384

prime radical

C₂³×C₄₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
82. C₄₇	#8827	semidirect

H ≅ G/N

type

82. C₄₇

#8827

semidirect

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

subgroup	index
#20243526	2
#20242147	2
#20242144	2
C₂×C₄₇×#322	2
#8827	47

subgroup

index

C₂×C₄₇×#322

Subgroups — #26110096

Conjugacy classes of subgroups

Complements

Maximal subgroups