Group #26147325 — subgroups

Subgroups — #26147325

Order 12032.

1462 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 411 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×#15688991	C₂	1	2	yes	maximal
2	#20241659	C₂	1	4	yes	maximal
3	#20241656	C₂	1	8	yes	maximal
4	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15688991	—	2	8	no	—
6	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
7	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
8	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
9	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₄₇	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	8	yes	complemented
12	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×Dic₄₇	C₂³	1	8	yes	—
15	C₄₇⋊#95	—	2	16	no	—
16	C₄₇⋊#95	—	2	8	no	—
17	C₄₇⋊#95	—	4	16	no	—
18	C₂²×Q₈×C₄₇	—	2	8	no	—
19	C₂²×Q₈×C₄₇	—	4	8	no	—
20	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
22	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈×C₄₇	—	2	32	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
26	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₇	D₄	1	8	yes	—
29	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
30	C₂×Q₈×C₄₇	—	2	16	no	—
31	C₂×Q₈×C₄₇	—	4	16	no	—
32	C₂×Q₈×C₄₇	—	4	32	no	—
33	C₂²×C₄×C₄₇	—	2	4	no	—
34	C₂²×C₄×C₄₇	—	2	8	no	—
35	C₂²×C₄×C₄₇	—	2	8	no	—
36	C₂²×C₄×C₄₇	—	2	16	no	—
37	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
38	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
42	C₂²×Dic₄₇	—	2	8	no	—
43	C₂²×Dic₄₇	—	2	16	no	—
44	C₂²×Dic₄₇	—	2	32	no	—
45	C₄⋊C₄×C₄₇	—	2	32	no	—
46	C₄²×C₄₇	C₂²⋊C₄	1	16	yes	—
47	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
48	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
49	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
50	C₂³×C₄₇	2₋¹⁺⁴	1	8	yes	—
51	Q₈×C₄₇	—	4	32	no	—
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	—	2	8	no	—
54	C₂×C₄×C₄₇	—	2	8	no	—
55	C₂×C₄×C₄₇	—	2	16	no	—
56	C₂×C₄×C₄₇	—	2	8	no	—
57	C₂×C₄×C₄₇	—	2	16	no	—
58	C₂×C₄×C₄₇	—	4	16	no	—
59	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
60	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
61	C₂×Dic₄₇	—	4	8	no	—
62	C₂×Dic₄₇	—	4	16	no	—
63	C₂×Dic₄₇	—	4	32	no	—
64	#14663	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄₇	#518	1	2	yes	—
67	C₂²×C₄₇	#519	1	4	yes	—
68	C₂²×C₄₇	#535	1	4	yes	—
69	C₂²×C₄₇	#536	1	8	yes	—
70	C₂²×C₄₇	#563	1	16	yes	—
71	C₄×C₄₇	—	2	4	no	—
72	C₄×C₄₇	—	2	8	no	—
73	C₄×C₄₇	—	4	8	no	—
74	Dic₄₇	—	8	8	no	—
75	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
76	C₂×#383	—	47	2	no	—
77	#1446	—	47	4	no	—
78	#1445	—	47	8	no	—
79	C₂×C₄₇	#2309	1	1	yes	characteristic
80	C₂×C₄₇	#2310	1	2	yes	—
81	C₂×C₄₇	#2381	1	4	yes	—
82	C₂×C₄₇	#2821	1	8	yes	—
83	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
84	C₂×C₄⋊Q₈	—	2	8	no	—
85	C₂×C₄⋊Q₈	Dic₄₇	1	8	yes	complemented
86	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
87	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
88	#383	—	94	8	no	—
89	C₂×#95	—	47	8	no	—
90	C₂×#95	—	47	4	no	—
91	C₂×#95	—	47	16	no	—
92	C₄₇	#14663	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₈	—	2	8	no	—
94	C₂²×Q₈	—	4	8	no	—
95	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
96	C₂³×C₄	—	47	8	no	—
97	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
98	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
99	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
100	C₄⋊Q₈	—	2	32	no	—
101	C₂×C₄⋊C₄	—	2	16	no	—
102	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
103	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
104	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	—
105	#95	—	94	16	no	—
106	#95	—	94	8	no	—
107	#95	—	188	16	no	—
108	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
109	C₂×Q₈	—	2	16	no	—
110	C₂×Q₈	—	4	16	no	—
111	C₂×Q₈	—	4	32	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	2	16	no	—
116	C₂²×C₄	—	94	8	no	—
117	C₂²×C₄	—	94	16	no	—
118	C₂²×C₄	—	94	32	no	—
119	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
120	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
121	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
124	C₄⋊C₄	—	2	32	no	—
125	C₄²	C₂²⋊Dic₄₇	1	16	yes	—
126	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
127	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
128	C₂³	#13094069	1	4	yes	—
129	C₂³	#13094072	1	8	yes	—
130	Q₈	—	4	32	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	16	no	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	2	16	no	—
137	C₂×C₄	—	4	16	no	—
138	C₂×C₄	—	188	32	no	—
139	C₂×C₄	—	188	8	no	—
140	C₂×C₄	—	188	16	no	—
141	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
142	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
143	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
144	C₂²	#15689598	1	2	yes	—
145	C₂²	#15689601	1	4	yes	—
146	C₂²	#15689690	1	4	yes	—
147	C₂²	#15689697	1	8	yes	—
148	C₂²	#15689872	1	16	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	8	no	—
151	C₄	—	4	8	no	—
152	C₄	—	376	8	no	—
153	C₂	#20245970	1	1	yes	characteristic
154	C₂	#20245974	1	2	yes	—
155	C₂	#20246569	1	4	yes	—
156	C₂	#20251447	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688991

C₂

yes

maximal

#20241659

C₂

yes

maximal

#20241656

C₂

yes

maximal

C₂²×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

#15688991

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂³×Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₇

—

C₂×C₄⋊Q₈×C₄₇

C₄

yes

complemented

C₂²×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×Dic₄₇

C₂³

yes

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂⁴×C₄₇

C₂⁴

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂³×C₄₇

C₂²×D₄

yes

—

C₂³×C₄₇

H(5,𝔽₂)

yes

—

C₂³×C₄₇

2₋¹⁺⁴

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

#14663

—

maximal, Sylow

C₂²×C₄₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#518

yes

—

C₂²×C₄₇

#519

yes

—

C₂²×C₄₇

#535

yes

—

C₂²×C₄₇

#536

yes

—

C₂²×C₄₇

#563

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

C₂²×C₄⋊Q₈

D₄₇

yes

—

—

—

C₂×C₄₇

#2309

yes

characteristic

C₂×C₄₇

#2310

yes

—

C₂×C₄₇

#2381

yes

—

C₂×C₄₇

#2821

yes

—

C₂³×Q₈

C₂×D₄₇

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

Dic₄₇

yes

complemented

C₂²×C₄⋊C₄

C₂×D₄₇

yes

—

C₂²×C₄²

C₂×D₄₇

yes

characteristic

#383

—

C₂×#95

—

C₂×#95

—

C₂×#95

—

C₄₇

#14663

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×Dic₄₇

yes

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂³×C₄

C₂²×D₄₇

yes

characteristic

100

C₄⋊Q₈

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

103

C₂×C₄²

C₂×Dic₄₇

yes

—

104

C₂×C₄²

C₂²⋊D₄₇

yes

—

105

#95

—

106

#95

—

107

#95

—

188

—

108

C₂⁴

C₂³×D₄₇

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²×Dic₄₇

yes

—

120

C₂²×C₄

C₂²×Dic₄₇

yes

—

121

C₂²×C₄

C₂²×Dic₄₇

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

123

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

124

C₄⋊C₄

—

125

C₄²

C₂²⋊Dic₄₇

yes

—

126

C₂³

C₂³×Dic₄₇

yes

characteristic

127

C₂³

C₂²×C₂²⋊D₄₇

yes

—

128

C₂³

#13094069

yes

—

129

C₂³

#13094072

yes

—

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

188

—

139

C₂×C₄

—

188

—

140

C₂×C₄

—

188

—

141

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

142

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

143

C₂²

C₂²×C₂²⋊Dic₄₇

yes

characteristic

144

C₂²

#15689598

yes

—

145

C₂²

#15689601

yes

—

146

C₂²

#15689690

yes

—

147

C₂²

#15689697

yes

—

148

C₂²

#15689872

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

376

—

153

C₂

#20245970

yes

characteristic

154

C₂

#20245974

yes

—

155

C₂

#20246569

yes

—

156

C₂

#20251447

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#14663
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₇

commutator

C₂³×C₄₇

C₂³×C₄

Fitting

C₂²×C₄⋊Q₈×C₄₇

C₂

socle, prime radical, cosocle

C₂⁴×C₄₇

C₂⁴

hypercenter

C₂²×C₄⋊Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

N	H ≅ G/N	type
11. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
85. C₂×C₄⋊Q₈	Dic₄₇	semidirect
92. C₄₇	#14663	semidirect

H ≅ G/N

type

11. C₂×C₄⋊Q₈×C₄₇

C₄

semidirect

85. C₂×C₄⋊Q₈

Dic₄₇

semidirect

92. C₄₇

#14663

semidirect

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

subgroup	index
C₂×#15688991	2
#20241659	2
#20241656	2
C₂²×C₄⋊Q₈×C₄₇	2
#14663	47

subgroup

index

C₂²×C₄⋊Q₈×C₄₇

Subgroups — #26147325

Conjugacy classes of subgroups

Complements

Maximal subgroups