Group #26147629 — subgroups

Subgroups — #26147629

Order 12032.

1422 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1422 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	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241619	C₂	1	12	yes	maximal
4	C₂×C₄₇×#383	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₇⋊C₄	C₂²	1	4	yes	—
6	C₂²×C₄×Dic₄₇	C₂²	1	6	yes	—
7	#15688991	C₄	1	8	yes	complemented
8	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
9	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
10	C₄²×Dic₄₇	—	2	4	no	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#383	—	2	4	no	—
13	C₂×#95×C₄₇	C₂²	1	6	yes	—
14	C₂³×Dic₄₇	C₂³	1	4	yes	—
15	C₂³×Dic₄₇	C₂³	1	4	yes	—
16	C₂×Dic₄₇⋊C₄	—	2	24	no	—
17	C₂×C₄×Dic₄₇	—	2	12	no	—
18	C₂×C₄×Dic₄₇	—	2	24	no	—
19	C₄₇⋊#95	—	2	24	no	—
20	C₄₇⋊#95	C₂×C₄	1	24	yes	—
21	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
22	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
23	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₄₇	D₄	1	4	yes	—
25	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
26	#95×C₄₇	—	2	12	no	—
27	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	—	2	12	no	—
29	C₂²×C₄×C₄₇	—	2	4	no	—
30	C₂²×C₄×C₄₇	—	2	12	no	—
31	C₂²×C₄×C₄₇	—	2	12	no	—
32	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
33	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
34	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
35	C₂²×Dic₄₇	—	2	4	no	—
36	C₂²×Dic₄₇	—	2	12	no	—
37	C₂²×Dic₄₇	—	2	12	no	—
38	C₂²×Dic₄₇	—	2	12	no	—
39	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
40	C₂²×Dic₄₇	#41	1	24	yes	—
41	C₄²×C₄₇	C₄⋊C₄	1	16	yes	complemented
42	Dic₄₇⋊C₄	—	4	16	no	—
43	C₄×Dic₄₇	—	2	48	no	—
44	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
45	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
46	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
47	C₂³×C₄₇	C₂×#41	1	12	yes	—
48	C₂×C₄×C₄₇	—	2	12	no	—
49	C₂×C₄×C₄₇	—	2	12	no	—
50	C₂×C₄×C₄₇	—	2	12	no	—
51	C₂×C₄×C₄₇	—	4	12	no	—
52	C₂×C₄×C₄₇	—	4	4	no	—
53	C₂×C₄×C₄₇	—	4	12	no	—
54	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
55	C₂×Dic₄₇	—	2	24	no	—
56	C₂×Dic₄₇	—	2	24	no	—
57	C₂×Dic₄₇	—	2	24	no	—
58	C₂×Dic₄₇	—	4	4	no	—
59	C₂×Dic₄₇	—	4	12	no	—
60	C₂×Dic₄₇	#126	1	16	yes	—
61	#14710	—	47	1	no	maximal, Sylow
62	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄₇	C₄×#41	1	6	yes	—
64	C₂²×C₄₇	#525	1	6	yes	—
65	C₂²×C₄₇	C₂×#126	1	4	yes	—
66	C₂²×C₄₇	#529	1	12	yes	—
67	C₂²×C₄₇	#533	1	6	yes	—
68	C₄×C₄₇	—	2	12	no	—
69	C₄×C₄₇	—	8	4	no	—
70	Dic₄₇	—	2	16	no	—
71	Dic₄₇	—	4	8	no	—
72	C₂×#383	—	47	1	no	—
73	C₂×#383	D₄₇	1	1	yes	characteristic
74	C₂×C₄³	—	47	1	no	—
75	#1436	—	47	12	no	—
76	C₂×C₄₇	#2296	1	3	yes	—
77	C₂×C₄₇	C₄×#126	1	4	yes	—
78	C₂×C₄₇	#2852	1	4	yes	—
79	C₂×C₄₇	#2863	1	4	yes	—
80	C₂²×C₄⋊C₄	—	47	4	no	—
81	C₂²×C₄²	—	47	6	no	—
82	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
83	#383	—	2	4	no	—
84	#383	—	47	8	no	—
85	C₂×#95	—	47	6	no	—
86	C₂×#95	—	47	12	no	—
87	C₂×#95	C₂×D₄₇	1	6	yes	—
88	C₄³	—	94	4	no	—
89	C₄₇	#14710	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	—	47	4	no	—
91	C₂³×C₄	—	47	4	no	—
92	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
93	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
94	C₂×C₄⋊C₄	—	94	24	no	—
95	C₂×C₄²	—	94	24	no	—
96	C₂×C₄²	—	94	12	no	—
97	C₂×C₄²	Dic₉₄	1	4	yes	—
98	C₂×C₄²	C₄×D₄₇	1	4	yes	—
99	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
100	#95	—	2	12	no	—
101	#95	—	47	24	no	—
102	#95	—	94	24	no	—
103	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	12	no	—
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	12	no	—
107	C₂²×C₄	—	2	12	no	—
108	C₂²×C₄	—	47	24	no	—
109	C₂²×C₄	—	47	8	no	—
110	C₂²×C₄	—	94	4	no	—
111	C₂²×C₄	—	94	12	no	—
112	C₂²×C₄	—	94	12	no	—
113	C₂²×C₄	—	94	12	no	—
114	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
115	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
116	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
117	C₄⋊C₄	—	188	16	no	—
118	C₄²	—	94	48	no	—
119	C₄²	Dic₄₇⋊C₄	1	16	yes	complemented
120	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
121	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
122	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
123	C₂³	#41×D₄₇	1	12	yes	—
124	C₂×C₄	—	2	12	no	—
125	C₂×C₄	—	2	12	no	—
126	C₂×C₄	—	2	12	no	—
127	C₂×C₄	—	4	12	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	4	12	no	—
130	C₂×C₄	—	47	16	no	—
131	C₂×C₄	—	94	24	no	—
132	C₂×C₄	—	94	24	no	—
133	C₂×C₄	—	94	24	no	—
134	C₂×C₄	—	188	4	no	—
135	C₂×C₄	—	188	12	no	—
136	C₂×C₄	C₂×Dic₄₇⋊C₄	1	12	yes	—
137	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
138	C₂²	#15689591	1	6	yes	—
139	C₂²	#15689630	1	6	yes	—
140	C₂²	#126×D₄₇	1	4	yes	—
141	C₂²	#15689656	1	12	yes	—
142	C₂²	#15689676	1	6	yes	—
143	C₄	—	2	12	no	—
144	C₄	—	8	4	no	—
145	C₄	—	94	16	no	—
146	C₄	—	188	8	no	—
147	C₂	#20245870	1	3	yes	—
148	C₂	#20245936	1	4	yes	—
149	C₂	#20251700	1	4	yes	—
150	C₂	#20251768	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688991

C₂

yes

maximal, characteristic

C₂×C₄²×Dic₄₇

C₂

yes

maximal, characteristic

#20241619

C₂

yes

maximal

C₂×C₄₇×#383

C₂

yes

maximal, characteristic

C₂²×Dic₄₇⋊C₄

C₂²

yes

—

C₂²×C₄×Dic₄₇

C₂²

yes

—

#15688991

C₄

yes

complemented

C₂×C₄₇⋊#95

C₂²

yes

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₄²×Dic₄₇

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₄₇×#383

—

C₂×#95×C₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

—

C₄₇⋊#95

—

C₄₇⋊#95

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

Q₈

yes

—

#95×C₄₇

—

C₂⁴×C₄₇

C₂⁴

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

#41

yes

—

C₄²×C₄₇

C₄⋊C₄

yes

complemented

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂³×C₄₇

C₂²×D₄

yes

characteristic

C₂³×C₄₇

C₂²×Q₈

yes

characteristic

C₂³×C₄₇

C₂×#41

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#126

yes

—

#14710

—

maximal, Sylow

C₂²×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

C₄×#41

yes

—

C₂²×C₄₇

#525

yes

—

C₂²×C₄₇

C₂×#126

yes

—

C₂²×C₄₇

#529

yes

—

C₂²×C₄₇

#533

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

C₂×#383

—

C₂×#383

D₄₇

yes

characteristic

C₂×C₄³

—

#1436

—

C₂×C₄₇

#2296

yes

—

C₂×C₄₇

C₄×#126

yes

—

C₂×C₄₇

#2852

yes

—

C₂×C₄₇

#2863

yes

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

C₂²×C₄²

C₂×D₄₇

yes

characteristic

#383

—

#383

—

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₄₇

yes

—

C₄³

—

C₄₇

#14710

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

Dic₉₄

yes

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₂²⋊D₄₇

yes

—

100

#95

—

101

#95

—

102

#95

—

103

C₂⁴

C₂³×D₄₇

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×Dic₉₄

yes

—

115

C₂²×C₄

C₂×C₄×D₄₇

yes

—

116

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

117

C₄⋊C₄

—

188

—

118

C₄²

—

119

C₄²

Dic₄₇⋊C₄

yes

complemented

120

C₂³

C₂²×C₄×D₄₇

yes

characteristic

121

C₂³

C₂²×C₂²⋊D₄₇

yes

characteristic

122

C₂³

C₂²×Dic₉₄

yes

characteristic

123

C₂³

#41×D₄₇

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

188

—

135

C₂×C₄

—

188

—

136

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

—

137

C₂²

C₂²×Dic₄₇⋊C₄

yes

characteristic

138

C₂²

#15689591

yes

—

139

C₂²

#15689630

yes

—

140

C₂²

#126×D₄₇

yes

—

141

C₂²

#15689656

yes

—

142

C₂²

#15689676

yes

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

188

—

147

C₂

#20245870

yes

—

148

C₂

#20245936

yes

—

149

C₂

#20251700

yes

—

150

C₂

#20251768

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#383	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#383	D₄₇
hypocenter	C₄₇	#14710
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#383

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₇

commutator

C₂³×C₄₇

C₂³×C₄

Fitting

C₂×C₄₇×#383

C₂

socle, prime radical, cosocle

C₂⁴×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ C₂×#383

N	H ≅ G/N	type
7. #15688991	C₄	semidirect
41. C₄²×C₄₇	C₄⋊C₄	semidirect
89. C₄₇	#14710	semidirect
119. C₄²	Dic₄₇⋊C₄	semidirect

H ≅ G/N

type

7. #15688991

C₄

semidirect

41. C₄²×C₄₇

C₄⋊C₄

semidirect

89. C₄₇

#14710

semidirect

119. C₄²

Dic₄₇⋊C₄

semidirect

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

subgroup	index
C₂×#15688991	2
C₂×C₄²×Dic₄₇	2
#20241619	2
C₂×C₄₇×#383	2
#14710	47

subgroup

index

C₂×#15688991

C₂×C₄²×Dic₄₇

#20241619

C₂×C₄₇×#383

#14710

Subgroups — #26147629

Conjugacy classes of subgroups

Complements

Maximal subgroups