Group #26102804 — subgroups

Subgroups — C₄₇⋊#7014 #26102804

Order 12032.

338 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 159 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	#20243134	C₂	1	2	yes	maximal
2	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	—
4	#15688878	—	2	2	no	—
5	#15688878	—	2	2	no	—
6	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
7	C₄³×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#336	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	2	4	no	—
10	C₄×C₄₇⋊C₈	—	2	4	no	—
11	C₄×C₄₇⋊C₈	C₈	1	8	yes	complemented
12	C₂²×C₈×C₄₇	—	2	2	no	—
13	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	D₄	1	2	yes	—
17	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
18	C₄×C₈×C₄₇	—	2	2	no	—
19	C₄×C₈×C₄₇	C₈	1	4	yes	complemented
20	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	—	2	4	no	—
26	C₂×C₈×C₄₇	—	4	4	no	—
27	C₄²×C₄₇	—	4	1	no	—
28	C₄²×C₄₇	—	4	2	no	—
29	C₄²×C₄₇	—	4	1	no	—
30	C₄²×C₄₇	C₂×C₈	1	4	yes	—
31	C₄²×C₄₇	C₂×C₈	1	2	yes	—
32	C₄²×C₄₇	M₁₆	1	4	yes	—
33	C₄²×C₄₇	M₁₆	1	2	yes	—
34	C₂×C₄₇⋊C₈	—	2	8	no	—
35	C₂×C₄₇⋊C₈	—	4	4	no	—
36	C₂×C₄₇⋊C₈	—	4	4	no	—
37	C₂³×C₄₇	#95	1	1	yes	characteristic
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	1	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#97	1	2	yes	—
48	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
50	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
51	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
53	C₈×C₄₇	—	4	4	no	—
54	C₄₇⋊C₈	—	4	8	no	—
55	#7014	—	47	1	no	maximal, Sylow
56	C₂²×C₄₇	#336	1	2	yes	—
57	C₂²×C₄₇	#336	1	1	yes	characteristic
58	C₂²×C₄₇	H(3,ℤ/4)	1	2	yes	—
59	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
60	C₂²×C₄₇	#338	1	1	yes	characteristic
61	C₄×C₄₇	—	2	4	no	—
62	C₄×C₄₇	—	2	2	no	—
63	C₄×C₄₇	—	2	4	no	—
64	C₄×C₄₇	—	2	2	no	—
65	C₄×C₄₇	C₈²	1	1	yes	characteristic
66	C₄×C₄₇	#322	1	1	yes	characteristic
67	C₄×C₄₇	#322	1	2	yes	—
68	#1744	—	47	2	no	—
69	#1744	D₄₇	1	1	yes	characteristic
70	C₂×C₄₇	#1273	1	1	yes	characteristic
71	C₂×C₄₇	#1274	1	2	yes	—
72	C₂×C₄₇	#1274	1	1	yes	characteristic
73	C₂×C₄₇	#1275	1	2	yes	—
74	C₂×C₄₇	#1275	1	1	yes	characteristic
75	C₂×C₄×C₈	—	47	4	no	—
76	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
77	C₄³	C₂×D₄₇	1	1	yes	characteristic
78	#336	—	2	2	no	—
79	#336	—	94	2	no	—
80	#336	—	94	2	no	—
81	C₄₇	#7014	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	—	2	2	no	—
83	C₂²×C₈	—	94	4	no	—
84	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
85	C₂×C₄²	C₄×D₄₇	1	2	yes	—
86	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
87	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
88	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	47	8	no	—
91	C₄×C₈	—	94	4	no	—
92	C₄×C₈	C₄₇⋊C₈	1	4	yes	complemented
93	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
94	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
95	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
96	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
97	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	4	4	no	—
100	C₂×C₈	—	94	8	no	—
101	C₂×C₈	—	188	4	no	—
102	C₂×C₈	—	188	4	no	—
103	C₄²	—	4	1	no	—
104	C₄²	—	4	1	no	—
105	C₄²	—	4	2	no	—
106	C₄²	C₈×D₄₇	1	4	yes	—
107	C₄²	#10625324	1	4	yes	—
108	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
109	C₄²	#10625329	1	2	yes	—
110	C₂³	C₄₇⋊#95	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
119	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
120	C₂×C₄	#13093934	1	2	yes	—
121	C₂×C₄	#13093935	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
123	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
124	C₂×C₄	Dic₄₇⋊C₈	1	4	yes	—
125	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
126	C₈	—	4	4	no	—
127	C₈	—	188	8	no	—
128	C₂²	#15688877	1	2	yes	—
129	C₂²	#15688878	1	1	yes	characteristic
130	C₂²	#15688879	1	2	yes	—
131	C₂²	#15688880	1	1	yes	characteristic
132	C₂²	C₄₇⋊#338	1	1	yes	characteristic
133	C₄	—	2	4	no	—
134	C₄	—	2	2	no	—
135	C₄	—	2	4	no	—
136	C₄	—	2	2	no	—
137	C₄	C₈×C₄₇⋊C₈	1	1	yes	characteristic
138	C₄	#15688844	1	1	yes	characteristic
139	C₄	#15688845	1	2	yes	—
140	C₂	C₄₇⋊#1273	1	1	yes	characteristic
141	C₂	#20241216	1	2	yes	—
142	C₂	#20241217	1	1	yes	characteristic
143	C₂	#20241218	1	2	yes	—
144	C₂	#20241219	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243134

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₈

C₄

yes

—

#15688878

—

#15688878

—

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₈

yes

complemented

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

Q₈

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₈

yes

complemented

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×C₈

yes

—

C₂×C₄×C₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7014

—

maximal, Sylow

C₂²×C₄₇

#336

yes

—

C₂²×C₄₇

#336

yes

characteristic

C₂²×C₄₇

H(3,ℤ/4)

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₈²

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

#1744

—

#1744

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1274

yes

characteristic

C₂×C₄₇

#1275

yes

—

C₂×C₄₇

#1275

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₄³

C₂×D₄₇

yes

—

—

—

C₄₇

#7014

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

Dic₉₄

yes

characteristic

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

C₂²⋊D₄₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄₇⋊C₈

yes

complemented

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

C₂²×C₄

Dic₄₇⋊C₄

yes

—

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

188

—

102

C₂×C₈

—

188

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₈×D₄₇

yes

—

107

C₄²

#10625324

yes

—

108

C₄²

C₂×C₄₇⋊C₈

yes

—

109

C₄²

#10625329

yes

—

110

C₂³

C₄₇⋊#95

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₈×Dic₄₇

yes

—

119

C₂×C₄

C₄×C₄₇⋊C₈

yes

characteristic

120

C₂×C₄

#13093934

yes

—

121

C₂×C₄

#13093935

yes

characteristic

122

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

—

123

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

124

C₂×C₄

Dic₄₇⋊C₈

yes

—

125

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

126

C₈

—

127

C₈

—

188

—

128

C₂²

#15688877

yes

—

129

C₂²

#15688878

yes

characteristic

130

C₂²

#15688879

yes

—

131

C₂²

#15688880

yes

characteristic

132

C₂²

C₄₇⋊#338

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

C₈×C₄₇⋊C₈

yes

characteristic

138

C₄

#15688844

yes

characteristic

139

C₄

#15688845

yes

—

140

C₂

C₄₇⋊#1273

yes

characteristic

141

C₂

#20241216

yes

—

142

C₂

#20241217

yes

characteristic

143

C₂

#20241218

yes

—

144

C₂

#20241219

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₄₇
commutator	C₄×C₄₇	C₈²
Frattini	C₄³	C₂×D₄₇
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	#95
cosocle	C₄³×C₄₇	C₂²
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#7014
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₄×Dic₄₇

commutator

C₄×C₄₇

C₈²

Frattini

C₄³

C₂×D₄₇

Fitting

C₄₇×#1744

C₂

socle, prime radical

C₂³×C₄₇

#95

cosocle

C₄³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ #1744

N	H ≅ G/N	type
11. C₄×C₄₇⋊C₈	C₈	semidirect
19. C₄×C₈×C₄₇	C₈	semidirect
81. C₄₇	#7014	semidirect
92. C₄×C₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

11. C₄×C₄₇⋊C₈

C₈

semidirect

19. C₄×C₈×C₄₇

C₈

semidirect

81. C₄₇

#7014

semidirect

92. C₄×C₈

C₄₇⋊C₈

semidirect

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

subgroup	index
#20243134	2
C₄₇×#1744	2
#7014	47

subgroup

index

#20243134

C₄₇×#1744

#7014

Subgroups — C₄₇⋊#7014 #26102804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47⋊#7014 #26102804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇⋊#7014 #26102804