Group #26242832 — subgroups

Subgroups — #26242832

Order 12032.

912 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 215 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	C₄×#13094021	C₂	1	1	yes	maximal, characteristic
2	#20243684	C₂	1	12	yes	maximal, complemented
3	#20243531	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#10625324	C₂²	1	4	yes	—
6	C₂×#13094021	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₈₈	C₂²	1	6	yes	—
8	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	12	yes	—
9	#15689025	C₄	1	6	yes	—
10	#15688993	C₄	1	1	yes	characteristic
11	#15688878	C₂²	1	6	yes	—
12	C₄³×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#336	C₂²	1	6	yes	—
14	C₂²×D₁₈₈	C₂×C₄	1	6	yes	—
15	C₂²×C₄×D₄₇	C₂³	1	4	yes	—
16	C₂×#10625324	—	4	12	no	—
17	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
18	#13094021	—	2	4	no	—
19	C₄×D₁₈₈	—	4	6	no	—
20	C₄×D₁₈₈	—	4	6	no	—
21	C₂×C₄⋊Dic₄₇	C₂×C₄	1	6	yes	—
22	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	12	yes	—
23	(C₂×D₄₇)⋊C₈	—	4	12	no	—
24	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
25	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
28	C₂³×D₄₇	C₂²×C₄	1	4	yes	—
29	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
30	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
31	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
32	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
33	C₂×D₁₈₈	—	2	12	no	—
34	C₂×D₁₈₈	—	4	6	no	—
35	C₂×D₁₈₈	—	4	6	no	—
36	C₂×C₄×D₄₇	—	2	12	no	—
37	C₂×C₄×D₄₇	—	2	12	no	—
38	C₂×C₈×C₄₇	—	2	12	no	—
39	C₄²×C₄₇	—	2	6	no	—
40	C₄²×C₄₇	—	4	3	no	—
41	C₄²×C₄₇	M₁₆	1	4	yes	complemented
42	(C₂×D₄₇)⋊C₄	—	4	12	no	—
43	C₄⋊Dic₄₇	—	4	6	no	—
44	C₂×C₄₇⋊C₈	—	2	12	no	—
45	#10625324	—	4	16	no	—
46	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
47	C₂²×D₄₇	—	2	12	no	—
48	C₂²×D₄₇	—	2	4	no	—
49	C₂²×D₄₇	—	2	12	no	—
50	C₂×C₄×C₄₇	—	2	3	no	—
51	C₂×C₄×C₄₇	—	2	3	no	—
52	C₂×C₄×C₄₇	—	2	3	no	—
53	C₂×C₄×C₄₇	—	2	6	no	—
54	C₂×C₄×C₄₇	C₂×M₁₆	1	6	yes	—
55	C₂×C₄×C₄₇	H(5,𝔽₂)	1	6	yes	—
56	C₂×Dic₄₇	—	2	12	no	—
57	D₁₈₈	—	4	24	no	—
58	C₄×D₄₇	—	4	4	no	—
59	C₄×D₄₇	—	4	12	no	—
60	C₈×C₄₇	—	4	4	no	—
61	C₄₇⋊C₈	—	4	4	no	—
62	#23357	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	#518	1	3	yes	—
64	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
65	C₂²×C₄₇	#568	1	3	yes	—
66	C₂×D₄₇	—	4	4	no	—
67	C₂×D₄₇	—	4	12	no	—
68	C₂×D₄₇	—	4	12	no	—
69	C₄×C₄₇	—	2	6	no	—
70	C₄×C₄₇	—	4	3	no	—
71	C₄×C₄₇	#560	1	4	yes	—
72	Dic₄₇	—	4	4	no	—
73	C₄×#127	—	47	1	no	—
74	#1870	—	47	12	no	—
75	#1839	—	47	1	no	—
76	#1839	D₄₇	1	1	yes	characteristic, complemented
77	C₂×C₄₇	#2380	1	1	yes	characteristic
78	C₂×C₄₇	#2975	1	3	yes	—
79	C₂×C₄₇	#2980	1	3	yes	—
80	D₄₇	—	8	4	no	—
81	C₂²×M₁₆	—	47	4	no	—
82	C₂×#127	—	47	1	no	—
83	C₂×C₄×D₄	—	47	6	no	—
84	C₂×C₂²⋊C₈	—	47	12	no	—
85	#390	—	47	6	no	—
86	#384	—	47	1	no	—
87	C₄³	C₂×D₄₇	1	1	yes	characteristic
88	#336	—	47	6	no	—
89	#336	C₂×D₄₇	1	6	yes	—
90	C₄₇	#23357	1	1	yes	characteristic, complemented, Sylow
91	C₂²×D₄	—	47	6	no	—
92	C₂³×C₄	—	47	4	no	—
93	C₂×M₁₆	—	188	12	no	—
94	C₂²×C₈	—	47	4	no	—
95	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
96	#127	—	94	4	no	—
97	C₄×D₄	—	188	6	no	—
98	C₄×D₄	—	188	6	no	—
99	C₂×C₄⋊C₄	—	47	6	no	—
100	C₂×C₂²⋊C₄	—	47	12	no	—
101	C₂×C₄²	C₄×D₄₇	1	3	yes	—
102	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
104	C₂²⋊C₈	—	188	12	no	—
105	C₂⁴	—	47	4	no	—
106	C₂×D₄	—	94	12	no	—
107	C₂×D₄	—	188	6	no	—
108	C₂×D₄	—	188	6	no	—
109	C₂²×C₄	—	47	4	no	—
110	C₂²×C₄	—	94	12	no	—
111	C₂²×C₄	—	94	12	no	—
112	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
113	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
114	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
115	M₁₆	—	188	16	no	—
116	C₂×C₈	—	2	12	no	—
117	C₂×C₈	—	94	12	no	—
118	C₄⋊C₄	—	188	6	no	—
119	C₂²⋊C₄	—	188	12	no	—
120	C₄²	—	2	6	no	—
121	C₄²	—	4	3	no	—
122	C₄²	#10625324	1	4	yes	complemented
123	C₂³	—	94	12	no	—
124	C₂³	—	94	4	no	—
125	C₂³	—	94	12	no	—
126	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
127	D₄	—	188	24	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	3	no	—
131	C₂×C₄	—	2	6	no	—
132	C₂×C₄	—	94	12	no	—
133	C₂×C₄	—	188	4	no	—
134	C₂×C₄	—	188	12	no	—
135	C₂×C₄	C₂×#10625324	1	6	yes	—
136	C₂×C₄	#13094070	1	6	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	188	4	no	—
139	C₂²	—	188	12	no	—
140	C₂²	—	188	4	no	—
141	C₂²	—	188	12	no	—
142	C₂²	#15689597	1	3	yes	—
143	C₂²	C₂²×#10625324	1	1	yes	characteristic
144	C₂²	#15689889	1	3	yes	—
145	C₄	—	2	6	no	—
146	C₄	—	4	3	no	—
147	C₄	—	188	4	no	—
148	C₄	#15689864	1	4	yes	—
149	C₂	—	376	4	no	—
150	C₂	#20246564	1	1	yes	characteristic
151	C₂	#20252428	1	3	yes	—
152	C₂	#20252458	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13094021

C₂

yes

maximal, characteristic

#20243684

C₂

yes

maximal, complemented

#20243531

C₂

yes

maximal, characteristic, complemented

C₄₇×#1839

C₂

yes

maximal, characteristic, complemented

C₂²×#10625324

C₂²

yes

—

C₂×#13094021

C₄

yes

characteristic

C₂×C₄×D₁₈₈

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₈

C₂²

yes

—

#15689025

C₄

yes

—

#15688993

C₄

yes

characteristic

#15688878

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₂²

yes

—

C₂²×D₁₈₈

C₂×C₄

yes

—

C₂²×C₄×D₄₇

C₂³

yes

—

C₂×#10625324

—

C₂²×C₄₇⋊C₈

C₂³

yes

—

#13094021

—

C₄×D₁₈₈

—

C₄×D₁₈₈

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

(C₂×D₄₇)⋊C₈

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

—

C₂³×D₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

C₂²×C₄

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

M₁₆

yes

complemented

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

H(5,𝔽₂)

yes

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#23357

—

maximal, Sylow

C₂²×C₄₇

#518

yes

—

C₂²×C₄₇

C₂²×M₁₆

yes

characteristic

C₂²×C₄₇

#568

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#560

yes

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×C₄₇

#2380

yes

characteristic

C₂×C₄₇

#2975

yes

—

C₂×C₄₇

#2980

yes

—

D₄₇

—

C₂²×M₁₆

—

C₂×#127

—

C₂×C₄×D₄

—

C₂×C₂²⋊C₈

—

#390

—

#384

—

C₄³

C₂×D₄₇

yes

characteristic

#336

—

#336

C₂×D₄₇

yes

—

C₄₇

#23357

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂×M₁₆

—

188

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₇

yes

—

#127

—

C₄×D₄

—

188

—

C₄×D₄

—

188

—

C₂×C₄⋊C₄

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₄²

C₄×D₄₇

yes

—

102

C₂×C₄²

C₄×D₄₇

yes

characteristic

103

C₂×C₄²

C₂²×D₄₇

yes

—

104

C₂²⋊C₈

—

188

—

105

C₂⁴

—

106

C₂×D₄

—

107

C₂×D₄

—

188

—

108

C₂×D₄

—

188

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×C₄×D₄₇

yes

—

113

C₂²×C₄

C₂×C₄×D₄₇

yes

—

114

C₂²×C₄

C₂³×D₄₇

yes

characteristic

115

M₁₆

—

188

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

188

—

119

C₂²⋊C₄

—

188

—

120

C₄²

—

121

C₄²

—

122

C₄²

#10625324

yes

complemented

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

C₂²×C₄×D₄₇

yes

characteristic

127

D₄

—

188

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

188

—

134

C₂×C₄

—

188

—

135

C₂×C₄

C₂×#10625324

yes

—

136

C₂×C₄

#13094070

yes

—

137

C₈

—

138

C₈

—

188

—

139

C₂²

—

188

—

140

C₂²

—

188

—

141

C₂²

—

188

—

142

C₂²

#15689597

yes

—

143

C₂²

C₂²×#10625324

yes

characteristic

144

C₂²

#15689889

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

188

—

148

C₄

#15689864

yes

—

149

C₂

—

376

—

150

C₂

#20246564

yes

characteristic

151

C₂

#20252428

yes

—

152

C₂

#20252458

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1839	C₂
prime radical	C₂×#13094021	C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1839	D₄₇
hypocenter	C₄₇	#23357
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂³×C₄

Fitting

C₄₇×#1839

C₂

prime radical

C₂×#13094021

C₄

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ #1839

N	H ≅ G/N	type
2. #20243684	C₂	semidirect
3. #20243531	C₂	semidirect
4. C₄₇×#1839	C₂	semidirect
41. C₄²×C₄₇	M₁₆	semidirect
76. #1839	D₄₇	semidirect
90. C₄₇	#23357	semidirect
122. C₄²	#10625324	semidirect

H ≅ G/N

type

2. #20243684

C₂

semidirect

3. #20243531

C₂

semidirect

4. C₄₇×#1839

C₂

semidirect

41. C₄²×C₄₇

M₁₆

semidirect

76. #1839

D₄₇

semidirect

90. C₄₇

#23357

semidirect

122. C₄²

#10625324

semidirect

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

subgroup	index
C₄×#13094021	2
#20243684	2
#20243531	2
C₄₇×#1839	2
#23357	47

subgroup

index

C₄₇×#1839

Subgroups — #26242832

Conjugacy classes of subgroups

Complements

Maximal subgroups