Group #20249500 — subgroups

Subgroups — #20249500

Order 6016.

612 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13094022	C₂	1	2	yes	maximal, complemented
2	C₂×#13094013	C₂	1	2	yes	maximal
3	C₂×#13094012	C₂	1	1	yes	maximal, characteristic
4	#15689058	C₂	1	8	yes	maximal, complemented
5	#15688969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
8	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
9	#13094022	—	2	8	no	—
10	#13094013	—	2	8	no	—
11	#13094012	—	2	4	no	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	8	yes	—
13	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
14	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
18	C₄⋊Q₈×C₄₇	—	2	4	no	—
19	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂³×D₄₇	C₂³	1	2	yes	—
22	C₂×Q₈×C₄₇	—	2	4	no	—
23	C₂×Q₈×C₄₇	—	2	4	no	—
24	C₂×Q₈×C₄₇	—	2	4	no	—
25	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₄₇	C₂³	1	2	yes	—
29	C₂²×Dic₄₇	C₂³	1	4	yes	—
30	C₂×D₁₈₈	—	2	2	no	—
31	C₂×D₁₈₈	—	2	4	no	—
32	C₂×Dic₉₄	—	2	2	no	—
33	C₂×Dic₉₄	—	2	4	no	—
34	C₄⋊C₄×C₄₇	—	2	8	no	—
35	C₄²×C₄₇	D₄	1	4	yes	—
36	(C₂×D₄₇)⋊C₄	—	2	8	no	—
37	(C₂×D₄₇)⋊C₄	—	2	16	no	—
38	Dic₄₇⋊C₄	—	2	16	no	—
39	C₄×Dic₄₇	D₄	1	8	yes	—
40	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
41	C₂²×D₄₇	—	2	4	no	—
42	C₂²×D₄₇	—	2	2	no	—
43	C₂²×D₄₇	—	2	8	no	—
44	Q₈×C₄₇	—	4	8	no	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×Dic₄₇	—	2	8	no	—
53	C₂×Dic₄₇	—	2	4	no	—
54	C₂×Dic₄₇	—	2	2	no	—
55	C₂×Dic₄₇	C₂×D₄	1	8	yes	—
56	D₁₈₈	—	4	4	no	—
57	Dic₉₄	—	4	4	no	—
58	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
59	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
60	C₂²×C₄₇	2₋¹⁺⁴	1	4	yes	—
61	C₂×D₄₇	—	4	2	no	—
62	C₂×D₄₇	—	4	4	no	—
63	C₂×D₄₇	—	4	8	no	—
64	C₄×C₄₇	—	2	4	no	—
65	C₄×C₄₇	—	4	4	no	—
66	C₄×C₄₇	#127	1	4	yes	—
67	Dic₄₇	—	2	8	no	—
68	Dic₄₇	—	4	2	no	—
69	#2658	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	C₂×#127	1	1	yes	characteristic
71	C₂×C₄₇	#536	1	2	yes	—
72	C₂×C₄₇	#536	1	4	yes	—
73	D₄₇	—	8	2	no	—
74	C₂×C₄⋊Q₈	—	47	2	no	—
75	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
76	C₂×#124	—	47	2	no	—
77	C₂×#124	—	47	1	no	—
78	#399	—	47	8	no	—
79	#376	—	47	1	no	—
80	C₄₇	#2658	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	47	1	no	—
82	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
83	C₂²×D₄	—	47	1	no	—
84	C₄⋊Q₈	—	2	4	no	—
85	C₄⋊Q₈	—	94	8	no	—
86	#124	—	94	8	no	—
87	#124	—	94	4	no	—
88	C₂×C₄⋊C₄	—	47	8	no	—
89	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
90	C₂×C₂²⋊C₄	—	47	8	no	—
91	C₂×C₂²⋊C₄	—	47	4	no	—
92	C₂×C₄²	—	47	2	no	—
93	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
94	#95	—	47	4	no	—
95	C₂⁴	—	47	2	no	—
96	C₂×Q₈	—	2	4	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	94	4	no	—
100	C₂×Q₈	—	94	2	no	—
101	C₂×D₄	—	94	4	no	—
102	C₂×D₄	—	94	2	no	—
103	C₂²×C₄	—	47	4	no	—
104	C₂²×C₄	—	47	2	no	—
105	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
106	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
107	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
108	C₄⋊C₄	—	2	8	no	—
109	C₄⋊C₄	—	94	16	no	—
110	C₂²⋊C₄	—	94	16	no	—
111	C₂²⋊C₄	—	94	8	no	—
112	C₄²	—	47	8	no	—
113	C₄²	C₂²⋊D₄₇	1	4	yes	—
114	C₂³	—	94	4	no	—
115	C₂³	—	94	2	no	—
116	C₂³	—	94	8	no	—
117	C₂³	C₂³×D₄₇	1	1	yes	characteristic
118	Q₈	—	4	8	no	—
119	Q₈	—	188	4	no	—
120	D₄	—	188	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	47	8	no	—
126	C₂×C₄	—	94	8	no	—
127	C₂×C₄	—	94	4	no	—
128	C₂×C₄	—	94	2	no	—
129	C₂×C₄	D₄×D₄₇	1	4	yes	—
130	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
131	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
132	C₂²	—	188	2	no	—
133	C₂²	—	188	4	no	—
134	C₂²	—	188	8	no	—
135	C₂²	C₂×D₄×D₄₇	1	2	yes	—
136	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
137	C₂²	#13094072	1	4	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	4	4	no	—
140	C₄	—	94	8	no	—
141	C₄	—	188	2	no	—
142	C₄	#13094020	1	4	yes	—
143	C₂	—	376	2	no	—
144	C₂	C₂×#13094020	1	1	yes	characteristic
145	C₂	#15689697	1	2	yes	—
146	C₂	#15689698	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094022

C₂

yes

maximal, complemented

C₂×#13094013

C₂

yes

maximal

C₂×#13094012

C₂

yes

maximal, characteristic

#15689058

C₂

yes

maximal, complemented

#15688969

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂²

yes

—

—

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

D₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#127

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#536

yes

—

D₄₇

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic, complemented

—

—

—

—

C₄₇

#2658

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₄₇

yes

—

C₂²×D₄

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#124

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#95

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₄₇

yes

—

106

C₂²×C₄

C₂²×D₄₇

yes

—

107

C₂²×C₄

C₂²×D₄₇

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

C₂²⋊D₄₇

yes

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂³×D₄₇

yes

characteristic

118

Q₈

—

119

Q₈

—

188

—

120

D₄

—

188

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₄×D₄₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

131

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

132

C₂²

—

188

—

133

C₂²

—

188

—

134

C₂²

—

188

—

135

C₂²

C₂×D₄×D₄₇

yes

—

136

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

137

C₂²

#13094072

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

188

—

142

C₄

#13094020

yes

—

143

C₂

—

376

—

144

C₂

C₂×#13094020

yes

characteristic

145

C₂

#15689697

yes

—

146

C₂

#15689698

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#2658
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×#13094022	C₂	semidirect
4. #15689058	C₂	semidirect
5. #15688969	C₂	semidirect
6. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
75. C₂×C₄⋊Q₈	D₄₇	semidirect
80. C₄₇	#2658	semidirect

H ≅ G/N

type

1. C₂×#13094022

C₂

semidirect

4. #15689058

C₂

semidirect

5. #15688969

C₂

semidirect

6. C₂×C₄⋊Q₈×C₄₇

C₂

semidirect

75. C₂×C₄⋊Q₈

D₄₇

semidirect

80. C₄₇

#2658

semidirect

Subgroups — #20249500

Conjugacy classes of subgroups

Complements