Group #20246111 — subgroups

Subgroups — #20246111

Order 6016.

740 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 197 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×#13094021	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093986	C₂	1	2	yes	maximal, complemented
3	#15689024	C₂	1	2	yes	maximal, complemented
4	#15689001	C₂	1	8	yes	maximal, complemented
5	#15688969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₈₈	C₂²	1	2	yes	—
8	C₂²×D₁₈₈	C₂²	1	4	yes	—
9	C₂²×C₄×D₄₇	C₂²	1	4	yes	—
10	#13094021	C₄	1	8	yes	complemented
11	#13093986	—	2	8	no	—
12	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
13	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	#95×C₄₇	C₂²	1	4	yes	—
20	C₂³×D₄₇	C₂³	1	4	yes	—
21	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₇	C₂³	1	2	yes	—
25	C₂²×Dic₄₇	C₂³	1	2	yes	—
26	C₂×D₁₈₈	—	2	4	no	—
27	C₂×D₁₈₈	—	2	4	no	—
28	C₂×D₁₈₈	—	4	8	no	—
29	C₂×D₁₈₈	C₂×C₄	1	8	yes	complemented
30	C₂×D₁₈₈	C₂×C₄	1	16	yes	complemented
31	C₂×C₄×D₄₇	—	2	8	no	—
32	C₂×C₄×D₄₇	—	4	8	no	—
33	C₄⋊C₄×C₄₇	—	2	4	no	—
34	C₄²×C₄₇	C₂×C₄	1	4	yes	complemented
35	(C₂×D₄₇)⋊C₄	—	2	8	no	—
36	(C₂×D₄₇)⋊C₄	—	4	8	no	—
37	C₄×Dic₄₇	—	2	2	no	—
38	C₄×Dic₄₇	D₄	1	4	yes	complemented
39	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
40	C₂²×D₄₇	—	2	4	no	—
41	C₂²×D₄₇	—	2	4	no	—
42	C₂²×D₄₇	—	2	8	no	—
43	C₂²×D₄₇	—	2	8	no	—
44	C₂²×D₄₇	C₂²×C₄	1	8	yes	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	—	2	8	no	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₇	#41	1	2	yes	—
52	C₂×Dic₄₇	—	2	4	no	—
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	—	2	4	no	—
55	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
56	D₁₈₈	—	2	16	no	—
57	D₁₈₈	—	4	16	no	—
58	C₄×D₄₇	—	4	16	no	—
59	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
61	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
62	C₂²×C₄₇	H(5,𝔽₂)	1	4	yes	—
63	C₂×D₄₇	—	2	8	no	—
64	C₂×D₄₇	—	2	16	no	—
65	C₂×D₄₇	—	4	4	no	—
66	C₂×D₄₇	—	4	4	no	—
67	C₂×D₄₇	—	4	8	no	—
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	—	4	4	no	—
70	C₄×C₄₇	C₄×D₄	1	4	yes	—
71	Dic₄₇	—	2	4	no	—
72	Dic₄₇	—	4	2	no	—
73	#2328	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
75	C₂×C₄₇	#518	1	2	yes	—
76	C₂×C₄₇	#535	1	2	yes	—
77	C₂×C₄₇	#540	1	2	yes	—
78	D₄₇	—	4	8	no	—
79	C₂×#127	—	47	1	no	—
80	C₂×C₄×D₄	—	47	2	no	—
81	#390	—	47	2	no	—
82	#386	—	47	8	no	—
83	#382	D₄₇	1	1	yes	characteristic, complemented
84	#376	—	47	1	no	—
85	C₄₇	#2328	1	1	yes	characteristic, complemented, Sylow
86	C₂²×D₄	—	47	2	no	—
87	C₂²×D₄	—	47	4	no	—
88	C₂³×C₄	—	47	4	no	—
89	#127	—	47	8	no	—
90	C₄×D₄	—	94	8	no	—
91	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
92	C₂×C₂²⋊C₄	—	47	4	no	—
93	C₂×C₂²⋊C₄	—	47	8	no	—
94	C₂×C₄²	—	47	1	no	—
95	C₂×C₄²	—	47	1	no	—
96	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
97	#95	—	47	4	no	—
98	#95	C₂×D₄₇	1	4	yes	—
99	C₂⁴	—	47	4	no	—
100	C₂×D₄	—	47	8	no	—
101	C₂×D₄	—	47	16	no	—
102	C₂×D₄	—	94	4	no	—
103	C₂×D₄	—	94	4	no	—
104	C₂×D₄	—	188	8	no	—
105	C₂²×C₄	—	47	2	no	—
106	C₂²×C₄	—	47	2	no	—
107	C₂²×C₄	—	94	8	no	—
108	C₂²×C₄	—	188	8	no	—
109	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
110	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
112	C₄⋊C₄	—	2	4	no	—
113	C₂²⋊C₄	—	94	8	no	—
114	C₂²⋊C₄	—	188	8	no	—
115	C₄²	—	47	4	no	—
116	C₄²	—	94	2	no	—
117	C₄²	C₄×D₄₇	1	4	yes	complemented
118	C₂³	—	47	8	no	—
119	C₂³	—	94	4	no	—
120	C₂³	—	94	4	no	—
121	C₂³	—	94	8	no	—
122	C₂³	—	94	8	no	—
123	C₂³	C₂³×D₄₇	1	1	yes	characteristic
124	D₄	—	94	16	no	—
125	D₄	—	188	16	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	8	no	—
129	C₂×C₄	—	47	4	no	—
130	C₂×C₄	—	94	4	no	—
131	C₂×C₄	—	94	2	no	—
132	C₂×C₄	—	94	4	no	—
133	C₂×C₄	—	188	16	no	—
134	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
136	C₂×C₄	D₄×D₄₇	1	2	yes	—
137	C₂×C₄	#10625354	1	2	yes	—
138	C₂²	—	94	8	no	—
139	C₂²	—	94	16	no	—
140	C₂²	—	188	8	no	—
141	C₂²	—	188	4	no	—
142	C₂²	—	188	4	no	—
143	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
144	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
145	C₂²	C₂×#10625354	1	1	yes	characteristic
146	C₂²	#13094070	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	4	4	no	—
149	C₄	—	94	4	no	—
150	C₄	—	188	2	no	—
151	C₄	#13093986	1	4	yes	—
152	C₂	—	188	8	no	—
153	C₂	C₂×#13093986	1	1	yes	characteristic
154	C₂	#15689597	1	2	yes	—
155	C₂	#15689691	1	2	yes	—
156	C₂	#15689729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094021

C₂

yes

maximal, characteristic

C₂×#13093986

C₂

yes

maximal, complemented

#15689024

C₂

yes

maximal, complemented

#15689001

C₂

yes

maximal, complemented

#15688969

C₂

yes

maximal, characteristic, complemented

C₄₇×#382

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

—

C₂²×D₁₈₈

C₂²

yes

—

C₂²×C₄×D₄₇

C₂²

yes

—

#13094021

C₄

yes

complemented

#13093986

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

C₂³×D₄₇

C₂³

yes

—

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×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

complemented

C₂×D₁₈₈

C₂×C₄

yes

complemented

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

complemented

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

complemented

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

Dic₄₇

—

Dic₄₇

—

#2328

—

maximal, Sylow

C₂×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂×C₄₇

#518

yes

—

C₂×C₄₇

#535

yes

—

C₂×C₄₇

#540

yes

—

D₄₇

—

C₂×#127

—

C₂×C₄×D₄

—

#390

—

#386

—

#382

D₄₇

yes

characteristic, complemented

#376

—

C₄₇

#2328

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

#127

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#95

—

#95

C₂×D₄₇

yes

—

C₂⁴

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

188

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

188

—

109

C₂²×C₄

C₂²×D₄₇

yes

—

110

C₂²×C₄

C₂²×D₄₇

yes

—

111

C₂²×C₄

C₂²×D₄₇

yes

characteristic

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

188

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₄×D₄₇

yes

complemented

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

C₂³×D₄₇

yes

characteristic

124

D₄

—

125

D₄

—

188

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

188

—

134

C₂×C₄

C₂×C₄×D₄₇

yes

—

135

C₂×C₄

C₂×C₄×D₄₇

yes

—

136

C₂×C₄

D₄×D₄₇

yes

—

137

C₂×C₄

#10625354

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

188

—

141

C₂²

—

188

—

142

C₂²

—

188

—

143

C₂²

C₂²×C₄×D₄₇

yes

characteristic

144

C₂²

C₂×D₄×D₄₇

yes

characteristic

145

C₂²

C₂×#10625354

yes

characteristic

146

C₂²

#13094070

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

188

—

151

C₄

#13093986

yes

—

152

C₂

—

188

—

153

C₂

C₂×#13093986

yes

characteristic

154

C₂

#15689597

yes

—

155

C₂

#15689691

yes

—

156

C₂

#15689729

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#382	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	#382	D₄₇
hypocenter	C₄₇	#2328
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#382

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
2. C₂×#13093986	C₂	semidirect
3. #15689024	C₂	semidirect
4. #15689001	C₂	semidirect
5. #15688969	C₂	semidirect
6. C₄₇×#382	C₂	semidirect
10. #13094021	C₄	semidirect
29. C₂×D₁₈₈	C₂×C₄	semidirect
30. C₂×D₁₈₈	C₂×C₄	semidirect
34. C₄²×C₄₇	C₂×C₄	semidirect
38. C₄×Dic₄₇	D₄	semidirect
83. #382	D₄₇	semidirect
85. C₄₇	#2328	semidirect
117. C₄²	C₄×D₄₇	semidirect

H ≅ G/N

type

2. C₂×#13093986

C₂

semidirect

3. #15689024

C₂

semidirect

4. #15689001

C₂

semidirect

5. #15688969

C₂

semidirect

6. C₄₇×#382

C₂

semidirect

10. #13094021

C₄

semidirect

29. C₂×D₁₈₈

C₂×C₄

semidirect

30. C₂×D₁₈₈

C₂×C₄

semidirect

34. C₄²×C₄₇

C₂×C₄

semidirect

38. C₄×Dic₄₇

D₄

semidirect

83. #382

D₄₇

semidirect

85. C₄₇

#2328

semidirect

117. C₄²

C₄×D₄₇

semidirect

Subgroups — #20246111

Conjugacy classes of subgroups

Complements