Group #20254212 — subgroups

Subgroups — #20254212

Order 6016.

410 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13094003	C₂	1	2	yes	maximal
2	#15689454	C₂	1	4	yes	maximal, complemented
3	#15689381	C₂	1	4	yes	maximal, complemented
4	#15689281	C₂	1	4	yes	maximal, complemented
5	C₄₇×#407	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₂²	1	2	yes	—
7	C₂×Dic₁₈₈	C₂²	1	4	yes	complemented
8	#13094003	—	2	2	no	—
9	#13094003	—	2	2	no	—
10	#13094003	C₂²	1	4	yes	—
11	#13094003	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₄	—	2	2	no	—
14	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
15	#13093956	C₂²	1	4	yes	complemented
16	#13093949	C₂²	1	8	yes	complemented
17	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₄₇	C₂²	1	4	yes	—
20	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	—	4	2	no	—
22	C₂²×C₄×C₄₇	D₄	1	2	yes	—
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
26	C₂²×Dic₄₇	—	2	2	no	—
27	C₂²×Dic₄₇	—	2	2	no	—
28	C₂×Dic₉₄	—	2	2	no	—
29	C₂×Dic₉₄	—	4	4	no	—
30	C₂×Dic₉₄	C₂³	1	4	yes	—
31	M₁₆×C₄₇	—	4	2	no	—
32	C₂×C₈×C₄₇	C₂³	1	4	yes	—
33	C₂²⋊Dic₄₇	—	2	4	no	—
34	C₂²⋊Dic₄₇	—	2	4	no	—
35	C₄⋊Dic₄₇	—	2	2	no	—
36	C₄⋊Dic₄₇	C₂³	1	4	yes	—
37	Dic₄₇⋊C₄	—	2	4	no	—
38	Dic₄₇⋊C₄	—	2	4	no	—
39	Dic₁₈₈	—	4	4	no	—
40	C₂³×C₄₇	—	2	4	no	—
41	C₂³×C₄₇	—	4	1	no	—
42	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂³×C₄₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	—	4	1	no	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₄₇	—	2	4	no	—
52	C₂×Dic₄₇	—	2	4	no	—
53	C₂×Dic₄₇	—	4	4	no	—
54	C₂×Dic₄₇	—	4	4	no	—
55	Dic₉₄	—	2	8	no	—
56	Dic₉₄	—	4	4	no	—
57	C₈×C₄₇	—	2	4	no	—
58	C₂²×C₄₇	—	2	2	no	—
59	C₂²×C₄₇	—	2	2	no	—
60	C₂²×C₄₇	—	2	1	no	—
61	C₂²×C₄₇	—	2	2	no	—
62	C₂²×C₄₇	—	4	4	no	—
63	C₂²×C₄₇	#137	1	4	yes	—
64	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
68	Dic₄₇	—	4	4	no	—
69	Dic₄₇	—	4	4	no	—
70	#3195	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	—	2	4	no	—
72	C₂×C₄₇	—	4	1	no	—
73	C₂×C₄₇	#534	1	1	yes	characteristic
74	C₂×C₄₇	C₂×#137	1	2	yes	—
75	C₂×C₂²⋊Q₈	—	47	2	no	—
76	#483	—	47	4	no	—
77	#470	—	47	4	no	—
78	#451	—	47	4	no	—
79	#407	D₄₇	1	1	yes	characteristic
80	C₄₇	#3195	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	47	2	no	—
82	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
83	C₂×Q₁₆	—	47	4	no	—
84	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
85	C₂²⋊Q₈	—	47	4	no	—
86	C₂²⋊Q₈	—	47	4	no	—
87	C₂²⋊Q₈	—	94	2	no	—
88	C₂²⋊Q₈	—	94	2	no	—
89	C₂×C₄⋊C₄	—	47	2	no	—
90	C₂×C₄⋊C₄	—	94	2	no	—
91	C₂×C₂²⋊C₄	—	94	2	no	—
92	#106	—	47	4	no	—
93	#103	—	47	8	no	—
94	C₂²⋊C₈	C₂×D₄₇	1	4	yes	—
95	C₂⁴	D₁₈₈	1	1	yes	characteristic
96	C₂×Q₈	—	47	4	no	—
97	C₂×Q₈	—	94	2	no	—
98	C₂×Q₈	—	188	4	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	94	2	no	—
101	C₂²×C₄	—	94	2	no	—
102	C₂²×C₄	D₁₈₈	1	2	yes	—
103	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
105	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
106	Q₁₆	—	188	4	no	—
107	M₁₆	—	4	2	no	—
108	C₂×C₈	C₂²×D₄₇	1	4	yes	—
109	C₄⋊C₄	—	47	4	no	—
110	C₄⋊C₄	—	94	4	no	—
111	C₄⋊C₄	—	94	4	no	—
112	C₄⋊C₄	—	94	2	no	—
113	C₂²⋊C₄	—	94	4	no	—
114	C₂²⋊C₄	—	94	4	no	—
115	C₂³	—	2	4	no	—
116	C₂³	—	4	1	no	—
117	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
118	C₂³	C₂×D₁₈₈	1	2	yes	—
119	Q₈	—	94	8	no	—
120	Q₈	—	188	4	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	94	4	no	—
126	C₂×C₄	—	94	4	no	—
127	C₂×C₄	—	188	4	no	—
128	C₂×C₄	—	188	4	no	—
129	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
130	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
131	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	4	no	—
138	C₂²	#13094057	1	4	yes	—
139	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	—	188	4	no	—
143	C₄	—	188	4	no	—
144	C₄	#13094069	1	2	yes	—
145	C₂	—	2	4	no	—
146	C₂	—	4	1	no	—
147	C₂	#15689685	1	1	yes	characteristic
148	C₂	C₂×#13094057	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094003

C₂

yes

maximal

#15689454

C₂

yes

maximal, complemented

#15689381

C₂

yes

maximal, complemented

#15689281

C₂

yes

maximal, complemented

C₄₇×#407

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

C₂×Dic₁₈₈

C₂²

yes

—

—

C₂²

yes

—

#13094003

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×C₂²⋊Dic₄₇

—

#13093956

C₂²

yes

complemented

#13093949

C₂²

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₁₈₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂³×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#137

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

H(5,𝔽₂)

yes

—

Dic₄₇

—

Dic₄₇

—

#3195

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#534

yes

characteristic

C₂×C₄₇

C₂×#137

yes

—

C₂×C₂²⋊Q₈

—

—

—

—

D₄₇

yes

characteristic

C₄₇

#3195

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

#106

—

#103

—

C₂²⋊C₈

C₂×D₄₇

yes

—

C₂⁴

D₁₈₈

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

D₁₈₈

yes

—

103

C₂²×C₄

D₁₈₈

yes

characteristic

104

C₂²×C₄

C₂²×D₄₇

yes

—

105

C₂²×C₄

C₂²×D₄₇

yes

characteristic

106

Q₁₆

—

188

—

107

M₁₆

—

108

C₂×C₈

C₂²×D₄₇

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂×D₁₈₈

yes

characteristic

118

C₂³

C₂×D₁₈₈

yes

—

119

Q₈

—

120

Q₈

—

188

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

188

—

128

C₂×C₄

—

188

—

129

C₂×C₄

C₂×D₁₈₈

yes

characteristic

130

C₂×C₄

C₂×D₁₈₈

yes

—

131

C₂×C₄

C₂³×D₄₇

yes

characteristic

132

C₈

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#13094057

yes

—

139

C₂²

C₂²×D₁₈₈

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

188

—

143

C₄

—

188

—

144

C₄

#13094069

yes

—

145

C₂

—

146

C₂

—

147

C₂

#15689685

yes

characteristic

148

C₂

C₂×#13094057

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#407	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#407	D₄₇
hypocenter	C₄₇	#3195
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂²×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#407

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. #15689454	C₂	semidirect
3. #15689381	C₂	semidirect
4. #15689281	C₂	semidirect
7. C₂×Dic₁₈₈	C₂²	semidirect
15. #13093956	C₂²	semidirect
16. #13093949	C₂²	semidirect
80. C₄₇	#3195	semidirect

H ≅ G/N

type

2. #15689454

C₂

semidirect

3. #15689381

C₂

semidirect

4. #15689281

C₂

semidirect

7. C₂×Dic₁₈₈

C₂²

semidirect

15. #13093956

C₂²

semidirect

16. #13093949

C₂²

semidirect

80. C₄₇

#3195

semidirect

Subgroups — #20254212

Conjugacy classes of subgroups

Complements