Group #20242231 — subgroups

Subgroups — #20242231

Order 6016.

166 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 69 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×#10625336	C₂	1	1	yes	maximal, characteristic
2	#15689202	C₂	1	1	yes	maximal, characteristic
3	#15688873	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688850	C₂	1	1	yes	maximal, characteristic
6	#15688844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625336	C₄	1	1	yes	characteristic
9	C₂×#10625324	—	2	1	no	—
10	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#13093955	C₄	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#13093948	C₄	1	1	yes	characteristic
14	#13093944	C₄	1	1	yes	characteristic
15	(C₂×D₄₇)⋊C₈	—	2	1	no	—
16	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₄₇	—	2	1	no	—
20	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₄₇	—	2	1	no	—
22	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₄₇	—	2	1	no	—
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#10625336	—	2	2	no	—
34	(C₂×D₄₇)⋊C₄	—	2	1	no	—
35	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄₇⋊C₈	—	2	1	no	—
37	C₂×C₄₇⋊C₈	—	2	1	no	—
38	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
40	#10625324	—	4	2	no	—
41	C₂²×D₄₇	—	2	1	no	—
42	Q₈×C₄₇	—	2	1	no	—
43	Q₈×C₄₇	M₁₆	1	2	yes	complemented
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
49	C₂×Dic₄₇	—	2	1	no	—
50	D₁₈₈	—	2	1	no	—
51	D₁₈₈	—	2	1	no	—
52	C₄×D₄₇	—	4	1	no	—
53	C₄×D₄₇	—	4	1	no	—
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	—	4	1	no	—
56	C₈×C₄₇	SD₁₆	1	2	yes	complemented
57	C₄₇⋊C₈	—	2	1	no	—
58	C₄₇⋊C₈	—	2	1	no	—
59	C₄₇⋊C₈	—	4	1	no	—
60	C₄₇⋊C₈	—	4	1	no	—
61	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
62	C₂×D₄₇	—	4	1	no	—
63	C₂×D₄₇	—	4	1	no	—
64	C₂×D₄₇	—	4	1	no	—
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	4	1	no	—
68	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
69	C₄×C₄₇	#131	1	1	yes	characteristic
70	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₄₇	#135	1	1	yes	characteristic
72	Dic₄₇	—	4	1	no	—
73	#1583	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	#435	1	1	yes	characteristic
75	C₂×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₄₇	#444	1	1	yes	characteristic
77	D₄₇	—	8	1	no	—
78	C₈×Q₈	D₄₇	1	1	yes	characteristic, complemented
79	C₄×SD₁₆	—	47	1	no	—
80	#436	—	47	1	no	—
81	#334	—	47	1	no	—
82	Q₈⋊C₈	—	47	1	no	—
83	D₄⋊C₈	—	47	1	no	—
84	#322	—	47	1	no	—
85	C₄₇	#1583	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	47	1	no	—
87	C₂×M₁₆	—	94	1	no	—
88	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
89	C₄×D₄	—	47	1	no	—
90	#106	—	47	1	no	—
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	47	1	no	—
93	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
94	#103	—	47	1	no	—
95	#102	—	47	1	no	—
96	C₂²⋊C₈	—	94	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	47	1	no	—
99	C₄×C₈	—	47	1	no	—
100	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
101	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
102	C₂×D₄	—	47	1	no	—
103	C₂²×C₄	—	94	1	no	—
104	SD₁₆	—	94	2	no	—
105	M₁₆	—	188	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	47	1	no	—
108	C₂×C₈	—	47	1	no	—
109	C₂×C₈	—	94	1	no	—
110	C₂×C₈	—	94	1	no	—
111	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	47	1	no	—
115	C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
116	C₂²⋊C₄	—	94	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₄₇	1	1	yes	characteristic
119	C₂³	—	94	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	#10625324	1	2	yes	complemented
122	D₄	—	94	1	no	—
123	D₄	—	94	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	94	1	no	—
127	C₂×C₄	—	188	1	no	—
128	C₂×C₄	—	188	1	no	—
129	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₂×C₄	#10625350	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	94	1	no	—
135	C₈	—	94	1	no	—
136	C₈	—	188	1	no	—
137	C₈	—	188	1	no	—
138	C₈	#10625336	1	2	yes	complemented
139	C₂²	—	188	1	no	—
140	C₂²	—	188	1	no	—
141	C₂²	—	188	1	no	—
142	C₂²	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	4	1	no	—
146	C₄	—	188	1	no	—
147	C₄	C₂×#10625324	1	1	yes	characteristic
148	C₄	#13094030	1	1	yes	characteristic
149	C₄	C₂×#10625336	1	1	yes	characteristic
150	C₄	#13094044	1	1	yes	characteristic
151	C₂	—	376	1	no	—
152	C₂	#15689194	1	1	yes	characteristic
153	C₂	C₄×#10625336	1	1	yes	characteristic
154	C₂	#15689243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10625336

C₂

yes

maximal, characteristic

#15689202

C₂

yes

maximal, characteristic

#15688873

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

#15688850

C₂

yes

maximal, characteristic

#15688844

C₂

yes

maximal, characteristic, complemented

C₈×Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625336

C₄

yes

characteristic

C₂×#10625324

—

C₄×D₁₈₈

C₂²

yes

characteristic

#13093955

C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093948

C₄

yes

characteristic

#13093944

C₄

yes

characteristic

(C₂×D₄₇)⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×D₁₈₈

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625336

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

#10625324

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

M₁₆

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

SD₁₆

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

#131

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

#135

yes

characteristic

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₄×SD₁₆

yes

characteristic

C₂×C₄₇

#444

yes

characteristic

D₄₇

—

C₈×Q₈

D₄₇

yes

characteristic, complemented

C₄×SD₁₆

—

#436

—

#334

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₄₇

#1583

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₄₇

yes

characteristic

101

C₂×Q₈

C₄×D₄₇

yes

characteristic

102

C₂×D₄

—

103

C₂²×C₄

—

104

SD₁₆

—

105

M₁₆

—

188

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

112

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₄×D₄₇

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂²×D₄₇

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

#10625324

yes

complemented

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

188

—

128

C₂×C₄

—

188

—

129

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

130

C₂×C₄

#10625350

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

188

—

137

C₈

—

188

—

138

C₈

#10625336

yes

complemented

139

C₂²

—

188

—

140

C₂²

—

188

—

141

C₂²

—

188

—

142

C₂²

C₄×C₂²⋊D₄₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

188

—

147

C₄

C₂×#10625324

yes

characteristic

148

C₄

#13094030

yes

characteristic

149

C₄

C₂×#10625336

yes

characteristic

150

C₄

#13094044

yes

characteristic

151

C₂

—

376

—

152

C₂

#15689194

yes

characteristic

153

C₂

C₄×#10625336

yes

characteristic

154

C₂

#15689243

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₈×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₄×D₄
hypercenter	C₈×Q₈	D₄₇
hypocenter	C₄₇	#1583
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₈×Q₈×C₄₇

C₂

socle

C₂²×C₄₇

C₄×D₄

hypercenter

C₈×Q₈

D₄₇

hypocenter

C₄₇

#1583

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

N	H ≅ G/N	type
3. #15688873	C₂	semidirect
4. Q₈⋊(C₄₇⋊C₈)	C₂	semidirect
6. #15688844	C₂	semidirect
7. C₈×Q₈×C₄₇	C₂	semidirect
43. Q₈×C₄₇	M₁₆	semidirect
56. C₈×C₄₇	SD₁₆	semidirect
78. C₈×Q₈	D₄₇	semidirect
85. C₄₇	#1583	semidirect
121. Q₈	#10625324	semidirect
138. C₈	#10625336	semidirect

H ≅ G/N

type

3. #15688873

C₂

semidirect

4. Q₈⋊(C₄₇⋊C₈)

C₂

semidirect

6. #15688844

C₂

semidirect

7. C₈×Q₈×C₄₇

C₂

semidirect

43. Q₈×C₄₇

M₁₆

semidirect

56. C₈×C₄₇

SD₁₆

semidirect

78. C₈×Q₈

D₄₇

semidirect

85. C₄₇

#1583

semidirect

121. Q₈

#10625324

semidirect

138. C₈

#10625336

semidirect

Subgroups — #20242231

Conjugacy classes of subgroups

Complements