Group #20245095 — subgroups

Subgroups — #20245095

Order 6016.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689161	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688927	C₂	1	2	yes	maximal, complemented
4	#15688924	C₂	1	2	yes	maximal, complemented
5	C₂×#135×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#13093983	—	2	1	no	—
7	C₂×C₄₇⋊C₁₆	—	2	2	no	—
8	C₂×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	#13093958	C₂²	1	2	yes	—
10	#13093956	—	2	1	no	—
11	C₂×#41×C₄₇	—	2	1	no	—
12	#135×C₄₇	—	2	2	no	—
13	#135×C₄₇	C₄	1	4	yes	complemented
14	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×SD₁₆×C₄₇	—	2	1	no	—
16	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	#41×C₄₇	—	2	4	no	—
19	#41×C₄₇	—	4	2	no	—
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×D₄×C₄₇	—	2	1	no	—
22	C₂×D₄×C₄₇	—	4	1	no	—
23	C₂²×C₄×C₄₇	—	4	1	no	—
24	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
25	Q₁₆×C₄₇	—	2	1	no	—
26	Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
27	SD₁₆×C₄₇	—	2	4	no	—
28	D₈×C₄₇	—	2	1	no	—
29	D₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	D₄	1	2	yes	—
33	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₄₇	—	4	1	no	—
35	C₄⋊Dic₄₇	—	2	2	no	—
36	C₄×Dic₄₇	—	4	1	no	—
37	C₄₇⋊C₁₆	—	2	2	no	—
38	C₂³×C₄₇	—	4	1	no	—
39	C₂³×C₄₇	D₈	1	1	yes	characteristic
40	Q₈×C₄₇	—	2	2	no	—
41	Q₈×C₄₇	—	4	1	no	—
42	D₄×C₄₇	—	2	2	no	—
43	D₄×C₄₇	—	4	4	no	—
44	D₄×C₄₇	—	4	1	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	4	4	no	—
47	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₄₇	D₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₄₇	—	4	2	no	—
52	C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
53	C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	—	4	2	no	—
57	C₂²×C₄₇	—	8	2	no	—
58	C₂²×C₄₇	#102	1	2	yes	—
59	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
60	C₄×C₄₇	—	4	2	no	—
61	C₄×C₄₇	#102	1	2	yes	—
62	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
64	Dic₄₇	—	8	2	no	—
65	#2138	—	47	1	no	maximal, Sylow
66	C₂×C₄₇	—	2	2	no	—
67	C₂×C₄₇	—	8	2	no	—
68	C₂×C₄₇	C₂×#102	1	1	yes	characteristic
69	C₂×C₄₇	#508	1	2	yes	—
70	C₂×#135	D₄₇	1	1	yes	characteristic
71	C₂²×C₁₆	—	47	1	no	—
72	#427	—	47	1	no	—
73	#358	—	47	2	no	—
74	#357	—	47	2	no	—
75	C₄₇	#2138	1	1	yes	characteristic, complemented, Sylow
76	C₂×#41	—	2	1	no	—
77	#135	—	2	2	no	—
78	#135	Dic₄₇	1	4	yes	complemented
79	C₂×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
80	C₂×SD₁₆	—	2	1	no	—
81	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
83	#117	—	94	1	no	—
84	C₂×C₁₆	—	47	2	no	—
85	C₂×C₁₆	—	94	2	no	—
86	#107	—	47	2	no	—
87	#106	—	94	1	no	—
88	#41	—	2	4	no	—
89	#41	—	4	2	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	4	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
95	Q₁₆	—	2	1	no	—
96	Q₁₆	C₂×Dic₄₇	1	2	yes	—
97	SD₁₆	—	2	4	no	—
98	D₈	—	2	1	no	—
99	D₈	C₂×Dic₄₇	1	2	yes	—
100	C₂×C₈	C₂×Dic₄₇	1	2	yes	—
101	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
103	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
104	C₄⋊C₄	—	94	2	no	—
105	C₂²⋊C₄	—	188	1	no	—
106	C₄²	—	188	1	no	—
107	C₁₆	—	94	2	no	—
108	C₂³	—	4	1	no	—
109	C₂³	#10625334	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	—	4	1	no	—
112	D₄	—	2	2	no	—
113	D₄	—	4	4	no	—
114	D₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	188	2	no	—
118	C₂×C₄	#10625334	1	1	yes	characteristic
119	C₂×C₄	#10625335	1	2	yes	—
120	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
122	C₈	C₂²⋊Dic₄₇	1	2	yes	—
123	C₈	C₂²×Dic₄₇	1	1	yes	characteristic
124	C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂²	—	2	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	#13093945	1	2	yes	—
129	C₂²	C₂×#10625334	1	1	yes	characteristic
130	C₄	—	4	2	no	—
131	C₄	—	376	2	no	—
132	C₄	#13093945	1	2	yes	—
133	C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
134	C₄	C₂×#10625335	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	8	2	no	—
137	C₂	C₂×#13093945	1	1	yes	characteristic
138	C₂	#15689553	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#15689161

C₂

yes

maximal, characteristic, complemented

#15688927

C₂

yes

maximal, complemented

#15688924

C₂

yes

maximal, complemented

C₂×#135×C₄₇

C₂

yes

maximal, characteristic

#13093983

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂²

yes

—

#13093958

C₂²

yes

—

#13093956

—

C₂×#41×C₄₇

—

#135×C₄₇

—

#135×C₄₇

C₄

yes

complemented

C₂×Q₁₆×C₄₇

C₂²

yes

characteristic

C₂×SD₁₆×C₄₇

—

C₂×D₈×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

Q₁₆×C₄₇

—

Q₁₆×C₄₇

C₂×C₄

yes

—

SD₁₆×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

D₈

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

D₈

yes

characteristic

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₈×C₄₇

C₂²⋊C₄

yes

—

C₈×C₄₇

C₂²×C₄

yes

characteristic

C₈×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#102

yes

—

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#102

yes

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

Dic₄₇

—

#2138

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#102

yes

characteristic

C₂×C₄₇

#508

yes

—

C₂×#135

D₄₇

yes

characteristic

C₂²×C₁₆

—

#427

—

#358

—

#357

—

C₄₇

#2138

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

#135

Dic₄₇

yes

complemented

C₂×Q₁₆

C₂×D₄₇

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂×D₄₇

yes

characteristic

C₂²×C₈

C₂×D₄₇

yes

characteristic

#117

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

Q₁₆

—

Q₁₆

C₂×Dic₄₇

yes

—

SD₁₆

—

D₈

—

D₈

C₂×Dic₄₇

yes

—

100

C₂×C₈

C₂×Dic₄₇

yes

—

101

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

102

C₂×C₈

C₂²⋊D₄₇

yes

—

103

C₂×C₈

C₂²×D₄₇

yes

characteristic

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

188

—

106

C₄²

—

188

—

107

C₁₆

—

108

C₂³

—

109

C₂³

#10625334

yes

characteristic

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

188

—

118

C₂×C₄

#10625334

yes

characteristic

119

C₂×C₄

#10625335

yes

—

120

C₂×C₄

C₂²⋊Dic₄₇

yes

—

121

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

122

C₈

C₂²⋊Dic₄₇

yes

—

123

C₈

C₂²×Dic₄₇

yes

characteristic

124

C₈

C₂×C₂²⋊D₄₇

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#13093945

yes

—

129

C₂²

C₂×#10625334

yes

characteristic

130

C₄

—

131

C₄

—

376

—

132

C₄

#13093945

yes

—

133

C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

134

C₄

C₂×#10625335

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

C₂×#13093945

yes

characteristic

138

C₂

#15689553

yes

—

	subgroup	G/H
center	C₂×C₄	#10625334
commutator	C₈×C₄₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₂×#135×C₄₇	C₂
socle	C₂²×C₄₇	C₂×D₈
hypercenter	C₂×#135	D₄₇
hypocenter	C₄₇	#2138
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

subgroup

G/H

center

C₂×C₄

#10625334

commutator

C₈×C₄₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₂×#135×C₄₇

C₂

socle

C₂²×C₄₇

C₂×D₈

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂²×C₄₇⋊C₁₆	C₂	semidirect
2. #15689161	C₂	semidirect
3. #15688927	C₂	semidirect
4. #15688924	C₂	semidirect
13. #135×C₄₇	C₄	semidirect
75. C₄₇	#2138	semidirect
78. #135	Dic₄₇	semidirect

H ≅ G/N

type

1. C₂²×C₄₇⋊C₁₆

C₂

semidirect

2. #15689161

C₂

semidirect

3. #15688927

C₂

semidirect

4. #15688924

C₂

semidirect

13. #135×C₄₇

C₄

semidirect

75. C₄₇

#2138

semidirect

78. #135

Dic₄₇

semidirect

Subgroups — #20245095

Conjugacy classes of subgroups

Complements