Group #20253941 — subgroups

Subgroups — #20253941

Order 6016.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13094016	C₂	1	1	yes	maximal, characteristic
2	#15689529	C₂	1	4	yes	maximal, complemented
3	#15689522	C₂	1	4	yes	maximal, complemented
4	#15689166	C₂	1	4	yes	maximal
5	C₄×#10625329	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#533	C₂	1	1	yes	maximal, characteristic
7	C₂×#10625329	C₂²	1	2	yes	—
8	#13094016	—	2	2	no	—
9	#13094016	C₂²	1	2	yes	—
10	#13094016	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₄	—	2	2	no	—
13	#13093957	C₂²	1	8	yes	—
14	#13093955	C₂²	1	8	yes	—
15	#13093935	C₂²	1	2	yes	—
16	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
18	#125×C₄₇	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₄₇	—	2	2	no	—
20	C₄×Q₈×C₄₇	—	2	2	no	—
21	#117×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	—	2	2	no	—
24	C₂²×C₄×C₄₇	D₄	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	—	2	2	no	—
27	C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
30	C₂²⋊C₄×C₄₇	—	2	2	no	—
31	C₄²×C₄₇	—	2	2	no	—
32	C₄²×C₄₇	D₄	1	2	yes	—
33	C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	C₄²×C₄₇	C₂³	1	1	yes	characteristic
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	#10625329	Q₈	1	8	yes	complemented
40	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
41	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₄₇	—	4	4	no	—
43	C₂×C₄×C₄₇	—	2	4	no	—
44	C₂×C₄×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₄₇	—	2	4	no	—
51	C₂×Dic₄₇	—	4	4	no	—
52	C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
53	C₂²×C₄₇	—	2	2	no	—
54	C₂²×C₄₇	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₄₇	—	2	2	no	—
57	C₄×C₄₇	—	2	2	no	—
58	C₄×C₄₇	—	4	4	no	—
59	C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
60	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
61	Dic₄₇	—	4	4	no	—
62	#3166	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	—	2	2	no	—
64	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄₇	#575	1	2	yes	—
66	#533	D₄₇	1	1	yes	characteristic
67	C₂×#125	—	47	1	no	—
68	#501	—	47	4	no	—
69	#499	—	47	4	no	—
70	#428	—	47	4	no	—
71	C₄×M₁₆	—	47	1	no	—
72	C₄₇	#3166	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	—	47	2	no	—
74	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
75	#125	—	47	2	no	—
76	#125	—	47	2	no	—
77	#125	—	94	2	no	—
78	#125	C₂×D₄₇	1	2	yes	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	#117	C₂×D₄₇	1	2	yes	—
82	C₂×C₄⋊C₄	—	47	2	no	—
83	C₂×C₄⋊C₄	—	94	2	no	—
84	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
85	#107	—	47	8	no	—
86	#106	—	47	8	no	—
87	#97	—	47	2	no	—
88	C₄×C₈	—	47	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂²×C₄	—	94	2	no	—
91	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
92	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
93	M₁₆	—	47	8	no	—
94	C₂×C₈	—	47	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	47	4	no	—
98	C₄⋊C₄	—	94	4	no	—
99	C₄⋊C₄	—	94	4	no	—
100	C₄⋊C₄	—	94	2	no	—
101	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₄₇	1	2	yes	—
105	C₄²	C₂²×D₄₇	1	1	yes	characteristic
106	C₄²	C₂²×D₄₇	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
108	Q₈	—	4	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	94	4	no	—
112	C₂×C₄	—	188	4	no	—
113	C₂×C₄	Q₈×D₄₇	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
118	C₈	—	47	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#13094022	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	188	4	no	—
126	C₄	#13094022	1	2	yes	—
127	C₄	C₂×Q₈×D₄₇	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#13094022	1	1	yes	characteristic
130	C₂	#15689921	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094016

C₂

yes

maximal, characteristic

#15689529

C₂

yes

maximal, complemented

#15689522

C₂

yes

maximal, complemented

#15689166

C₂

yes

maximal

C₄×#10625329

C₂

yes

maximal, characteristic

C₄₇×#533

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

—

#13094016

—

#13094016

C₂²

yes

—

#13094016

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

—

#13093957

C₂²

yes

—

#13093955

C₂²

yes

—

#13093935

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

#125×C₄₇

C₂²

yes

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

#10625329

Q₈

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×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

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

C₂×Q₈

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄⋊Q₈

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄⋊Q₈

yes

—

C₄×C₄₇

C₂²×Q₈

yes

—

Dic₄₇

—

#3166

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄₇

#575

yes

—

#533

D₄₇

yes

—

—

—

—

C₄×M₁₆

—

C₄₇

#3166

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

—

—

—

—

C₂×D₄₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

—

—

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂²×D₄₇

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₂²⋊D₄₇

yes

—

105

C₄²

C₂²×D₄₇

yes

characteristic

106

C₄²

C₂²×D₄₇

yes

characteristic

107

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

188

—

113

C₂×C₄

Q₈×D₄₇

yes

—

114

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

115

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

116

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

117

C₂×C₄

C₂³×D₄₇

yes

characteristic

118

C₈

—

119

C₂²

—

120

C₂²

#13094022

yes

—

121

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

188

—

126

C₄

#13094022

yes

—

127

C₄

C₂×Q₈×D₄₇

yes

—

128

C₂

—

129

C₂

C₂×#13094022

yes

characteristic

130

C₂

#15689921

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#533	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#533	D₄₇
hypocenter	C₄₇	#3166
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#533

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #15689529	C₂	semidirect
3. #15689522	C₂	semidirect
39. #10625329	Q₈	semidirect
72. C₄₇	#3166	semidirect

H ≅ G/N

type

2. #15689529

C₂

semidirect

3. #15689522

C₂

semidirect

39. #10625329

Q₈

semidirect

72. C₄₇

#3166

semidirect

Subgroups — #20253941

Conjugacy classes of subgroups

Complements