Group #20244191 — subgroups

Subgroups — #20244191

Order 6016.

332 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13093948	C₂	1	4	yes	maximal
2	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#15688993	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₄₇	—	2	2	no	—
7	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	#13093948	—	2	8	no	—
9	C₄×C₄₇⋊C₈	—	2	2	no	—
10	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₄₇	—	2	2	no	—
12	C₄⋊Q₈×C₄₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₄₇	—	2	2	no	—
16	C₂×Q₈×C₄₇	—	2	4	no	—
17	C₂×Q₈×C₄₇	—	2	4	no	—
18	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	D₄	1	2	yes	—
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₇	—	2	2	no	—
23	C₄⋊C₄×C₄₇	—	2	8	no	—
24	C₄²×C₄₇	C₂×C₄	1	2	yes	—
25	C₄²×C₄₇	D₄	1	2	yes	—
26	C₄⋊Dic₄₇	—	2	4	no	—
27	C₄⋊Dic₄₇	—	4	4	no	—
28	C₂×C₄₇⋊C₈	—	2	2	no	—
29	C₂×C₄₇⋊C₈	—	2	2	no	—
30	C₂×C₄₇⋊C₈	D₄	1	4	yes	—
31	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₄₇	—	2	8	no	—
33	Q₈×C₄₇	—	4	4	no	—
34	C₂×C₄×C₄₇	—	2	4	no	—
35	C₂×C₄×C₄₇	—	4	2	no	—
36	C₂×C₄×C₄₇	—	4	2	no	—
37	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₄₇	SD₁₆	1	4	yes	—
39	C₂×C₄×C₄₇	Q₁₆	1	4	yes	—
40	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	#41	1	2	yes	—
44	C₂×Dic₄₇	—	4	2	no	—
45	C₂×Dic₄₇	—	4	4	no	—
46	C₄₇⋊C₈	—	2	4	no	—
47	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄₇	#124	1	1	yes	characteristic
49	C₂²×C₄₇	#127	1	1	yes	characteristic
50	C₂²×C₄₇	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₄₇	C₂×Q₁₆	1	2	yes	—
52	C₄×C₄₇	—	4	4	no	—
53	C₄×C₄₇	#103	1	8	yes	complemented
54	C₄×C₄₇	C₄×D₄	1	2	yes	—
55	C₄×C₄₇	#121	1	2	yes	—
56	Dic₄₇	—	8	2	no	—
57	#1956	—	47	1	no	maximal, Sylow
58	C₂×C₄₇	#390	1	1	yes	characteristic
59	C₂×C₄₇	C₂×#103	1	2	yes	—
60	C₂×C₄₇	#487	1	2	yes	—
61	C₂×C₄₇	#492	1	1	yes	characteristic
62	C₂×C₄₇	#494	1	1	yes	characteristic
63	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
64	C₂×#103	—	47	4	no	—
65	C₂×C₄×C₈	—	47	1	no	—
66	#384	—	47	1	no	—
67	C₄₇	#1956	1	1	yes	characteristic, complemented, Sylow
68	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
69	C₂²×C₈	—	47	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	Dic₄₇	1	4	yes	complemented
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	47	2	no	—
74	C₂×C₄⋊C₄	—	94	2	no	—
75	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
76	#103	—	94	8	no	—
77	C₄×C₈	—	94	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	4	no	—
81	C₂×Q₈	C₂×Dic₄₇	1	4	yes	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	94	2	no	—
84	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
86	C₂×C₈	—	47	4	no	—
87	C₂×C₈	—	94	2	no	—
88	C₂×C₈	—	94	2	no	—
89	C₄⋊C₄	—	2	8	no	—
90	C₄⋊C₄	—	94	4	no	—
91	C₄⋊C₄	—	188	4	no	—
92	C₄²	C₂×Dic₄₇	1	2	yes	—
93	C₄²	C₂²⋊D₄₇	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
95	Q₈	—	2	8	no	—
96	Q₈	—	4	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	188	2	no	—
101	C₂×C₄	—	188	4	no	—
102	C₂×C₄	#10625336	1	4	yes	—
103	C₂×C₄	#10625337	1	4	yes	—
104	C₂×C₄	C₂²⋊Dic₄₇	1	4	yes	—
105	C₂×C₄	D₄×D₄₇	1	2	yes	—
106	C₂×C₄	#10625352	1	2	yes	—
107	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
109	C₈	—	94	4	no	—
110	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
111	C₂²	#13094011	1	1	yes	characteristic
112	C₂²	#13094020	1	1	yes	characteristic
113	C₂²	C₂×#10625336	1	2	yes	—
114	C₂²	C₂×#10625337	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	—	376	2	no	—
117	C₄	#13093948	1	8	yes	complemented
118	C₄	D₄×Dic₄₇	1	2	yes	—
119	C₄	#13093995	1	2	yes	—
120	C₂	#15689026	1	1	yes	characteristic
121	C₂	C₂×#13093948	1	2	yes	—
122	C₂	#15689473	1	2	yes	—
123	C₂	#15689497	1	1	yes	characteristic
124	C₂	#15689502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093948

C₂

yes

maximal

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic

#15688993

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

#13093948

—

C₄×C₄₇⋊C₈

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×SD₁₆

yes

—

C₂²×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

#103

yes

complemented

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#487

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊Q₈

D₄₇

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

#384

—

C₄₇

#1956

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄²

C₂×Dic₄₇

yes

—

C₄²

C₂²⋊D₄₇

yes

—

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

188

—

101

C₂×C₄

—

188

—

102

C₂×C₄

#10625336

yes

—

103

C₂×C₄

#10625337

yes

—

104

C₂×C₄

C₂²⋊Dic₄₇

yes

—

105

C₂×C₄

D₄×D₄₇

yes

—

106

C₂×C₄

#10625352

yes

—

107

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

109

C₈

—

110

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

111

C₂²

#13094011

yes

characteristic

112

C₂²

#13094020

yes

characteristic

113

C₂²

C₂×#10625336

yes

—

114

C₂²

C₂×#10625337

yes

—

115

C₄

—

116

C₄

—

376

—

117

C₄

#13093948

yes

complemented

118

C₄

D₄×Dic₄₇

yes

—

119

C₄

#13093995

yes

—

120

C₂

#15689026

yes

characteristic

121

C₂

C₂×#13093948

yes

—

122

C₂

#15689473

yes

—

123

C₂

#15689497

yes

characteristic

124

C₂

#15689502

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#1956
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

#1956

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₄₇	C₄	semidirect
53. C₄×C₄₇	#103	semidirect
67. C₄₇	#1956	semidirect
71. C₄⋊Q₈	Dic₄₇	semidirect
117. C₄	#13093948	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₄₇

C₄

semidirect

53. C₄×C₄₇

#103

semidirect

67. C₄₇

#1956

semidirect

71. C₄⋊Q₈

Dic₄₇

semidirect

117. C₄

#13093948

semidirect

Subgroups — #20244191

Conjugacy classes of subgroups

Complements