Group #20242774 — subgroups

Subgroups — #20242774

Order 6016.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#15689325	C₂	1	2	yes	maximal
2	C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#15688876	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688869	C₂	1	2	yes	maximal, complemented
5	C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625336	—	2	2	no	—
7	C₂×C₈×D₄₇	—	2	1	no	—
8	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	#13093948	—	2	2	no	—
12	(C₂×D₄₇)⋊C₈	—	2	1	no	—
13	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
14	#107×C₄₇	—	2	1	no	—
15	#107×C₄₇	—	2	1	no	—
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₄₇	—	2	2	no	—
18	C₂×D₁₈₈	D₄	1	1	yes	characteristic
19	C₂×C₄×D₄₇	—	2	1	no	—
20	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₄⋊C₄×C₄₇	—	2	2	no	—
24	C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	#10625336	—	4	4	no	—
26	(C₂×D₄₇)⋊C₄	—	2	1	no	—
27	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
28	C₂×C₄₇⋊C₈	—	2	1	no	—
29	C₂×C₄₇⋊C₈	—	2	2	no	—
30	C₈×D₄₇	—	4	2	no	—
31	C₂²×D₄₇	—	2	1	no	—
32	Q₈×C₄₇	—	4	2	no	—
33	Q₈×C₄₇	—	4	2	no	—
34	C₂×C₄×C₄₇	—	4	2	no	—
35	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
38	C₂×Dic₄₇	—	2	1	no	—
39	D₁₈₈	—	2	1	no	—
40	D₁₈₈	D₈	1	2	yes	—
41	C₄×D₄₇	—	4	1	no	—
42	C₄×D₄₇	—	4	1	no	—
43	C₈×C₄₇	—	2	1	no	—
44	C₈×C₄₇	SD₁₆	1	2	yes	complemented
45	C₄₇⋊C₈	—	4	2	no	—
46	C₄₇⋊C₈	—	4	1	no	—
47	C₂²×C₄₇	#121	1	1	yes	characteristic
48	C₂×D₄₇	—	2	2	no	—
49	C₂×D₄₇	—	4	1	no	—
50	C₂×D₄₇	—	4	1	no	—
51	C₄×C₄₇	—	2	1	no	—
52	C₄×C₄₇	—	8	2	no	—
53	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
54	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
55	C₄×C₄₇	#135	1	1	yes	characteristic
56	C₄×C₄₇	#137	1	1	yes	characteristic
57	Dic₄₇	—	4	1	no	—
58	#1673	—	47	1	no	maximal, Sylow
59	C₂×C₄₇	#461	1	1	yes	characteristic
60	C₂×C₄₇	#466	1	1	yes	characteristic
61	C₂×C₄₇	#473	1	1	yes	characteristic
62	D₄₇	—	4	2	no	—
63	#500	D₄₇	1	1	yes	characteristic, complemented
64	#461	—	47	2	no	—
65	C₈×D₄	—	47	1	no	—
66	#335	—	47	1	no	—
67	#332	—	47	2	no	—
68	C₄₇	#1673	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₁₆	—	94	2	no	—
70	C₂²×C₈	—	94	1	no	—
71	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
72	C₄×D₄	—	47	1	no	—
73	#107	—	2	1	no	—
74	#107	—	2	1	no	—
75	C₄⋊C₈	—	47	1	no	—
76	C₄⋊C₈	—	47	2	no	—
77	#103	—	94	2	no	—
78	C₂²⋊C₈	—	94	1	no	—
79	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
80	C₂×Q₈	—	2	2	no	—
81	C₂×D₄	—	47	1	no	—
82	C₂²×C₄	—	94	1	no	—
83	SD₁₆	—	188	4	no	—
84	C₂×C₈	—	94	2	no	—
85	C₂×C₈	—	94	1	no	—
86	C₂×C₈	—	188	2	no	—
87	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	47	1	no	—
92	C₂²⋊C₄	—	94	1	no	—
93	C₄²	C₂²×D₄₇	1	1	yes	characteristic
94	C₂³	—	94	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	4	2	no	—
97	D₄	—	47	2	no	—
98	D₄	—	94	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	94	1	no	—
101	C₂×C₄	—	188	1	no	—
102	C₂×C₄	—	188	1	no	—
103	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
104	C₂×C₄	#10625352	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	188	2	no	—
108	C₈	—	188	1	no	—
109	C₈	#10625336	1	2	yes	complemented
110	C₂²	—	94	2	no	—
111	C₂²	—	188	1	no	—
112	C₂²	—	188	1	no	—
113	C₂²	#13093995	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	8	2	no	—
116	C₄	—	188	1	no	—
117	C₄	D₈×D₄₇	1	1	yes	characteristic
118	C₄	C₂×#10625336	1	1	yes	characteristic
119	C₄	#13094043	1	1	yes	characteristic
120	C₄	#13094055	1	1	yes	characteristic
121	C₂	—	188	2	no	—
122	C₂	#15689325	1	1	yes	characteristic
123	C₂	#15689356	1	1	yes	characteristic
124	C₂	#15689395	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689325

C₂

yes

maximal

C₈×D₁₈₈

C₂

yes

maximal, characteristic

#15688876

C₂

yes

maximal, characteristic, complemented

#15688869

C₂

yes

maximal, complemented

C₄₇×#500

C₂

yes

maximal, characteristic, complemented

C₂×#10625336

—

C₂×C₈×D₄₇

—

C₄×D₁₈₈

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093948

—

(C₂×D₄₇)⋊C₈

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

#107×C₄₇

—

#107×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625336

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₈×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

D₈

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

SD₁₆

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#500

D₄₇

yes

characteristic, complemented

#461

—

C₈×D₄

—

#335

—

#332

—

C₄₇

#1673

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

SD₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

188

—

102

C₂×C₄

—

188

—

103

C₂×C₄

D₄×D₄₇

yes

characteristic

104

C₂×C₄

#10625352

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

106

C₈

—

107

C₈

—

188

—

108

C₈

—

188

—

109

C₈

#10625336

yes

complemented

110

C₂²

—

111

C₂²

—

188

—

112

C₂²

—

188

—

113

C₂²

#13093995

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

188

—

117

C₄

D₈×D₄₇

yes

characteristic

118

C₄

C₂×#10625336

yes

characteristic

119

C₄

#13094043

yes

characteristic

120

C₄

#13094055

yes

characteristic

121

C₂

—

188

—

122

C₂

#15689325

yes

characteristic

123

C₂

#15689356

yes

characteristic

124

C₂

#15689395

yes

characteristic

	subgroup	G/H
center	C₂²	#13093995
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#500	C₂
socle	C₂²×C₄₇	#121
hypercenter	#500	D₄₇
hypocenter	C₄₇	#1673
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#13093995

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#500

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
3. #15688876	C₂	semidirect
4. #15688869	C₂	semidirect
5. C₄₇×#500	C₂	semidirect
44. C₈×C₄₇	SD₁₆	semidirect
63. #500	D₄₇	semidirect
68. C₄₇	#1673	semidirect
109. C₈	#10625336	semidirect

H ≅ G/N

type

3. #15688876

C₂

semidirect

4. #15688869

C₂

semidirect

5. C₄₇×#500

C₂

semidirect

44. C₈×C₄₇

SD₁₆

semidirect

63. #500

D₄₇

semidirect

68. C₄₇

#1673

semidirect

109. C₈

#10625336

semidirect

Subgroups — #20242774

Conjugacy classes of subgroups

Complements