Group #20242531 — subgroups

Subgroups — #20242531

Order 6016.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#15689839	C₂	1	1	yes	maximal, characteristic
2	#15689410	C₂	1	2	yes	maximal, complemented
3	#15688872	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688852	C₂	1	2	yes	maximal, complemented
5	C₄₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	#13094013	—	2	2	no	—
7	#13093999	—	2	2	no	—
8	C₄×D₁₈₈	C₂²	1	2	yes	—
9	#13093982	—	2	2	no	—
10	#13093955	—	2	2	no	—
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	#13093944	—	2	2	no	—
13	C₂×Q₁₆×C₄₇	—	2	2	no	—
14	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₄₇	—	2	1	no	—
18	C₂×Q₈×C₄₇	D₄	1	2	yes	—
19	C₂×D₁₈₈	—	2	2	no	—
20	C₂×D₁₈₈	D₄	1	2	yes	—
21	C₂×C₄×D₄₇	—	2	2	no	—
22	Q₁₆×C₄₇	—	4	4	no	—
23	C₂×C₈×C₄₇	—	2	1	no	—
24	C₄⋊C₄×C₄₇	—	2	2	no	—
25	C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	(C₂×D₄₇)⋊C₄	—	2	4	no	—
28	(C₂×D₄₇)⋊C₄	—	2	2	no	—
29	C₄⋊Dic₄₇	D₄	1	2	yes	—
30	C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×D₄₇	—	2	2	no	—
33	Q₈×C₄₇	—	4	2	no	—
34	Q₈×C₄₇	—	4	2	no	—
35	C₂×C₄×C₄₇	—	2	2	no	—
36	C₂×C₄×C₄₇	—	4	1	no	—
37	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₄₇	—	2	2	no	—
40	D₁₈₈	—	2	2	no	—
41	D₁₈₈	—	2	2	no	—
42	D₁₈₈	—	8	2	no	—
43	C₄×D₄₇	—	4	2	no	—
44	C₄×D₄₇	—	4	2	no	—
45	C₈×C₄₇	—	2	2	no	—
46	C₄₇⋊C₈	—	4	2	no	—
47	C₂²×C₄₇	#120	1	1	yes	characteristic
48	C₂×D₄₇	—	4	2	no	—
49	C₂×D₄₇	—	4	2	no	—
50	C₂×D₄₇	—	4	2	no	—
51	C₄×C₄₇	—	2	1	no	—
52	C₄×C₄₇	—	4	2	no	—
53	C₄×C₄₇	—	8	1	no	—
54	C₄×C₄₇	#135	1	2	yes	—
55	C₄×C₄₇	#137	1	2	yes	—
56	Dic₄₇	—	4	2	no	—
57	#1634	—	47	1	no	maximal, Sylow
58	C₂×C₄₇	#452	1	2	yes	—
59	C₂×C₄₇	#453	1	1	yes	characteristic
60	D₄₇	—	8	2	no	—
61	#555	—	47	1	no	—
62	#494	D₄₇	1	1	yes	characteristic, complemented
63	#476	—	47	2	no	—
64	#333	—	47	1	no	—
65	D₄⋊C₈	—	47	2	no	—
66	C₄₇	#1634	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
70	#124	—	94	2	no	—
71	#121	—	94	2	no	—
72	C₄×D₄	—	47	2	no	—
73	#117	—	94	2	no	—
74	#106	—	94	2	no	—
75	C₄⋊C₈	—	47	2	no	—
76	#102	—	94	2	no	—
77	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₄₇	1	2	yes	—
80	C₂×D₄	—	47	2	no	—
81	C₂×D₄	—	94	2	no	—
82	C₂²×C₄	—	94	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	94	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	47	2	no	—
89	C₂²⋊C₄	—	94	4	no	—
90	C₂²⋊C₄	—	94	2	no	—
91	C₄²	—	94	2	no	—
92	C₄²	C₂²×D₄₇	1	1	yes	characteristic
93	C₂³	—	94	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	94	2	no	—
97	D₄	—	94	2	no	—
98	D₄	—	376	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	94	2	no	—
102	C₂×C₄	—	188	2	no	—
103	C₂×C₄	—	188	2	no	—
104	C₂×C₄	D₄×D₄₇	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	188	2	no	—
108	C₂²	—	188	2	no	—
109	C₂²	—	188	2	no	—
110	C₂²	—	188	2	no	—
111	C₂²	#13093992	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	188	2	no	—
116	C₄	#13094047	1	2	yes	—
117	C₄	#13094056	1	2	yes	—
118	C₂	—	376	2	no	—
119	C₂	#15689284	1	2	yes	—
120	C₂	#15689290	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689839

C₂

yes

maximal, characteristic

#15689410

C₂

yes

maximal, complemented

#15688872

C₂

yes

maximal, characteristic, complemented

#15688852

C₂

yes

maximal, complemented

C₄₇×#494

C₂

yes

maximal, characteristic, complemented

#13094013

—

#13093999

—

C₄×D₁₈₈

C₂²

yes

—

#13093982

—

#13093955

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093944

—

C₂×Q₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

—

C₂×C₄×D₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

#120

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#135

yes

—

C₄×C₄₇

#137

yes

—

Dic₄₇

—

#1634

—

maximal, Sylow

C₂×C₄₇

#452

yes

—

C₂×C₄₇

#453

yes

characteristic

D₄₇

—

#555

—

#494

D₄₇

yes

characteristic, complemented

#476

—

#333

—

D₄⋊C₈

—

C₄₇

#1634

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

#124

—

#121

—

C₄×D₄

—

#117

—

#106

—

C₄⋊C₈

—

#102

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

376

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

188

—

103

C₂×C₄

—

188

—

104

C₂×C₄

D₄×D₄₇

yes

—

105

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

106

C₈

—

107

C₈

—

188

—

108

C₂²

—

188

—

109

C₂²

—

188

—

110

C₂²

—

188

—

111

C₂²

#13093992

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

188

—

116

C₄

#13094047

yes

—

117

C₄

#13094056

yes

—

118

C₂

—

376

—

119

C₂

#15689284

yes

—

120

C₂

#15689290

yes

characteristic

	subgroup	G/H
center	C₂²	#13093992
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#494	C₂
socle	C₂²×C₄₇	#120
hypercenter	#494	D₄₇
hypocenter	C₄₇	#1634
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#13093992

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#494

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #15689410	C₂	semidirect
3. #15688872	C₂	semidirect
4. #15688852	C₂	semidirect
5. C₄₇×#494	C₂	semidirect
62. #494	D₄₇	semidirect
66. C₄₇	#1634	semidirect

H ≅ G/N

type

2. #15689410

C₂

semidirect

3. #15688872

C₂

semidirect

4. #15688852

C₂

semidirect

5. C₄₇×#494

C₂

semidirect

62. #494

D₄₇

semidirect

66. C₄₇

#1634

semidirect

Subgroups — #20242531

Conjugacy classes of subgroups

Complements