Group #20249117 — subgroups

Subgroups — #20249117

Order 6016.

816 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13094003	C₂	1	3	yes	maximal, complemented
2	#15689042	C₂	1	2	yes	maximal, complemented
3	#15689037	C₂	1	6	yes	maximal, complemented
4	#15688983	C₂	1	3	yes	maximal, complemented
5	C₂²×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
7	#13094003	—	2	12	no	—
8	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
9	C₂×Dic₄₇⋊C₄	C₂²	1	6	yes	—
10	C₂×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
11	C₂×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
12	C₄₇⋊#95	C₂²	1	6	yes	—
13	C₂×#41×C₄₇	—	2	12	no	—
14	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₄₇	C₂²	1	3	yes	—
16	C₂³×C₄×C₄₇	C₂²	1	3	yes	—
17	C₂⁴×C₄₇	C₂³	1	3	yes	—
18	#41×C₄₇	—	2	32	no	—
19	C₂×Q₈×C₄₇	—	2	6	no	—
20	C₂×D₄×C₄₇	—	2	12	no	—
21	C₂×D₄×C₄₇	—	2	6	no	—
22	C₂²×C₄×C₄₇	—	2	12	no	—
23	C₂²×C₄×C₄₇	D₄	1	12	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	C₂³	1	6	yes	—
27	C₂²×Dic₄₇	C₂³	1	2	yes	—
28	C₂×Dic₉₄	—	2	6	no	—
29	C₂²⋊Dic₄₇	—	2	12	no	—
30	C₂²⋊Dic₄₇	—	4	6	no	—
31	C₄⋊Dic₄₇	—	2	6	no	—
32	Dic₄₇⋊C₄	—	2	12	no	—
33	C₂³×C₄₇	—	2	6	no	—
34	C₂³×C₄₇	—	2	3	no	—
35	C₂³×C₄₇	—	2	6	no	—
36	C₂³×C₄₇	—	2	3	no	—
37	C₂³×C₄₇	C₂×D₄	1	6	yes	—
38	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
39	Q₈×C₄₇	—	2	8	no	—
40	D₄×C₄₇	—	2	24	no	—
41	C₂×C₄×C₄₇	—	2	6	no	—
42	C₂×C₄×C₄₇	—	2	24	no	—
43	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
44	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
45	C₂×Dic₄₇	—	2	6	no	—
46	C₂×Dic₄₇	—	2	6	no	—
47	C₂×Dic₄₇	—	2	6	no	—
48	C₂×Dic₄₇	—	2	6	no	—
49	Dic₉₄	—	4	4	no	—
50	C₂²×C₄₇	—	2	6	no	—
51	C₂²×C₄₇	—	2	6	no	—
52	C₂²×C₄₇	—	2	6	no	—
53	C₂²×C₄₇	—	4	6	no	—
54	C₂²×C₄₇	—	4	6	no	—
55	C₂²×C₄₇	C₂²×D₄	1	3	yes	—
56	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂²×C₄₇	2₋¹⁺⁴	1	3	yes	—
58	C₄×C₄₇	—	2	6	no	—
59	C₄×C₄₇	#120	1	4	yes	—
60	Dic₄₇	—	4	2	no	—
61	Dic₄₇	—	4	6	no	—
62	#2616	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	—	4	6	no	—
64	C₂×C₄₇	C₂×#120	1	1	yes	characteristic
65	C₂×C₄₇	#536	1	3	yes	—
66	C₂×C₄₇	#537	1	3	yes	—
67	C₂²×#41	D₄₇	1	1	yes	characteristic
68	C₂×C₂²⋊Q₈	—	47	3	no	—
69	#395	—	47	2	no	—
70	#394	—	47	6	no	—
71	#380	—	47	3	no	—
72	C₄₇	#2616	1	1	yes	characteristic, complemented, Sylow
73	C₂×#41	—	2	12	no	—
74	C₂²×Q₈	—	47	1	no	—
75	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
76	C₂²×D₄	C₂×D₄₇	1	3	yes	—
77	C₂³×C₄	C₂×D₄₇	1	3	yes	—
78	C₂²⋊Q₈	—	94	12	no	—
79	C₂×C₄⋊C₄	—	47	6	no	—
80	C₂×C₄⋊C₄	—	47	3	no	—
81	C₂×C₂²⋊C₄	—	47	6	no	—
82	C₂×C₂²⋊C₄	—	47	6	no	—
83	#95	—	47	6	no	—
84	C₂⁴	C₂²×D₄₇	1	3	yes	—
85	#41	—	2	32	no	—
86	C₂×Q₈	—	2	6	no	—
87	C₂×Q₈	—	94	6	no	—
88	C₂×D₄	—	2	12	no	—
89	C₂×D₄	—	2	6	no	—
90	C₂²×C₄	—	2	12	no	—
91	C₂²×C₄	—	47	6	no	—
92	C₂²×C₄	—	47	2	no	—
93	C₂²×C₄	C₂²⋊D₄₇	1	12	yes	—
94	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
95	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
96	C₄⋊C₄	—	94	12	no	—
97	C₄⋊C₄	—	94	6	no	—
98	C₂²⋊C₄	—	94	12	no	—
99	C₂²⋊C₄	—	188	6	no	—
100	C₂³	—	2	6	no	—
101	C₂³	—	2	3	no	—
102	C₂³	—	2	6	no	—
103	C₂³	—	2	3	no	—
104	C₂³	C₂×C₂²⋊D₄₇	1	6	yes	—
105	C₂³	C₂³×D₄₇	1	1	yes	characteristic
106	Q₈	—	2	8	no	—
107	Q₈	—	188	4	no	—
108	D₄	—	2	24	no	—
109	C₂×C₄	—	2	6	no	—
110	C₂×C₄	—	2	24	no	—
111	C₂×C₄	—	94	6	no	—
112	C₂×C₄	—	94	6	no	—
113	C₂×C₄	—	94	6	no	—
114	C₂×C₄	—	94	6	no	—
115	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
116	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
117	C₂²	—	2	6	no	—
118	C₂²	—	2	6	no	—
119	C₂²	—	2	6	no	—
120	C₂²	—	4	6	no	—
121	C₂²	—	4	6	no	—
122	C₂²	C₂²×C₂²⋊D₄₇	1	3	yes	—
123	C₂²	#13094070	1	1	yes	characteristic
124	C₂²	#13094071	1	3	yes	—
125	C₄	—	2	6	no	—
126	C₄	—	188	2	no	—
127	C₄	—	188	6	no	—
128	C₄	#13093993	1	4	yes	—
129	C₂	—	4	6	no	—
130	C₂	C₂×#13093993	1	1	yes	characteristic
131	C₂	#15689694	1	3	yes	—
132	C₂	#15689701	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094003

C₂

yes

maximal, complemented

#15689042

C₂

yes

maximal, complemented

#15689037

C₂

yes

maximal, complemented

#15688983

C₂

yes

maximal, complemented

C₂²×#41×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

characteristic

#13094003

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂×#41×C₄₇

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂²×D₄×C₄₇

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

—

Dic₄₇

—

Dic₄₇

—

#2616

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#120

yes

characteristic

C₂×C₄₇

#536

yes

—

C₂×C₄₇

#537

yes

—

C₂²×#41

D₄₇

yes

characteristic

C₂×C₂²⋊Q₈

—

#395

—

#394

—

#380

—

C₄₇

#2616

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂²×D₄

C₂×D₄₇

yes

—

C₂³×C₄

C₂×D₄₇

yes

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

C₂²×D₄₇

yes

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

188

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

C₂×C₂²⋊D₄₇

yes

—

105

C₂³

C₂³×D₄₇

yes

characteristic

106

Q₈

—

107

Q₈

—

188

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

116

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂²×C₂²⋊D₄₇

yes

—

123

C₂²

#13094070

yes

characteristic

124

C₂²

#13094071

yes

—

125

C₄

—

126

C₄

—

188

—

127

C₄

—

188

—

128

C₄

#13093993

yes

—

129

C₂

—

130

C₂

C₂×#13093993

yes

characteristic

131

C₂

#15689694

yes

—

132

C₂

#15689701

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂²×#41×C₄₇	C₂
hypercenter	C₂²×#41	D₄₇
hypocenter	C₄₇	#2616
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂²×#41×C₄₇

C₂

hypercenter

C₂²×#41

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂²×#41

N	H ≅ G/N	type
1. C₂×#13094003	C₂	semidirect
2. #15689042	C₂	semidirect
3. #15689037	C₂	semidirect
4. #15688983	C₂	semidirect
72. C₄₇	#2616	semidirect

H ≅ G/N

type

1. C₂×#13094003

C₂

semidirect

2. #15689042

C₂

semidirect

3. #15689037

C₂

semidirect

4. #15688983

C₂

semidirect

72. C₄₇

#2616

semidirect

Subgroups — #20249117

Conjugacy classes of subgroups

Complements