Group #15689850 — subgroups

Subgroups — #15689850

Order 3008.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#13094023	C₂	1	2	yes	maximal
2	#13094023	C₂	1	1	yes	maximal, characteristic
3	#13094015	C₂	1	1	yes	maximal, characteristic
4	#13094014	C₂	1	2	yes	maximal
5	#13094014	C₂	1	2	yes	maximal
6	C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic
8	#13093989	C₂	1	1	yes	maximal, characteristic
9	#13093989	C₂	1	2	yes	maximal
10	#13093989	C₂	1	1	yes	maximal, characteristic
11	#125×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₉₄	C₂²	1	2	yes	—
13	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
14	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
19	C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
24	Dic₄₇⋊C₄	C₂²	1	2	yes	—
25	Dic₄₇⋊C₄	C₂²	1	2	yes	—
26	Dic₄₇⋊C₄	C₂²	1	2	yes	—
27	Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
28	Dic₄₇⋊C₄	C₂²	1	2	yes	—
29	Dic₄₇⋊C₄	C₂²	1	2	yes	—
30	C₄×Dic₄₇	C₂²	1	2	yes	—
31	C₄×Dic₄₇	C₂²	1	2	yes	—
32	C₄×Dic₄₇	C₂²	1	2	yes	—
33	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂³	1	2	yes	—
37	C₂×C₄×C₄₇	C₂³	1	2	yes	—
38	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
41	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₄₇	C₂³	1	2	yes	—
43	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₄₇	C₂³	1	2	yes	—
45	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
46	Dic₉₄	—	2	4	no	—
47	Dic₉₄	—	2	2	no	—
48	Dic₉₄	Q₈	1	4	yes	complemented
49	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₄×C₄₇	—	2	2	no	—
51	C₄×C₄₇	—	2	1	no	—
52	C₄×C₄₇	—	2	1	no	—
53	C₄×C₄₇	—	2	2	no	—
54	C₄×C₄₇	C₂×Q₈	1	2	yes	—
55	Dic₄₇	—	2	2	no	—
56	Dic₄₇	—	2	1	no	—
57	Dic₄₇	—	2	1	no	—
58	Dic₄₇	—	2	1	no	—
59	Dic₄₇	C₂×Q₈	1	4	yes	—
60	Dic₄₇	#41	1	2	yes	—
61	C₂×C₄₇	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
63	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
64	#557	—	47	1	no	maximal, Sylow
65	C₄₇	#557	1	1	yes	characteristic, complemented, Sylow
66	C₄⋊Q₈	—	47	2	no	—
67	C₄⋊Q₈	—	47	1	no	—
68	#125	—	47	2	no	—
69	#125	—	47	2	no	—
70	#125	—	47	1	no	—
71	#125	D₄₇	1	1	yes	characteristic
72	C₄×Q₈	—	47	1	no	—
73	C₄×Q₈	—	47	2	no	—
74	C₄×Q₈	—	47	1	no	—
75	C₄×Q₈	—	47	1	no	—
76	C₄×Q₈	—	47	1	no	—
77	C₂×Q₈	—	47	1	no	—
78	C₂×Q₈	—	47	2	no	—
79	C₂×Q₈	—	47	1	no	—
80	C₄⋊C₄	—	47	1	no	—
81	C₄⋊C₄	—	47	2	no	—
82	C₄⋊C₄	—	47	2	no	—
83	C₄⋊C₄	—	47	1	no	—
84	C₄⋊C₄	—	47	2	no	—
85	C₄⋊C₄	—	47	2	no	—
86	C₄⋊C₄	—	47	2	no	—
87	C₄⋊C₄	—	47	1	no	—
88	C₄⋊C₄	—	47	1	no	—
89	C₄⋊C₄	—	47	2	no	—
90	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
91	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
92	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
93	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
94	C₄²	—	47	2	no	—
95	C₄²	—	47	2	no	—
96	C₄²	—	47	1	no	—
97	C₄²	—	47	2	no	—
98	C₄²	—	47	1	no	—
99	C₄²	C₂×D₄₇	1	1	yes	characteristic
100	Q₈	—	47	4	no	—
101	Q₈	—	94	4	no	—
102	Q₈	—	94	2	no	—
103	C₂×C₄	—	47	2	no	—
104	C₂×C₄	—	47	1	no	—
105	C₂×C₄	—	47	1	no	—
106	C₂×C₄	—	47	1	no	—
107	C₂×C₄	—	47	1	no	—
108	C₂×C₄	—	47	2	no	—
109	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
110	C₂×C₄	C₂²×D₄₇	1	2	yes	—
111	C₂×C₄	C₂²×D₄₇	1	2	yes	—
112	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	C₂²	C₂³×D₄₇	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	47	4	no	—
120	C₄	—	47	2	no	—
121	C₄	—	94	1	no	—
122	C₄	—	94	2	no	—
123	C₄	—	94	1	no	—
124	C₄	—	94	1	no	—
125	C₄	Q₈×D₄₇	1	2	yes	—
126	C₂	C₂×Q₈×D₄₇	1	1	yes	characteristic
127	C₂	#41×D₄₇	1	1	yes	characteristic
128	C₂	#13094071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094023

C₂

yes

maximal

#13094023

C₂

yes

maximal, characteristic

#13094015

C₂

yes

maximal, characteristic

#13094014

C₂

yes

maximal

#13094014

C₂

yes

maximal

C₄×Dic₉₄

C₂

yes

maximal, characteristic

C₄×Dic₉₄

C₂

yes

maximal, characteristic

#13093989

C₂

yes

maximal, characteristic

#13093989

C₂

yes

maximal

#13093989

C₂

yes

maximal, characteristic

#125×C₄₇

C₂

yes

maximal, characteristic

C₂×Dic₉₄

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

characteristic

C₂×Dic₉₄

C₂²

yes

characteristic

C₄⋊C₄×C₄₇

C₂²

yes

—

C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₄⋊C₄×C₄₇

C₂²

yes

—

C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

characteristic

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

characteristic

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

characteristic

C₄×Dic₄₇

C₂²

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

characteristic

Dic₉₄

—

Dic₉₄

—

Dic₉₄

Q₈

yes

complemented

C₂²×C₄₇

C₂⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×Q₈

yes

—

Dic₄₇

#41

yes

—

C₂×C₄₇

C₂²×Q₈

yes

characteristic

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

2₋¹⁺⁴

yes

—

C₄₇

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₇

yes

—

C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₄⋊C₄

C₂×D₄₇

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄₇

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²×D₄₇

yes

characteristic

110

C₂×C₄

C₂²×D₄₇

yes

—

111

C₂×C₄

C₂²×D₄₇

yes

—

112

C₂×C₄

C₂²×D₄₇

yes

characteristic

113

C₂×C₄

C₂²×D₄₇

yes

characteristic

114

C₂²

C₂³×D₄₇

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

Q₈×D₄₇

yes

—

126

C₂

C₂×Q₈×D₄₇

yes

characteristic

127

C₂

#41×D₄₇

yes

characteristic

128

C₂

#13094071

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, prime radical, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#125×C₄₇	C₂
hypercenter	#125	D₄₇
hypocenter	C₄₇	#557
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₇

commutator, socle, prime radical, cosocle

C₂²×C₄₇

C₂⁴

Fitting

#125×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
48. Dic₉₄	Q₈	semidirect
65. C₄₇	#557	semidirect

H ≅ G/N

type

48. Dic₉₄

Q₈

semidirect

65. C₄₇

#557

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13094023	2
#13094015	2
#13094014	2
C₄×Dic₉₄	2
#13093989	2
#125×C₄₇	2
#557	47

subgroup

index

C₄×Dic₉₄

#125×C₄₇

Subgroups — #15689850

Conjugacy classes of subgroups

Complements

Maximal subgroups