Group #10604985 — subgroups

Subgroups — #10604985

Order 640.

280 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 94 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×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#4004	C₂	1	1	yes	maximal, characteristic
3	C₂×#4002	C₂	1	1	yes	maximal, characteristic
4	#64918	C₂	1	4	yes	maximal
5	C₂²×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₂²×#170	C₂²	1	2	yes	—
7	C₂²×#170	C₂²	1	2	yes	—
8	Q₈×Dic₅	—	2	2	no	—
9	Q₈×Dic₅	—	2	2	no	—
10	C₂×C₄⋊Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅	C₄	1	2	yes	—
12	C₂×C₄×Dic₅	C₂²	1	2	yes	—
13	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	#4004	—	2	2	no	—
15	#4002	—	2	2	no	—
16	#1985	—	5	1	no	maximal, Sylow
17	C₂×C₅×Q₈	—	2	2	no	—
18	C₂×C₅×Q₈	—	2	4	no	—
19	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
21	C₂²×Dic₅	C₂×C₄	1	2	yes	—
22	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₅	C₂³	1	1	yes	characteristic
24	C₂×#170	—	2	2	no	—
25	C₂×#170	—	2	2	no	—
26	C₂×#170	—	2	2	no	—
27	C₂×#170	—	2	4	no	—
28	C₂×#170	Q₈	1	4	yes	complemented
29	C₄⋊Dic₅	—	2	2	no	—
30	C₄⋊Dic₅	—	2	4	no	—
31	C₄×Dic₅	—	2	2	no	—
32	C₄×Dic₅	—	2	2	no	—
33	C₄×Dic₅	D₄	1	2	yes	—
34	C₄×Dic₅	D₄	1	2	yes	—
35	C₂×C₄×Q₈	—	5	1	no	—
36	C₂×C₄⋊C₈	—	5	1	no	—
37	C₂×#97	—	5	1	no	—
38	#336	—	5	4	no	—
39	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₅×Q₈	—	2	8	no	—
41	C₂×C₄×C₅	—	2	2	no	—
42	C₂×C₄×C₅	—	2	2	no	—
43	C₂×C₄×C₅	—	2	2	no	—
44	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₅	M₁₆	1	4	yes	—
46	C₂×Dic₅	—	2	2	no	—
47	C₂×Dic₅	—	2	2	no	—
48	C₂×Dic₅	—	2	4	no	—
49	C₂×Dic₅	C₂²⋊C₄	1	2	yes	—
50	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅	C₂×Q₈	1	2	yes	—
53	C₂×Dic₅	#41	1	2	yes	—
54	#170	—	2	4	no	—
55	#170	—	4	2	no	—
56	C₂²×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂²×C₈	—	5	2	no	—
58	C₂²×C₈	—	5	2	no	—
59	C₄×Q₈	—	10	2	no	—
60	C₄×Q₈	—	10	2	no	—
61	C₂×C₄⋊C₄	—	5	2	no	—
62	C₂×C₄⋊C₄	—	5	1	no	—
63	C₂×C₄²	—	5	1	no	—
64	C₂×C₄²	—	5	2	no	—
65	C₄⋊C₈	—	10	2	no	—
66	#97	—	10	2	no	—
67	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₅	C₄×Q₈	1	2	yes	—
69	C₂²×C₅	C₂×M₁₆	1	2	yes	—
70	C₂²×C₅	#131	1	2	yes	—
71	C₄×C₅	—	2	2	no	—
72	C₄×C₅	—	2	4	no	—
73	Dic₅	—	2	2	no	—
74	Dic₅	—	4	2	no	—
75	Dic₅	C₂²⋊Q₈	1	2	yes	—
76	Dic₅	#124	1	1	yes	characteristic
77	Dic₅	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×Q₈	—	2	4	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂²×C₄	—	5	1	no	—
81	C₂²×C₄	—	5	1	no	—
82	C₂²×C₄	—	5	2	no	—
83	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
84	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
85	C₂×C₈	—	5	4	no	—
86	C₂×C₈	—	10	2	no	—
87	C₂×C₈	—	10	2	no	—
88	C₂×C₈	—	10	2	no	—
89	C₂×C₈	—	10	4	no	—
90	C₄⋊C₄	—	10	4	no	—
91	C₄⋊C₄	—	10	2	no	—
92	C₄²	—	5	2	no	—
93	C₄²	—	5	2	no	—
94	C₄²	—	10	2	no	—
95	C₄²	—	10	2	no	—
96	C₂×C₅	#391	1	1	yes	characteristic
97	C₂×C₅	#407	1	1	yes	characteristic
98	C₂×C₅	#408	1	1	yes	characteristic
99	C₂×C₅	#446	1	4	yes	—
100	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
101	Q₈	—	2	8	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	5	2	no	—
106	C₂×C₄	—	5	2	no	—
107	C₂×C₄	—	5	1	no	—
108	C₂×C₄	—	5	1	no	—
109	C₂×C₄	—	5	2	no	—
110	C₂×C₄	—	10	2	no	—
111	C₂×C₄	—	10	4	no	—
112	C₂×C₄	—	10	2	no	—
113	C₂×C₄	#704	1	4	yes	—
114	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
115	C₈	—	10	4	no	—
116	C₈	—	20	2	no	—
117	C₅	#1985	1	1	yes	characteristic, complemented, Sylow
118	C₂²	#4133	1	2	yes	—
119	C₂²	Q₈×AGL(1,𝔽₅)	1	2	yes	—
120	C₂²	C₂×#704	1	2	yes	—
121	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	2	no	—
124	C₄	—	5	2	no	—
125	C₄	—	5	1	no	—
126	C₄	—	5	1	no	—
127	C₄	—	10	2	no	—
128	C₄	—	20	2	no	—
129	C₂	#65787	1	1	yes	characteristic
130	C₂	#65788	1	4	yes	—
131	C₂	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
132	C₂	#65798	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅

C₂

yes

maximal, characteristic

C₂×#4004

C₂

yes

maximal, characteristic

C₂×#4002

C₂

yes

maximal, characteristic

#64918

C₂

yes

maximal

C₂²×C₅×Q₈

C₄

yes

characteristic

C₂²×#170

C₂²

yes

—

C₂²×#170

C₂²

yes

—

Q₈×Dic₅

—

Q₈×Dic₅

—

C₂×C₄⋊Dic₅

C₄

yes

characteristic

C₂×C₄⋊Dic₅

C₄

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×C₄

yes

—

C₂²×Dic₅

C₂×C₄

yes

—

C₂²×Dic₅

C₂×C₄

yes

characteristic

C₂²×Dic₅

C₂³

yes

—

—

—

—

Q₈

yes

complemented

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

D₄

yes

—

C₄×Dic₅

D₄

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

C₂³×C₅

C₂²×C₄

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

M₁₆

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂²⋊C₄

yes

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

characteristic

C₂×Dic₅

C₂×Q₈

yes

—

C₂×Dic₅

#41

yes

—

#170

—

#170

—

C₂²×Q₈

AGL(1,𝔽₅)

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

#97

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅

C₄×Q₈

yes

—

C₂²×C₅

C₂×M₁₆

yes

—

C₂²×C₅

#131

yes

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

C₂²⋊Q₈

yes

—

Dic₅

#124

yes

characteristic

Dic₅

C₄⋊Q₈

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

—

100

C₂³

C₂²×AGL(1,𝔽₅)

yes

characteristic

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#704

yes

—

114

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

—

115

C₈

—

116

C₈

—

117

C₅

#1985

yes

characteristic, complemented, Sylow

118

C₂²

#4133

yes

—

119

C₂²

Q₈×AGL(1,𝔽₅)

yes

—

120

C₂²

C₂×#704

yes

—

121

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₂

#65787

yes

characteristic

130

C₂

#65788

yes

—

131

C₂

(C₂×AGL(1,𝔽₅))⋊Q₈

yes

characteristic

132

C₂

#65798

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×AGL(1,𝔽₅)
commutator, socle, prime radical	C₂³×C₅	C₂²×C₄
Fitting	C₂²×C₅×Q₈	C₄
cosocle	C₂²×Dic₅	C₂³
hypercenter	C₂²×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#1985
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂²×AGL(1,𝔽₅)

commutator, socle, prime radical

C₂³×C₅

C₂²×C₄

Fitting

C₂²×C₅×Q₈

C₄

cosocle

C₂²×Dic₅

C₂³

C₂²×Q₈

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
28. C₂×#170	Q₈	semidirect
117. C₅	#1985	semidirect

H ≅ G/N

type

28. C₂×#170

Q₈

semidirect

117. C₅

#1985

semidirect

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

subgroup	index
C₂×Q₈×Dic₅	2
C₂×#4004	2
C₂×#4002	2
#64918	2
#1985	5

subgroup

index

C₂×Q₈×Dic₅

Subgroups — #10604985

Conjugacy classes of subgroups

Complements

Maximal subgroups