Group #60655 — subgroups

Subgroups — C₂×#3160 #60655

Order 256.

971 conjugacy classes of subgroups; only the 583 normal subgroups are shown.

normal subgroups only

This group has 971 conjugacy classes of subgroups — too many to draw. Only its 583 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×#533	C₂	1	2	yes	maximal
2	#3160	C₂	1	16	yes	maximal, direct product factor
3	C₂×#500	C₂	1	2	yes	maximal, complemented
4	C₂×#499	C₂	1	4	yes	maximal, complemented
5	C₂×#498	C₂	1	2	yes	maximal, complemented
6	C₂×#427	C₂	1	4	yes	maximal
7	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₈	C₂²	1	2	yes	—
9	#533	—	2	8	no	—
10	#533	C₂²	1	16	yes	—
11	C₂×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂×#125	C₂²	1	4	yes	—
13	C₂×C₂²⋊Q₈	—	2	4	no	—
14	C₂×C₄×Q₈	—	2	4	no	—
15	C₂×#117	C₂²	1	4	yes	—
16	C₂²×C₄²	C₂²	1	1	yes	characteristic
17	#500	C₂²	1	16	yes	complemented
18	#499	C₂²	1	32	yes	complemented
19	#498	C₂²	1	16	yes	complemented
20	#427	C₂²	1	32	yes	—
21	C₂×#107	C₂²	1	8	yes	—
22	C₂×#106	C₂²	1	8	yes	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂²	1	8	yes	—
27	C₂²×Q₈	—	2	4	no	—
28	C₂³×C₄	D₄	1	2	yes	—
29	C₂³×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₈	Q₈	1	16	yes	complemented
31	C₂²×C₈	C₂³	1	2	yes	—
32	C₂²×C₈	C₂³	1	2	yes	—
33	C₂²×C₈	C₂³	1	8	yes	—
34	C₄⋊Q₈	—	2	8	no	—
35	C₄⋊Q₈	C₂³	1	16	yes	—
36	#125	—	2	8	no	—
37	#125	C₂³	1	16	yes	—
38	C₂²⋊Q₈	—	2	32	no	—
39	C₄×Q₈	—	2	32	no	—
40	#117	—	2	8	no	—
41	#117	C₂³	1	16	yes	—
42	C₂×C₄⋊C₄	—	2	8	no	—
43	C₂×C₄⋊C₄	—	2	4	no	—
44	C₂×C₄⋊C₄	C₂³	1	8	yes	—
45	C₂×C₂²⋊C₄	—	2	4	no	—
46	C₂×C₄²	—	2	4	no	—
47	C₂×C₄²	D₄	1	4	yes	—
48	C₂×C₄²	D₄	1	2	yes	—
49	C₂×C₄²	C₂³	1	1	yes	characteristic
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	C₂×C₄²	C₂³	1	4	yes	—
52	#107	C₂³	1	32	yes	—
53	#106	C₂³	1	32	yes	—
54	C₄×C₈	C₂³	1	4	yes	—
55	C₄×C₈	C₂³	1	8	yes	—
56	C₄×C₈	C₂³	1	4	yes	—
57	C₂⁴	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈	—	2	16	no	—
59	C₂×Q₈	—	4	16	no	—
60	C₂²×C₄	—	2	8	no	—
61	C₂²×C₄	—	2	8	no	—
62	C₂²×C₄	C₂×D₄	1	2	yes	—
63	C₂²×C₄	C₂×D₄	1	8	yes	—
64	C₂²×C₄	C₂×D₄	1	2	yes	—
65	C₂²×C₄	C₂×D₄	1	2	yes	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄	C₂×Q₈	1	8	yes	—
68	C₂²×C₄	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄	C₂⁴	1	2	yes	—
70	C₂×C₈	C₂×Q₈	1	8	yes	—
71	C₂×C₈	C₂×Q₈	1	8	yes	—
72	C₂×C₈	C₂×Q₈	1	32	yes	complemented
73	C₂×C₈	C₂⁴	1	4	yes	—
74	C₂×C₈	C₂⁴	1	4	yes	—
75	C₄⋊C₄	—	2	32	no	—
76	C₄⋊C₄	—	2	16	no	—
77	C₄⋊C₄	—	2	8	no	—
78	C₄⋊C₄	C₂⁴	1	16	yes	—
79	C₂²⋊C₄	—	2	16	no	—
80	C₄²	—	2	16	no	—
81	C₄²	C₂×D₄	1	8	yes	—
82	C₄²	C₂×D₄	1	2	yes	—
83	C₄²	C₂×D₄	1	2	yes	—
84	C₄²	C₂⁴	1	2	yes	—
85	C₄²	C₂⁴	1	2	yes	—
86	C₂³	—	2	4	no	—
87	C₂³	C₄⋊Q₈	1	4	yes	—
88	C₂³	C₂²×D₄	1	2	yes	—
89	C₂³	C₂²×D₄	1	1	yes	characteristic
90	Q₈	—	4	32	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	16	no	—
93	C₂×C₄	—	2	16	no	—
94	C₂×C₄	—	4	16	no	—
95	C₂×C₄	C₄⋊Q₈	1	8	yes	—
96	C₂×C₄	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄	#135	1	8	yes	—
98	C₂×C₄	C₂²×D₄	1	4	yes	—
99	C₂×C₄	C₂²×D₄	1	2	yes	—
100	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₄	1	4	yes	—
102	C₂×C₄	C₂²×Q₈	1	4	yes	—
103	C₂×C₄	C₂²×Q₈	1	8	yes	—
104	C₂×C₄	C₂⁵	1	1	yes	characteristic
105	C₈	C₂²×Q₈	1	8	yes	—
106	C₈	C₂²×Q₈	1	8	yes	—
107	C₂²	—	2	8	no	—
108	C₂²	—	2	4	no	—
109	C₂²	C₂×C₄⋊Q₈	1	4	yes	—
110	C₂²	C₂×C₄⋊Q₈	1	2	yes	—
111	C₂²	C₂×#135	1	4	yes	—
112	C₂²	C₂³×D₄	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	4	16	no	—
115	C₄	C₂×C₄⋊Q₈	1	2	yes	—
116	C₄	C₂×C₄⋊Q₈	1	4	yes	—
117	C₄	C₂×#135	1	8	yes	—
118	C₄	C₂³×Q₈	1	2	yes	—
119	C₂	—	2	4	no	—
120	C₂	#3160	1	4	yes	direct product factor
121	C₂	C₂²×C₄⋊Q₈	1	1	yes	characteristic
122	C₂	C₂²×#135	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#533

C₂

yes

maximal

#3160

C₂

yes

maximal, direct product factor

C₂×#500

C₂

yes

maximal, complemented

C₂×#499

C₂

yes

maximal, complemented

C₂×#498

C₂

yes

maximal, complemented

C₂×#427

C₂

yes

maximal

C₂²×C₄×C₈

C₂

yes

maximal, characteristic

C₂³×C₈

C₂²

yes

—

#533

—

#533

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×#125

C₂²

yes

—

C₂×C₂²⋊Q₈

—

C₂×C₄×Q₈

—

C₂×#117

C₂²

yes

—

C₂²×C₄²

C₂²

yes

characteristic

#500

C₂²

yes

complemented

#499

C₂²

yes

complemented

#498

C₂²

yes

complemented

#427

C₂²

yes

—

C₂×#107

C₂²

yes

—

C₂×#106

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂²×Q₈

—

C₂³×C₄

D₄

yes

—

C₂³×C₄

C₂³

yes

characteristic

C₂²×C₈

Q₈

yes

complemented

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

#125

—

#125

C₂³

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

—

#107

C₂³

yes

—

#106

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

C₂×D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

C₂²×C₄

C₂⁴

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

complemented

C₂×C₈

C₂⁴

yes

—

C₂×C₈

C₂⁴

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂⁴

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

—

C₄²

C₂⁴

yes

—

C₄²

C₂⁴

yes

—

C₂³

—

C₂³

C₄⋊Q₈

yes

—

C₂³

C₂²×D₄

yes

—

C₂³

C₂²×D₄

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Q₈

yes

—

C₂×C₄

C₄⋊Q₈

yes

—

C₂×C₄

#135

yes

—

C₂×C₄

C₂²×D₄

yes

—

C₂×C₄

C₂²×D₄

yes

—

100

C₂×C₄

C₂²×D₄

yes

characteristic

101

C₂×C₄

C₂²×D₄

yes

—

102

C₂×C₄

C₂²×Q₈

yes

—

103

C₂×C₄

C₂²×Q₈

yes

—

104

C₂×C₄

C₂⁵

yes

characteristic

105

C₈

C₂²×Q₈

yes

—

106

C₈

C₂²×Q₈

yes

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₄⋊Q₈

yes

—

110

C₂²

C₂×C₄⋊Q₈

yes

—

111

C₂²

C₂×#135

yes

—

112

C₂²

C₂³×D₄

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

C₂×C₄⋊Q₈

yes

—

116

C₄

C₂×C₄⋊Q₈

yes

—

117

C₄

C₂×#135

yes

—

118

C₄

C₂³×Q₈

yes

—

119

C₂

—

120

C₂

#3160

yes

direct product factor

121

C₂

C₂²×C₄⋊Q₈

yes

characteristic

122

C₂

C₂²×#135

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂³	C₂²×D₄
prime radical	C₂⁴	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁵

socle

C₂³

C₂²×D₄

prime radical

C₂⁴

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #3160	C₂	direct (N × H)
3. C₂×#500	C₂	semidirect
4. C₂×#499	C₂	semidirect
5. C₂×#498	C₂	semidirect
17. #500	C₂²	semidirect
18. #499	C₂²	semidirect
19. #498	C₂²	semidirect
30. C₂²×C₈	Q₈	semidirect
72. C₂×C₈	C₂×Q₈	semidirect
120. C₂	#3160	direct (N × H)

H ≅ G/N

type

2. #3160

C₂

direct (N × H)

3. C₂×#500

C₂

semidirect

4. C₂×#499

C₂

semidirect

5. C₂×#498

C₂

semidirect

17. #500

C₂²

semidirect

18. #499

C₂²

semidirect

19. #498

C₂²

semidirect

30. C₂²×C₈

Q₈

semidirect

72. C₂×C₈

C₂×Q₈

semidirect

120. C₂

#3160

direct (N × H)

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

subgroup	index
C₂×#533	2
#3160	2
C₂×#500	2
C₂×#499	2
C₂×#498	2
C₂×#427	2
C₂²×C₄×C₈	2

subgroup

index

C₂²×C₄×C₈

Subgroups — C₂×#3160 #60655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#3160 #60655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3160 #60655