Group #52171 — subgroups

Subgroups — C₂×#2736 #52171

Order 256.

1691 conjugacy classes of subgroups; only the 463 normal subgroups are shown.

normal subgroups only

This group has 1691 conjugacy classes of subgroups — too many to draw. Only its 463 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	#2736	C₂	1	16	yes	maximal, direct product factor
2	C₂×#397	C₂	1	3	yes	maximal, complemented
3	C₂×#393	C₂	1	3	yes	maximal, complemented
4	C₂×#388	C₂	1	6	yes	maximal, complemented
5	C₂×#379	C₂	1	3	yes	maximal
6	C₂⁶	C₂²	1	1	yes	characteristic
7	C₂³×Q₈	C₂²	1	1	yes	characteristic, complemented
8	C₂²×C₄⋊C₄	C₂²	1	3	yes	complemented
9	C₂²×C₂²⋊C₄	C₂²	1	6	yes	—
10	C₂²×C₂²⋊C₄	C₂²	1	3	yes	—
11	C₂²×C₂²⋊C₄	C₂²	1	6	yes	—
12	C₂²×C₂²⋊C₄	C₂²	1	3	yes	—
13	C₂²×C₄²	C₂²	1	2	yes	complemented
14	#397	C₂²	1	24	yes	complemented
15	#393	C₂²	1	24	yes	complemented
16	#388	C₂²	1	48	yes	complemented
17	#379	C₂²	1	24	yes	—
18	C₂×#95	C₂²	1	6	yes	complemented
19	C₂×#95	C₂²	1	3	yes	complemented
20	C₂×#95	C₂²	1	1	yes	characteristic, complemented
21	C₂⁵	—	2	12	no	—
22	C₂⁵	—	4	6	no	—
23	C₂⁵	—	4	2	no	—
24	C₂⁵	C₂³	1	3	yes	—
25	C₂⁵	C₂³	1	4	yes	—
26	C₂²×Q₈	—	4	6	no	—
27	C₂²×Q₈	C₂³	1	4	yes	complemented
28	C₂³×C₄	C₂³	1	3	yes	—
29	C₂³×C₄	C₂³	1	6	yes	—
30	C₂³×C₄	C₂³	1	3	yes	—
31	C₂×C₄⋊C₄	—	4	6	no	—
32	C₂×C₄⋊C₄	C₂³	1	12	yes	complemented
33	C₂×C₂²⋊C₄	—	2	24	no	—
34	C₂×C₂²⋊C₄	—	4	6	no	—
35	C₂×C₂²⋊C₄	—	4	6	no	—
36	C₂×C₂²⋊C₄	—	4	12	no	—
37	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
38	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
39	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
40	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
41	C₂×C₄²	—	4	4	no	—
42	C₂×C₄²	C₂³	1	8	yes	complemented
43	#95	C₂³	1	12	yes	complemented
44	#95	C₂³	1	24	yes	complemented
45	#95	C₂³	1	4	yes	complemented
46	C₂⁴	—	2	12	no	—
47	C₂⁴	—	2	6	no	—
48	C₂⁴	—	2	6	no	—
49	C₂⁴	—	2	12	no	—
50	C₂⁴	—	2	24	no	—
51	C₂⁴	—	4	12	no	—
52	C₂⁴	—	4	4	no	—
53	C₂⁴	—	4	48	no	—
54	C₂⁴	—	4	48	no	—
55	C₂⁴	—	4	16	no	—
56	C₂⁴	#41	1	12	yes	—
57	C₂⁴	C₂⁴	1	6	yes	—
58	C₂⁴	C₂⁴	1	1	yes	characteristic
59	C₂×Q₈	—	4	12	no	—
60	C₂×Q₈	—	4	16	no	—
61	C₂²×C₄	—	2	12	no	—
62	C₂²×C₄	—	2	12	no	—
63	C₂²×C₄	—	2	12	no	—
64	C₂²×C₄	—	2	12	no	—
65	C₂²×C₄	—	2	6	no	—
66	C₂²×C₄	—	2	6	no	—
67	C₂²×C₄	—	2	12	no	—
68	C₂²×C₄	C₂⁴	1	6	yes	—
69	C₂²×C₄	C₂⁴	1	6	yes	—
70	C₂²×C₄	C₂⁴	1	12	yes	—
71	C₄⋊C₄	—	4	12	no	—
72	C₂²⋊C₄	—	2	48	no	—
73	C₂²⋊C₄	—	4	12	no	—
74	C₂²⋊C₄	—	4	24	no	—
75	C₂²⋊C₄	—	4	12	no	—
76	C₄²	—	4	8	no	—
77	C₂³	—	2	24	no	—
78	C₂³	—	2	48	no	—
79	C₂³	—	2	12	no	—
80	C₂³	—	2	6	no	—
81	C₂³	—	2	6	no	—
82	C₂³	—	2	12	no	—
83	C₂³	—	4	48	no	—
84	C₂³	—	4	48	no	—
85	C₂³	—	4	128	no	—
86	C₂³	—	4	16	no	—
87	C₂³	—	4	24	no	—
88	C₂³	—	4	12	no	—
89	C₂³	—	4	12	no	—
90	C₂³	C₂×#41	1	12	yes	—
91	C₂³	C₂×#41	1	6	yes	—
92	C₂³	H(5,𝔽₂)	1	2	yes	—
93	C₂³	H(5,𝔽₂)	1	6	yes	—
94	C₂³	C₂⁵	1	1	yes	characteristic
95	Q₈	—	4	16	no	—
96	C₂×C₄	—	2	12	no	—
97	C₂×C₄	—	2	12	no	—
98	C₂×C₄	—	2	12	no	—
99	C₂×C₄	—	2	12	no	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	12	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	4	24	no	—
104	C₂×C₄	—	4	12	no	—
105	C₂×C₄	—	4	12	no	—
106	C₂²	—	2	12	no	—
107	C₂²	—	2	24	no	—
108	C₂²	—	4	64	no	—
109	C₂²	—	4	12	no	—
110	C₂²	—	4	6	no	—
111	C₂²	—	4	48	no	—
112	C₂²	—	4	6	no	—
113	C₂²	#538	1	12	yes	—
114	C₂²	#551	1	12	yes	—
115	C₂²	H(3,𝔽₄)	1	4	yes	—
116	C₂²	C₂²×#41	1	3	yes	—
117	C₂²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²	C₂×H(5,𝔽₂)	1	3	yes	—
119	C₄	—	4	12	no	—
120	C₄	—	4	6	no	—
121	C₄	—	4	6	no	—
122	C₂	—	4	12	no	—
123	C₂	#2736	1	8	yes	direct product factor
124	C₂	C₂×#538	1	3	yes	—
125	C₂	C₂×#551	1	3	yes	—
126	C₂	C₂×H(3,𝔽₄)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2736

C₂

yes

maximal, direct product factor

C₂×#397

C₂

yes

maximal, complemented

C₂×#393

C₂

yes

maximal, complemented

C₂×#388

C₂

yes

maximal, complemented

C₂×#379

C₂

yes

maximal

C₂⁶

C₂²

yes

characteristic

C₂³×Q₈

C₂²

yes

characteristic, complemented

C₂²×C₄⋊C₄

C₂²

yes

complemented

C₂²×C₂²⋊C₄

C₂²

yes

—

C₂²×C₂²⋊C₄

C₂²

yes

—

C₂²×C₂²⋊C₄

C₂²

yes

—

C₂²×C₂²⋊C₄

C₂²

yes

—

C₂²×C₄²

C₂²

yes

complemented

#397

C₂²

yes

complemented

#393

C₂²

yes

complemented

#388

C₂²

yes

complemented

#379

C₂²

yes

—

C₂×#95

C₂²

yes

complemented

C₂×#95

C₂²

yes

complemented

C₂×#95

C₂²

yes

characteristic, complemented

C₂⁵

—

C₂⁵

—

C₂⁵

—

C₂⁵

C₂³

yes

—

C₂⁵

C₂³

yes

—

C₂²×Q₈

—

C₂²×Q₈

C₂³

yes

complemented

C₂³×C₄

C₂³

yes

—

C₂³×C₄

C₂³

yes

—

C₂³×C₄

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³

yes

complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂³

yes

—

C₂×C₂²⋊C₄

C₂³

yes

—

C₂×C₂²⋊C₄

C₂³

yes

—

C₂×C₂²⋊C₄

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

complemented

#95

C₂³

yes

complemented

#95

C₂³

yes

complemented

#95

C₂³

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

#41

yes

—

C₂⁴

yes

—

C₂⁴

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

128

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×#41

yes

—

C₂³

C₂×#41

yes

—

C₂³

H(5,𝔽₂)

yes

—

C₂³

H(5,𝔽₂)

yes

—

C₂³

C₂⁵

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

#538

yes

—

114

C₂²

#551

yes

—

115

C₂²

H(3,𝔽₄)

yes

—

116

C₂²

C₂²×#41

yes

—

117

C₂²

C₂×H(5,𝔽₂)

yes

characteristic

118

C₂²

C₂×H(5,𝔽₂)

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₂

—

123

C₂

#2736

yes

direct product factor

124

C₂

C₂×#538

yes

—

125

C₂

C₂×#551

yes

—

126

C₂

C₂×H(3,𝔽₄)

yes

characteristic

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂⁶	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁵

prime radical

C₂⁶

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂⁴ ◁ G

LECS

G ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
1. #2736	C₂	direct (N × H)
2. C₂×#397	C₂	semidirect
3. C₂×#393	C₂	semidirect
4. C₂×#388	C₂	semidirect
7. C₂³×Q₈	C₂²	semidirect
8. C₂²×C₄⋊C₄	C₂²	semidirect
13. C₂²×C₄²	C₂²	semidirect
14. #397	C₂²	semidirect
15. #393	C₂²	semidirect
16. #388	C₂²	semidirect
18. C₂×#95	C₂²	semidirect
19. C₂×#95	C₂²	semidirect
20. C₂×#95	C₂²	semidirect
27. C₂²×Q₈	C₂³	semidirect
32. C₂×C₄⋊C₄	C₂³	semidirect
42. C₂×C₄²	C₂³	semidirect
43. #95	C₂³	semidirect
44. #95	C₂³	semidirect
45. #95	C₂³	semidirect
123. C₂	#2736	direct (N × H)

H ≅ G/N

type

1. #2736

C₂

direct (N × H)

2. C₂×#397

C₂

semidirect

3. C₂×#393

C₂

semidirect

4. C₂×#388

C₂

semidirect

7. C₂³×Q₈

C₂²

semidirect

8. C₂²×C₄⋊C₄

C₂²

semidirect

13. C₂²×C₄²

C₂²

semidirect

14. #397

C₂²

semidirect

15. #393

C₂²

semidirect

16. #388

C₂²

semidirect

18. C₂×#95

C₂²

semidirect

19. C₂×#95

C₂²

semidirect

20. C₂×#95

C₂²

semidirect

27. C₂²×Q₈

C₂³

semidirect

32. C₂×C₄⋊C₄

C₂³

semidirect

42. C₂×C₄²

C₂³

semidirect

43. #95

C₂³

semidirect

44. #95

C₂³

semidirect

45. #95

C₂³

semidirect

123. C₂

#2736

direct (N × H)

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

subgroup	index
#2736	2
C₂×#397	2
C₂×#393	2
C₂×#388	2
C₂×#379	2

subgroup

index

Subgroups — C₂×#2736 #52171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#2736 #52171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2736 #52171