Group #62757 — subgroups

Subgroups — C₂×#3549 #62757

Order 256.

4401 conjugacy classes of subgroups; only the 2863 normal subgroups are shown.

normal subgroups only

This group has 4401 conjugacy classes of subgroups — too many to draw. Only its 2863 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	#3549	C₂	1	32	yes	maximal, direct product factor
2	C₂×H(3,𝔽₄)	C₂	1	4	yes	maximal, complemented
3	C₂×#555	C₂	1	6	yes	maximal, complemented
4	C₂×D₄²	C₂	1	6	yes	maximal, complemented
5	C₂×#535	C₂	1	12	yes	maximal, complemented
6	C₂×#532	C₂	1	3	yes	maximal, complemented
7	C₂²×#41	C₂²	1	6	yes	complemented
8	C₂³×D₄	C₂²	1	12	yes	complemented
9	H(3,𝔽₄)	C₂²	1	64	yes	complemented
10	#555	C₂²	1	96	yes	complemented
11	D₄²	C₂²	1	96	yes	complemented
12	#535	C₂²	1	192	yes	complemented
13	#532	C₂²	1	48	yes	complemented
14	C₂×C₄⋊Q₈	C₂²	1	3	yes	complemented
15	C₂×#127	C₂²	1	3	yes	—
16	C₂×#127	C₂²	1	8	yes	complemented
17	C₂×#124	C₂²	1	6	yes	complemented
18	C₂×#124	C₂²	1	24	yes	complemented
19	C₂×#121	C₂²	1	12	yes	complemented
20	C₂×#121	C₂²	1	24	yes	complemented
21	C₂×#120	C₂²	1	8	yes	complemented
22	C₂×#120	C₂²	1	24	yes	complemented
23	C₂×C₄×D₄	C₂²	1	12	yes	complemented
24	C₂×#117	C₂²	1	12	yes	complemented
25	C₂²×C₄²	C₂²	1	1	yes	characteristic, complemented
26	C₂⁵	C₂³	1	8	yes	—
27	C₂×#41	—	4	24	no	—
28	C₂×#41	C₂³	1	48	yes	complemented
29	C₂²×Q₈	C₂³	1	6	yes	—
30	C₂²×D₄	—	4	48	no	—
31	C₂²×D₄	C₂³	1	12	yes	—
32	C₂²×D₄	C₂³	1	6	yes	—
33	C₂²×D₄	C₂³	1	24	yes	—
34	C₂²×D₄	C₂³	1	12	yes	—
35	C₂²×D₄	C₂³	1	96	yes	complemented
36	C₂³×C₄	C₂³	1	12	yes	—
37	C₂³×C₄	C₂³	1	3	yes	—
38	C₄⋊Q₈	C₂³	1	24	yes	complemented
39	#127	C₂³	1	24	yes	—
40	#127	C₂³	1	64	yes	complemented
41	#124	C₂³	1	48	yes	complemented
42	#124	C₂³	1	192	yes	complemented
43	#121	C₂³	1	96	yes	complemented
44	#121	C₂³	1	192	yes	complemented
45	#120	C₂³	1	64	yes	complemented
46	#120	C₂³	1	192	yes	complemented
47	C₄×D₄	C₂³	1	96	yes	complemented
48	#117	C₂³	1	96	yes	complemented
49	C₂×C₄⋊C₄	C₂³	1	12	yes	—
50	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
51	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
52	C₂×C₄²	C₂³	1	8	yes	—
53	C₂×C₄²	C₂³	1	3	yes	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	C₂×C₄²	C₂³	1	8	yes	complemented
56	C₂⁴	—	4	48	no	—
57	C₂⁴	C₂⁴	1	12	yes	—
58	C₂⁴	C₂⁴	1	1	yes	characteristic
59	C₂⁴	C₂⁴	1	32	yes	—
60	#41	—	4	96	no	—
61	C₂×Q₈	—	4	12	no	—
62	C₂×Q₈	C₂⁴	1	24	yes	—
63	C₂×D₄	—	2	48	no	—
64	C₂×D₄	—	4	192	no	—
65	C₂×D₄	—	4	48	no	—
66	C₂×D₄	—	4	24	no	—
67	C₂×D₄	—	4	12	no	—
68	C₂×D₄	C₂⁴	1	48	yes	—
69	C₂×D₄	C₂⁴	1	24	yes	—
70	C₂×D₄	C₂⁴	1	96	yes	—
71	C₂×D₄	C₂⁴	1	48	yes	—
72	C₂²×C₄	—	2	48	no	—
73	C₂²×C₄	—	2	12	no	—
74	C₂²×C₄	C₂⁴	1	12	yes	—
75	C₂²×C₄	C₂⁴	1	48	yes	—
76	C₂²×C₄	C₂⁴	1	3	yes	—
77	C₂²×C₄	C₂⁴	1	12	yes	—
78	C₂²×C₄	C₂⁴	1	3	yes	—
79	C₄⋊C₄	C₂⁴	1	48	yes	—
80	C₂²⋊C₄	C₂⁴	1	96	yes	—
81	C₂²⋊C₄	C₂⁴	1	96	yes	—
82	C₄²	C₂⁴	1	12	yes	—
83	C₄²	C₂⁴	1	4	yes	—
84	C₄²	C₂⁴	1	32	yes	—
85	C₂³	—	2	6	no	—
86	C₂³	—	2	24	no	—
87	C₂³	—	2	48	no	—
88	C₂³	—	4	96	no	—
89	C₂³	—	8	64	no	—
90	C₂³	C₂⁵	1	24	yes	—
91	C₂³	C₂⁵	1	2	yes	—
92	C₂³	C₂⁵	1	1	yes	characteristic
93	Q₈	—	4	24	no	—
94	D₄	—	2	96	no	—
95	D₄	—	4	96	no	—
96	D₄	—	4	48	no	—
97	D₄	—	4	24	no	—
98	C₂×C₄	—	2	24	no	—
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	2	96	no	—
101	C₂×C₄	—	2	24	no	—
102	C₂×C₄	H(5,𝔽₂)	1	12	yes	—
103	C₂×C₄	C₂⁵	1	6	yes	—
104	C₂×C₄	C₂⁵	1	24	yes	—
105	C₂×C₄	C₂⁵	1	6	yes	—
106	C₂²	—	2	6	no	—
107	C₂²	—	2	24	no	—
108	C₂²	—	2	48	no	—
109	C₂²	—	4	4	no	—
110	C₂²	—	4	48	no	—
111	C₂²	—	8	64	no	—
112	C₂²	C₂×H(5,𝔽₂)	1	6	yes	—
113	C₂²	C₂⁶	1	1	yes	characteristic
114	C₄	—	2	24	no	—
115	C₄	—	2	6	no	—
116	C₄	C₂×H(5,𝔽₂)	1	12	yes	—
117	C₂	—	4	2	no	—
118	C₂	—	4	24	no	—
119	C₂	#3549	1	4	yes	direct product factor
120	C₂	C₂²×H(5,𝔽₂)	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3549

C₂

yes

maximal, direct product factor

C₂×H(3,𝔽₄)

C₂

yes

maximal, complemented

C₂×#555

C₂

yes

maximal, complemented

C₂×D₄²

C₂

yes

maximal, complemented

C₂×#535

C₂

yes

maximal, complemented

C₂×#532

C₂

yes

maximal, complemented

C₂²×#41

C₂²

yes

complemented

C₂³×D₄

C₂²

yes

complemented

H(3,𝔽₄)

C₂²

yes

complemented

#555

C₂²

yes

complemented

D₄²

C₂²

yes

complemented

#535

C₂²

192

yes

complemented

#532

C₂²

yes

complemented

C₂×C₄⋊Q₈

C₂²

yes

complemented

C₂×#127

C₂²

yes

—

C₂×#127

C₂²

yes

complemented

C₂×#124

C₂²

yes

complemented

C₂×#124

C₂²

yes

complemented

C₂×#121

C₂²

yes

complemented

C₂×#121

C₂²

yes

complemented

C₂×#120

C₂²

yes

complemented

C₂×#120

C₂²

yes

complemented

C₂×C₄×D₄

C₂²

yes

complemented

C₂×#117

C₂²

yes

complemented

C₂²×C₄²

C₂²

yes

characteristic, complemented

C₂⁵

C₂³

yes

—

C₂×#41

—

C₂×#41

C₂³

yes

complemented

C₂²×Q₈

C₂³

yes

—

C₂²×D₄

—

C₂²×D₄

C₂³

yes

—

C₂²×D₄

C₂³

yes

—

C₂²×D₄

C₂³

yes

—

C₂²×D₄

C₂³

yes

—

C₂²×D₄

C₂³

yes

complemented

C₂³×C₄

C₂³

yes

—

C₂³×C₄

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

complemented

#127

C₂³

yes

—

#127

C₂³

yes

complemented

#124

C₂³

yes

complemented

#124

C₂³

192

yes

complemented

#121

C₂³

yes

complemented

#121

C₂³

192

yes

complemented

#120

C₂³

yes

complemented

#120

C₂³

192

yes

complemented

C₄×D₄

C₂³

yes

complemented

#117

C₂³

yes

complemented

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₂²⋊C₄

C₂³

yes

—

C₂×C₂²⋊C₄

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

complemented

C₂⁴

—

C₂⁴

yes

—

C₂⁴

yes

characteristic

C₂⁴

yes

—

#41

—

C₂×Q₈

—

C₂×Q₈

C₂⁴

yes

—

C₂×D₄

—

C₂×D₄

—

192

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂⁴

yes

—

C₂×D₄

C₂⁴

yes

—

C₂×D₄

C₂⁴

yes

—

C₂×D₄

C₂⁴

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₂²×C₄

C₂⁴

yes

—

C₄⋊C₄

C₂⁴

yes

—

C₂²⋊C₄

C₂⁴

yes

—

C₂²⋊C₄

C₂⁴

yes

—

C₄²

C₂⁴

yes

—

C₄²

C₂⁴

yes

—

C₄²

C₂⁴

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁵

yes

—

C₂³

C₂⁵

yes

—

C₂³

C₂⁵

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

H(5,𝔽₂)

yes

—

103

C₂×C₄

C₂⁵

yes

—

104

C₂×C₄

C₂⁵

yes

—

105

C₂×C₄

C₂⁵

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

C₂×H(5,𝔽₂)

yes

—

113

C₂²

C₂⁶

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

C₂×H(5,𝔽₂)

yes

—

117

C₂

—

118

C₂

—

119

C₂

#3549

yes

direct product factor

120

C₂

C₂²×H(5,𝔽₂)

yes

—

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

subgroup

G/H

center, socle

C₂³

C₂⁵

commutator, Frattini, cosocle

C₂²

C₂⁶

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ G

LECS

G ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3549	C₂	direct (N × H)
2. C₂×H(3,𝔽₄)	C₂	semidirect
3. C₂×#555	C₂	semidirect
4. C₂×D₄²	C₂	semidirect
5. C₂×#535	C₂	semidirect
6. C₂×#532	C₂	semidirect
7. C₂²×#41	C₂²	semidirect
8. C₂³×D₄	C₂²	semidirect
9. H(3,𝔽₄)	C₂²	semidirect
10. #555	C₂²	semidirect
11. D₄²	C₂²	semidirect
12. #535	C₂²	semidirect
13. #532	C₂²	semidirect
14. C₂×C₄⋊Q₈	C₂²	semidirect
16. C₂×#127	C₂²	semidirect
17. C₂×#124	C₂²	semidirect
18. C₂×#124	C₂²	semidirect
19. C₂×#121	C₂²	semidirect
20. C₂×#121	C₂²	semidirect
21. C₂×#120	C₂²	semidirect
22. C₂×#120	C₂²	semidirect
23. C₂×C₄×D₄	C₂²	semidirect
24. C₂×#117	C₂²	semidirect
25. C₂²×C₄²	C₂²	semidirect
28. C₂×#41	C₂³	semidirect
35. C₂²×D₄	C₂³	semidirect
38. C₄⋊Q₈	C₂³	semidirect
40. #127	C₂³	semidirect
41. #124	C₂³	semidirect
42. #124	C₂³	semidirect
43. #121	C₂³	semidirect
44. #121	C₂³	semidirect
45. #120	C₂³	semidirect
46. #120	C₂³	semidirect
47. C₄×D₄	C₂³	semidirect
48. #117	C₂³	semidirect
55. C₂×C₄²	C₂³	semidirect
119. C₂	#3549	direct (N × H)

H ≅ G/N

type

1. #3549

C₂

direct (N × H)

2. C₂×H(3,𝔽₄)

C₂

semidirect

3. C₂×#555

C₂

semidirect

4. C₂×D₄²

C₂

semidirect

5. C₂×#535

C₂

semidirect

6. C₂×#532

C₂

semidirect

7. C₂²×#41

C₂²

semidirect

8. C₂³×D₄

C₂²

semidirect

9. H(3,𝔽₄)

C₂²

semidirect

10. #555

C₂²

semidirect

11. D₄²

C₂²

semidirect

12. #535

C₂²

semidirect

13. #532

C₂²

semidirect

14. C₂×C₄⋊Q₈

C₂²

semidirect

16. C₂×#127

C₂²

semidirect

17. C₂×#124

C₂²

semidirect

18. C₂×#124

C₂²

semidirect

19. C₂×#121

C₂²

semidirect

20. C₂×#121

C₂²

semidirect

21. C₂×#120

C₂²

semidirect

22. C₂×#120

C₂²

semidirect

23. C₂×C₄×D₄

C₂²

semidirect

24. C₂×#117

C₂²

semidirect

25. C₂²×C₄²

C₂²

semidirect

28. C₂×#41

C₂³

semidirect

35. C₂²×D₄

C₂³

semidirect

38. C₄⋊Q₈

C₂³

semidirect

40. #127

C₂³

semidirect

41. #124

C₂³

semidirect

42. #124

C₂³

semidirect

43. #121

C₂³

semidirect

44. #121

C₂³

semidirect

45. #120

C₂³

semidirect

46. #120

C₂³

semidirect

47. C₄×D₄

C₂³

semidirect

48. #117

C₂³

semidirect

55. C₂×C₄²

C₂³

semidirect

119. C₂

#3549

direct (N × H)

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

subgroup	index
#3549	2
C₂×H(3,𝔽₄)	2
C₂×#555	2
C₂×D₄²	2
C₂×#535	2
C₂×#532	2

subgroup

index

C₂×H(3,𝔽₄)

C₂×D₄²

Subgroups — C₂×#3549 #62757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#3549 #62757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3549 #62757