Group #3469 — subgroups

Subgroups — C₂×#551 #3469

Order 128.

664 conjugacy classes of subgroups; only the 404 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 404 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	#551	C₂	1	16	yes	maximal, direct product factor
2	C₂×#126	C₂	1	2	yes	maximal, complemented
3	C₂×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#123	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#123	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
7	C₂×#120	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×D₄	C₂	1	2	yes	maximal, complemented
9	C₂×#117	C₂	1	2	yes	maximal, complemented
10	C₂²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₂⁵	C₂²	1	1	yes	characteristic
12	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄	C₂²	1	2	yes	—
14	C₂²×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₄	C₂²	1	2	yes	—
16	C₂³×C₄	C₂²	1	2	yes	—
17	C₂³×C₄	C₂²	1	1	yes	characteristic, complemented
18	#126	C₂²	1	16	yes	complemented
19	#124	C₂²	1	8	yes	complemented
20	#123	C₂²	1	8	yes	complemented
21	#123	C₂²	1	16	yes	complemented
22	C₂²⋊Q₈	C₂²	1	16	yes	complemented
23	#120	C₂²	1	8	yes	—
24	C₄×D₄	C₂²	1	16	yes	complemented
25	#117	C₂²	1	16	yes	complemented
26	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
27	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
28	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
29	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
30	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄	C₂²	1	16	yes	complemented
37	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
38	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₄²	C₂²	1	2	yes	complemented
41	C₂⁴	—	2	8	no	—
42	C₂⁴	—	4	2	no	—
43	C₂⁴	C₂³	1	1	yes	characteristic
44	C₂⁴	C₂³	1	1	yes	characteristic
45	C₂⁴	C₂³	1	4	yes	—
46	C₂⁴	C₂³	1	2	yes	—
47	C₂×Q₈	—	4	2	no	—
48	C₂×Q₈	C₂³	1	4	yes	complemented
49	C₂×D₄	—	2	8	no	—
50	C₂×D₄	—	4	2	no	—
51	C₂×D₄	C₂³	1	8	yes	—
52	C₂×D₄	C₂³	1	4	yes	—
53	C₂²×C₄	—	2	8	no	—
54	C₂²×C₄	—	4	4	no	—
55	C₂²×C₄	—	4	2	no	—
56	C₂²×C₄	C₂³	1	8	yes	—
57	C₂²×C₄	C₂³	1	2	yes	—
58	C₂²×C₄	C₂³	1	2	yes	—
59	C₂²×C₄	C₂³	1	2	yes	—
60	C₂²×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₄	C₂³	1	2	yes	—
62	C₂²×C₄	C₂³	1	2	yes	—
63	C₂²×C₄	C₂³	1	8	yes	—
64	C₂²×C₄	C₂³	1	4	yes	complemented
65	C₄⋊C₄	C₂³	1	8	yes	complemented
66	C₄⋊C₄	C₂³	1	8	yes	complemented
67	C₄⋊C₄	C₂³	1	8	yes	complemented
68	C₄⋊C₄	C₂³	1	8	yes	complemented
69	C₂²⋊C₄	C₂³	1	8	yes	—
70	C₂²⋊C₄	C₂³	1	8	yes	—
71	C₂²⋊C₄	C₂³	1	8	yes	—
72	C₂²⋊C₄	C₂³	1	8	yes	—
73	C₂²⋊C₄	C₂³	1	8	yes	—
74	C₂²⋊C₄	C₂³	1	4	yes	—
75	C₂²⋊C₄	C₂³	1	8	yes	complemented
76	C₂²⋊C₄	C₂³	1	4	yes	complemented
77	C₄²	C₂³	1	4	yes	complemented
78	C₄²	C₂³	1	8	yes	complemented
79	C₂³	—	2	4	no	—
80	C₂³	—	2	4	no	—
81	C₂³	—	2	4	no	—
82	C₂³	—	2	16	no	—
83	C₂³	—	2	4	no	—
84	C₂³	—	2	2	no	—
85	C₂³	—	2	2	no	—
86	C₂³	—	4	4	no	—
87	C₂³	—	4	16	no	—
88	C₂³	#41	1	8	yes	—
89	C₂³	C₂⁴	1	2	yes	—
90	C₂³	C₂⁴	1	2	yes	—
91	C₂³	C₂⁴	1	4	yes	—
92	C₂³	C₂⁴	1	1	yes	characteristic
93	Q₈	—	4	4	no	—
94	D₄	—	2	16	no	—
95	D₄	—	4	4	no	—
96	C₂×C₄	—	2	16	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	8	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂⁴	1	4	yes	—
106	C₂×C₄	C₂⁴	1	4	yes	—
107	C₂×C₄	C₂⁴	1	4	yes	—
108	C₂×C₄	C₂⁴	1	2	yes	—
109	C₂×C₄	C₂⁴	1	4	yes	—
110	C₂×C₄	C₂⁴	1	4	yes	—
111	C₂²	—	2	16	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	2	4	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	16	no	—
119	C₂²	—	4	4	no	—
120	C₂²	—	4	4	no	—
121	C₂²	C₂×#41	1	8	yes	—
122	C₂²	C₂×#41	1	4	yes	—
123	C₂²	H(5,𝔽₂)	1	2	yes	—
124	C₂²	C₂⁵	1	1	yes	characteristic
125	C₄	—	2	4	no	—
126	C₄	—	2	4	no	—
127	C₄	—	2	4	no	—
128	C₄	—	2	4	no	—
129	C₄	—	2	4	no	—
130	C₄	—	2	2	no	—
131	C₂	—	2	8	no	—
132	C₂	—	4	2	no	—
133	C₂	—	4	2	no	—
134	C₂	#551	1	4	yes	direct product factor
135	C₂	C₂²×#41	1	2	yes	—
136	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#551

C₂

yes

maximal, direct product factor

C₂×#126

C₂

yes

maximal, complemented

C₂×#124

C₂

yes

maximal, characteristic, complemented

C₂×#123

C₂

yes

maximal, characteristic, complemented

C₂×#123

C₂

yes

maximal, complemented

C₂×C₂²⋊Q₈

C₂

yes

maximal, complemented

C₂×#120

C₂

yes

maximal, characteristic

C₂×C₄×D₄

C₂

yes

maximal, complemented

C₂×#117

C₂

yes

maximal, complemented

C₂²×C₂²⋊C₄

C₂

yes

maximal, complemented

C₂⁵

C₂²

yes

characteristic

C₂²×Q₈

C₂²

yes

characteristic, complemented

C₂²×D₄

C₂²

yes

—

C₂²×D₄

C₂²

yes

characteristic

C₂³×C₄

C₂²

yes

—

C₂³×C₄

C₂²

yes

—

C₂³×C₄

C₂²

yes

characteristic, complemented

#126

C₂²

yes

complemented

#124

C₂²

yes

complemented

#123

C₂²

yes

complemented

#123

C₂²

yes

complemented

C₂²⋊Q₈

C₂²

yes

complemented

#120

C₂²

yes

—

C₄×D₄

C₂²

yes

complemented

#117

C₂²

yes

complemented

C₂×C₄⋊C₄

C₂²

yes

complemented

C₂×C₄⋊C₄

C₂²

yes

complemented

C₂×C₄⋊C₄

C₂²

yes

complemented

C₂×C₄⋊C₄

C₂²

yes

complemented

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

characteristic

C₂×C₂²⋊C₄

C₂²

yes

complemented

C₂×C₂²⋊C₄

C₂²

yes

complemented

C₂×C₂²⋊C₄

C₂²

yes

characteristic, complemented

C₂×C₄²

C₂²

yes

characteristic, complemented

C₂×C₄²

C₂²

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

C₂³

yes

characteristic

C₂⁴

C₂³

yes

characteristic

C₂⁴

C₂³

yes

—

C₂⁴

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

C₂³

yes

complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂³

yes

—

C₂×D₄

C₂³

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

complemented

C₄⋊C₄

C₂³

yes

complemented

C₄⋊C₄

C₂³

yes

complemented

C₄⋊C₄

C₂³

yes

complemented

C₄⋊C₄

C₂³

yes

complemented

C₂²⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

complemented

C₂²⋊C₄

C₂³

yes

complemented

C₄²

C₂³

yes

complemented

C₄²

C₂³

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#41

yes

—

C₂³

C₂⁴

yes

—

C₂³

C₂⁴

yes

—

C₂³

C₂⁴

yes

—

C₂³

C₂⁴

yes

characteristic

Q₈

—

D₄

—

D₄

—

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

C₂⁴

yes

—

106

C₂×C₄

C₂⁴

yes

—

107

C₂×C₄

C₂⁴

yes

—

108

C₂×C₄

C₂⁴

yes

—

109

C₂×C₄

C₂⁴

yes

—

110

C₂×C₄

C₂⁴

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×#41

yes

—

122

C₂²

C₂×#41

yes

—

123

C₂²

H(5,𝔽₂)

yes

—

124

C₂²

C₂⁵

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₂

—

132

C₂

—

133

C₂

—

134

C₂

#551

yes

direct product factor

135

C₂

C₂²×#41

yes

—

136

C₂

C₂×H(5,𝔽₂)

yes

characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂×#120	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂³

C₂⁴

commutator, Frattini, cosocle

C₂²

C₂⁵

C₂

G ▷ C₂² ▷ C₁

C₁ ◁ C₂³ ◁ G

G ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #551	C₂	direct (N × H)
2. C₂×#126	C₂	semidirect
3. C₂×#124	C₂	semidirect
4. C₂×#123	C₂	semidirect
5. C₂×#123	C₂	semidirect
6. C₂×C₂²⋊Q₈	C₂	semidirect
8. C₂×C₄×D₄	C₂	semidirect
9. C₂×#117	C₂	semidirect
10. C₂²×C₂²⋊C₄	C₂	semidirect
12. C₂²×Q₈	C₂²	semidirect
17. C₂³×C₄	C₂²	semidirect
18. #126	C₂²	semidirect
19. #124	C₂²	semidirect
20. #123	C₂²	semidirect
21. #123	C₂²	semidirect
22. C₂²⋊Q₈	C₂²	semidirect
24. C₄×D₄	C₂²	semidirect
25. #117	C₂²	semidirect
26. C₂×C₄⋊C₄	C₂²	semidirect
27. C₂×C₄⋊C₄	C₂²	semidirect
28. C₂×C₄⋊C₄	C₂²	semidirect
29. C₂×C₄⋊C₄	C₂²	semidirect
36. C₂×C₂²⋊C₄	C₂²	semidirect
37. C₂×C₂²⋊C₄	C₂²	semidirect
38. C₂×C₂²⋊C₄	C₂²	semidirect
39. C₂×C₄²	C₂²	semidirect
40. C₂×C₄²	C₂²	semidirect
48. C₂×Q₈	C₂³	semidirect
64. C₂²×C₄	C₂³	semidirect
65. C₄⋊C₄	C₂³	semidirect
66. C₄⋊C₄	C₂³	semidirect
67. C₄⋊C₄	C₂³	semidirect
68. C₄⋊C₄	C₂³	semidirect
75. C₂²⋊C₄	C₂³	semidirect
76. C₂²⋊C₄	C₂³	semidirect
77. C₄²	C₂³	semidirect
78. C₄²	C₂³	semidirect
134. C₂	#551	direct (N × H)

H ≅ G/N

type

1. #551

C₂

direct (N × H)

2. C₂×#126

C₂

semidirect

3. C₂×#124

C₂

semidirect

4. C₂×#123

C₂

semidirect

5. C₂×#123

C₂

semidirect

6. C₂×C₂²⋊Q₈

C₂

semidirect

8. C₂×C₄×D₄

C₂

semidirect

9. C₂×#117

C₂

semidirect

10. C₂²×C₂²⋊C₄

C₂

semidirect

12. C₂²×Q₈

C₂²

semidirect

17. C₂³×C₄

C₂²

semidirect

18. #126

C₂²

semidirect

19. #124

C₂²

semidirect

20. #123

C₂²

semidirect

21. #123

C₂²

semidirect

22. C₂²⋊Q₈

C₂²

semidirect

24. C₄×D₄

C₂²

semidirect

25. #117

C₂²

semidirect

26. C₂×C₄⋊C₄

C₂²

semidirect

27. C₂×C₄⋊C₄

C₂²

semidirect

28. C₂×C₄⋊C₄

C₂²

semidirect

29. C₂×C₄⋊C₄

C₂²

semidirect

36. C₂×C₂²⋊C₄

C₂²

semidirect

37. C₂×C₂²⋊C₄

C₂²

semidirect

38. C₂×C₂²⋊C₄

C₂²

semidirect

39. C₂×C₄²

C₂²

semidirect

40. C₂×C₄²

C₂²

semidirect

48. C₂×Q₈

C₂³

semidirect

64. C₂²×C₄

C₂³

semidirect

65. C₄⋊C₄

C₂³

semidirect

66. C₄⋊C₄

C₂³

semidirect

67. C₄⋊C₄

C₂³

semidirect

68. C₄⋊C₄

C₂³

semidirect

75. C₂²⋊C₄

C₂³

semidirect

76. C₂²⋊C₄

C₂³

semidirect

77. C₄²

C₂³

semidirect

78. C₄²

C₂³

semidirect

134. C₂

#551

direct (N × H)

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

subgroup	index
#551	2
C₂×#126	2
C₂×#124	2
C₂×#123	2
C₂×C₂²⋊Q₈	2
C₂×#120	2
C₂×C₄×D₄	2
C₂×#117	2
C₂²×C₂²⋊C₄	2

subgroup

index

C₂×C₂²⋊Q₈

C₂×C₄×D₄

C₂²×C₂²⋊C₄

Subgroups — C₂×#551 #3469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#551 #3469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#551 #3469