Group #2994 — subgroups

Subgroups — #2994

Order 128.

182 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 128 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	#529	C₂	1	1	yes	maximal, characteristic
2	#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#436	C₂	1	1	yes	maximal, characteristic, complemented
5	#435	C₂	1	2	yes	maximal, complemented
6	C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄	C₂	1	2	yes	maximal, complemented
8	#433	C₂	1	2	yes	maximal, complemented
9	#431	C₂	1	1	yes	maximal, characteristic, complemented
10	#424	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆	C₂²	1	2	yes	—
13	C₂²×C₈	C₂²	1	2	yes	—
14	C₂²×C₈	C₂²	1	2	yes	—
15	#126	C₄	1	2	yes	—
16	#125	C₄	1	1	yes	characteristic
17	#124	C₄	1	1	yes	characteristic
18	#123	C₄	1	2	yes	—
19	C₂²⋊Q₈	C₄	1	1	yes	characteristic
20	#121	C₄	1	1	yes	characteristic
21	C₄×Q₈	C₂²	1	1	yes	characteristic
22	C₄×D₄	C₂²	1	2	yes	—
23	C₄×D₄	C₂²	1	1	yes	characteristic
24	#117	C₂²	1	2	yes	—
25	C₂×C₄²	C₂²	1	1	yes	characteristic
26	C₄⋊C₈	C₂²	1	2	yes	—
27	C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₄⋊C₈	C₂²	1	2	yes	—
31	C₄⋊C₈	C₂²	1	2	yes	—
32	C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂²⋊C₈	C₂²	1	2	yes	—
36	C₂²⋊C₈	C₂²	1	2	yes	—
37	#97	C₂²	1	2	yes	—
38	C₄×C₈	C₂²	1	1	yes	characteristic
39	C₄×C₈	C₂²	1	1	yes	characteristic
40	C₄×C₈	C₂²	1	2	yes	—
41	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄	C₂×C₄	1	2	yes	—
43	C₂×D₄	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄	C₂³	1	1	yes	characteristic
47	C₂²×C₄	C₂³	1	2	yes	—
48	M₁₆	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₈	C₂³	1	2	yes	—
55	C₂×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₈	C₂³	1	2	yes	—
57	C₄⋊C₄	C₂×C₄	1	2	yes	—
58	C₄⋊C₄	C₂×C₄	1	2	yes	—
59	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄	C₂×C₄	1	2	yes	—
68	C₄²	C₂×C₄	1	1	yes	characteristic
69	C₄²	C₂×C₄	1	1	yes	characteristic
70	C₄²	C₂³	1	1	yes	characteristic
71	C₄²	C₂³	1	2	yes	—
72	C₄²	C₂³	1	1	yes	characteristic
73	C₂³	C₂²×C₄	1	2	yes	—
74	C₂³	C₂²×C₄	1	1	yes	characteristic
75	Q₈	—	2	2	no	—
76	D₄	—	2	4	no	—
77	D₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄	C₂²×C₄	1	2	yes	—
89	C₂×C₄	C₂²×C₄	1	2	yes	—
90	C₂×C₄	C₂²×C₄	1	2	yes	—
91	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄	C₂⁴	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	2	2	no	—
95	C₈	—	2	2	no	—
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	2	2	no	—
104	C₂²	#131	1	2	yes	—
105	C₂²	C₂³×C₄	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	2	no	—
113	C₄	#131	1	2	yes	—
114	C₄	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂	—	2	2	no	—
117	C₂	—	4	2	no	—
118	C₂	#520	1	1	yes	characteristic
119	C₂	C₂×#131	1	1	yes	characteristic
120	C₂	C₂×#131	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#529

C₂

yes

maximal, characteristic

#446

C₂

yes

maximal, characteristic, complemented

C₈×Q₈

C₂

yes

maximal, characteristic, complemented

#436

C₂

yes

maximal, characteristic, complemented

#435

C₂

yes

maximal, complemented

C₈×D₄

C₂

yes

maximal, characteristic, complemented

C₈×D₄

C₂

yes

maximal, complemented

#433

C₂

yes

maximal, complemented

#431

C₂

yes

maximal, characteristic, complemented

#424

C₂

yes

maximal, complemented

C₂×C₄⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×M₁₆

C₂²

yes

—

C₂²×C₈

C₂²

yes

—

C₂²×C₈

C₂²

yes

—

#126

C₄

yes

—

#125

C₄

yes

characteristic

#124

C₄

yes

characteristic

#123

C₄

yes

—

C₂²⋊Q₈

C₄

yes

characteristic

#121

C₄

yes

characteristic

C₄×Q₈

C₂²

yes

characteristic

C₄×D₄

C₂²

yes

—

C₄×D₄

C₂²

yes

characteristic

#117

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

—

C₄⋊C₈

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

—

C₄⋊C₈

C₂²

yes

—

C₄⋊C₈

C₂²

yes

characteristic

C₂²⋊C₈

C₂²

yes

characteristic

C₂²⋊C₈

C₂²

yes

characteristic

C₂²⋊C₈

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

#97

C₂²

yes

—

C₄×C₈

C₂²

yes

characteristic

C₄×C₈

C₂²

yes

characteristic

C₄×C₈

C₂²

yes

—

C₂×Q₈

C₂×C₄

yes

characteristic

C₂×D₄

C₂×C₄

yes

—

C₂×D₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₄

C₂×C₄

yes

characteristic

C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₄

C₂×C₄

yes

characteristic

C₄⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₄²

C₂×C₄

yes

characteristic

C₄²

C₂×C₄

yes

characteristic

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

—

C₄²

C₂³

yes

characteristic

C₂³

C₂²×C₄

yes

—

C₂³

C₂²×C₄

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

#131

yes

—

105

C₂²

C₂³×C₄

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#131

yes

—

114

C₄

H(5,𝔽₂)

yes

characteristic

115

C₄

2₋¹⁺⁴

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

#520

yes

characteristic

119

C₂

C₂×#131

yes

characteristic

120

C₂

C₂×#131

yes

characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator, socle	C₂²	C₂³×C₄
prime radical	#121	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂×C₄

C₂⁴

commutator, socle

C₂²

C₂³×C₄

C₄

G ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ G

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #446	C₂	semidirect
3. C₈×Q₈	C₂	semidirect
4. #436	C₂	semidirect
5. #435	C₂	semidirect
6. C₈×D₄	C₂	semidirect
7. C₈×D₄	C₂	semidirect
8. #433	C₂	semidirect
9. #431	C₂	semidirect
10. #424	C₂	semidirect
11. C₂×C₄⋊C₈	C₂	semidirect

H ≅ G/N

type

2. #446

C₂

semidirect

3. C₈×Q₈

C₂

semidirect

4. #436

C₂

semidirect

5. #435

C₂

semidirect

6. C₈×D₄

C₂

semidirect

7. C₈×D₄

C₂

semidirect

8. #433

C₂

semidirect

9. #431

C₂

semidirect

10. #424

C₂

semidirect

11. C₂×C₄⋊C₈

C₂

semidirect

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

subgroup	index
#529	2
#446	2
C₈×Q₈	2
#436	2
#435	2
C₈×D₄	2
#433	2
#431	2
#424	2
C₂×C₄⋊C₈	2

subgroup

index

C₈×Q₈

C₈×D₄

C₂×C₄⋊C₈

Subgroups — #2994

Conjugacy classes of subgroups

Complements

Maximal subgroups