Group #2808 — subgroups

Subgroups — #2808

Order 128.

214 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 88 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	#397	C₂	1	2	yes	maximal, complemented
2	#394	C₂	1	2	yes	maximal
3	#393	C₂	1	2	yes	maximal, complemented
4	#391	C₂	1	1	yes	maximal, characteristic, complemented
5	#390	C₂	1	1	yes	maximal, characteristic
6	#389	C₂	1	1	yes	maximal, characteristic, complemented
7	#388	C₂	1	2	yes	maximal, complemented
8	#388	C₂	1	2	yes	maximal, complemented
9	#387	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄²	C₂²	1	2	yes	—
24	C₂×C₄²	C₂²	1	1	yes	characteristic
25	#95	C₂²	1	2	yes	—
26	#95	C₂²	1	2	yes	—
27	#95	C₂²	1	2	yes	—
28	#95	C₂²	1	2	yes	—
29	#95	C₂²	1	2	yes	—
30	#95	C₂²	1	2	yes	—
31	C₂⁴	C₂³	1	2	yes	—
32	C₂×Q₈	—	2	2	no	—
33	C₂×Q₈	—	4	1	no	—
34	C₂×Q₈	—	4	1	no	—
35	C₂×D₄	—	2	2	no	—
36	C₂×D₄	—	4	1	no	—
37	C₂×D₄	—	4	1	no	—
38	C₂²×C₄	C₂³	1	2	yes	—
39	C₂²×C₄	C₂³	1	2	yes	—
40	C₂²×C₄	C₂³	1	2	yes	—
41	C₂²×C₄	C₂³	1	2	yes	—
42	C₂²×C₄	C₂³	1	2	yes	—
43	C₂²×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₄	C₂³	1	1	yes	characteristic
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	2	4	no	—
48	C₄⋊C₄	—	4	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄⋊C₄	—	4	1	no	—
51	C₂²⋊C₄	—	2	4	no	—
52	C₂²⋊C₄	—	2	4	no	—
53	C₂²⋊C₄	—	4	2	no	—
54	C₂²⋊C₄	—	4	2	no	—
55	C₂²⋊C₄	—	4	2	no	—
56	C₂²⋊C₄	—	4	2	no	—
57	C₄²	—	4	1	no	—
58	C₄²	—	4	2	no	—
59	C₂³	—	2	2	no	—
60	C₂³	—	2	2	no	—
61	C₂³	—	2	2	no	—
62	C₂³	—	2	2	no	—
63	C₂³	—	2	2	no	—
64	C₂³	—	2	2	no	—
65	C₂³	—	2	2	no	—
66	C₂³	C₂⁴	1	1	yes	characteristic
67	Q₈	—	4	4	no	—
68	D₄	—	4	4	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	#41	1	2	yes	—
91	C₂×C₄	#41	1	2	yes	—
92	C₂×C₄	#41	1	2	yes	—
93	C₂²	—	4	2	no	—
94	C₂²	—	4	2	no	—
95	C₂²	—	4	2	no	—
96	C₂²	—	4	2	no	—
97	C₂²	—	4	2	no	—
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	C₂×#41	1	1	yes	characteristic
101	C₂²	C₂×#41	1	1	yes	characteristic
102	C₂²	C₂×#41	1	1	yes	characteristic
103	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	2	2	no	—
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	—	4	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₂	—	8	2	no	—
116	C₂	#538	1	1	yes	characteristic
117	C₂	#538	1	1	yes	characteristic
118	C₂	#539	1	1	yes	characteristic
119	C₂	#555	1	1	yes	characteristic
120	C₂	#556	1	1	yes	characteristic
121	C₂	#559	1	1	yes	characteristic
122	C₂	#562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#397

C₂

yes

maximal, complemented

#394

C₂

yes

maximal

#393

C₂

yes

maximal, complemented

#391

C₂

yes

maximal, characteristic, complemented

#390

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic, complemented

#388

C₂

yes

maximal, complemented

#388

C₂

yes

maximal, complemented

#387

C₂

yes

maximal, complemented

C₂²×Q₈

C₂²

yes

characteristic

C₂²×D₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

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

—

C₂×C₄²

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

C₂⁴

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂³

yes

—

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

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁴

yes

characteristic

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#41

yes

—

C₂×C₄

#41

yes

—

C₂×C₄

#41

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂×#41

yes

characteristic

101

C₂²

C₂×#41

yes

characteristic

102

C₂²

C₂×#41

yes

characteristic

103

C₂²

H(5,𝔽₂)

yes

characteristic

104

C₂²

H(5,𝔽₂)

yes

characteristic

105

C₂²

H(5,𝔽₂)

yes

characteristic

106

C₂²

2₋¹⁺⁴

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₂

—

116

C₂

#538

yes

characteristic

117

C₂

#538

yes

characteristic

118

C₂

#539

yes

characteristic

119

C₂

#555

yes

characteristic

120

C₂

#556

yes

characteristic

121

C₂

#559

yes

characteristic

122

C₂

#562

yes

characteristic

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

subgroup

G/H

center, commutator, Frattini, socle, cosocle

C₂³

C₂⁴

prime radical

C₂²×D₄

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ G

LECS

G ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
1. #397	C₂	semidirect
3. #393	C₂	semidirect
4. #391	C₂	semidirect
6. #389	C₂	semidirect
7. #388	C₂	semidirect
8. #388	C₂	semidirect
9. #387	C₂	semidirect

H ≅ G/N

type

1. #397

C₂

semidirect

3. #393

C₂

semidirect

4. #391

C₂

semidirect

6. #389

C₂

semidirect

7. #388

C₂

semidirect

8. #388

C₂

semidirect

9. #387

C₂

semidirect

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

subgroup	index
#397	2
#394	2
#393	2
#391	2
#390	2
#389	2
#388	2
#387	2

subgroup

index

Subgroups — #2808

Conjugacy classes of subgroups

Complements

Maximal subgroups