Group #2604 — subgroups

Subgroups — #2604

Order 128.

235 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 100 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	C₂×#126	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#400	C₂	1	1	yes	maximal, characteristic, complemented
4	#399	C₂	1	1	yes	maximal, characteristic
5	#395	C₂	1	1	yes	maximal, characteristic, complemented
6	#393	C₂	1	1	yes	maximal, characteristic
7	#389	C₂	1	2	yes	maximal, complemented
8	#388	C₂	1	2	yes	maximal
9	#388	C₂	1	2	yes	maximal
10	#387	C₂	1	2	yes	maximal, complemented
11	#376	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈	C₂²	1	1	yes	characteristic
13	#126	—	2	4	no	—
14	C₄×Q₈	—	2	4	no	—
15	C₂×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄²	C₂²	1	2	yes	—
28	C₂×C₄²	C₂²	1	2	yes	—
29	C₂×C₄²	C₂²	1	1	yes	characteristic
30	#95	C₂²	1	2	yes	—
31	#95	C₂²	1	1	yes	characteristic
32	#95	C₂²	1	1	yes	characteristic
33	#95	C₂²	1	2	yes	—
34	#95	C₂²	1	2	yes	—
35	#95	C₂²	1	2	yes	—
36	#95	C₂²	1	2	yes	—
37	C₂⁴	C₂³	1	1	yes	characteristic
38	C₂×Q₈	—	2	2	no	—
39	C₂×Q₈	—	2	4	no	—
40	C₂²×C₄	C₂³	1	2	yes	—
41	C₂²×C₄	C₂³	1	1	yes	characteristic
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	C₂²×C₄	C₂³	1	1	yes	characteristic
47	C₂²×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₄	C₂³	1	2	yes	—
49	C₂²×C₄	C₂³	1	2	yes	—
50	C₄⋊C₄	—	2	4	no	—
51	C₄⋊C₄	—	2	2	no	—
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	2	4	no	—
57	C₂²⋊C₄	—	2	2	no	—
58	C₂²⋊C₄	—	2	2	no	—
59	C₂²⋊C₄	—	2	2	no	—
60	C₂²⋊C₄	—	2	4	no	—
61	C₂²⋊C₄	—	2	4	no	—
62	C₄²	—	2	4	no	—
63	C₄²	—	2	4	no	—
64	C₄²	D₄	1	4	yes	—
65	C₂³	—	2	1	no	—
66	C₂³	—	2	2	no	—
67	C₂³	—	2	2	no	—
68	C₂³	—	2	1	no	—
69	C₂³	—	2	1	no	—
70	C₂³	C₂⁴	1	1	yes	characteristic
71	Q₈	—	2	8	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	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	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	C₂×D₄	1	2	yes	—
94	C₂×C₄	C₂×D₄	1	2	yes	—
95	C₂×C₄	C₂×D₄	1	2	yes	—
96	C₂×C₄	#41	1	4	yes	—
97	C₂×C₄	#41	1	4	yes	—
98	C₂²	—	4	1	no	—
99	C₂²	—	4	1	no	—
100	C₂²	—	4	1	no	—
101	C₂²	—	4	2	no	—
102	C₂²	—	4	2	no	—
103	C₂²	C₂²×D₄	1	1	yes	characteristic
104	C₂²	C₂×#41	1	2	yes	—
105	C₂²	C₂×#41	1	2	yes	—
106	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
107	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	2	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	4	no	—
113	C₄	—	4	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	1	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	2	no	—
118	C₂	—	8	1	no	—
119	C₂	#525	1	1	yes	characteristic
120	C₂	#532	1	1	yes	characteristic
121	C₂	#536	1	1	yes	characteristic
122	C₂	#552	1	2	yes	—
123	C₂	#556	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#126

C₂

yes

maximal, characteristic

C₂×C₄×Q₈

C₂

yes

maximal, characteristic, complemented

#400

C₂

yes

maximal, characteristic, complemented

#399

C₂

yes

maximal, characteristic

#395

C₂

yes

maximal, characteristic, complemented

#393

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, complemented

#388

C₂

yes

maximal

#388

C₂

yes

maximal

#387

C₂

yes

maximal, complemented

#376

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈

C₂²

yes

characteristic

#126

—

C₄×Q₈

—

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×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

characteristic

C₂×C₂²⋊C₄

C₂²

yes

characteristic

C₂×C₂²⋊C₄

C₂²

yes

characteristic

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²

C₂²

yes

—

C₂×C₄²

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

#95

C₂²

yes

—

#95

C₂²

yes

characteristic

#95

C₂²

yes

characteristic

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

C₂⁴

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

D₄

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁴

yes

characteristic

Q₈

—

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

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

#41

yes

—

C₂×C₄

#41

yes

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂²×D₄

yes

characteristic

104

C₂²

C₂×#41

yes

—

105

C₂²

C₂×#41

yes

—

106

C₂²

2₋¹⁺⁴

yes

characteristic

107

C₂²

2₋¹⁺⁴

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₂

—

119

C₂

#525

yes

characteristic

120

C₂

#532

yes

characteristic

121

C₂

#536

yes

characteristic

122

C₂

#552

yes

—

123

C₂

#556

yes

—

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

subgroup

G/H

center, commutator, Frattini, socle, cosocle

C₂³

C₂⁴

prime radical

C₂⁴

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ G

LECS

G ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
2. C₂×C₄×Q₈	C₂	semidirect
3. #400	C₂	semidirect
5. #395	C₂	semidirect
7. #389	C₂	semidirect
10. #387	C₂	semidirect
11. #376	C₂	semidirect

H ≅ G/N

type

2. C₂×C₄×Q₈

C₂

semidirect

3. #400

C₂

semidirect

5. #395

C₂

semidirect

7. #389

C₂

semidirect

10. #387

C₂

semidirect

11. #376

C₂

semidirect

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

subgroup	index
C₂×#126	2
C₂×C₄×Q₈	2
#400	2
#399	2
#395	2
#393	2
#389	2
#388	2
#387	2
#376	2

subgroup

index

C₂×C₄×Q₈

Subgroups — #2604

Conjugacy classes of subgroups

Complements

Maximal subgroups