Group #3097 — subgroups

Subgroups — #3097

Order 128.

193 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 92 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	#536	C₂	1	1	yes	maximal, characteristic
2	#520	C₂	1	1	yes	maximal, characteristic
3	#485	C₂	1	2	yes	maximal, complemented
4	#485	C₂	1	2	yes	maximal, complemented
5	#484	C₂	1	2	yes	maximal, complemented
6	#483	C₂	1	2	yes	maximal, complemented
7	#428	C₂	1	1	yes	maximal, characteristic, complemented
8	#427	C₂	1	1	yes	maximal, characteristic, complemented
9	#419	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#103	C₂	1	1	yes	maximal, characteristic, complemented
11	#408	C₂	1	1	yes	maximal, characteristic
12	C₂×#41	C₂²	1	1	yes	characteristic
13	C₂²×Q₈	C₂²	1	1	yes	characteristic
14	C₂×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈	—	2	1	no	—
17	#124	—	2	1	no	—
18	#123	—	2	2	no	—
19	C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₄×Q₈	—	2	1	no	—
22	C₄×D₄	—	2	2	no	—
23	C₄×D₄	—	2	1	no	—
24	#117	C₂²	1	1	yes	characteristic
25	#117	C₂²	1	2	yes	—
26	#117	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄	C₂²	1	2	yes	—
29	#107	C₂²	1	1	yes	characteristic
30	#107	C₂²	1	1	yes	characteristic
31	#107	C₂²	1	2	yes	—
32	#106	C₂²	1	1	yes	characteristic
33	#106	C₂²	1	1	yes	characteristic
34	#106	C₂²	1	2	yes	—
35	#103	C₂²	1	2	yes	—
36	#103	C₂²	1	2	yes	—
37	#103	C₂²	1	2	yes	—
38	#103	C₂²	1	2	yes	—
39	C₂²⋊C₈	C₂²	1	2	yes	—
40	C₂²⋊C₈	C₂²	1	2	yes	—
41	#41	—	2	2	no	—
42	#41	—	2	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	—	2	2	no	—
45	C₂×Q₈	—	2	2	no	—
46	C₂×Q₈	D₄	1	1	yes	characteristic
47	C₂×Q₈	C₂³	1	2	yes	—
48	C₂×D₄	D₄	1	2	yes	—
49	C₂×D₄	D₄	1	1	yes	characteristic
50	C₂²×C₄	—	2	1	no	—
51	C₂²×C₄	—	2	2	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	C₂³	1	2	yes	—
54	C₂²×C₄	C₂³	1	1	yes	characteristic
55	M₁₆	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₈	C₂³	1	2	yes	—
59	C₂×C₈	C₂³	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	2	1	no	—
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	C₂³	1	2	yes	—
65	C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄	—	2	2	no	—
70	C₂²⋊C₄	—	2	1	no	—
71	C₂²⋊C₄	—	2	1	no	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₂²⋊C₄	—	2	2	no	—
74	C₄²	—	2	2	no	—
75	C₄²	—	2	1	no	—
76	C₄²	—	2	1	no	—
77	C₂³	C₂×D₄	1	2	yes	—
78	C₂³	C₂×D₄	1	1	yes	characteristic
79	Q₈	—	2	2	no	—
80	Q₈	—	2	2	no	—
81	Q₈	—	2	2	no	—
82	Q₈	—	4	2	no	—
83	D₄	—	2	4	no	—
84	D₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	8	1	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	C₂×D₄	1	2	yes	—
97	C₂×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄	#41	1	2	yes	—
99	C₂×C₄	#41	1	2	yes	—
100	C₂×C₄	C₂⁴	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	2	2	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	#123	1	2	yes	—
110	C₂²	C₂²×D₄	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	4	2	no	—
113	C₄	—	4	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	1	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	1	no	—
118	C₄	#123	1	2	yes	—
119	C₄	C₂×#41	1	1	yes	characteristic
120	C₄	C₂×#41	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	4	2	no	—
123	C₂	C₂×#123	1	1	yes	characteristic
124	C₂	#577	1	1	yes	characteristic
125	C₂	#578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#536

C₂

yes

maximal, characteristic

#520

C₂

yes

maximal, characteristic

#485

C₂

yes

maximal, complemented

#485

C₂

yes

maximal, complemented

#484

C₂

yes

maximal, complemented

#483

C₂

yes

maximal, complemented

#428

C₂

yes

maximal, characteristic, complemented

#427

C₂

yes

maximal, characteristic, complemented

#419

C₂

yes

maximal, characteristic, complemented

C₂×#103

C₂

yes

maximal, characteristic, complemented

#408

C₂

yes

maximal, characteristic

C₂×#41

C₂²

yes

characteristic

C₂²×Q₈

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

characteristic

C₂²×C₈

C₂²

yes

characteristic

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

C₂²

yes

characteristic

#117

C₂²

yes

—

#117

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

—

#107

C₂²

yes

characteristic

#107

C₂²

yes

characteristic

#107

C₂²

yes

—

#106

C₂²

yes

characteristic

#106

C₂²

yes

characteristic

#106

C₂²

yes

—

#103

C₂²

yes

—

#103

C₂²

yes

—

#103

C₂²

yes

—

#103

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄

yes

characteristic

C₂×Q₈

C₂³

yes

—

C₂×D₄

D₄

yes

—

C₂×D₄

D₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

characteristic

C₄⋊C₄

C₂³

yes

characteristic

C₄⋊C₄

C₂³

yes

characteristic

C₄⋊C₄

C₂³

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂×D₄

yes

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

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×D₄

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

#41

yes

—

C₂×C₄

#41

yes

—

100

C₂×C₄

C₂⁴

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#123

yes

—

110

C₂²

C₂²×D₄

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#123

yes

—

119

C₄

C₂×#41

yes

characteristic

120

C₄

C₂×#41

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

C₂×#123

yes

characteristic

124

C₂

#577

yes

characteristic

125

C₂

#578

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

prime radical

C₂×#41

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #485	C₂	semidirect
4. #485	C₂	semidirect
5. #484	C₂	semidirect
6. #483	C₂	semidirect
7. #428	C₂	semidirect
8. #427	C₂	semidirect
9. #419	C₂	semidirect
10. C₂×#103	C₂	semidirect

H ≅ G/N

type

3. #485

C₂

semidirect

4. #485

C₂

semidirect

5. #484

C₂

semidirect

6. #483

C₂

semidirect

7. #428

C₂

semidirect

8. #427

C₂

semidirect

9. #419

C₂

semidirect

10. C₂×#103

C₂

semidirect

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

subgroup	index
#536	2
#520	2
#485	2
#484	2
#483	2
#428	2
#427	2
#419	2
C₂×#103	2
#408	2

subgroup

index

Subgroups — #3097

Conjugacy classes of subgroups

Complements

Maximal subgroups