Group #3374 — subgroups

Subgroups — #3374

Order 128.

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

normal subgroups only

This group has 163 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	#557	C₂	1	1	yes	maximal, characteristic
2	#557	C₂	1	1	yes	maximal, characteristic
3	#491	C₂	1	2	yes	maximal
4	#487	C₂	1	1	yes	maximal, characteristic
5	#479	C₂	1	2	yes	maximal
6	#475	C₂	1	1	yes	maximal, characteristic
7	#462	C₂	1	2	yes	maximal
8	#462	C₂	1	1	yes	maximal, characteristic
9	#446	C₂	1	1	yes	maximal, characteristic
10	#441	C₂	1	2	yes	maximal
11	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆	C₂²	1	2	yes	—
14	C₄⋊Q₈	—	2	1	no	—
15	C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈	C₂²	1	2	yes	—
17	C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	#125	—	2	2	no	—
19	#125	—	2	1	no	—
20	#125	—	2	2	no	—
21	#125	C₂²	1	2	yes	—
22	C₄×Q₈	—	2	1	no	—
23	C₄×Q₈	—	2	1	no	—
24	C₄×Q₈	C₂²	1	1	yes	characteristic
25	C₄×Q₈	C₂²	1	1	yes	characteristic
26	C₄×Q₈	C₂²	1	2	yes	—
27	C₄×Q₈	C₂²	1	2	yes	—
28	C₄×Q₈	C₂²	1	1	yes	characteristic
29	#107	C₂²	1	1	yes	characteristic
30	#106	C₂²	1	2	yes	—
31	C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₄⋊C₈	C₂²	1	2	yes	—
33	#103	C₂²	1	1	yes	characteristic
34	#103	C₂²	1	1	yes	characteristic
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	#97	C₂²	1	2	yes	—
40	C₄×C₈	C₂²	1	1	yes	characteristic
41	C₂×Q₈	—	2	1	no	—
42	C₂×Q₈	D₄	1	1	yes	characteristic
43	C₂×Q₈	C₂³	1	1	yes	characteristic
44	C₂×Q₈	C₂³	1	2	yes	—
45	C₂×Q₈	C₂³	1	1	yes	characteristic
46	Q₁₆	—	2	2	no	—
47	Q₁₆	—	2	4	no	—
48	C₂×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₈	C₂³	1	2	yes	—
51	C₄⋊C₄	—	2	2	no	—
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	2	1	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₄⋊C₄	D₄	1	2	yes	—
64	C₄⋊C₄	C₂³	1	2	yes	—
65	C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₄²	—	2	2	no	—
68	C₄²	—	2	2	no	—
69	C₄²	—	2	1	no	—
70	C₄²	—	2	1	no	—
71	C₄²	C₂³	1	1	yes	characteristic
72	C₄²	C₂³	1	2	yes	—
73	Q₈	—	2	2	no	—
74	Q₈	—	2	1	no	—
75	Q₈	—	2	1	no	—
76	Q₈	—	2	2	no	—
77	Q₈	—	2	1	no	—
78	Q₈	—	2	2	no	—
79	Q₈	—	4	2	no	—
80	Q₈	#41	1	2	yes	—
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	1	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₂×D₄	1	2	yes	—
88	C₂×C₄	C₂×D₄	1	2	yes	—
89	C₂×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄	C₂⁴	1	1	yes	characteristic
92	C₈	—	2	1	no	—
93	C₈	—	2	2	no	—
94	C₈	—	2	1	no	—
95	C₂²	C₂²×D₄	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	—	4	2	no	—
101	C₄	—	4	1	no	—
102	C₄	—	4	2	no	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	#137	1	2	yes	—
106	C₄	C₂×#41	1	1	yes	characteristic
107	C₄	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂	#548	1	1	yes	characteristic
109	C₂	C₂×#137	1	1	yes	characteristic
110	C₂	#578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#557

C₂

yes

maximal, characteristic

#557

C₂

yes

maximal, characteristic

#491

C₂

yes

maximal

#487

C₂

yes

maximal, characteristic

#479

C₂

yes

maximal

#475

C₂

yes

maximal, characteristic

#462

C₂

yes

maximal

#462

C₂

yes

maximal, characteristic

#446

C₂

yes

maximal, characteristic

#441

C₂

yes

maximal

C₄×Q₁₆

C₂

yes

maximal, characteristic

C₂×Q₁₆

C₂²

yes

characteristic

C₂×Q₁₆

C₂²

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²

yes

characteristic

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

—

—

C₂²

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂²

yes

characteristic

C₄×Q₈

C₂²

yes

characteristic

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

characteristic

#107

C₂²

yes

characteristic

#106

C₂²

yes

—

C₄⋊C₈

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

—

#103

C₂²

yes

characteristic

#103

C₂²

yes

characteristic

#103

C₂²

yes

—

#103

C₂²

yes

—

#103

C₂²

yes

—

#103

C₂²

yes

—

#97

C₂²

yes

—

C₄×C₈

C₂²

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄

yes

characteristic

C₂×Q₈

C₂³

yes

characteristic

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

characteristic

C₄⋊C₄

D₄

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

characteristic

C₄⋊C₄

C₂³

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

#41

yes

—

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

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

C₂²×D₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#137

yes

—

106

C₄

C₂×#41

yes

characteristic

107

C₄

2₋¹⁺⁴

yes

characteristic

108

C₂

#548

yes

characteristic

109

C₂

C₂×#137

yes

characteristic

110

C₂

#578

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#557	2
#491	2
#487	2
#479	2
#475	2
#462	2
#446	2
#441	2
C₄×Q₁₆	2

subgroup

index

C₄×Q₁₆

Subgroups — #3374

Conjugacy classes of subgroups

Maximal subgroups