Group #2726 — subgroups

Subgroups — #2726

Order 128.

208 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 104 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×#125	C₂	1	2	yes	maximal
2	#398	C₂	1	2	yes	maximal
3	#398	C₂	1	2	yes	maximal
4	#398	C₂	1	2	yes	maximal
5	#389	C₂	1	1	yes	maximal, characteristic
6	#389	C₂	1	1	yes	maximal, characteristic
7	#389	C₂	1	2	yes	maximal
8	#389	C₂	1	1	yes	maximal, characteristic
9	#387	C₂	1	2	yes	maximal
10	#125	—	2	8	no	—
11	C₂×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
22	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄²	C₂²	1	2	yes	—
25	C₂×C₄²	C₂²	1	1	yes	characteristic
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	#95	C₂²	1	2	yes	—
32	C₂²×C₄	C₂³	1	2	yes	—
33	C₂²×C₄	C₂³	1	2	yes	—
34	C₂²×C₄	C₂³	1	2	yes	—
35	C₂²×C₄	C₂³	1	2	yes	—
36	C₂²×C₄	C₂³	1	2	yes	—
37	C₂²×C₄	C₂³	1	1	yes	characteristic
38	C₂²×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₄	C₂³	1	2	yes	—
40	C₂²×C₄	C₂³	1	1	yes	characteristic
41	C₄⋊C₄	—	2	4	no	—
42	C₄⋊C₄	—	2	4	no	—
43	C₄⋊C₄	—	2	4	no	—
44	C₄⋊C₄	—	2	4	no	—
45	C₄⋊C₄	—	2	4	no	—
46	C₄⋊C₄	—	2	4	no	—
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	4	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄⋊C₄	—	4	1	no	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄⋊C₄	D₄	1	4	yes	—
53	C₄⋊C₄	Q₈	1	4	yes	—
54	C₄²	—	2	4	no	—
55	C₄²	—	4	1	no	—
56	C₂³	C₂⁴	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	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	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	C₂×D₄	1	4	yes	—
80	C₂×C₄	C₂×D₄	1	2	yes	—
81	C₂×C₄	C₂×Q₈	1	2	yes	—
82	C₂×C₄	C₂×Q₈	1	4	yes	—
83	C₂×C₄	#41	1	2	yes	—
84	C₂²	C₂²×D₄	1	1	yes	characteristic
85	C₂²	C₂²×Q₈	1	1	yes	characteristic
86	C₂²	C₂×#41	1	1	yes	characteristic
87	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
89	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
90	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	2	4	no	—
95	C₄	—	2	4	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	4	2	no	—
99	C₄	—	4	2	no	—
100	C₂	#537	1	1	yes	characteristic
101	C₂	#539	1	1	yes	characteristic
102	C₂	#544	1	1	yes	characteristic
103	C₂	#547	1	1	yes	characteristic
104	C₂	D₄×Q₈	1	1	yes	characteristic
105	C₂	#557	1	1	yes	characteristic
106	C₂	#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125

C₂

yes

maximal

#398

C₂

yes

maximal

#398

C₂

yes

maximal

#398

C₂

yes

maximal

#389

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal

#389

C₂

yes

maximal, characteristic

#387

C₂

yes

maximal

#125

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

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

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

characteristic

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

—

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

—

C₂²×C₄

C₂³

yes

characteristic

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

—

C₄⋊C₄

Q₈

yes

—

C₄²

—

C₄²

—

C₂³

C₂⁴

yes

characteristic

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×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

#41

yes

—

C₂²

C₂²×D₄

yes

characteristic

C₂²

C₂²×Q₈

yes

characteristic

C₂²

C₂×#41

yes

characteristic

C₂²

H(5,𝔽₂)

yes

characteristic

C₂²

2₋¹⁺⁴

yes

characteristic

C₂²

2₋¹⁺⁴

yes

characteristic

C₂²

2₋¹⁺⁴

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

100

C₂

#537

yes

characteristic

101

C₂

#539

yes

characteristic

102

C₂

#544

yes

characteristic

103

C₂

#547

yes

characteristic

104

C₂

D₄×Q₈

yes

characteristic

105

C₂

#557

yes

characteristic

106

C₂

#564

yes

characteristic

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

subgroup

G/H

center, commutator, Frattini, socle, prime radical, cosocle

C₂³

C₂⁴

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ G

LECS

G ▷ C₂³ ▷ C₁

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

subgroup	index
C₂×#125	2
#398	2
#389	2
#387	2

subgroup

index

Subgroups — #2726

Conjugacy classes of subgroups

Maximal subgroups