Group #2805 — subgroups

Subgroups — #2805

Order 128.

184 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 96 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	#400	C₂	1	2	yes	maximal
2	#400	C₂	1	2	yes	maximal
3	#395	C₂	1	2	yes	maximal
4	#391	C₂	1	1	yes	maximal, characteristic
5	#389	C₂	1	1	yes	maximal, characteristic
6	#389	C₂	1	1	yes	maximal, characteristic
7	#389	C₂	1	2	yes	maximal
8	#387	C₂	1	2	yes	maximal
9	#387	C₂	1	2	yes	maximal
10	C₂²×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄	C₂²	1	2	yes	—
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	2	yes	—
16	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄²	C₂²	1	2	yes	—
21	C₂×C₄²	C₂²	1	1	yes	characteristic
22	#95	C₂²	1	2	yes	—
23	#95	C₂²	1	2	yes	—
24	#95	C₂²	1	2	yes	—
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	C₂×Q₈	—	2	2	no	—
32	C₂×Q₈	—	4	1	no	—
33	C₂×Q₈	—	4	1	no	—
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	2	yes	—
38	C₂²×C₄	C₂³	1	2	yes	—
39	C₂²×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₄	C₂³	1	2	yes	—
41	C₂²×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₄	C₂³	1	1	yes	characteristic
43	C₄⋊C₄	—	2	4	no	—
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	—	2	4	no	—
46	C₄⋊C₄	—	4	2	no	—
47	C₄⋊C₄	—	4	1	no	—
48	C₄⋊C₄	—	4	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₄⋊C₄	Q₈	1	4	yes	complemented
52	C₄²	—	4	1	no	—
53	C₄²	—	4	2	no	—
54	C₂³	C₂⁴	1	1	yes	characteristic
55	Q₈	—	4	4	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	2	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	2	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×Q₈	1	2	yes	—
80	C₂×C₄	C₂×Q₈	1	4	yes	—
81	C₂×C₄	#41	1	2	yes	—
82	C₂×C₄	#41	1	2	yes	—
83	C₂²	C₂²×Q₈	1	1	yes	characteristic
84	C₂²	C₂×#41	1	1	yes	characteristic
85	C₂²	C₂×#41	1	1	yes	characteristic
86	C₂²	H(5,𝔽₂)	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	C₄	—	2	2	no	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	2	4	no	—
94	C₄	—	4	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	4	2	no	—
99	C₂	#538	1	1	yes	characteristic
100	C₂	#539	1	1	yes	characteristic
101	C₂	#544	1	1	yes	characteristic
102	C₂	#554	1	1	yes	characteristic
103	C₂	#556	1	1	yes	characteristic
104	C₂	#557	1	1	yes	characteristic
105	C₂	#563	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#400

C₂

yes

maximal

#400

C₂

yes

maximal

#395

C₂

yes

maximal

#391

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal, characteristic

#389

C₂

yes

maximal

#387

C₂

yes

maximal

#387

C₂

yes

maximal

C₂²×Q₈

C₂²

yes

characteristic

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

—

C₂×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄⋊C₄

C₂²

yes

—

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

—

#95

C₂²

yes

—

#95

C₂²

yes

—

#95

C₂²

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Q₈

yes

complemented

C₄²

—

C₄²

—

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×C₄

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

C₂×Q₈

yes

—

C₂×C₄

#41

yes

—

C₂×C₄

#41

yes

—

C₂²

C₂²×Q₈

yes

characteristic

C₂²

C₂×#41

yes

characteristic

C₂²

C₂×#41

yes

C₂²

yes

C₂²

yes

C₂²

2₋¹⁺⁴

yes

characteristic

C₂²

2₋¹⁺⁴

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

#538

yes

characteristic

100

C₂

#539

yes

characteristic

101

C₂

#544

yes

characteristic

102

C₂

#554

yes

characteristic

103

C₂

#556

yes

characteristic

104

C₂

#557

yes

characteristic

105

C₂

#563

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

N	H ≅ G/N	type
51. C₄⋊C₄	Q₈	semidirect

H ≅ G/N

type

51. C₄⋊C₄

Q₈

semidirect

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

subgroup	index
#400	2
#395	2
#391	2
#389	2
#387	2

subgroup

index

Subgroups — #2805

Conjugacy classes of subgroups

Complements

Maximal subgroups