Group #13185 — subgroups

Subgroups — #13185

Order 256.

153 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 153 conjugacy classes of subgroups — too many to draw. Only its 77 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×#365	C₂	1	2	yes	maximal
2	#2150	C₂	1	1	yes	maximal, characteristic
3	#1848	C₂	1	1	yes	maximal, characteristic
4	C₄×#106	C₂	1	1	yes	maximal, characteristic
5	#1383	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂×#108	—	2	1	no	—
8	C₂×#108	C₂²	1	2	yes	—
9	C₂×#106	C₂²	1	1	yes	characteristic
10	C₂×#106	C₂²	1	1	yes	characteristic
11	#423	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄	—	2	1	no	—
14	#365	—	2	4	no	—
15	C₄⋊C₁₆	C₄	1	2	yes	complemented
16	C₄⋊C₁₆	C₄	1	2	yes	complemented
17	#340	—	2	1	no	—
18	C₂×M₁₆	—	2	2	no	—
19	C₂²×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄	—	2	1	no	—
22	C₂×C₄⋊C₄	—	2	1	no	—
23	C₂×C₄²	—	4	1	no	—
24	C₂×C₄²	D₄	1	1	yes	characteristic
25	M₃₂	—	2	2	no	—
26	M₃₂	—	2	2	no	—
27	C₂×C₁₆	C₂×C₄	1	4	yes	—
28	#108	—	2	2	no	—
29	#108	—	2	2	no	—
30	#108	—	4	2	no	—
31	#106	—	2	1	no	—
32	#106	—	2	1	no	—
33	#106	D₄	1	2	yes	—
34	#106	Q₈	1	2	yes	—
35	C₄⋊C₈	—	4	1	no	—
36	C₄⋊C₈	—	4	1	no	—
37	C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈	Q₈	1	1	yes	characteristic
40	C₄×C₈	Q₈	1	1	yes	characteristic
41	#95	—	4	1	no	—
42	C₂²×C₄	—	4	1	no	—
43	C₂²×C₄	—	4	1	no	—
44	C₂²×C₄	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	M₁₆	—	4	2	no	—
48	M₁₆	—	4	2	no	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	4	2	no	—
52	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈	#41	1	1	yes	characteristic
59	C₂×C₈	#41	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	4	1	no	—
63	C₄⋊C₄	—	4	1	no	—
64	C₄²	—	8	2	no	—
65	C₄²	C₄⋊C₄	1	1	yes	characteristic
66	C₄²	C₄⋊C₄	1	1	yes	characteristic
67	C₄²	SD₁₆	1	1	yes	characteristic
68	C₄²	SD₁₆	1	1	yes	characteristic
69	C₁₆	—	2	4	no	—
70	C₂³	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	8	1	no	—
77	C₂×C₄	—	8	1	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	#106	1	1	yes	characteristic
80	C₂×C₄	#106	1	1	yes	characteristic
81	C₂×C₄	#106	1	1	yes	characteristic
82	C₂×C₄	#106	1	1	yes	characteristic
83	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄	#121	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×C₄	#137	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	8	2	no	—
92	C₈	C₄×D₄	1	1	yes	characteristic
93	C₈	C₄×Q₈	1	1	yes	characteristic
94	C₈	#125	1	1	yes	characteristic
95	C₈	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	C₂×#106	1	1	yes	characteristic
99	C₂²	#461	1	1	yes	characteristic
100	C₂²	#476	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	8	2	no	—
105	C₄	—	8	2	no	—
106	C₄	#365	1	2	yes	complemented
107	C₄	#389	1	1	yes	characteristic
108	C₄	#428	1	1	yes	characteristic
109	C₄	#460	1	1	yes	characteristic
110	C₄	#475	1	1	yes	characteristic
111	C₂	—	2	1	no	—
112	C₂	—	2	1	no	—
113	C₂	#1948	1	1	yes	characteristic
114	C₂	C₂×#365	1	1	yes	characteristic
115	C₂	#2155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#365

C₂

yes

maximal

#2150

C₂

yes

maximal, characteristic

#1848

C₂

yes

maximal, characteristic

C₄×#106

C₂

yes

maximal, characteristic

#1383

C₂

yes

maximal

C₂×M₃₂

C₂²

yes

—

C₂×#108

—

C₂×#108

C₂²

yes

—

C₂×#106

C₂²

yes

characteristic

C₂×#106

C₂²

yes

characteristic

#423

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

—

#365

—

C₄⋊C₁₆

C₄

yes

complemented

C₄⋊C₁₆

C₄

yes

complemented

#340

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

—

#106

Q₈

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

SD₁₆

yes

characteristic

C₂²×C₄

SD₁₆

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄⋊C₄

yes

characteristic

C₄²

C₄⋊C₄

yes

characteristic

C₄²

SD₁₆

yes

characteristic

C₄²

SD₁₆

yes

characteristic

C₁₆

—

C₂³

C₂×SD₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

#121

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

C₄×D₄

yes

characteristic

C₈

C₄×Q₈

yes

characteristic

C₈

#125

yes

characteristic

C₈

C₄⋊Q₈

yes

characteristic

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

100

C₂²

#476

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#365

yes

complemented

107

C₄

#389

yes

characteristic

108

C₄

#428

yes

characteristic

109

C₄

#460

yes

characteristic

110

C₄

#475

yes

characteristic

111

C₂

—

112

C₂

—

113

C₂

#1948

yes

characteristic

114

C₂

C₂×#365

yes

characteristic

115

C₂

#2155

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#106
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#106

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

C₂×SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
15. C₄⋊C₁₆	C₄	semidirect
16. C₄⋊C₁₆	C₄	semidirect
106. C₄	#365	semidirect

H ≅ G/N

type

15. C₄⋊C₁₆

C₄

semidirect

16. C₄⋊C₁₆

C₄

semidirect

106. C₄

#365

semidirect

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

subgroup	index
C₂×#365	2
#2150	2
#1848	2
C₄×#106	2
#1383	2

subgroup

index

Subgroups — #13185

Conjugacy classes of subgroups

Complements

Maximal subgroups