Group #13199 — subgroups

Subgroups — #13199

Order 256.

203 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 69 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	#2955	C₂	1	1	yes	maximal, characteristic
2	#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#350	C₂	1	2	yes	maximal, complemented
4	#2115	C₂	1	2	yes	maximal, complemented
5	#2110	C₂	1	1	yes	maximal, characteristic, complemented
6	#575	C₄	1	1	yes	characteristic
7	C₂×#131	C₂²	1	2	yes	—
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	2	yes	—
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	#444	—	2	2	no	—
12	#443	—	2	2	no	—
13	#430	C₄	1	1	yes	characteristic
14	#421	C₄	1	2	yes	—
15	#405	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆	—	2	2	no	—
17	#350	—	2	4	no	—
18	#349	—	2	4	no	—
19	#347	—	2	2	no	—
20	C₂×#41	C₂×C₄	1	2	yes	—
21	#137	—	2	2	no	—
22	AGL(1,ℤ/8)	—	2	2	no	—
23	#135	—	2	2	no	—
24	#131	—	2	2	no	—
25	#131	—	2	2	no	—
26	#131	—	2	2	no	—
27	#131	—	2	2	no	—
28	C₂×M₁₆	—	2	2	no	—
29	C₂×M₁₆	C₂×C₄	1	2	yes	—
30	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	—	2	2	no	—
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	#117	C₂×C₄	1	1	yes	characteristic
34	M₃₂	—	2	2	no	—
35	M₃₂	—	2	2	no	—
36	M₃₂	D₄	1	4	yes	—
37	C₂×C₁₆	—	2	2	no	—
38	C₂×C₁₆	—	2	2	no	—
39	C₂×C₁₆	—	2	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	#108	—	2	2	no	—
42	#104	—	2	4	no	—
43	#97	D₄	1	2	yes	—
44	C₄×C₈	D₄	1	2	yes	—
45	#41	—	2	2	no	—
46	#41	—	2	2	no	—
47	#41	—	2	2	no	—
48	#41	—	2	2	no	—
49	C₂×Q₈	M₁₆	1	2	yes	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	M₁₆	1	2	yes	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
54	Q₁₆	—	2	2	no	—
55	SD₁₆	—	2	4	no	—
56	D₈	—	2	2	no	—
57	M₁₆	—	2	2	no	—
58	M₁₆	—	2	2	no	—
59	M₁₆	—	2	2	no	—
60	M₁₆	—	4	2	no	—
61	M₁₆	—	4	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈	C₂×D₄	1	2	yes	—
68	C₂×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈	#41	1	2	yes	—
70	C₄⋊C₄	—	2	1	no	—
71	C₂²⋊C₄	—	2	1	no	—
72	C₄²	C₂²⋊C₄	1	2	yes	—
73	C₁₆	—	2	4	no	—
74	C₁₆	—	4	1	no	—
75	C₁₆	—	4	1	no	—
76	C₂³	—	2	2	no	—
77	C₂³	#131	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	2	2	no	—
80	D₄	—	2	2	no	—
81	D₄	—	2	2	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄	C₄×D₄	1	2	yes	—
90	C₂×C₄	C₂×M₁₆	1	2	yes	—
91	C₂×C₄	#131	1	1	yes	characteristic
92	C₈	—	2	2	no	—
93	C₈	—	4	2	no	—
94	C₈	#121	1	2	yes	—
95	C₈	#124	1	1	yes	characteristic
96	C₈	#127	1	1	yes	characteristic
97	C₂²	—	4	1	no	—
98	C₂²	—	4	2	no	—
99	C₂²	—	8	2	no	—
100	C₂²	#408	1	1	yes	characteristic
101	C₂²	#435	1	2	yes	—
102	C₄	—	4	2	no	—
103	C₄	—	4	2	no	—
104	C₄	#390	1	1	yes	characteristic
105	C₄	#407	1	1	yes	characteristic
106	C₄	#435	1	2	yes	—
107	C₂	—	2	1	no	—
108	C₂	—	2	2	no	—
109	C₂	—	8	2	no	—
110	C₂	#1955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2955

C₂

yes

maximal, characteristic

#2151

C₂

yes

maximal, characteristic, complemented

C₂×#350

C₂

yes

maximal, complemented

#2115

C₂

yes

maximal, complemented

#2110

C₂

yes

maximal, characteristic, complemented

#575

C₄

yes

characteristic

C₂×#131

C₂²

yes

—

C₂×M₃₂

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

—

C₄

yes

characteristic

#421

C₄

yes

—

#405

C₂²

yes

characteristic

C₄⋊C₁₆

—

#350

—

#349

—

#347

—

C₂×#41

C₂×C₄

yes

—

—

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂³

yes

characteristic

#117

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

D₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#104

—

#97

D₄

yes

—

C₄×C₈

D₄

yes

—

—

—

—

—

C₂×Q₈

M₁₆

yes

—

C₂×D₄

—

C₂×D₄

M₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊C₄

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

—

C₂³

#131

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

#131

yes

characteristic

C₈

—

C₈

—

C₈

#121

yes

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

100

C₂²

#408

yes

characteristic

101

C₂²

#435

yes

—

102

C₄

—

103

C₄

—

104

C₄

#390

yes

characteristic

105

C₄

#407

yes

characteristic

106

C₄

#435

yes

—

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

#1955

yes

characteristic

	subgroup	G/H
center	C₈	#124
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1955
prime radical	#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

C₈

C₂²×C₄

C₂²×C₈

C₂³

C₂

C₄

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2151	C₂	semidirect
3. C₂×#350	C₂	semidirect
4. #2115	C₂	semidirect
5. #2110	C₂	semidirect

H ≅ G/N

type

2. #2151

C₂

semidirect

3. C₂×#350

C₂

semidirect

4. #2115

C₂

semidirect

5. #2110

C₂

semidirect

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

subgroup	index
#2955	2
#2151	2
C₂×#350	2
#2115	2
#2110	2

subgroup

index

Subgroups — #13199

Conjugacy classes of subgroups

Complements

Maximal subgroups