Group #27187516 — subgroups

Subgroups — #27187516

Order 12928.

162 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 74 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×#15692329	C₂	1	2	yes	maximal, complemented
2	C₄⋊#13108077	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#13108077	C₂	1	2	yes	maximal, complemented
4	C₄×#13108077	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#11717579	C₂	1	1	yes	maximal, characteristic
6	#15692329	—	2	4	no	—
7	C₂×#13108077	C₂²	1	1	yes	characteristic
8	C₂×#13108077	C₂²	1	2	yes	—
9	C₂×#13108077	C₂²	1	1	yes	characteristic
10	C₂²×#11717579	C₂²	1	2	yes	—
11	C₄⋊#11717579	C₄	1	1	yes	characteristic
12	C₂²⋊#11717579	C₄	1	2	yes	—
13	C₄×#11717579	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₀₁	C₄	1	1	yes	characteristic
15	#13108077	—	2	2	no	—
16	#13108077	—	2	1	no	—
17	#13108077	D₄	1	2	yes	complemented
18	C₂×#11717579	—	2	2	no	—
19	C₂×#11717579	—	2	2	no	—
20	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
21	C₂×#11717579	C₂×C₄	1	2	yes	—
22	C₂×#11717579	C₂³	1	1	yes	characteristic
23	C₂×D₄×C₁₀₁	C₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
25	C₄⋊C₄×C₁₀₁	C₈	1	1	yes	characteristic
26	C₂²⋊C₄×C₁₀₁	C₈	1	2	yes	—
27	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₀₁	C₂×C₈	1	2	yes	—
29	D₄×C₁₀₁	—	2	2	no	—
30	C₂×C₄×C₁₀₁	—	2	2	no	—
31	C₂×C₄×C₁₀₁	—	2	2	no	—
32	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₁	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
36	#11717579	—	2	2	no	—
37	#11717579	—	2	1	no	—
38	#11717579	C₂×D₄	1	1	yes	characteristic
39	#11717579	#41	1	1	yes	characteristic
40	C₂²×C₁₀₁	—	2	2	no	—
41	C₂²×C₁₀₁	—	2	2	no	—
42	C₂²×C₁₀₁	—	2	2	no	—
43	C₂²×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₀₁	—	2	2	no	—
45	C₄×C₁₀₁	—	2	1	no	—
46	C₄×C₁₀₁	M₃₂	1	2	yes	complemented
47	C₄×C₁₀₁	C₄×D₄	1	1	yes	characteristic
48	C₄×C₁₀₁	#131	1	1	yes	characteristic
49	C₂×C₁₀₁	—	4	2	no	—
50	C₂×C₁₀₁	C₈×D₄	1	1	yes	characteristic
51	C₂×C₁₀₁	C₂×M₃₂	1	1	yes	characteristic
52	C₂×C₁₀₁	#504	1	1	yes	characteristic
53	#2169	—	101	1	no	maximal, Sylow
54	C₁₀₁	#2169	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	101	2	no	—
56	C₈×D₄	—	101	1	no	—
57	C₄⋊C₁₆	—	101	1	no	—
58	C₂²⋊C₁₆	—	101	2	no	—
59	C₄×C₁₆	—	101	1	no	—
60	C₂²×C₈	—	101	2	no	—
61	C₄×D₄	#87938	1	1	yes	characteristic
62	M₃₂	—	202	4	no	—
63	C₂×C₁₆	—	101	1	no	—
64	C₂×C₁₆	—	101	1	no	—
65	C₂×C₁₆	—	101	2	no	—
66	C₄⋊C₈	—	101	1	no	—
67	C₂²⋊C₈	—	101	2	no	—
68	C₄×C₈	—	101	1	no	—
69	C₂×D₄	#11717581	1	1	yes	characteristic
70	C₂²×C₄	C₂×#87938	1	2	yes	—
71	C₂×C₈	—	101	1	no	—
72	C₂×C₈	—	101	1	no	—
73	C₂×C₈	—	101	2	no	—
74	C₂×C₈	—	202	2	no	—
75	C₂×C₈	—	202	2	no	—
76	C₄⋊C₄	#11717581	1	1	yes	characteristic
77	C₂²⋊C₄	#11717581	1	2	yes	—
78	C₄²	C₂×#87938	1	1	yes	characteristic
79	C₁₆	—	101	2	no	—
80	C₁₆	—	202	1	no	—
81	C₁₆	—	202	2	no	—
82	C₂³	C₂×#11717581	1	2	yes	—
83	D₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂×#11717581	1	1	yes	characteristic
87	C₂×C₄	C₂×#11717581	1	2	yes	—
88	C₂×C₄	C₂×#11717581	1	1	yes	characteristic
89	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
90	C₈	—	101	1	no	—
91	C₈	—	101	1	no	—
92	C₈	—	202	2	no	—
93	C₈	—	202	1	no	—
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂²×#11717581	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	#15692329	1	2	yes	complemented
101	C₄	D₄×#87938	1	1	yes	characteristic
102	C₄	#15692342	1	1	yes	characteristic
103	C₂	—	4	2	no	—
104	C₂	D₄×#11717581	1	1	yes	characteristic
105	C₂	C₂×#15692329	1	1	yes	characteristic
106	C₂	#20266235	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692329

C₂

yes

maximal, complemented

C₄⋊#13108077

C₂

yes

maximal, characteristic, complemented

C₂²⋊#13108077

C₂

yes

maximal, complemented

C₄×#13108077

C₂

yes

maximal, characteristic, complemented

D₄×#11717579

C₂

yes

maximal, characteristic

#15692329

—

C₂×#13108077

C₂²

yes

characteristic

C₂×#13108077

C₂²

yes

—

C₂×#13108077

C₂²

yes

characteristic

C₂²×#11717579

C₂²

yes

—

C₄⋊#11717579

C₄

yes

characteristic

C₂²⋊#11717579

C₄

yes

—

C₄×#11717579

C₂²

yes

characteristic

C₄×D₄×C₁₀₁

C₄

yes

—

—

D₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#11717579

C₂×C₄

yes

—

C₂×#11717579

C₂³

yes

characteristic

C₂×D₄×C₁₀₁

C₈

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

C₈

yes

characteristic

C₂²⋊C₄×C₁₀₁

C₈

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂³×C₁₀₁

C₂×C₈

yes

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₁

C₂×C₈

yes

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

—

C₂×D₄

yes

yes

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²×C₈

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

M₃₂

yes

complemented

C₄×C₁₀₁

C₄×D₄

yes

characteristic

C₄×C₁₀₁

#131

yes

characteristic

C₂×C₁₀₁

—

C₂×C₁₀₁

C₈×D₄

yes

characteristic

C₂×C₁₀₁

C₂×M₃₂

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#2169

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

101

—

C₈×D₄

—

101

—

C₄⋊C₁₆

—

101

—

C₂²⋊C₁₆

—

101

—

C₄×C₁₆

—

101

—

C₂²×C₈

—

101

—

C₄×D₄

#87938

yes

characteristic

M₃₂

—

202

—

C₂×C₁₆

—

101

—

C₂×C₁₆

—

101

—

C₂×C₁₆

—

101

—

C₄⋊C₈

—

101

—

C₂²⋊C₈

—

101

—

C₄×C₈

—

101

—

C₂×D₄

#11717581

yes

characteristic

C₂²×C₄

C₂×#87938

yes

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

#11717581

yes

characteristic

C₂²⋊C₄

#11717581

yes

—

C₄²

C₂×#87938

yes

characteristic

C₁₆

—

101

—

C₁₆

—

202

—

C₁₆

—

202

—

C₂³

C₂×#11717581

yes

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#11717581

yes

characteristic

C₂×C₄

C₂×#11717581

yes

—

C₂×C₄

C₂×#11717581

yes

characteristic

C₂×C₄

C₂²×#87938

yes

characteristic

C₈

—

101

—

C₈

—

101

—

C₈

—

202

—

C₈

—

202

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×#11717581

yes

characteristic

C₄

—

C₄

—

100

C₄

#15692329

yes

complemented

101

C₄

D₄×#87938

yes

characteristic

102

C₄

#15692342

yes

characteristic

103

C₂

—

104

C₂

D₄×#11717581

yes

characteristic

105

C₂

C₂×#15692329

yes

characteristic

106

C₂

#20266235

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#87938
commutator, socle	C₂²×C₁₀₁	C₂²×C₈
Fitting	C₄×D₄×C₁₀₁	C₄
prime radical	C₂×D₄×C₁₀₁	C₈
cosocle	C₂×#11717579	C₂³
hypercenter	C₄×D₄	#87938
hypocenter	C₁₀₁	#2169
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#87938

commutator, socle

C₂²×C₁₀₁

C₂²×C₈

Fitting

C₄×D₄×C₁₀₁

C₄

prime radical

C₂×D₄×C₁₀₁

C₈

C₂³

C₄×D₄

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂×C₄ ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#15692329	C₂	semidirect
2. C₄⋊#13108077	C₂	semidirect
3. C₂²⋊#13108077	C₂	semidirect
4. C₄×#13108077	C₂	semidirect
17. #13108077	D₄	semidirect
46. C₄×C₁₀₁	M₃₂	semidirect
54. C₁₀₁	#2169	semidirect
100. C₄	#15692329	semidirect

H ≅ G/N

type

1. C₂×#15692329

C₂

semidirect

2. C₄⋊#13108077

C₂

semidirect

3. C₂²⋊#13108077

C₂

semidirect

4. C₄×#13108077

C₂

semidirect

17. #13108077

D₄

semidirect

46. C₄×C₁₀₁

M₃₂

semidirect

54. C₁₀₁

#2169

semidirect

100. C₄

#15692329

semidirect

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

subgroup	index
C₂×#15692329	2
C₄⋊#13108077	2
C₂²⋊#13108077	2
C₄×#13108077	2
D₄×#11717579	2
#2169	101

subgroup

index

101

Subgroups — #27187516

Conjugacy classes of subgroups

Complements

Maximal subgroups