Group #27197195 — subgroups

Subgroups — #27197195

Order 13184.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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×Dic₈₂₄	C₂	1	2	yes	maximal
2	#20268171	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₄₁₂	C₂	1	1	yes	maximal, characteristic
4	#20267596	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₁₂	—	2	1	no	—
7	C₂×Dic₄₁₂	C₂²	1	2	yes	—
8	C₂×Dic₄₁₂	C₂²	1	1	yes	characteristic
9	#15693021	—	2	1	no	—
10	C₄×Dic₂₀₆	—	2	1	no	—
11	Dic₈₂₄	—	2	4	no	—
12	#15692955	C₂²	1	1	yes	characteristic
13	#15692946	—	2	1	no	—
14	C₂×C₁₆×C₁₀₃	C₂²	1	2	yes	—
15	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₀₆	—	2	1	no	—
17	C₂×Dic₂₀₆	—	2	2	no	—
18	Dic₄₁₂	—	2	2	no	—
19	Dic₄₁₂	—	2	1	no	—
20	Dic₄₁₂	—	2	2	no	—
21	Dic₄₁₂	—	4	2	no	—
22	Dic₄₁₂	D₄	1	2	yes	—
23	C₄⋊Dic₁₀₃	—	2	1	no	—
24	C₄⋊Dic₁₀₃	—	4	1	no	—
25	C₄⋊Dic₁₀₃	—	4	1	no	—
26	Dic₁₀₃⋊C₄	—	4	1	no	—
27	C₄×Dic₁₀₃	—	4	1	no	—
28	C₂×C₈×C₁₀₃	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
30	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
31	C₁₆×C₁₀₃	—	2	2	no	—
32	C₂×C₄×C₁₀₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₃	—	4	1	no	—
36	C₂×Dic₁₀₃	—	4	1	no	—
37	C₂×Dic₁₀₃	—	4	2	no	—
38	Dic₂₀₆	—	2	2	no	—
39	Dic₂₀₆	—	4	1	no	—
40	Dic₂₀₆	—	4	2	no	—
41	Dic₂₀₆	—	4	2	no	—
42	C₈×C₁₀₃	—	2	1	no	—
43	C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₀₃	#41	1	1	yes	characteristic
45	C₂²×C₁₀₃	C₂×D₈	1	1	yes	characteristic
46	C₄×C₁₀₃	—	2	1	no	—
47	C₄×C₁₀₃	Q₃₂	1	2	yes	complemented
48	C₄×C₁₀₃	#121	1	1	yes	characteristic
49	C₄×C₁₀₃	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₁₀₃	—	4	2	no	—
51	Dic₁₀₃	—	8	2	no	—
52	Dic₁₀₃	—	8	1	no	—
53	C₂×C₁₀₃	#459	1	1	yes	characteristic
54	C₂×C₁₀₃	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₁₀₃	#510	1	1	yes	characteristic
56	#2209	—	103	1	no	maximal, Sylow
57	C₁₀₃	#2209	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₃₂	—	103	2	no	—
59	#494	—	103	1	no	—
60	C₄×Q₁₆	—	103	1	no	—
61	C₄⋊C₁₆	D₁₀₃	1	1	yes	characteristic
62	#358	—	103	2	no	—
63	C₂×Q₁₆	—	103	1	no	—
64	C₂×Q₁₆	—	103	2	no	—
65	C₂×Q₁₆	—	206	1	no	—
66	C₄⋊Q₈	—	206	1	no	—
67	C₄×Q₈	—	206	1	no	—
68	Q₃₂	—	206	4	no	—
69	C₂×C₁₆	C₂×D₁₀₃	1	2	yes	—
70	#107	—	103	1	no	—
71	#103	—	206	1	no	—
72	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
73	C₂×Q₈	—	206	2	no	—
74	C₂×Q₈	—	206	1	no	—
75	Q₁₆	—	103	2	no	—
76	Q₁₆	—	412	2	no	—
77	Q₁₆	—	206	2	no	—
78	Q₁₆	—	206	2	no	—
79	Q₁₆	—	206	1	no	—
80	C₂×C₈	D₄₁₂	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
82	C₄⋊C₄	—	412	1	no	—
83	C₄⋊C₄	—	412	1	no	—
84	C₄⋊C₄	—	412	1	no	—
85	C₄⋊C₄	—	206	1	no	—
86	C₄²	—	412	1	no	—
87	C₄²	D₄₁₂	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	412	2	no	—
90	Q₈	—	412	2	no	—
91	Q₈	—	412	1	no	—
92	Q₈	—	206	2	no	—
93	C₂×C₄	—	412	1	no	—
94	C₂×C₄	—	412	2	no	—
95	C₂×C₄	—	412	1	no	—
96	C₂×C₄	D₈₂₄	1	1	yes	characteristic
97	C₂×C₄	D₈₂₄	1	1	yes	characteristic
98	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₁₀₃	1	1	yes	characteristic
101	C₈	#13110106	1	1	yes	characteristic
102	C₂²	C₂×D₈₂₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	824	1	no	—
105	C₄	—	824	2	no	—
106	C₄	—	412	2	no	—
107	C₄	Dic₈₂₄	1	2	yes	complemented
108	C₄	#15692996	1	1	yes	characteristic
109	C₄	#15693049	1	1	yes	characteristic
110	C₂	#20267987	1	1	yes	characteristic
111	C₂	C₂×Dic₈₂₄	1	1	yes	characteristic
112	C₂	#20268234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₈₂₄

C₂

yes

maximal

#20268171

C₂

yes

maximal, characteristic

C₄×Dic₄₁₂

C₂

yes

maximal, characteristic

#20267596

C₂

yes

maximal

C₄⋊C₁₆×C₁₀₃

C₂

yes

maximal, characteristic

C₂×Dic₄₁₂

—

C₂×Dic₄₁₂

C₂²

yes

—

C₂×Dic₄₁₂

C₂²

yes

characteristic

#15693021

—

C₄×Dic₂₀₆

—

Dic₈₂₄

—

#15692955

C₂²

yes

characteristic

#15692946

—

C₂×C₁₆×C₁₀₃

C₂²

yes

—

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

Dic₄₁₂

—

Dic₄₁₂

—

Dic₄₁₂

—

Dic₄₁₂

—

Dic₄₁₂

D₄

yes

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

Dic₁₀₃⋊C₄

—

C₄×Dic₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

characteristic

C₂×C₈×C₁₀₃

C₂³

yes

characteristic

C₄²×C₁₀₃

D₄

yes

characteristic

C₁₆×C₁₀₃

—

C₂×C₄×C₁₀₃

D₈

yes

characteristic

C₂×C₄×C₁₀₃

D₈

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₈×C₁₀₃

C₂×D₄

yes

characteristic

C₈×C₁₀₃

#41

yes

characteristic

C₂²×C₁₀₃

C₂×D₈

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

Q₃₂

yes

C₄×C₁₀₃

yes

C₄×C₁₀₃

yes

Dic₁₀₃

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

#459

yes

characteristic

C₂×C₁₀₃

C₂×Q₃₂

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#2209

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

103

—

#494

—

103

—

C₄×Q₁₆

—

103

—

C₄⋊C₁₆

D₁₀₃

yes

characteristic

#358

—

103

—

C₂×Q₁₆

—

103

—

C₂×Q₁₆

—

103

—

C₂×Q₁₆

—

206

—

C₄⋊Q₈

—

206

—

C₄×Q₈

—

206

—

Q₃₂

—

206

—

C₂×C₁₆

C₂×D₁₀₃

yes

—

#107

—

103

—

#103

—

206

—

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

Q₁₆

—

103

—

Q₁₆

—

412

—

Q₁₆

—

206

—

Q₁₆

—

206

—

Q₁₆

—

206

—

C₂×C₈

D₄₁₂

yes

characteristic

C₂×C₈

C₂²×D₁₀₃

yes

characteristic

C₄⋊C₄

—

412

—

C₄⋊C₄

—

412

—

C₄⋊C₄

—

412

—

C₄⋊C₄

—

206

—

C₄²

—

412

—

C₄²

D₄₁₂

yes

characteristic

C₁₆

—

Q₈

—

412

—

Q₈

—

412

—

Q₈

—

412

—

Q₈

—

206

—

C₂×C₄

—

412

—

C₂×C₄

—

412

—

C₂×C₄

—

412

—

C₂×C₄

D₈₂₄

yes

characteristic

C₂×C₄

D₈₂₄

yes

characteristic

C₂×C₄

C₂×D₄₁₂

yes

characteristic

C₈

—

100

C₈

D₄×D₁₀₃

yes

characteristic

101

C₈

#13110106

yes

characteristic

102

C₂²

C₂×D₈₂₄

yes

characteristic

103

C₄

—

104

C₄

—

824

—

105

C₄

—

824

—

106

C₄

—

412

—

107

C₄

Dic₈₂₄

yes

complemented

108

C₄

#15692996

yes

characteristic

109

C₄

#15693049

yes

characteristic

110

C₂

#20267987

yes

characteristic

111

C₂

C₂×Dic₈₂₄

yes

characteristic

112

C₂

#20268234

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₂₄
commutator, cosocle	C₂×C₈×C₁₀₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₃
Fitting	C₄⋊C₁₆×C₁₀₃	C₂
socle, prime radical	C₂²×C₁₀₃	C₂×D₈
hypercenter	C₄⋊C₁₆	D₁₀₃
hypocenter	C₁₀₃	#2209
LCS	G ▷ C₂×C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₈₂₄

commutator, cosocle

C₂×C₈×C₁₀₃

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₃

Fitting

C₄⋊C₁₆×C₁₀₃

C₂

socle, prime radical

C₂²×C₁₀₃

C₂×D₈

hypercenter

C₄⋊C₁₆

D₁₀₃

hypocenter

C₁₀₃

#2209

LCS

G ▷ C₂×C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₁₀₃	Q₃₂	semidirect
57. C₁₀₃	#2209	semidirect
107. C₄	Dic₈₂₄	semidirect

H ≅ G/N

type

47. C₄×C₁₀₃

Q₃₂

semidirect

57. C₁₀₃

#2209

semidirect

107. C₄

Dic₈₂₄

semidirect

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

subgroup	index
C₂×Dic₈₂₄	2
#20268171	2
C₄×Dic₄₁₂	2
#20267596	2
C₄⋊C₁₆×C₁₀₃	2
#2209	103

subgroup

index

C₂×Dic₈₂₄

#20268171

C₄×Dic₄₁₂

#20267596

C₄⋊C₁₆×C₁₀₃

#2209

103

Subgroups — #27197195

Conjugacy classes of subgroups

Complements

Maximal subgroups