Group #27196899 — subgroups

Subgroups — #27196899

Order 13184.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15692961	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₀₃	C₂	1	1	yes	maximal, characteristic
3	Dic₁₀₃⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₀₃⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₃	C₄	1	1	yes	characteristic
7	#15692961	—	2	2	no	—
8	C₂×C₁₀₃⋊C₁₆	C₂²	1	2	yes	—
9	C₈×Dic₁₀₃	C₄	1	2	yes	—
10	C₈×Dic₁₀₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
12	M₃₂×C₁₀₃	—	2	2	no	—
13	C₂×C₁₆×C₁₀₃	C₂²	1	2	yes	—
14	C₂²×Dic₁₀₃	C₈	1	2	yes	—
15	C₂×C₁₀₃⋊C₈	—	2	2	no	—
16	C₂×C₁₀₃⋊C₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₀₃	C₈	1	1	yes	characteristic
18	C₄×Dic₁₀₃	C₈	1	1	yes	characteristic
19	C₄×Dic₁₀₃	C₂×C₄	1	2	yes	—
20	C₁₀₃⋊C₁₆	—	2	2	no	—
21	C₂²×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
24	C₂×C₈×C₁₀₃	Q₈	1	2	yes	—
25	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
26	C₁₆×C₁₀₃	—	2	2	no	—
27	C₂³×C₁₀₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₃	—	2	4	no	—
33	C₂×Dic₁₀₃	C₂×C₈	1	2	yes	—
34	C₂×Dic₁₀₃	C₂×C₈	1	2	yes	—
35	C₈×C₁₀₃	C₄⋊C₄	1	2	yes	—
36	C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
38	C₁₀₃⋊C₈	—	2	2	no	—
39	C₂²×C₁₀₃	—	2	2	no	—
40	C₂²×C₁₀₃	C₄⋊C₈	1	2	yes	—
41	C₂²×C₁₀₃	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₀₃	C₄⋊C₈	1	2	yes	—
43	C₄×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₁₀₃	C₂×M₁₆	1	1	yes	characteristic
45	Dic₁₀₃	—	2	2	no	—
46	Dic₁₀₃	M₃₂	1	4	yes	complemented
47	C₂×C₁₀₃	—	2	2	no	—
48	C₂×C₁₀₃	C₂×C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₁₀₃	C₂×M₃₂	1	2	yes	—
50	#2150	—	103	1	no	maximal, Sylow
51	C₁₀₃	#2150	1	1	yes	characteristic, complemented, Sylow
52	C₂×M₃₂	—	103	1	no	—
53	C₂×M₃₂	D₁₀₃	1	1	yes	characteristic
54	C₂×C₄×C₈	—	103	1	no	—
55	C₄⋊C₁₆	—	103	4	no	—
56	C₂²×C₈	—	103	1	no	—
57	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
58	C₂×C₄²	—	103	1	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	206	2	no	—
61	C₂×C₁₆	—	103	2	no	—
62	C₂×C₁₆	C₂×D₁₀₃	1	2	yes	—
63	C₄×C₈	—	103	2	no	—
64	C₄×C₈	—	103	2	no	—
65	C₂²×C₄	—	103	2	no	—
66	C₂²×C₄	C₄×D₁₀₃	1	1	yes	characteristic
67	C₂×C₈	—	103	2	no	—
68	C₂×C₈	—	206	2	no	—
69	C₂×C₈	Dic₂₀₆	1	2	yes	—
70	C₂×C₈	C₄×D₁₀₃	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊D₁₀₃	1	2	yes	—
72	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
73	C₄²	—	103	1	no	—
74	C₄²	—	103	1	no	—
75	C₄²	—	103	2	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	206	2	no	—
78	C₂³	C₈×D₁₀₃	1	1	yes	characteristic
79	C₂×C₄	—	103	2	no	—
80	C₂×C₄	—	103	2	no	—
81	C₂×C₄	—	206	4	no	—
82	C₂×C₄	Dic₁₀₃⋊C₄	1	2	yes	—
83	C₂×C₄	C₈×D₁₀₃	1	1	yes	characteristic
84	C₂×C₄	#13110088	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
86	C₈	—	206	2	no	—
87	C₈	Dic₁₀₃⋊C₄	1	2	yes	—
88	C₈	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
89	C₈	C₂×Dic₂₀₆	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	Dic₁₀₃⋊C₈	1	2	yes	—
92	C₂²	C₂×C₈×D₁₀₃	1	1	yes	characteristic
93	C₄	—	103	4	no	—
94	C₄	—	206	2	no	—
95	C₄	Dic₁₀₃⋊C₈	1	2	yes	—
96	C₄	C₂×Dic₁₀₃⋊C₄	1	1	yes	characteristic
97	C₄	C₂×#13110088	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×Dic₁₀₃⋊C₈	1	1	yes	characteristic
100	C₂	M₃₂×D₁₀₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692961

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₀₃

C₂

yes

maximal, characteristic

Dic₁₀₃⋊C₁₆

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₁₀₃

C₄

yes

characteristic

#15692961

—

C₂×C₁₀₃⋊C₁₆

C₂²

yes

—

C₈×Dic₁₀₃

C₄

yes

—

C₈×Dic₁₀₃

C₂²

yes

—

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

M₃₂×C₁₀₃

—

C₂×C₁₆×C₁₀₃

C₂²

yes

—

C₂²×Dic₁₀₃

C₈

yes

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₀₃

C₈

yes

characteristic

C₄×Dic₁₀₃

C₈

yes

characteristic

C₄×Dic₁₀₃

C₂×C₄

yes

—

C₁₀₃⋊C₁₆

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₃

D₄

yes

—

C₂×C₈×C₁₀₃

Q₈

yes

—

C₂×C₈×C₁₀₃

C₂³

yes

characteristic

C₁₆×C₁₀₃

—

C₂³×C₁₀₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₃

M₁₆

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

C₂×C₈

yes

—

C₂×Dic₁₀₃

C₂×C₈

yes

—

C₈×C₁₀₃

C₄⋊C₄

yes

—

C₈×C₁₀₃

C₂×D₄

yes

characteristic

C₈×C₁₀₃

C₂×Q₈

yes

characteristic

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₄⋊C₈

yes

—

C₂²×C₁₀₃

C₂²×C₈

yes

characteristic

C₄×C₁₀₃

C₄⋊C₈

yes

—

C₄×C₁₀₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₃

C₂×M₁₆

yes

characteristic

Dic₁₀₃

—

Dic₁₀₃

M₃₂

yes

complemented

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₀₃

C₂×M₃₂

yes

—

#2150

—

103

maximal, Sylow

C₁₀₃

#2150

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

103

—

C₂×M₃₂

D₁₀₃

yes

characteristic

C₂×C₄×C₈

—

103

—

C₄⋊C₁₆

—

103

—

C₂²×C₈

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

C₂×C₄²

—

103

—

M₃₂

—

M₃₂

—

206

—

C₂×C₁₆

—

103

—

C₂×C₁₆

C₂×D₁₀₃

yes

—

C₄×C₈

—

103

—

C₄×C₈

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

C₄×D₁₀₃

yes

characteristic

C₂×C₈

—

103

—

C₂×C₈

—

206

—

C₂×C₈

Dic₂₀₆

yes

—

C₂×C₈

C₄×D₁₀₃

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₃

yes

—

C₂×C₈

C₂²×D₁₀₃

yes

characteristic

C₄²

—

103

—

C₄²

—

103

—

C₄²

—

103

—

C₁₆

—

C₁₆

—

206

—

C₂³

C₈×D₁₀₃

yes

characteristic

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

206

—

C₂×C₄

Dic₁₀₃⋊C₄

yes

—

C₂×C₄

C₈×D₁₀₃

yes

characteristic

C₂×C₄

#13110088

yes

—

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

C₈

—

206

—

C₈

Dic₁₀₃⋊C₄

yes

—

C₈

C₂×C₂²⋊D₁₀₃

yes

characteristic

C₈

C₂×Dic₂₀₆

yes

characteristic

C₂²

—

C₂²

Dic₁₀₃⋊C₈

yes

—

C₂²

C₂×C₈×D₁₀₃

yes

characteristic

C₄

—

103

—

C₄

—

206

—

C₄

Dic₁₀₃⋊C₈

yes

—

C₄

C₂×Dic₁₀₃⋊C₄

yes

characteristic

C₄

C₂×#13110088

yes

characteristic

C₂

—

C₂

C₂×Dic₁₀₃⋊C₈

yes

characteristic

100

C₂

M₃₂×D₁₀₃

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂²×C₈
Fitting	C₂×M₃₂×C₁₀₃	C₂
prime radical	C₂³×C₁₀₃	C₂×C₈
cosocle	C₂×C₈×C₁₀₃	C₂³
hypercenter	C₂×M₃₂	D₁₀₃
hypocenter	C₁₀₃	#2150
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₃

commutator, socle

C₂²×C₁₀₃

C₂²×C₈

Fitting

C₂×M₃₂×C₁₀₃

C₂

prime radical

C₂³×C₁₀₃

C₂×C₈

cosocle

C₂×C₈×C₁₀₃

C₂³

hypercenter

C₂×M₃₂

D₁₀₃

C₁₀₃

G ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. Dic₁₀₃⋊C₁₆	C₂	semidirect
46. Dic₁₀₃	M₃₂	semidirect
51. C₁₀₃	#2150	semidirect

H ≅ G/N

type

3. Dic₁₀₃⋊C₁₆

C₂

semidirect

46. Dic₁₀₃

M₃₂

semidirect

51. C₁₀₃

#2150

semidirect

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

subgroup	index
C₂×#15692961	2
C₂×C₈×Dic₁₀₃	2
Dic₁₀₃⋊C₁₆	2
C₂×M₃₂×C₁₀₃	2
#2150	103

subgroup

index

C₂×#15692961

C₂×C₈×Dic₁₀₃

Dic₁₀₃⋊C₁₆

C₂×M₃₂×C₁₀₃

#2150

103

Subgroups — #27196899

Conjugacy classes of subgroups

Complements

Maximal subgroups