Group #27194284 — subgroups

Subgroups — #27194284

Order 13184.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20268170	C₂	1	1	yes	maximal, characteristic
2	#20268086	C₂	1	2	yes	maximal
3	#20267540	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₁₀₃⋊C₈)	C₂	1	2	yes	maximal
5	Q₈²×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13110095	—	2	2	no	—
7	#15693021	C₂²	1	1	yes	characteristic
8	#15692954	—	2	2	no	—
9	C₄⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	—
10	#15692945	—	2	2	no	—
11	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₃	—	2	2	no	—
13	C₄⋊Q₈×C₁₀₃	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₀₃	—	2	2	no	—
16	C₄×Q₈×C₁₀₃	C₂²	1	2	yes	—
17	C₂×Dic₂₀₆	—	2	1	no	—
18	#13110095	—	4	4	no	—
19	C₂×C₁₀₃⋊C₈	—	2	1	no	—
20	C₂×C₁₀₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₀₃	—	2	2	no	—
22	C₂×Q₈×C₁₀₃	—	2	2	no	—
23	C₂×Q₈×C₁₀₃	D₄	1	2	yes	—
24	C₂×Q₈×C₁₀₃	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₀₃	—	2	4	no	—
26	C₄⋊C₄×C₁₀₃	—	2	2	no	—
27	C₄⋊C₄×C₁₀₃	—	2	2	no	—
28	C₄⋊C₄×C₁₀₃	D₄	1	2	yes	—
29	C₄²×C₁₀₃	—	2	2	no	—
30	C₄²×C₁₀₃	—	2	2	no	—
31	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
32	Q₈×C₁₀₃	—	2	2	no	—
33	Q₈×C₁₀₃	—	4	2	no	—
34	Q₈×C₁₀₃	—	8	2	no	—
35	Q₈×C₁₀₃	Q₁₆	1	4	yes	complemented
36	C₂×C₄×C₁₀₃	—	2	2	no	—
37	C₂×C₄×C₁₀₃	—	2	2	no	—
38	C₂×C₄×C₁₀₃	—	2	2	no	—
39	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
41	C₂×Dic₁₀₃	—	4	1	no	—
42	Dic₂₀₆	—	4	2	no	—
43	C₁₀₃⋊C₈	—	2	2	no	—
44	C₁₀₃⋊C₈	—	4	2	no	—
45	C₂²×C₁₀₃	#120	1	1	yes	characteristic
46	C₄×C₁₀₃	—	2	4	no	—
47	C₄×C₁₀₃	—	2	1	no	—
48	C₄×C₁₀₃	—	4	2	no	—
49	C₄×C₁₀₃	—	4	2	no	—
50	C₄×C₁₀₃	C₂×Q₁₆	1	2	yes	—
51	C₄×C₁₀₃	#137	1	2	yes	—
52	Dic₁₀₃	—	8	1	no	—
53	C₂×C₁₀₃	#451	1	2	yes	—
54	C₂×C₁₀₃	#453	1	1	yes	characteristic
55	#1633	—	103	1	no	maximal, Sylow
56	C₁₀₃	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	D₁₀₃	1	1	yes	characteristic
58	#494	—	103	1	no	—
59	#477	—	103	2	no	—
60	#333	—	103	1	no	—
61	Q₈⋊C₈	—	103	2	no	—
62	C₂×Q₁₆	—	206	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	103	1	no	—
66	C₄⋊Q₈	C₂×D₁₀₃	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₁₀₃	1	2	yes	—
69	#107	—	206	2	no	—
70	C₄⋊C₈	—	103	2	no	—
71	#103	—	206	2	no	—
72	C₄×C₈	—	103	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	206	1	no	—
75	C₂×Q₈	C₂²⋊D₁₀₃	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₁₀₃	1	2	yes	—
77	Q₁₆	—	412	4	no	—
78	C₂×C₈	—	206	2	no	—
79	C₂×C₈	—	206	1	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	206	2	no	—
84	C₄⋊C₄	C₂²⋊D₁₀₃	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	412	2	no	—
92	Q₈	#13110095	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	412	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	2	yes	—
99	C₈	—	412	2	no	—
100	C₈	—	206	2	no	—
101	C₂²	#15692990	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	824	1	no	—
107	C₄	C₂×#13110095	1	2	yes	—
108	C₄	#15693052	1	2	yes	—
109	C₂	#20267947	1	2	yes	—
110	C₂	#20267959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268170

C₂

yes

maximal, characteristic

#20268086

C₂

yes

maximal

#20267540

C₂

yes

maximal, characteristic

Q₈⋊(C₁₀₃⋊C₈)

C₂

yes

maximal

Q₈²×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110095

—

#15693021

C₂²

yes

characteristic

#15692954

—

C₄⋊(C₁₀₃⋊C₈)

C₂²

yes

—

#15692945

—

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₃

—

C₄⋊Q₈×C₁₀₃

—

C₄⋊Q₈×C₁₀₃

C₂²

yes

characteristic

C₄×Q₈×C₁₀₃

—

C₄×Q₈×C₁₀₃

C₂²

yes

—

C₂×Dic₂₀₆

—

#13110095

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×Q₈×C₁₀₃

D₄

yes

—

C₂×Q₈×C₁₀₃

D₄

yes

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

—

C₄²×C₁₀₃

—

C₄²×C₁₀₃

C₂³

yes

characteristic

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

Q₈×C₁₀₃

Q₁₆

yes

complemented

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

#120

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×Q₁₆

yes

—

C₄×C₁₀₃

#137

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

#451

yes

—

C₂×C₁₀₃

#453

yes

characteristic

#1633

—

103

maximal, Sylow

C₁₀₃

#1633

yes

characteristic, complemented, Sylow

Q₈²

D₁₀₃

yes

characteristic

#494

—

103

—

#477

—

103

—

#333

—

103

—

Q₈⋊C₈

—

103

—

C₂×Q₁₆

—

206

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

C₂×D₁₀₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₁₀₃

yes

—

#107

—

206

—

C₄⋊C₈

—

103

—

#103

—

206

—

C₄×C₈

—

103

—

C₂×Q₈

—

C₂×Q₈

—

206

—

C₂×Q₈

C₂²⋊D₁₀₃

yes

—

C₂×Q₈

C₂²⋊D₁₀₃

yes

—

Q₁₆

—

412

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

206

—

C₄⋊C₄

C₂²⋊D₁₀₃

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₀₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

412

—

Q₈

#13110095

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

412

—

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

—

C₈

—

412

—

100

C₈

—

206

—

101

C₂²

#15692990

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

824

—

107

C₄

C₂×#13110095

yes

—

108

C₄

#15693052

yes

—

109

C₂

#20267947

yes

—

110

C₂

#20267959

yes

characteristic

	subgroup	G/H
center	C₂²	#15692990
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting	Q₈²×C₁₀₃	C₂
socle, prime radical	C₂²×C₁₀₃	#120
hypercenter	Q₈²	D₁₀₃
hypocenter	C₁₀₃	#1633
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#15692990

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting

Q₈²×C₁₀₃

C₂

socle, prime radical

C₂²×C₁₀₃

#120

hypercenter

Q₈²

D₁₀₃

hypocenter

C₁₀₃

#1633

LCS

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
35. Q₈×C₁₀₃	Q₁₆	semidirect
56. C₁₀₃	#1633	semidirect
92. Q₈	#13110095	semidirect

H ≅ G/N

type

35. Q₈×C₁₀₃

Q₁₆

semidirect

56. C₁₀₃

#1633

semidirect

92. Q₈

#13110095

semidirect

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

subgroup	index
#20268170	2
#20268086	2
#20267540	2
Q₈⋊(C₁₀₃⋊C₈)	2
Q₈²×C₁₀₃	2
#1633	103

subgroup

index

#20268170

#20268086

#20267540

Q₈⋊(C₁₀₃⋊C₈)

Q₈²×C₁₀₃

#1633

103

Subgroups — #27194284

Conjugacy classes of subgroups

Complements

Maximal subgroups