Group #28336385 — subgroups

Subgroups — #28336385

Order 13696.

564 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812028	C₂	1	4	yes	maximal, complemented
2	#20292591	C₂	1	2	yes	maximal
3	C₂³×Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133802	C₂²	1	4	yes	—
5	#16812028	—	2	8	no	—
6	C₂²⋊(C₁₀₇⋊C₈)	—	2	4	no	—
7	C₂²×Q₈×C₁₀₇	—	2	8	no	—
8	C₂²×Q₈×C₁₀₇	—	2	4	no	—
9	C₂²×Q₈×C₁₀₇	C₄	1	2	yes	—
10	C₂²×Q₈×C₁₀₇	C₂²	1	2	yes	—
11	C₂³×C₄×C₁₀₇	C₄	1	2	yes	—
12	C₂³×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
13	#13133802	—	2	8	no	—
14	C₂×C₁₀₇⋊C₈	—	2	4	no	—
15	C₂⁴×C₁₀₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₁₀₇	—	2	32	no	—
17	C₂×Q₈×C₁₀₇	—	2	8	no	—
18	C₂×Q₈×C₁₀₇	—	2	4	no	—
19	C₂×Q₈×C₁₀₇	—	2	4	no	—
20	C₂×Q₈×C₁₀₇	—	2	4	no	—
21	C₂×Q₈×C₁₀₇	D₄	1	8	yes	—
22	C₂²×C₄×C₁₀₇	—	2	4	no	—
23	C₂²×C₄×C₁₀₇	—	2	8	no	—
24	C₂²×C₄×C₁₀₇	—	2	4	no	—
25	C₂²×C₄×C₁₀₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₇	D₄	1	4	yes	—
28	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₂³×C₁₀₇	—	2	4	no	—
30	C₂³×C₁₀₇	—	2	2	no	—
31	C₂³×C₁₀₇	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
33	Q₈×C₁₀₇	—	2	16	no	—
34	Q₈×C₁₀₇	—	2	16	no	—
35	C₂×C₄×C₁₀₇	—	2	16	no	—
36	C₂×C₄×C₁₀₇	—	2	4	no	—
37	C₂×C₄×C₁₀₇	—	2	2	no	—
38	C₂×C₄×C₁₀₇	—	2	4	no	—
39	C₂×C₄×C₁₀₇	—	2	8	no	—
40	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	8	yes	—
41	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
44	C₁₀₇⋊C₈	—	4	4	no	—
45	C₂²×C₁₀₇	—	2	8	no	—
46	C₂²×C₁₀₇	—	2	2	no	—
47	C₂²×C₁₀₇	—	2	2	no	—
48	C₂²×C₁₀₇	—	2	2	no	—
49	C₂²×C₁₀₇	#101	1	4	yes	complemented
50	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	2	yes	—
51	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₁₀₇	—	2	8	no	—
53	C₄×C₁₀₇	—	2	2	no	—
54	C₄×C₁₀₇	#120	1	4	yes	—
55	C₂×C₁₀₇	—	2	4	no	—
56	C₂×C₁₀₇	—	2	2	no	—
57	C₂×C₁₀₇	#379	1	1	yes	characteristic
58	C₂×C₁₀₇	C₂×#101	1	2	yes	—
59	#1784	—	107	1	no	maximal, Sylow
60	C₁₀₇	#1784	1	1	yes	characteristic, complemented, Sylow
61	C₂³×Q₈	D₁₀₇	1	1	yes	characteristic
62	C₂×#101	—	107	4	no	—
63	#407	—	107	2	no	—
64	C₂²×Q₈	—	2	8	no	—
65	C₂²×Q₈	—	2	4	no	—
66	C₂²×Q₈	Dic₁₀₇	1	2	yes	—
67	C₂²×Q₈	C₂×D₁₀₇	1	2	yes	—
68	C₂³×C₄	Dic₁₀₇	1	2	yes	—
69	C₂³×C₄	C₂×D₁₀₇	1	1	yes	characteristic
70	C₂×M₁₆	—	107	4	no	—
71	#101	—	214	8	no	—
72	C₂²⋊C₈	—	214	4	no	—
73	C₂⁴	C₂×Dic₁₀₇	1	1	yes	characteristic
74	C₂×Q₈	—	2	32	no	—
75	C₂×Q₈	—	2	8	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	—	2	4	no	—
78	C₂×Q₈	—	2	4	no	—
79	C₂×Q₈	C₂²⋊D₁₀₇	1	8	yes	—
80	C₂²×C₄	—	2	4	no	—
81	C₂²×C₄	—	2	8	no	—
82	C₂²×C₄	—	2	4	no	—
83	C₂²×C₄	C₂×Dic₁₀₇	1	4	yes	—
84	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
85	C₂²×C₄	C₂²⋊D₁₀₇	1	4	yes	—
86	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
87	M₁₆	—	214	8	no	—
88	C₂×C₈	—	214	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	—	2	4	no	—
91	C₂³	C₂²⋊Dic₁₀₇	1	2	yes	—
92	C₂³	C₂²×Dic₁₀₇	1	1	yes	characteristic
93	Q₈	—	2	16	no	—
94	Q₈	—	2	16	no	—
95	C₂×C₄	—	2	16	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	C₂²⋊Dic₁₀₇	1	8	yes	—
101	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	4	yes	—
103	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
104	C₈	—	428	4	no	—
105	C₂²	—	2	8	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	#16812028	1	4	yes	complemented
110	C₂²	C₂×C₂²⋊Dic₁₀₇	1	2	yes	—
111	C₂²	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
112	C₄	—	2	8	no	—
113	C₄	—	2	2	no	—
114	C₄	#16812078	1	4	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	2	4	no	—
117	C₂	#20292483	1	1	yes	characteristic
118	C₂	C₂×#16812028	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812028

C₂

yes

maximal, complemented

#20292591

C₂

yes

maximal

C₂³×Q₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

#16812028

—

C₂²⋊(C₁₀₇⋊C₈)

—

C₂²×Q₈×C₁₀₇

—

C₂²×Q₈×C₁₀₇

—

C₂²×Q₈×C₁₀₇

C₄

yes

—

C₂²×Q₈×C₁₀₇

C₂²

yes

—

C₂³×C₄×C₁₀₇

C₄

yes

—

C₂³×C₄×C₁₀₇

C₂²

yes

characteristic

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂⁴×C₁₀₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₂²⋊C₄

yes

—

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#101

yes

complemented

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#120

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#379

yes

characteristic

C₂×C₁₀₇

C₂×#101

yes

—

#1784

—

107

maximal, Sylow

C₁₀₇

#1784

yes

characteristic, complemented, Sylow

C₂³×Q₈

D₁₀₇

yes

characteristic

C₂×#101

—

107

—

#407

—

107

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

Dic₁₀₇

yes

—

C₂²×Q₈

C₂×D₁₀₇

yes

—

C₂³×C₄

Dic₁₀₇

yes

—

C₂³×C₄

C₂×D₁₀₇

yes

characteristic

C₂×M₁₆

—

107

—

#101

—

214

—

C₂²⋊C₈

—

214

—

C₂⁴

C₂×Dic₁₀₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

214

—

C₂×C₈

—

214

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₁₀₇

yes

—

C₂³

C₂²×Dic₁₀₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

101

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

102

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

103

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

104

C₈

—

428

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#16812028

yes

complemented

110

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

—

111

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#16812078

yes

—

115

C₂

—

116

C₂

—

117

C₂

#20292483

yes

characteristic

118

C₂

C₂×#16812028

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₇
commutator	C₂³×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂³×Q₈×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₀₇	C₂×C₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂³×Q₈	D₁₀₇
hypocenter	C₁₀₇	#1784
LCS	G ▷ C₂³×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₇

commutator

C₂³×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₂³×Q₈×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₀₇

C₂×C₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂³×Q₈

D₁₀₇

hypocenter

C₁₀₇

#1784

LCS

G ▷ C₂³×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×Q₈

N	H ≅ G/N	type
1. C₂×#16812028	C₂	semidirect
49. C₂²×C₁₀₇	#101	semidirect
60. C₁₀₇	#1784	semidirect
109. C₂²	#16812028	semidirect

H ≅ G/N

type

1. C₂×#16812028

C₂

semidirect

49. C₂²×C₁₀₇

#101

semidirect

60. C₁₀₇

#1784

semidirect

109. C₂²

#16812028

semidirect

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

subgroup	index
C₂×#16812028	2
#20292591	2
C₂³×Q₈×C₁₀₇	2
#1784	107

subgroup

index

C₂×#16812028

#20292591

C₂³×Q₈×C₁₀₇

#1784

107

Subgroups — #28336385

Conjugacy classes of subgroups

Complements

Maximal subgroups