Group #28336368 — subgroups

Subgroups — #28336368

Order 13696.

220 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16812044	C₂	1	2	yes	maximal
2	C₄×#13133802	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812020	C₂	1	1	yes	maximal, characteristic
4	#20292391	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133802	C₂²	1	2	yes	—
7	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
9	#16812044	C₄	1	8	yes	complemented
10	#16812020	—	2	1	no	—
11	#16812020	C₄	1	4	yes	—
12	C₄×C₁₀₇⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₁₀₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	#97×C₁₀₇	—	2	1	no	—
16	C₄×C₈×C₁₀₇	—	2	1	no	—
17	#13133802	—	2	2	no	—
18	#13133802	C₂×C₄	1	4	yes	—
19	C₂×C₁₀₇⋊C₈	—	2	2	no	—
20	C₂×C₁₀₇⋊C₈	—	2	2	no	—
21	C₂×C₁₀₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₁₀₇⋊C₈	C₂×C₄	1	4	yes	—
23	C₂×C₁₀₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₀₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₇	—	2	2	no	—
27	M₁₆×C₁₀₇	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₀₇	—	2	2	no	—
29	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
31	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
32	C₂³×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₇	—	4	2	no	—
34	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₀₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	#41	1	2	yes	—
40	C₈×C₁₀₇	—	2	4	no	—
41	C₁₀₇⋊C₈	—	2	2	no	—
42	C₁₀₇⋊C₈	—	2	4	no	—
43	C₁₀₇⋊C₈	C₄²	1	4	yes	—
44	C₂²×C₁₀₇	—	2	2	no	—
45	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₇	C₄×Q₈	1	2	yes	—
47	C₄×C₁₀₇	—	2	2	no	—
48	C₄×C₁₀₇	—	2	2	no	—
49	C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₁₀₇	#117	1	1	yes	characteristic
51	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
52	C₂×C₁₀₇	—	2	2	no	—
53	C₂×C₁₀₇	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₁₀₇	#430	1	2	yes	—
55	#1778	—	107	1	no	maximal, Sylow
56	C₁₀₇	#1778	1	1	yes	characteristic, complemented, Sylow
57	C₂×#108	—	107	2	no	—
58	C₄×M₁₆	—	107	1	no	—
59	C₄×M₁₆	D₁₀₇	1	1	yes	characteristic
60	C₂×#97	—	107	1	no	—
61	#341	—	107	2	no	—
62	C₂×M₁₆	—	107	2	no	—
63	C₂×M₁₆	C₂×D₁₀₇	1	2	yes	—
64	C₂²×C₈	—	107	1	no	—
65	C₂²×C₈	—	107	1	no	—
66	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
67	#108	—	107	8	no	—
68	#97	—	2	1	no	—
69	#97	—	107	4	no	—
70	#97	—	214	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	214	1	no	—
73	C₂²×C₄	Dic₂₁₄	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	107	4	no	—
77	M₁₆	—	214	2	no	—
78	M₁₆	C₄×D₁₀₇	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	107	2	no	—
81	C₂×C₈	—	107	4	no	—
82	C₂×C₈	—	107	2	no	—
83	C₂×C₈	—	214	2	no	—
84	C₂×C₈	—	214	2	no	—
85	C₄²	C₄×D₁₀₇	1	2	yes	—
86	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
87	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
88	C₂³	C₂×Dic₂₁₄	1	1	yes	characteristic
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	Dic₁₀₇⋊C₄	1	2	yes	—
91	C₂×C₄	Dic₁₀₇⋊C₄	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
95	C₂×C₄	#13133818	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	107	4	no	—
98	C₈	—	214	2	no	—
99	C₈	—	214	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×Dic₁₀₇⋊C₄	1	1	yes	characteristic
102	C₂²	C₄×Dic₂₁₄	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	C₄²×D₁₀₇	1	1	yes	characteristic
106	C₄	#16812066	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₁₀₇	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×Dic₁₀₇⋊C₄	1	1	yes	characteristic
110	C₂	#20292675	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812044

C₂

yes

maximal

C₄×#13133802

C₂

yes

maximal, characteristic

C₂×#16812020

C₂

yes

maximal, characteristic

#20292391

C₂

yes

maximal

C₄×M₁₆×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

#16812044

C₄

yes

complemented

#16812020

—

#16812020

C₄

yes

—

C₄×C₁₀₇⋊C₈

—

C₂×M₁₆×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#97×C₁₀₇

—

C₄×C₈×C₁₀₇

—

#13133802

—

#13133802

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

M₁₆×C₁₀₇

—

M₁₆×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

—

C₄²×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

D₄

yes

characteristic

C₄²×C₁₀₇

D₄

yes

characteristic

C₂³×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

C₄²

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₇

C₄×Q₈

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×C₄²

yes

characteristic

C₄×C₁₀₇

#117

yes

characteristic

C₄×C₁₀₇

C₄×D₄

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₇

#430

yes

—

#1778

—

107

maximal, Sylow

C₁₀₇

#1778

yes

characteristic, complemented, Sylow

C₂×#108

—

107

—

C₄×M₁₆

—

107

—

C₄×M₁₆

D₁₀₇

yes

characteristic

C₂×#97

—

107

—

#341

—

107

—

C₂×M₁₆

—

107

—

C₂×M₁₆

C₂×D₁₀₇

yes

—

C₂²×C₈

—

107

—

C₂²×C₈

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#108

—

107

—

#97

—

#97

—

107

—

#97

—

214

—

C₄×C₈

—

C₄×C₈

—

214

—

C₂²×C₄

Dic₂₁₄

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

M₁₆

—

107

—

M₁₆

—

214

—

M₁₆

C₄×D₁₀₇

yes

—

C₂×C₈

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄²

C₄×D₁₀₇

yes

—

C₄²

C₂²⋊D₁₀₇

yes

characteristic

C₄²

C₂²⋊D₁₀₇

yes

characteristic

C₂³

C₂×Dic₂₁₄

yes

characteristic

C₂×C₄

—

C₂×C₄

Dic₁₀₇⋊C₄

yes

—

C₂×C₄

Dic₁₀₇⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

C₂×C₄

#13133818

yes

—

C₈

—

C₈

—

107

—

C₈

—

214

—

C₈

—

214

—

100

C₂²

—

101

C₂²

C₂×Dic₁₀₇⋊C₄

yes

characteristic

102

C₂²

C₄×Dic₂₁₄

yes

—

103

C₄

—

104

C₄

—

105

C₄

C₄²×D₁₀₇

yes

characteristic

106

C₄

#16812066

yes

characteristic

107

C₄

C₄×C₂²⋊D₁₀₇

yes

—

108

C₂

—

109

C₂

C₄×Dic₁₀₇⋊C₄

yes

characteristic

110

C₂

#20292675

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₇
commutator	C₄×C₁₀₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₄×M₁₆×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₄×M₁₆	D₁₀₇
hypocenter	C₁₀₇	#1778
LCS	G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₇

commutator

C₄×C₁₀₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₄×M₁₆×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₄×M₁₆

D₁₀₇

hypocenter

C₁₀₇

#1778

LCS

G ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #16812044	C₄	semidirect
56. C₁₀₇	#1778	semidirect

H ≅ G/N

type

9. #16812044

C₄

semidirect

56. C₁₀₇

#1778

semidirect

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

subgroup	index
C₂×#16812044	2
C₄×#13133802	2
C₂×#16812020	2
#20292391	2
C₄×M₁₆×C₁₀₇	2
#1778	107

subgroup

index

C₄×M₁₆×C₁₀₇

107

Subgroups — #28336368

Conjugacy classes of subgroups

Complements

Maximal subgroups