Group #28337281 — subgroups

Subgroups — #28337281

Order 13696.

308 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 109 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×#16812109	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812034	C₂	1	4	yes	maximal
3	#20292493	C₂	1	1	yes	maximal, characteristic
4	C₂×#97×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₄	C₂²	1	2	yes	—
6	#16812109	—	2	2	no	—
7	#16812109	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₀₇	—	2	2	no	—
9	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
10	#16812034	—	2	8	no	—
11	C₁₀₇⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
14	#97×C₁₀₇	—	2	2	no	—
15	C₂×Dic₂₁₄	—	2	4	no	—
16	C₂×Dic₂₁₄	—	2	4	no	—
17	C₂×Dic₂₁₄	—	2	2	no	—
18	C₂×Dic₂₁₄	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₀₇	—	2	2	no	—
20	C₂²×Dic₁₀₇	—	2	2	no	—
21	C₄⋊Dic₁₀₇	—	2	8	no	—
22	C₄⋊Dic₁₀₇	—	2	4	no	—
23	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
24	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₀₇	—	2	2	no	—
26	C₂×C₈×C₁₀₇	—	2	2	no	—
27	C₂×C₈×C₁₀₇	D₄	1	4	yes	—
28	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
29	C₄²×C₁₀₇	D₄	1	2	yes	—
30	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₇	—	2	4	no	—
32	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	4	yes	—
33	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₀₇	#41	1	2	yes	—
37	C₂×Dic₁₀₇	—	2	4	no	—
38	C₂×Dic₁₀₇	—	4	2	no	—
39	C₂×Dic₁₀₇	—	4	2	no	—
40	C₂×Dic₁₀₇	—	4	2	no	—
41	C₂×Dic₁₀₇	—	4	4	no	—
42	Dic₂₁₄	—	2	8	no	—
43	Dic₂₁₄	—	4	4	no	—
44	C₈×C₁₀₇	—	2	4	no	—
45	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₇	#124	1	1	yes	characteristic
47	C₂²×C₁₀₇	#127	1	1	yes	characteristic
48	C₂²×C₁₀₇	#137	1	4	yes	—
49	C₄×C₁₀₇	—	2	4	no	—
50	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
51	C₄×C₁₀₇	#121	1	2	yes	—
52	Dic₁₀₇	—	4	4	no	—
53	Dic₁₀₇	—	8	2	no	—
54	C₂×C₁₀₇	#390	1	1	yes	characteristic
55	C₂×C₁₀₇	#419	1	2	yes	—
56	C₂×C₁₀₇	#491	1	2	yes	—
57	C₂×C₁₀₇	#497	1	2	yes	—
58	#1960	—	107	1	no	maximal, Sylow
59	C₁₀₇	#1960	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊Q₈	—	107	1	no	—
61	C₂×#103	—	107	4	no	—
62	C₂×#97	D₁₀₇	1	1	yes	characteristic
63	#382	—	107	1	no	—
64	C₂²×Q₈	—	107	2	no	—
65	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
66	C₄⋊Q₈	—	107	4	no	—
67	C₄⋊Q₈	—	214	2	no	—
68	C₂×C₄⋊C₄	—	107	2	no	—
69	C₂×C₄⋊C₄	—	214	2	no	—
70	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
71	#103	—	214	8	no	—
72	#97	—	2	2	no	—
73	#95	—	214	2	no	—
74	C₂×Q₈	—	107	4	no	—
75	C₂×Q₈	—	214	4	no	—
76	C₂×Q₈	—	214	4	no	—
77	C₂×Q₈	—	214	2	no	—
78	C₂²×C₄	—	214	2	no	—
79	C₂²×C₄	—	214	2	no	—
80	C₂²×C₄	D₄₂₈	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	D₄₂₈	1	4	yes	—
85	C₄⋊C₄	—	214	4	no	—
86	C₄⋊C₄	—	214	8	no	—
87	C₄²	C₄×D₁₀₇	1	2	yes	—
88	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
89	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
90	Q₈	—	428	4	no	—
91	Q₈	—	214	8	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	428	2	no	—
94	C₂×C₄	—	428	4	no	—
95	C₂×C₄	—	428	2	no	—
96	C₂×C₄	—	428	2	no	—
97	C₂×C₄	—	214	4	no	—
98	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₄₂₈	1	2	yes	—
102	C₂×C₄	#13133818	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	C₂×(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
105	C₂²	#16812097	1	1	yes	characteristic
106	C₂²	#16812106	1	1	yes	characteristic
107	C₂²	#16812142	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	856	2	no	—
110	C₄	—	428	4	no	—
111	C₄	C₄×D₄₂₈	1	2	yes	—
112	C₄	#16812083	1	2	yes	—
113	C₂	#20292529	1	1	yes	characteristic
114	C₂	#20292637	1	2	yes	—
115	C₂	#20292996	1	2	yes	—
116	C₂	#20293021	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812109

C₂

yes

maximal, characteristic

C₂×#16812034

C₂

yes

maximal

#20292493

C₂

yes

maximal, characteristic

C₂×#97×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₂²

yes

—

#16812109

—

#16812109

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₇

—

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

#16812034

—

C₁₀₇⋊#95

—

C₂²×C₈×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#97×C₁₀₇

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

C₂×C₄

yes

—

C₂²×Dic₁₀₇

—

C₂²×Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄×D₄

yes

—

C₄×C₁₀₇

#121

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

#390

yes

characteristic

C₂×C₁₀₇

#419

yes

—

C₂×C₁₀₇

#491

yes

—

C₂×C₁₀₇

#497

yes

—

#1960

—

107

maximal, Sylow

C₁₀₇

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

107

—

C₂×#103

—

107

—

C₂×#97

D₁₀₇

yes

characteristic

#382

—

107

—

C₂²×Q₈

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

—

214

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

214

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#103

—

214

—

#97

—

#95

—

214

—

C₂×Q₈

—

107

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂²×C₄

—

214

—

C₂²×C₄

—

214

—

C₂²×C₄

D₄₂₈

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄₂₈

yes

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

C₄×D₁₀₇

yes

—

C₄²

C₂²⋊D₁₀₇

yes

—

C₂³

C₂×D₄₂₈

yes

characteristic

Q₈

—

428

—

Q₈

—

214

—

C₂×C₄

—

C₂×C₄

—

428

—

C₂×C₄

—

428

—

C₂×C₄

—

428

—

C₂×C₄

—

428

—

C₂×C₄

—

214

—

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

101

C₂×C₄

C₂×D₄₂₈

yes

—

102

C₂×C₄

#13133818

yes

—

103

C₈

—

104

C₂²

C₂×(C₂×D₁₀₇)⋊C₄

yes

characteristic

105

C₂²

#16812097

yes

characteristic

106

C₂²

#16812106

yes

characteristic

107

C₂²

#16812142

yes

—

108

C₄

—

109

C₄

—

856

—

110

C₄

—

428

—

111

C₄

C₄×D₄₂₈

yes

—

112

C₄

#16812083

yes

—

113

C₂

#20292529

yes

characteristic

114

C₂

#20292637

yes

—

115

C₂

#20292996

yes

—

116

C₂

#20293021

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₂₈
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂×#97×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×#97	D₁₀₇
hypocenter	C₁₀₇	#1960
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₄₂₈

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₂×#97×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂×D₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×#97

D₁₀₇

hypocenter

C₁₀₇

#1960

LCS

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #16812109	C₄	semidirect
59. C₁₀₇	#1960	semidirect

H ≅ G/N

type

7. #16812109

C₄

semidirect

59. C₁₀₇

#1960

semidirect

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

subgroup	index
C₂×#16812109	2
C₂×#16812034	2
#20292493	2
C₂×#97×C₁₀₇	2
#1960	107

subgroup

index

C₂×#97×C₁₀₇

107

Subgroups — #28337281

Conjugacy classes of subgroups

Complements

Maximal subgroups