Group #28337263 — subgroups

Subgroups — #28337263

Order 13696.

332 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 141 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	#20292497	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×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	—	2	2	no	—
10	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
11	#16812034	—	2	8	no	—
12	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×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	C₄⋊Dic₁₀₇	—	4	4	no	—
24	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₀₇	—	2	2	no	—
27	C₂×C₈×C₁₀₇	—	2	2	no	—
28	C₂×C₈×C₁₀₇	D₄	1	4	yes	—
29	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₇	D₄	1	2	yes	—
31	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	4	yes	—
33	C₂×C₄×C₁₀₇	SD₁₆	1	4	yes	—
34	C₂×C₄×C₁₀₇	Q₁₆	1	4	yes	—
35	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₀₇	#41	1	2	yes	—
39	C₂×Dic₁₀₇	—	2	4	no	—
40	C₂×Dic₁₀₇	—	4	2	no	—
41	C₂×Dic₁₀₇	—	4	2	no	—
42	C₂×Dic₁₀₇	—	4	2	no	—
43	C₂×Dic₁₀₇	—	4	4	no	—
44	Dic₂₁₄	—	2	8	no	—
45	Dic₂₁₄	—	4	4	no	—
46	C₈×C₁₀₇	—	2	4	no	—
47	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₀₇	#124	1	1	yes	characteristic
49	C₂²×C₁₀₇	#127	1	1	yes	characteristic
50	C₂²×C₁₀₇	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₁₀₇	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₀₇	#103	1	8	yes	complemented
53	C₄×C₁₀₇	C₄×D₄	1	2	yes	—
54	C₄×C₁₀₇	#121	1	2	yes	—
55	Dic₁₀₇	—	4	4	no	—
56	Dic₁₀₇	—	8	2	no	—
57	C₂×C₁₀₇	#390	1	1	yes	characteristic
58	C₂×C₁₀₇	C₂×#103	1	2	yes	—
59	C₂×C₁₀₇	#487	1	2	yes	—
60	C₂×C₁₀₇	#492	1	1	yes	characteristic
61	C₂×C₁₀₇	#494	1	1	yes	characteristic
62	#1956	—	107	1	no	maximal, Sylow
63	C₁₀₇	#1956	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	107	1	no	—
65	C₂×#103	—	107	4	no	—
66	C₂×C₄×C₈	D₁₀₇	1	1	yes	characteristic
67	#384	—	107	1	no	—
68	C₂²×Q₈	—	107	2	no	—
69	C₂²×C₈	C₂×D₁₀₇	1	2	yes	—
70	C₄⋊Q₈	—	107	4	no	—
71	C₄⋊Q₈	—	214	2	no	—
72	C₂×C₄⋊C₄	—	107	2	no	—
73	C₂×C₄⋊C₄	—	214	2	no	—
74	C₂×C₄⋊C₄	—	214	2	no	—
75	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
76	#103	—	214	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂×Q₈	—	107	4	no	—
79	C₂×Q₈	—	214	4	no	—
80	C₂×Q₈	—	214	4	no	—
81	C₂×Q₈	—	214	2	no	—
82	C₂²×C₄	—	214	2	no	—
83	C₂²×C₄	—	214	2	no	—
84	C₂²×C₄	D₄₂₈	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₄₂₈	1	4	yes	—
89	C₄⋊C₄	—	428	4	no	—
90	C₄⋊C₄	—	214	4	no	—
91	C₄⋊C₄	—	214	8	no	—
92	C₄²	C₄×D₁₀₇	1	2	yes	—
93	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
94	C₂³	C₂×D₄₂₈	1	1	yes	characteristic
95	Q₈	—	428	4	no	—
96	Q₈	—	214	8	no	—
97	C₂×C₄	—	428	2	no	—
98	C₂×C₄	—	428	4	no	—
99	C₂×C₄	—	428	2	no	—
100	C₂×C₄	—	428	2	no	—
101	C₂×C₄	—	214	4	no	—
102	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	4	yes	—
103	C₂×C₄	#13133807	1	4	yes	—
104	C₂×C₄	Dic₄₂₈	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
107	C₂×C₄	C₂×D₄₂₈	1	2	yes	—
108	C₂×C₄	#13133818	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₂²	C₂×(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
111	C₂²	#16812097	1	1	yes	characteristic
112	C₂²	#16812106	1	1	yes	characteristic
113	C₂²	C₂×#13133807	1	2	yes	—
114	C₂²	C₂×Dic₄₂₈	1	2	yes	—
115	C₄	—	856	2	no	—
116	C₄	—	428	4	no	—
117	C₄	#16812034	1	8	yes	complemented
118	C₄	C₄×D₄₂₈	1	2	yes	—
119	C₄	#16812083	1	2	yes	—
120	C₂	#20292529	1	1	yes	characteristic
121	C₂	C₂×#16812034	1	2	yes	—
122	C₂	#20292978	1	2	yes	—
123	C₂	#20293000	1	1	yes	characteristic
124	C₂	#20293007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812109

C₂

yes

maximal, characteristic

C₂×#16812034

C₂

yes

maximal

#20292497

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₂²

yes

—

#16812109

—

#16812109

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₇

—

C₂×C₄⋊Dic₁₀₇

—

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

#16812034

—

C₂²×C₈×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄×C₈×C₁₀₇

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

C₂×C₄

yes

—

C₂²×Dic₁₀₇

—

C₂²×Dic₁₀₇

—

C₄⋊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

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

SD₁₆

yes

—

C₂×C₄×C₁₀₇

Q₁₆

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

C₂×SD₁₆

yes

—

C₂²×C₁₀₇

C₂×Q₁₆

yes

—

C₄×C₁₀₇

#103

yes

complemented

C₄×C₁₀₇

C₄×D₄

yes

—

C₄×C₁₀₇

#121

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

#390

yes

characteristic

C₂×C₁₀₇

C₂×#103

yes

—

C₂×C₁₀₇

#487

yes

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#1956

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

107

—

C₂×#103

—

107

—

C₂×C₄×C₈

D₁₀₇

yes

characteristic

#384

—

107

—

C₂²×Q₈

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

—

214

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

214

—

C₂×C₄⋊C₄

—

214

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#103

—

214

—

C₄×C₈

—

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

—

428

—

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

—

428

—

C₂×C₄

—

428

—

C₂×C₄

—

428

—

100

C₂×C₄

—

428

—

101

C₂×C₄

—

214

—

102

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

103

C₂×C₄

#13133807

yes

—

104

C₂×C₄

Dic₄₂₈

yes

—

105

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

107

C₂×C₄

C₂×D₄₂₈

yes

—

108

C₂×C₄

#13133818

yes

—

109

C₈

—

110

C₂²

C₂×(C₂×D₁₀₇)⋊C₄

yes

characteristic

111

C₂²

#16812097

yes

characteristic

112

C₂²

#16812106

yes

characteristic

113

C₂²

C₂×#13133807

yes

—

114

C₂²

C₂×Dic₄₂₈

yes

—

115

C₄

—

856

—

116

C₄

—

428

—

117

C₄

#16812034

yes

complemented

118

C₄

C₄×D₄₂₈

yes

—

119

C₄

#16812083

yes

—

120

C₂

#20292529

yes

characteristic

121

C₂

C₂×#16812034

yes

—

122

C₂

#20292978

yes

—

123

C₂

#20293000

yes

characteristic

124

C₂

#20293007

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₂₈
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₂×C₄×C₈×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₇
hypocenter	C₁₀₇	#1956
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₄₂₈

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₂×C₄×C₈×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂×D₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₀₇

hypocenter

C₁₀₇

#1956

LCS

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #16812109	C₄	semidirect
52. C₄×C₁₀₇	#103	semidirect
63. C₁₀₇	#1956	semidirect
117. C₄	#16812034	semidirect

H ≅ G/N

type

7. #16812109

C₄

semidirect

52. C₄×C₁₀₇

#103

semidirect

63. C₁₀₇

#1956

semidirect

117. C₄

#16812034

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
#20292497	2
C₂×C₄×C₈×C₁₀₇	2
#1956	107

subgroup

index

C₂×C₄×C₈×C₁₀₇

107

Subgroups — #28337263

Conjugacy classes of subgroups

Complements

Maximal subgroups