Group #28337210 — subgroups

Subgroups — #28337210

Order 13696.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16812045	C₂	1	1	yes	maximal, characteristic
2	#20292654	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
4	#20292386	C₂	1	2	yes	maximal
5	#427×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133802	C₂²	1	2	yes	—
7	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
9	#16812045	—	2	2	no	—
10	C₄⋊(C₁₀₇⋊C₈)	—	2	4	no	—
11	C₈×Dic₁₀₇	—	2	2	no	—
12	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
13	#117×C₁₀₇	C₂²	1	2	yes	—
14	#107×C₁₀₇	C₄	1	2	yes	complemented
15	#106×C₁₀₇	C₄	1	2	yes	complemented
16	C₂²×Dic₁₀₇	—	2	1	no	—
17	#13133802	—	2	2	no	—
18	#13133802	—	2	2	no	—
19	C₂×C₁₀₇⋊C₈	—	2	1	no	—
20	C₂×C₁₀₇⋊C₈	—	2	1	no	—
21	C₂×C₁₀₇⋊C₈	—	2	2	no	—
22	C₂×C₁₀₇⋊C₈	D₄	1	2	yes	—
23	C₄×Dic₁₀₇	—	2	1	no	—
24	C₄×Dic₁₀₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₀₇	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
32	C₄²×C₁₀₇	—	2	2	no	—
33	C₂³×C₁₀₇	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₀₇	—	2	4	no	—
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	1	yes	characteristic
38	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
41	C₂×Dic₁₀₇	—	2	4	no	—
42	C₂×Dic₁₀₇	—	2	2	no	—
43	C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
44	C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
45	C₁₀₇⋊C₈	—	2	2	no	—
46	C₁₀₇⋊C₈	—	4	2	no	—
47	C₂²×C₁₀₇	—	2	2	no	—
48	C₂²×C₁₀₇	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₀₇	#125	1	1	yes	characteristic
51	C₄×C₁₀₇	—	4	4	no	—
52	C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₀₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₀₇	#121	1	1	yes	characteristic
55	C₄×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₀₇	—	2	4	no	—
57	C₂×C₁₀₇	—	2	1	no	—
58	C₂×C₁₀₇	—	2	1	no	—
59	C₂×C₁₀₇	#389	1	1	yes	characteristic
60	C₂×C₁₀₇	#443	1	2	yes	—
61	#1945	—	107	1	no	maximal, Sylow
62	C₁₀₇	#1945	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	—	107	1	no	—
64	#427	D₁₀₇	1	1	yes	characteristic
65	#424	—	107	2	no	—
66	C₂×C₄×C₈	—	107	1	no	—
67	#339	—	107	2	no	—
68	C₂×M₁₆	—	107	2	no	—
69	C₂²×C₈	—	107	1	no	—
70	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
71	#117	C₂×D₁₀₇	1	2	yes	—
72	C₂×C₄²	—	107	1	no	—
73	#108	—	214	2	no	—
74	#107	Dic₁₀₇	1	2	yes	complemented
75	#106	Dic₁₀₇	1	2	yes	complemented
76	C₄⋊C₈	—	214	4	no	—
77	C₄×C₈	—	214	2	no	—
78	C₂²×C₄	—	214	1	no	—
79	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
80	M₁₆	—	214	2	no	—
81	M₁₆	—	214	2	no	—
82	C₂×C₈	—	107	2	no	—
83	C₂×C₈	—	214	2	no	—
84	C₂×C₈	—	214	1	no	—
85	C₂×C₈	—	214	1	no	—
86	C₂×C₈	Dic₂₁₄	1	2	yes	—
87	C₂×C₈	D₄₂₈	1	2	yes	—
88	C₂×C₈	C₂×Dic₁₀₇	1	1	yes	characteristic
89	C₂×C₈	C₂×Dic₁₀₇	1	1	yes	characteristic
90	C₄⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	107	2	no	—
94	C₄²	—	214	1	no	—
95	C₂³	#13133817	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	214	4	no	—
98	C₂×C₄	—	214	2	no	—
99	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₀₇	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
104	C₂×C₄	#13133819	1	1	yes	characteristic
105	C₈	—	428	2	no	—
106	C₈	—	214	2	no	—
107	C₈	C₄⋊Dic₁₀₇	1	2	yes	—
108	C₈	C₄⋊Dic₁₀₇	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₀₇	1	1	yes	characteristic
111	C₂²	#16812089	1	1	yes	characteristic
112	C₂²	#16812100	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	214	4	no	—
115	C₄	C₂×C₄⋊Dic₁₀₇	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₀₇	1	1	yes	characteristic
117	C₄	#16812084	1	1	yes	characteristic
118	C₄	#16812108	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#20292526	1	1	yes	characteristic
122	C₂	#20292744	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812045

C₂

yes

maximal, characteristic

#20292654

C₂

yes

maximal

C₂×C₈×Dic₁₀₇

C₂

yes

maximal, characteristic

#20292386

C₂

yes

maximal

#427×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₇

C₂²

yes

characteristic

#16812045

—

C₄⋊(C₁₀₇⋊C₈)

—

C₈×Dic₁₀₇

—

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

#117×C₁₀₇

C₂²

yes

—

#107×C₁₀₇

C₄

yes

complemented

#106×C₁₀₇

C₄

yes

complemented

C₂²×Dic₁₀₇

—

#13133802

—

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

D₄

yes

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₇

D₄

yes

—

C₂×C₈×C₁₀₇

Q₈

yes

—

C₄⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

#41

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₈×C₁₀₇

C₄⋊C₄

yes

—

C₈×C₁₀₇

C₄⋊C₄

yes

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₄×D₄

yes

characteristic

C₂²×C₁₀₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₇

#125

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₇

C₄×Q₈

yes

characteristic

C₄×C₁₀₇

#121

yes

characteristic

C₄×C₁₀₇

C₄⋊Q₈

yes

characteristic

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#389

yes

characteristic

C₂×C₁₀₇

#443

yes

—

#1945

—

107

maximal, Sylow

C₁₀₇

#1945

yes

characteristic, complemented, Sylow

C₂×#108

—

107

—

#427

D₁₀₇

yes

characteristic

#424

—

107

—

C₂×C₄×C₈

—

107

—

#339

—

107

—

C₂×M₁₆

—

107

—

C₂²×C₈

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

#117

C₂×D₁₀₇

yes

—

C₂×C₄²

—

107

—

#108

—

214

—

#107

Dic₁₀₇

yes

complemented

#106

Dic₁₀₇

yes

complemented

C₄⋊C₈

—

214

—

C₄×C₈

—

214

—

C₂²×C₄

—

214

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

214

—

M₁₆

—

214

—

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₂×C₈

Dic₂₁₄

yes

—

C₂×C₈

D₄₂₈

yes

—

C₂×C₈

C₂×Dic₁₀₇

yes

characteristic

C₂×C₈

C₂×Dic₁₀₇

yes

characteristic

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

107

—

C₄²

—

214

—

C₂³

#13133817

yes

characteristic

C₂×C₄

—

C₂×C₄

—

214

—

C₂×C₄

—

214

—

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

100

C₂×C₄

D₄×D₁₀₇

yes

characteristic

101

C₂×C₄

C₂×D₄₂₈

yes

characteristic

102

C₂×C₄

Q₈×D₁₀₇

yes

characteristic

103

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

104

C₂×C₄

#13133819

yes

characteristic

105

C₈

—

428

—

106

C₈

—

214

—

107

C₈

C₄⋊Dic₁₀₇

yes

—

108

C₈

C₄⋊Dic₁₀₇

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₀₇

yes

characteristic

111

C₂²

#16812089

yes

characteristic

112

C₂²

#16812100

yes

characteristic

113

C₄

—

114

C₄

—

214

—

115

C₄

C₂×C₄⋊Dic₁₀₇

yes

characteristic

116

C₄

Q₈×Dic₁₀₇

yes

characteristic

117

C₄

#16812084

yes

characteristic

118

C₄

#16812108

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#20292526

yes

characteristic

122

C₂

#20292744

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	#427×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₄×D₄
prime radical	C₂³×C₁₀₇	#41
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	#427	D₁₀₇
hypocenter	C₁₀₇	#1945
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₇

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

#427×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₄×D₄

prime radical

C₂³×C₁₀₇

#41

cosocle

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₀₇	C₄	semidirect
15. #106×C₁₀₇	C₄	semidirect
62. C₁₀₇	#1945	semidirect
74. #107	Dic₁₀₇	semidirect
75. #106	Dic₁₀₇	semidirect

H ≅ G/N

type

14. #107×C₁₀₇

C₄

semidirect

15. #106×C₁₀₇

C₄

semidirect

62. C₁₀₇

#1945

semidirect

74. #107

Dic₁₀₇

semidirect

75. #106

Dic₁₀₇

semidirect

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

subgroup	index
C₂×#16812045	2
#20292654	2
C₂×C₈×Dic₁₀₇	2
#20292386	2
#427×C₁₀₇	2
#1945	107

subgroup

index

C₂×C₈×Dic₁₀₇

#427×C₁₀₇

107

Subgroups — #28337210

Conjugacy classes of subgroups

Complements

Maximal subgroups