Group #28335945 — subgroups

Subgroups — #28335945

Order 13696.

184 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20293004	C₂	1	1	yes	maximal, characteristic
2	#20292827	C₂	1	2	yes	maximal, complemented
3	#20292378	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292370	C₂	1	2	yes	maximal, complemented
5	#446×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133805	—	2	2	no	—
7	C₂×D₈₅₆	—	2	1	no	—
8	C₂×D₈₅₆	—	2	1	no	—
9	#16812106	C₂²	1	2	yes	—
10	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
11	#16812029	—	2	2	no	—
12	C₄×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₀₇	—	2	1	no	—
14	C₄⋊C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	#97×C₁₀₇	—	2	1	no	—
16	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
17	C₂×D₄₂₈	—	2	2	no	—
18	C₂×D₄₂₈	—	2	2	no	—
19	C₂×D₄₂₈	—	2	2	no	—
20	#13133805	—	4	4	no	—
21	D₈₅₆	—	4	2	no	—
22	D₈₅₆	—	4	2	no	—
23	C₂×C₁₀₇⋊C₈	—	2	2	no	—
24	C₂×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₁₀₇	—	2	1	no	—
26	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₇	—	2	1	no	—
29	C₄⋊C₄×C₁₀₇	D₄	1	1	yes	characteristic
30	C₄²×C₁₀₇	—	2	1	no	—
31	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
32	C₂²×D₁₀₇	—	4	2	no	—
33	Q₈×C₁₀₇	—	2	2	no	—
34	C₂×C₄×C₁₀₇	—	2	1	no	—
35	C₂×C₄×C₁₀₇	—	2	1	no	—
36	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
39	D₄₂₈	—	4	2	no	—
40	D₄₂₈	—	4	2	no	—
41	D₄₂₈	—	4	2	no	—
42	D₄₂₈	—	4	2	no	—
43	D₄₂₈	—	8	2	no	—
44	C₈×C₁₀₇	—	2	1	no	—
45	C₈×C₁₀₇	—	4	1	no	—
46	C₈×C₁₀₇	SD₁₆	1	2	yes	complemented
47	C₁₀₇⋊C₈	—	4	2	no	—
48	C₂²×C₁₀₇	#121	1	1	yes	characteristic
49	C₂×D₁₀₇	—	8	2	no	—
50	C₂×D₁₀₇	—	8	2	no	—
51	C₂×D₁₀₇	—	8	2	no	—
52	C₄×C₁₀₇	—	2	1	no	—
53	C₄×C₁₀₇	—	4	1	no	—
54	C₄×C₁₀₇	—	4	1	no	—
55	C₄×C₁₀₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₄×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
58	C₂×C₁₀₇	#460	1	1	yes	characteristic
59	C₂×C₁₀₇	#469	1	1	yes	characteristic
60	C₂×C₁₀₇	#472	1	1	yes	characteristic
61	D₁₀₇	—	16	2	no	—
62	#1688	—	107	1	no	maximal, Sylow
63	C₁₀₇	#1688	1	1	yes	characteristic, complemented, Sylow
64	#493	—	107	1	no	—
65	#460	—	107	2	no	—
66	#446	D₁₀₇	1	1	yes	characteristic, complemented
67	#334	—	107	1	no	—
68	#331	—	107	2	no	—
69	C₂×SD₁₆	—	214	2	no	—
70	C₂×D₈	—	214	1	no	—
71	C₂×D₈	—	214	1	no	—
72	#127	—	107	2	no	—
73	C₄×Q₈	C₂×D₁₀₇	1	1	yes	characteristic
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	—	107	2	no	—
76	C₄⋊C₈	C₂×D₁₀₇	1	1	yes	characteristic
77	#102	—	214	2	no	—
78	#97	—	2	1	no	—
79	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
80	C₂×Q₈	D₄₂₈	1	1	yes	characteristic
81	C₂×D₄	—	214	2	no	—
82	C₂×D₄	—	214	2	no	—
83	C₂×D₄	—	214	2	no	—
84	SD₁₆	—	428	4	no	—
85	D₈	—	428	2	no	—
86	D₈	—	428	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	214	2	no	—
89	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	D₄₂₈	1	1	yes	characteristic
93	C₄²	—	2	1	no	—
94	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
95	C₂³	—	428	2	no	—
96	Q₈	—	2	2	no	—
97	D₄	—	856	2	no	—
98	D₄	—	428	2	no	—
99	D₄	—	428	2	no	—
100	D₄	—	428	2	no	—
101	D₄	—	428	2	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
105	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
106	C₂×C₄	#13133818	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	4	1	no	—
109	C₈	—	428	2	no	—
110	C₈	#13133805	1	2	yes	complemented
111	C₂²	—	856	2	no	—
112	C₂²	—	856	2	no	—
113	C₂²	—	856	2	no	—
114	C₂²	#16812083	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	1	no	—
118	C₄	C₂×#13133805	1	1	yes	characteristic
119	C₄	#16812134	1	1	yes	characteristic
120	C₄	#16812137	1	2	yes	—
121	C₂	—	1712	2	no	—
122	C₂	#20292827	1	1	yes	characteristic
123	C₂	#20292880	1	1	yes	characteristic
124	C₂	#20292897	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293004

C₂

yes

maximal, characteristic

#20292827

C₂

yes

maximal, complemented

#20292378

C₂

yes

maximal, characteristic, complemented

#20292370

C₂

yes

maximal, complemented

#446×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×#13133805

—

C₂×D₈₅₆

—

C₂×D₈₅₆

—

#16812106

C₂²

yes

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812029

—

C₄×Q₈×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₇

—

C₄⋊C₈×C₁₀₇

C₂²

yes

characteristic

#97×C₁₀₇

—

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×D₄₂₈

—

C₂×D₄₂₈

—

C₂×D₄₂₈

—

#13133805

—

D₈₅₆

—

D₈₅₆

—

C₂×C₁₀₇⋊C₈

—

C₂×Q₈×C₁₀₇

D₄

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

characteristic

C₂×C₈×C₁₀₇

D₄

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

D₄

yes

characteristic

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

characteristic

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

SD₁₆

yes

complemented

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

#121

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×SD₁₆

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

D₁₀₇

—

#1688

—

107

maximal, Sylow

C₁₀₇

#1688

yes

characteristic, complemented, Sylow

#493

—

107

—

#460

—

107

—

#446

D₁₀₇

yes

characteristic, complemented

#334

—

107

—

#331

—

107

—

C₂×SD₁₆

—

214

—

C₂×D₈

—

214

—

C₂×D₈

—

214

—

#127

—

107

—

C₄×Q₈

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

107

—

C₄⋊C₈

C₂×D₁₀₇

yes

characteristic

#102

—

214

—

#97

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×Q₈

D₄₂₈

yes

characteristic

C₂×D₄

—

214

—

C₂×D₄

—

214

—

C₂×D₄

—

214

—

SD₁₆

—

428

—

D₈

—

428

—

D₈

—

428

—

C₂×C₈

—

C₂×C₈

—

214

—

C₂×C₈

C₂²⋊D₁₀₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₄₂₈

yes

characteristic

C₄²

—

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

—

428

—

Q₈

—

D₄

—

856

—

D₄

—

428

—

D₄

—

428

—

100

D₄

—

428

—

101

D₄

—

428

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

105

C₂×C₄

C₂×D₄₂₈

yes

characteristic

106

C₂×C₄

#13133818

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

428

—

110

C₈

#13133805

yes

complemented

111

C₂²

—

856

—

112

C₂²

—

856

—

113

C₂²

—

856

—

114

C₂²

#16812083

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂×#13133805

yes

characteristic

119

C₄

#16812134

yes

characteristic

120

C₄

#16812137

yes

—

121

C₂

—

1712

—

122

C₂

#20292827

yes

characteristic

123

C₂

#20292880

yes

characteristic

124

C₂

#20292897

yes

characteristic

	subgroup	G/H
center	C₂²	#16812083
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#446×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#121
prime radical	#20293004	C₂
hypercenter	#446	D₁₀₇
hypocenter	C₁₀₇	#1688
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#16812083

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#446×C₁₀₇

C₂

C₂²×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
2. #20292827	C₂	semidirect
3. #20292378	C₂	semidirect
4. #20292370	C₂	semidirect
5. #446×C₁₀₇	C₂	semidirect
46. C₈×C₁₀₇	SD₁₆	semidirect
63. C₁₀₇	#1688	semidirect
66. #446	D₁₀₇	semidirect
110. C₈	#13133805	semidirect

H ≅ G/N

type

2. #20292827

C₂

semidirect

3. #20292378

C₂

semidirect

4. #20292370

C₂

semidirect

5. #446×C₁₀₇

C₂

semidirect

46. C₈×C₁₀₇

SD₁₆

semidirect

63. C₁₀₇

#1688

semidirect

66. #446

D₁₀₇

semidirect

110. C₈

#13133805

semidirect

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

subgroup	index
#20293004	2
#20292827	2
#20292378	2
#20292370	2
#446×C₁₀₇	2
#1688	107

subgroup

index

#446×C₁₀₇

107

Subgroups — #28335945

Conjugacy classes of subgroups

Complements

Maximal subgroups