Group #28335520 — subgroups

Subgroups — #28335520

Order 13696.

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

normal subgroups only

This group has 240 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	#20293003	C₂	1	1	yes	maximal, characteristic
2	#20292827	C₂	1	2	yes	maximal, complemented
3	#20292371	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₁₀₇⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#559×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133805	—	2	2	no	—
7	C₂×#13133803	—	2	2	no	—
8	#16812106	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
10	#16812029	—	2	2	no	—
11	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
12	#127×C₁₀₇	C₂²	1	1	yes	characteristic
13	#124×C₁₀₇	—	2	2	no	—
14	#123×C₁₀₇	—	2	2	no	—
15	C₄×Q₈×C₁₀₇	C₂²	1	2	yes	—
16	C₄×D₄×C₁₀₇	—	2	2	no	—
17	C₂×D₄₂₈	—	2	1	no	—
18	C₂×D₄₂₈	—	2	2	no	—
19	#13133805	—	4	4	no	—
20	#13133803	—	4	4	no	—
21	C₂×C₁₀₇⋊C₈	—	2	1	no	—
22	C₂×C₁₀₇⋊C₈	—	2	2	no	—
23	C₂×Q₈×C₁₀₇	D₄	1	2	yes	—
24	C₂×D₄×C₁₀₇	—	2	2	no	—
25	C₂×D₄×C₁₀₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₇	—	2	2	no	—
27	C₄⋊C₄×C₁₀₇	—	2	2	no	—
28	C₄⋊C₄×C₁₀₇	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₀₇	—	2	4	no	—
30	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
31	C₄²×C₁₀₇	—	2	2	no	—
32	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₂³×C₁₀₇	—	2	2	no	—
34	C₂²×D₁₀₇	—	4	1	no	—
35	Q₈×C₁₀₇	—	2	2	no	—
36	Q₈×C₁₀₇	—	2	2	no	—
37	D₄×C₁₀₇	—	4	2	no	—
38	D₄×C₁₀₇	—	4	2	no	—
39	D₄×C₁₀₇	—	4	2	no	—
40	C₂×C₄×C₁₀₇	—	2	2	no	—
41	C₂×C₄×C₁₀₇	—	2	2	no	—
42	C₂×C₄×C₁₀₇	—	8	2	no	—
43	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
45	D₄₂₈	—	4	2	no	—
46	D₄₂₈	—	4	2	no	—
47	D₄₂₈	—	4	2	no	—
48	C₁₀₇⋊C₈	—	2	2	no	—
49	C₁₀₇⋊C₈	—	4	2	no	—
50	C₂²×C₁₀₇	—	4	4	no	—
51	C₂²×C₁₀₇	—	4	2	no	—
52	C₂²×C₁₀₇	#120	1	1	yes	characteristic
53	C₂×D₁₀₇	—	8	2	no	—
54	C₂×D₁₀₇	—	8	1	no	—
55	C₄×C₁₀₇	—	2	1	no	—
56	C₄×C₁₀₇	—	4	2	no	—
57	C₄×C₁₀₇	—	4	2	no	—
58	C₄×C₁₀₇	#135	1	2	yes	—
59	C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₁₀₇	—	8	2	no	—
61	C₂×C₁₀₇	#449	1	2	yes	—
62	C₂×C₁₀₇	#453	1	1	yes	characteristic
63	D₁₀₇	—	16	1	no	—
64	#1620	—	107	1	no	maximal, Sylow
65	C₁₀₇	#1620	1	1	yes	characteristic, complemented, Sylow
66	#559	D₁₀₇	1	1	yes	characteristic, complemented
67	#493	—	107	1	no	—
68	#460	—	107	2	no	—
69	#331	—	107	1	no	—
70	Q₈⋊C₈	—	107	2	no	—
71	C₂×SD₁₆	—	214	2	no	—
72	C₂×D₈	—	214	2	no	—
73	#127	—	107	1	no	—
74	#127	C₂×D₁₀₇	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₁₀₇	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	107	2	no	—
80	#102	—	214	2	no	—
81	C₄×C₈	—	107	1	no	—
82	C₂×Q₈	C₂²⋊D₁₀₇	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	214	2	no	—
85	C₂×D₄	—	214	1	no	—
86	C₂×D₄	C₂²⋊D₁₀₇	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	428	4	no	—
89	D₈	—	428	4	no	—
90	C₂×C₈	—	214	2	no	—
91	C₂×C₈	—	214	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₁₀₇	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	428	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	428	2	no	—
106	D₄	—	428	2	no	—
107	D₄	—	428	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
113	C₈	—	428	2	no	—
114	C₈	—	214	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	856	2	no	—
118	C₂²	—	856	1	no	—
119	C₂²	#16812078	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#16812129	1	2	yes	—
124	C₄	#16812134	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	1712	1	no	—
127	C₂	#20292772	1	2	yes	—
128	C₂	#20292795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293003

C₂

yes

maximal, characteristic

#20292827

C₂

yes

maximal, complemented

#20292371

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, complemented

#559×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×#13133805

—

C₂×#13133803

—

#16812106

C₂²

yes

characteristic

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812029

—

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

#127×C₁₀₇

C₂²

yes

characteristic

#124×C₁₀₇

—

#123×C₁₀₇

—

C₄×Q₈×C₁₀₇

C₂²

yes

—

C₄×D₄×C₁₀₇

—

C₂×D₄₂₈

—

C₂×D₄₂₈

—

#13133805

—

#13133803

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

D₄

yes

—

C₂²⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

—

C₂²×D₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#120

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#135

yes

—

C₄×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#449

yes

—

C₂×C₁₀₇

#453

yes

characteristic

D₁₀₇

—

#1620

—

107

maximal, Sylow

C₁₀₇

#1620

yes

characteristic, complemented, Sylow

#559

D₁₀₇

yes

characteristic, complemented

#493

—

107

—

#460

—

107

—

#331

—

107

—

Q₈⋊C₈

—

107

—

C₂×SD₁₆

—

214

—

C₂×D₈

—

214

—

#127

—

107

—

#127

C₂×D₁₀₇

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₀₇

yes

—

C₄×D₄

—

C₄⋊C₈

—

107

—

#102

—

214

—

C₄×C₈

—

107

—

C₂×Q₈

C₂²⋊D₁₀₇

yes

—

C₂×D₄

—

C₂×D₄

—

214

—

C₂×D₄

—

214

—

C₂×D₄

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

—

SD₁₆

—

428

—

D₈

—

428

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₀₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

—

C₂³

—

428

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

428

—

106

D₄

—

428

—

107

D₄

—

428

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

113

C₈

—

428

—

114

C₈

—

214

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

856

—

118

C₂²

—

856

—

119

C₂²

#16812078

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#16812129

yes

—

124

C₄

#16812134

yes

—

125

C₂

—

126

C₂

—

1712

—

127

C₂

#20292772

yes

—

128

C₂

#20292795

yes

characteristic

	subgroup	G/H
center	C₂²	#16812078
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#559×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#120
prime radical	#20293003	C₂
hypercenter	#559	D₁₀₇
hypocenter	C₁₀₇	#1620
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#16812078

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#559×C₁₀₇

C₂

C₂²×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #20292827	C₂	semidirect
3. #20292371	C₂	semidirect
4. Q₈⋊(C₁₀₇⋊C₈)	C₂	semidirect
5. #559×C₁₀₇	C₂	semidirect
65. C₁₀₇	#1620	semidirect
66. #559	D₁₀₇	semidirect

H ≅ G/N

type

2. #20292827

C₂

semidirect

3. #20292371

C₂

semidirect

4. Q₈⋊(C₁₀₇⋊C₈)

C₂

semidirect

5. #559×C₁₀₇

C₂

semidirect

65. C₁₀₇

#1620

semidirect

66. #559

D₁₀₇

semidirect

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

subgroup	index
#20293003	2
#20292827	2
#20292371	2
Q₈⋊(C₁₀₇⋊C₈)	2
#559×C₁₀₇	2
#1620	107

subgroup

index

#20293003

#20292827

#20292371

Q₈⋊(C₁₀₇⋊C₈)

#559×C₁₀₇

#1620

107

Subgroups — #28335520

Conjugacy classes of subgroups

Complements

Maximal subgroups