Group #28334357 — subgroups

Subgroups — #28334357

Order 13696.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#20292651	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133802	—	2	1	no	—
5	C₂²×C₁₀₇⋊C₈	—	2	1	no	—
6	C₄⋊(C₁₀₇⋊C₈)	—	2	1	no	—
7	C₄⋊(C₁₀₇⋊C₈)	—	2	1	no	—
8	C₄⋊(C₁₀₇⋊C₈)	C₄	1	2	yes	—
9	C₄⋊(C₁₀₇⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₀₇	—	2	1	no	—
12	C₂²⋊C₈×C₁₀₇	C₄	1	2	yes	—
13	C₄×C₈×C₁₀₇	—	2	1	no	—
14	#13133802	—	4	1	no	—
15	#13133802	—	4	1	no	—
16	C₂×C₁₀₇⋊C₈	—	2	2	no	—
17	C₂×C₁₀₇⋊C₈	—	2	2	no	—
18	C₂×C₁₀₇⋊C₈	—	4	1	no	—
19	C₂×C₁₀₇⋊C₈	—	4	1	no	—
20	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₇	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₁₀₇	—	2	2	no	—
24	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
27	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
28	C₂³×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₇	—	2	2	no	—
30	C₂×C₄×C₁₀₇	—	4	1	no	—
31	C₂×C₄×C₁₀₇	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₇	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₁₀₇	Q₁₆	1	1	yes	characteristic
41	C₈×C₁₀₇	—	4	2	no	—
42	C₁₀₇⋊C₈	—	4	2	no	—
43	C₁₀₇⋊C₈	—	4	2	no	—
44	C₂²×C₁₀₇	—	2	1	no	—
45	C₂²×C₁₀₇	—	2	1	no	—
46	C₂²×C₁₀₇	#95	1	1	yes	characteristic
47	C₂²×C₁₀₇	#106	1	1	yes	characteristic
48	C₂²×C₁₀₇	#107	1	1	yes	characteristic
49	C₄×C₁₀₇	—	2	1	no	—
50	C₄×C₁₀₇	—	2	1	no	—
51	C₄×C₁₀₇	—	2	1	no	—
52	C₄×C₁₀₇	#102	1	1	yes	characteristic
53	C₄×C₁₀₇	#102	1	1	yes	characteristic
54	C₄×C₁₀₇	#103	1	1	yes	characteristic
55	C₄×C₁₀₇	#103	1	1	yes	characteristic
56	C₄×C₁₀₇	#108	1	2	yes	—
57	C₂×C₁₀₇	—	2	1	no	—
58	C₂×C₁₀₇	—	2	1	no	—
59	C₂×C₁₀₇	#340	1	1	yes	characteristic
60	C₂×C₁₀₇	#341	1	1	yes	characteristic
61	C₂×C₁₀₇	#344	1	1	yes	characteristic
62	#1301	—	107	1	no	maximal, Sylow
63	C₁₀₇	#1301	1	1	yes	characteristic, complemented, Sylow
64	#431	D₁₀₇	1	1	yes	characteristic
65	#423	—	107	1	no	—
66	C₂×C₄⋊C₈	—	107	1	no	—
67	C₂×M₁₆	—	214	1	no	—
68	C₂²×C₈	—	214	1	no	—
69	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	107	2	no	—
72	C₄⋊C₈	—	107	2	no	—
73	C₄⋊C₈	—	214	1	no	—
74	C₄⋊C₈	—	214	1	no	—
75	C₂²⋊C₈	Dic₁₀₇	1	2	yes	—
76	C₄×C₈	—	2	1	no	—
77	C₂²×C₄	Dic₂₁₄	1	1	yes	characteristic
78	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
79	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
80	M₁₆	—	428	1	no	—
81	M₁₆	—	428	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	428	1	no	—
84	C₂×C₈	—	428	1	no	—
85	C₂×C₈	—	214	2	no	—
86	C₂×C₈	—	214	2	no	—
87	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
88	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
89	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
91	C₂³	C₄⋊Dic₁₀₇	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₁₀₇	1	1	yes	characteristic
95	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
96	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
98	C₂×C₄	Dic₁₀₇⋊C₄	1	1	yes	characteristic
99	C₂×C₄	Dic₁₀₇⋊C₄	1	1	yes	characteristic
100	C₂×C₄	D₈₅₆	1	1	yes	characteristic
101	C₂×C₄	#13133807	1	1	yes	characteristic
102	C₂×C₄	#13133807	1	1	yes	characteristic
103	C₂×C₄	Dic₄₂₈	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	428	2	no	—
106	C₈	—	428	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₁₀₇⋊#95	1	1	yes	characteristic
110	C₂²	#16812041	1	1	yes	characteristic
111	C₂²	#16812043	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#16812031	1	1	yes	characteristic
116	C₄	#16812031	1	1	yes	characteristic
117	C₄	#16812034	1	1	yes	characteristic
118	C₄	#16812034	1	1	yes	characteristic
119	C₄	#16812044	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#20292390	1	1	yes	characteristic
123	C₂	#20292391	1	1	yes	characteristic
124	C₂	#20292397	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20292651

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₀₇⋊C₈)

C₂

yes

maximal, characteristic

#431×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

—

C₂²×C₁₀₇⋊C₈

—

C₄⋊(C₁₀₇⋊C₈)

—

C₄⋊(C₁₀₇⋊C₈)

—

C₄⋊(C₁₀₇⋊C₈)

C₄

yes

—

C₄⋊(C₁₀₇⋊C₈)

C₄

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₇

—

C₂²⋊C₈×C₁₀₇

C₄

yes

—

C₄×C₈×C₁₀₇

—

#13133802

—

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

D₄

yes

characteristic

C₂²×C₄×C₁₀₇

Q₈

yes

characteristic

C₂×C₈×C₁₀₇

—

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

D₄

yes

characteristic

C₄²×C₁₀₇

D₄

yes

characteristic

C₂³×C₁₀₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄²

yes

characteristic

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

D₈

yes

characteristic

C₂×C₄×C₁₀₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₇

Q₁₆

yes

characteristic

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#1301

yes

characteristic, complemented, Sylow

#431

D₁₀₇

yes

characteristic

#423

—

107

—

C₂×C₄⋊C₈

—

107

—

C₂×M₁₆

—

214

—

C₂²×C₈

—

214

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

107

—

C₄⋊C₈

—

107

—

C₄⋊C₈

—

214

—

C₄⋊C₈

—

214

—

C₂²⋊C₈

Dic₁₀₇

yes

—

C₄×C₈

—

C₂²×C₄

Dic₂₁₄

yes

characteristic

C₂²×C₄

D₄₂₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

M₁₆

—

428

—

M₁₆

—

428

—

C₂×C₈

—

C₂×C₈

—

428

—

C₂×C₈

—

428

—

C₂×C₈

—

214

—

C₂×C₈

—

214

—

C₄²

C₄×D₁₀₇

yes

characteristic

C₄²

C₄×D₁₀₇

yes

characteristic

C₄²

C₂²⋊D₁₀₇

yes

characteristic

C₄²

C₂²⋊D₁₀₇

yes

characteristic

C₂³

C₄⋊Dic₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₇

yes

characteristic

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₀₇

yes

characteristic

C₂×C₄

Dic₁₀₇⋊C₄

yes

characteristic

C₂×C₄

Dic₁₀₇⋊C₄

yes

characteristic

100

C₂×C₄

D₈₅₆

yes

characteristic

101

C₂×C₄

#13133807

yes

characteristic

102

C₂×C₄

#13133807

yes

characteristic

103

C₂×C₄

Dic₄₂₈

yes

characteristic

104

C₈

—

105

C₈

—

428

—

106

C₈

—

428

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₁₀₇⋊#95

yes

characteristic

110

C₂²

#16812041

yes

characteristic

111

C₂²

#16812043

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#16812031

yes

characteristic

116

C₄

#16812031

yes

characteristic

117

C₄

#16812034

yes

characteristic

118

C₄

#16812034

yes

characteristic

119

C₄

#16812044

yes

—

120

C₂

—

121

C₂

—

122

C₂

#20292390

yes

characteristic

123

C₂

#20292391

yes

characteristic

124

C₂

#20292397

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₁₀₇)⋊C₄
commutator	C₂×C₄×C₁₀₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₀₇
Fitting	#431×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#95
prime radical	C₂³×C₁₀₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₀₇	C₂²
hypercenter	#431	D₁₀₇
hypocenter	C₁₀₇	#1301
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₁₀₇)⋊C₄

commutator

C₂×C₄×C₁₀₇

C₄²

Frattini

C₂×C₄²

C₂×D₁₀₇

Fitting

#431×C₁₀₇

C₂

socle

C₂²×C₁₀₇

#95

prime radical

C₂³×C₁₀₇

C₄⋊C₄

cosocle

C₂×C₄²×C₁₀₇

C₂²

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
63. C₁₀₇	#1301	semidirect

H ≅ G/N

type

63. C₁₀₇

#1301

semidirect

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

subgroup	index
#20292651	2
C₂×C₄⋊(C₁₀₇⋊C₈)	2
#431×C₁₀₇	2
#1301	107

subgroup

index

#20292651

C₂×C₄⋊(C₁₀₇⋊C₈)

#431×C₁₀₇

#1301

107

Subgroups — #28334357

Conjugacy classes of subgroups

Complements

Maximal subgroups