Group #28344823 — subgroups

Subgroups — #28344823

Order 13696.

472 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16812100	C₂	1	2	yes	maximal
2	#20292524	C₂	1	8	yes	maximal
3	#20292496	C₂	1	2	yes	maximal
4	#20292493	C₂	1	1	yes	maximal, characteristic
5	C₄²×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
6	#382×C₁₀₇	C₂	1	1	yes	maximal, characteristic
7	#16812100	—	2	8	no	—
8	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₀₇	C₂²	1	4	yes	—
10	C₂×Dic₁₀₇⋊C₄	C₂²	1	4	yes	—
11	C₂×Dic₁₀₇⋊C₄	C₂²	1	8	yes	—
12	C₂×C₄×Dic₁₀₇	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
15	C₁₀₇⋊#95	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
18	#95×C₁₀₇	C₂²	1	4	yes	—
19	C₂²×Dic₁₀₇	C₂³	1	4	yes	—
20	C₂²×Dic₁₀₇	C₂³	1	2	yes	—
21	C₂²×Dic₁₀₇	C₂³	1	2	yes	—
22	C₄⋊Dic₁₀₇	—	2	4	no	—
23	C₄⋊Dic₁₀₇	—	2	8	no	—
24	Dic₁₀₇⋊C₄	—	2	8	no	—
25	Dic₁₀₇⋊C₄	—	2	16	no	—
26	C₄×Dic₁₀₇	—	2	8	no	—
27	C₄×Dic₁₀₇	D₄	1	4	yes	—
28	C₄×Dic₁₀₇	Q₈	1	4	yes	—
29	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₇	—	2	4	no	—
33	C₄²×C₁₀₇	Q₈	1	4	yes	—
34	C₂³×C₁₀₇	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	—	2	4	no	—
36	C₂×C₄×C₁₀₇	—	2	4	no	—
37	C₂×C₄×C₁₀₇	—	2	8	no	—
38	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₀₇	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
42	C₂×Dic₁₀₇	—	2	4	no	—
43	C₂×Dic₁₀₇	—	2	4	no	—
44	C₂×Dic₁₀₇	—	2	8	no	—
45	C₂×Dic₁₀₇	C₂×D₄	1	4	yes	—
46	C₂×Dic₁₀₇	C₂×Q₈	1	4	yes	—
47	C₂×Dic₁₀₇	#41	1	8	yes	—
48	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂²×C₁₀₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₇	C₂×#41	1	4	yes	—
52	C₄×C₁₀₇	—	2	4	no	—
53	C₄×C₁₀₇	—	4	4	no	—
54	C₄×C₁₀₇	C₄⋊Q₈	1	4	yes	—
55	Dic₁₀₇	—	2	4	no	—
56	Dic₁₀₇	—	4	4	no	—
57	Dic₁₀₇	#125	1	8	yes	complemented
58	C₂×C₁₀₇	C₂×#125	1	2	yes	—
59	C₂×C₁₀₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₁₀₇	#532	1	2	yes	—
61	C₂×C₁₀₇	#533	1	2	yes	—
62	#2859	—	107	1	no	maximal, Sylow
63	C₁₀₇	#2859	1	1	yes	characteristic, complemented, Sylow
64	C₂×#125	—	107	2	no	—
65	#389	—	107	8	no	—
66	#384	—	107	2	no	—
67	#382	—	107	1	no	—
68	#382	D₁₀₇	1	1	yes	characteristic
69	C₄³	—	107	1	no	—
70	#125	—	214	8	no	—
71	C₂×C₄⋊C₄	—	107	4	no	—
72	C₂×C₄⋊C₄	—	107	2	no	—
73	C₂×C₄⋊C₄	—	107	4	no	—
74	C₂×C₄⋊C₄	—	107	8	no	—
75	C₂×C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
76	C₂×C₄²	—	107	1	no	—
77	C₂×C₄²	—	107	1	no	—
78	C₂×C₄²	—	107	4	no	—
79	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
80	#95	—	107	4	no	—
81	#95	C₂×D₁₀₇	1	4	yes	—
82	C₂²×C₄	—	107	2	no	—
83	C₂²×C₄	—	107	2	no	—
84	C₂²×C₄	—	107	4	no	—
85	C₂²×C₄	C₂²×D₁₀₇	1	4	yes	—
86	C₂²×C₄	C₂²×D₁₀₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
88	C₄⋊C₄	—	2	4	no	—
89	C₄⋊C₄	—	214	8	no	—
90	C₄⋊C₄	—	214	8	no	—
91	C₄⋊C₄	—	214	16	no	—
92	C₄⋊C₄	—	214	4	no	—
93	C₄²	—	107	4	no	—
94	C₄²	—	107	4	no	—
95	C₄²	—	214	8	no	—
96	C₄²	Dic₂₁₄	1	4	yes	—
97	C₂³	C₂³×D₁₀₇	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	107	4	no	—
102	C₂×C₄	—	107	8	no	—
103	C₂×C₄	—	107	4	no	—
104	C₂×C₄	—	214	4	no	—
105	C₂×C₄	—	214	8	no	—
106	C₂×C₄	—	214	4	no	—
107	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
108	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
109	C₂×C₄	C₂×Dic₂₁₄	1	4	yes	—
110	C₂×C₄	C₂×Dic₂₁₄	1	2	yes	—
111	C₂²	C₂×D₄×D₁₀₇	1	1	yes	characteristic
112	C₂²	C₂×Q₈×D₁₀₇	1	1	yes	characteristic
113	C₂²	C₂²×Dic₂₁₄	1	1	yes	characteristic
114	C₂²	#41×D₁₀₇	1	4	yes	—
115	C₄	—	2	4	no	—
116	C₄	—	4	4	no	—
117	C₄	—	107	8	no	—
118	C₄	—	428	4	no	—
119	C₄	—	214	4	no	—
120	C₄	#16812108	1	4	yes	—
121	C₂	#125×D₁₀₇	1	2	yes	—
122	C₂	C₂×#16812108	1	1	yes	characteristic
123	C₂	#20293176	1	2	yes	—
124	C₂	#20293180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812100

C₂

yes

maximal

#20292524

C₂

yes

maximal

#20292496

C₂

yes

maximal

#20292493

C₂

yes

maximal, characteristic

C₄²×Dic₁₀₇

C₂

yes

maximal, characteristic

#382×C₁₀₇

C₂

yes

maximal, characteristic

#16812100

—

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

C₂×Dic₁₀₇⋊C₄

C₂²

yes

—

C₂×Dic₁₀₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₇

C₂²

yes

characteristic

C₁₀₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#95×C₁₀₇

C₂²

yes

—

C₂²×Dic₁₀₇

C₂³

yes

—

C₂²×Dic₁₀₇

C₂³

yes

—

C₂²×Dic₁₀₇

C₂³

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

Dic₁₀₇⋊C₄

—

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

D₄

yes

—

C₄×Dic₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

Q₈

yes

—

C₂³×C₁₀₇

C₂⁴

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₂×D₄

yes

—

C₂×Dic₁₀₇

C₂×Q₈

yes

—

C₂×Dic₁₀₇

#41

yes

—

C₂²×C₁₀₇

C₂²×D₄

yes

characteristic

C₂²×C₁₀₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₇

C₂×#41

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₄⋊Q₈

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

#125

yes

complemented

C₂×C₁₀₇

C₂×#125

yes

—

C₂×C₁₀₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₇

#532

yes

—

C₂×C₁₀₇

#533

yes

—

#2859

—

107

maximal, Sylow

C₁₀₇

#2859

yes

characteristic, complemented, Sylow

C₂×#125

—

107

—

#389

—

107

—

#384

—

107

—

#382

—

107

—

#382

D₁₀₇

yes

characteristic

C₄³

—

107

—

#125

—

214

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

—

C₂×C₄²

—

107

—

C₂×C₄²

—

107

—

C₂×C₄²

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#95

—

107

—

#95

C₂×D₁₀₇

yes

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂²×D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

—

107

—

C₄²

—

107

—

C₄²

—

214

—

C₄²

Dic₂₁₄

yes

—

C₂³

C₂³×D₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

107

—

102

C₂×C₄

—

107

—

103

C₂×C₄

—

107

—

104

C₂×C₄

—

214

—

105

C₂×C₄

—

214

—

106

C₂×C₄

—

214

—

107

C₂×C₄

D₄×D₁₀₇

yes

—

108

C₂×C₄

Q₈×D₁₀₇

yes

—

109

C₂×C₄

C₂×Dic₂₁₄

yes

—

110

C₂×C₄

C₂×Dic₂₁₄

yes

—

111

C₂²

C₂×D₄×D₁₀₇

yes

characteristic

112

C₂²

C₂×Q₈×D₁₀₇

yes

characteristic

113

C₂²

C₂²×Dic₂₁₄

yes

characteristic

114

C₂²

#41×D₁₀₇

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

107

—

118

C₄

—

428

—

119

C₄

—

214

—

120

C₄

#16812108

yes

—

121

C₂

#125×D₁₀₇

yes

—

122

C₂

C₂×#16812108

yes

characteristic

123

C₂

#20293176

yes

—

124

C₂

#20293180

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₇
commutator, socle, prime radical, cosocle	C₂³×C₁₀₇	C₂⁴
Fitting	#382×C₁₀₇	C₂
hypercenter	#382	D₁₀₇
hypocenter	C₁₀₇	#2859
LCS	G ▷ C₂³×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₇

commutator, socle, prime radical, cosocle

C₂³×C₁₀₇

C₂⁴

Fitting

#382×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₂³×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
57. Dic₁₀₇	#125	semidirect
63. C₁₀₇	#2859	semidirect

H ≅ G/N

type

57. Dic₁₀₇

#125

semidirect

63. C₁₀₇

#2859

semidirect

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

subgroup	index
C₂×#16812100	2
#20292524	2
#20292496	2
#20292493	2
C₄²×Dic₁₀₇	2
#382×C₁₀₇	2
#2859	107

subgroup

index

C₄²×Dic₁₀₇

#382×C₁₀₇

#2859

107

Subgroups — #28344823

Conjugacy classes of subgroups

Complements

Maximal subgroups