Group #28344817 — subgroups

Subgroups — #28344817

Order 13696.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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×#16812109	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812096	C₂	1	4	yes	maximal, complemented
3	#20292530	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₄	C₂²	1	2	yes	—
7	#16812109	—	2	4	no	—
8	#16812096	—	2	16	no	—
9	C₂×C₄⋊Dic₁₀₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₁₀₇	C₂²	1	16	yes	—
11	C₂×C₄×Dic₁₀₇	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₀₇	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₀₇	C₂²	1	2	yes	—
14	C₂²×D₄×C₁₀₇	C₂²	1	4	yes	—
15	#127×C₁₀₇	—	2	4	no	—
16	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₄	—	2	8	no	—
18	C₂×Dic₂₁₄	—	2	4	no	—
19	C₂²×Dic₁₀₇	C₂³	1	4	yes	—
20	C₂²×Dic₁₀₇	C₂³	1	4	yes	—
21	C₄⋊Dic₁₀₇	—	2	8	no	—
22	C₂²⋊Dic₁₀₇	—	2	32	no	—
23	C₄×Dic₁₀₇	—	2	8	no	—
24	C₄×Dic₁₀₇	D₄	1	8	yes	complemented
25	C₂⁴×C₁₀₇	C₂³	1	4	yes	—
26	C₂×D₄×C₁₀₇	—	2	8	no	—
27	C₂×D₄×C₁₀₇	—	2	8	no	—
28	C₂×D₄×C₁₀₇	—	2	4	no	—
29	C₂×D₄×C₁₀₇	—	2	16	no	—
30	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
32	C₄²×C₁₀₇	D₄	1	4	yes	—
33	C₂³×C₁₀₇	—	2	4	no	—
34	C₂³×C₁₀₇	—	2	8	no	—
35	C₂³×C₁₀₇	—	2	16	no	—
36	C₂³×C₁₀₇	C₂⁴	1	1	yes	characteristic
37	D₄×C₁₀₇	—	4	16	no	—
38	D₄×C₁₀₇	—	4	8	no	—
39	C₂×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₁₀₇	#41	1	8	yes	—
43	C₂×Dic₁₀₇	—	2	8	no	—
44	C₂×Dic₁₀₇	—	2	8	no	—
45	C₂×Dic₁₀₇	—	2	4	no	—
46	C₂×Dic₁₀₇	C₂×D₄	1	8	yes	—
47	Dic₂₁₄	—	4	8	no	—
48	C₂²×C₁₀₇	—	4	8	no	—
49	C₂²×C₁₀₇	—	4	4	no	—
50	C₂²×C₁₀₇	—	4	16	no	—
51	C₂²×C₁₀₇	C₂²×D₄	1	2	yes	—
52	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
53	C₂²×C₁₀₇	C₂×#41	1	4	yes	—
54	C₄×C₁₀₇	#124	1	8	yes	complemented
55	C₄×C₁₀₇	#127	1	4	yes	—
56	Dic₁₀₇	—	2	8	no	—
57	Dic₁₀₇	—	4	4	no	—
58	C₂×C₁₀₇	—	8	4	no	—
59	C₂×C₁₀₇	C₂×#124	1	2	yes	—
60	C₂×C₁₀₇	C₂×#127	1	1	yes	characteristic
61	C₂×C₁₀₇	#532	1	4	yes	—
62	#2858	—	107	1	no	maximal, Sylow
63	C₁₀₇	#2858	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	107	1	no	—
65	C₂×#127	D₁₀₇	1	1	yes	characteristic
66	C₂×#124	—	107	4	no	—
67	#390	—	107	8	no	—
68	C₄³	—	107	1	no	—
69	C₂²×Q₈	—	107	2	no	—
70	C₂²×D₄	C₂×D₁₀₇	1	2	yes	—
71	C₂²×D₄	C₂×D₁₀₇	1	4	yes	—
72	C₄⋊Q₈	—	214	4	no	—
73	#127	—	2	4	no	—
74	#124	—	214	16	no	—
75	C₂×C₄⋊C₄	—	107	4	no	—
76	C₂×C₂²⋊C₄	—	107	16	no	—
77	C₂×C₄²	—	107	2	no	—
78	C₂×C₄²	—	107	4	no	—
79	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
80	C₂⁴	C₂²×D₁₀₇	1	4	yes	—
81	C₂×Q₈	—	214	8	no	—
82	C₂×Q₈	—	214	4	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	2	8	no	—
85	C₂×D₄	—	2	4	no	—
86	C₂×D₄	—	2	16	no	—
87	C₂²×C₄	—	107	4	no	—
88	C₂²×C₄	—	107	4	no	—
89	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₀₇	1	2	yes	—
91	C₄⋊C₄	—	214	8	no	—
92	C₂²⋊C₄	—	214	32	no	—
93	C₄²	—	107	8	no	—
94	C₄²	—	214	8	no	—
95	C₄²	C₂²⋊D₁₀₇	1	4	yes	—
96	C₂³	—	2	8	no	—
97	C₂³	—	2	4	no	—
98	C₂³	—	2	16	no	—
99	C₂³	C₂³×D₁₀₇	1	1	yes	characteristic
100	Q₈	—	428	8	no	—
101	D₄	—	4	16	no	—
102	D₄	—	4	8	no	—
103	C₂×C₄	—	107	8	no	—
104	C₂×C₄	—	214	8	no	—
105	C₂×C₄	—	214	4	no	—
106	C₂×C₄	—	214	8	no	—
107	C₂×C₄	D₄×D₁₀₇	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
110	C₂×C₄	#13133817	1	8	yes	—
111	C₂²	—	4	8	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	4	16	no	—
114	C₂²	C₂×D₄×D₁₀₇	1	2	yes	—
115	C₂²	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
116	C₂²	C₂×#13133817	1	4	yes	—
117	C₄	—	428	4	no	—
118	C₄	—	214	8	no	—
119	C₄	#16812096	1	8	yes	complemented
120	C₄	#16812105	1	4	yes	—
121	C₂	—	8	4	no	—
122	C₂	C₂×#16812096	1	2	yes	—
123	C₂	C₂×#16812105	1	1	yes	characteristic
124	C₂	#20293175	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812109

C₂

yes

maximal, characteristic, complemented

C₂×#16812096

C₂

yes

maximal, complemented

#20292530

C₂

yes

maximal, complemented

C₄²×Dic₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₂²

yes

—

#16812109

—

#16812096

—

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₇

C₂²

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

—

C₂²×D₄×C₁₀₇

C₂²

yes

—

C₂²×D₄×C₁₀₇

C₂²

yes

—

#127×C₁₀₇

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂²×Dic₁₀₇

C₂³

yes

—

C₂²×Dic₁₀₇

C₂³

yes

—

C₄⋊Dic₁₀₇

—

C₂²⋊Dic₁₀₇

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

D₄

yes

complemented

C₂⁴×C₁₀₇

C₂³

yes

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂²×C₄×C₁₀₇

C₂³

yes

—

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₂⁴

yes

characteristic

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₂×D₄

yes

—

Dic₂₁₄

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂²×D₄

yes

—

C₂²×C₁₀₇

C₂²×D₄

yes

characteristic

C₂²×C₁₀₇

C₂×#41

yes

—

C₄×C₁₀₇

#124

yes

complemented

C₄×C₁₀₇

#127

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×#124

yes

—

C₂×C₁₀₇

C₂×#127

yes

characteristic

C₂×C₁₀₇

#532

yes

—

#2858

—

107

maximal, Sylow

C₁₀₇

#2858

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

107

—

C₂×#127

D₁₀₇

yes

characteristic

C₂×#124

—

107

—

#390

—

107

—

C₄³

—

107

—

C₂²×Q₈

—

107

—

C₂²×D₄

C₂×D₁₀₇

yes

—

C₂²×D₄

C₂×D₁₀₇

yes

—

C₄⋊Q₈

—

214

—

#127

—

#124

—

214

—

C₂×C₄⋊C₄

—

107

—

C₂×C₂²⋊C₄

—

107

—

C₂×C₄²

—

107

—

C₂×C₄²

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₂⁴

C₂²×D₁₀₇

yes

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂²×C₄

C₂²×D₁₀₇

yes

—

C₄⋊C₄

—

214

—

C₂²⋊C₄

—

214

—

C₄²

—

107

—

C₄²

—

214

—

C₄²

C₂²⋊D₁₀₇

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₀₇

yes

characteristic

100

Q₈

—

428

—

101

D₄

—

102

D₄

—

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

C₂×C₂²⋊D₁₀₇

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

110

C₂×C₄

#13133817

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×D₄×D₁₀₇

yes

—

115

C₂²

C₂²×C₂²⋊D₁₀₇

yes

characteristic

116

C₂²

C₂×#13133817

yes

—

117

C₄

—

428

—

118

C₄

—

214

—

119

C₄

#16812096

yes

complemented

120

C₄

#16812105

yes

—

121

C₂

—

122

C₂

C₂×#16812096

yes

—

123

C₂

C₂×#16812105

yes

characteristic

124

C₂

#20293175

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₇
commutator, socle, cosocle	C₂³×C₁₀₇	C₂⁴
Fitting, prime radical	C₂×#127×C₁₀₇	C₂
hypercenter	C₂×#127	D₁₀₇
hypocenter	C₁₀₇	#2858
LCS	G ▷ C₂³×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₇

commutator, socle, cosocle

C₂³×C₁₀₇

C₂⁴

Fitting, prime radical

C₂×#127×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₂³×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#16812109	C₂	semidirect
2. C₂×#16812096	C₂	semidirect
3. #20292530	C₂	semidirect
4. C₄²×Dic₁₀₇	C₂	semidirect
24. C₄×Dic₁₀₇	D₄	semidirect
54. C₄×C₁₀₇	#124	semidirect
63. C₁₀₇	#2858	semidirect
119. C₄	#16812096	semidirect

H ≅ G/N

type

1. C₂×#16812109

C₂

semidirect

2. C₂×#16812096

C₂

semidirect

3. #20292530

C₂

semidirect

4. C₄²×Dic₁₀₇

C₂

semidirect

24. C₄×Dic₁₀₇

D₄

semidirect

54. C₄×C₁₀₇

#124

semidirect

63. C₁₀₇

#2858

semidirect

119. C₄

#16812096

semidirect

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

subgroup	index
C₂×#16812109	2
C₂×#16812096	2
#20292530	2
C₄²×Dic₁₀₇	2
C₂×#127×C₁₀₇	2
#2858	107

subgroup

index

C₂×#16812109

C₂×#16812096

#20292530

C₄²×Dic₁₀₇

C₂×#127×C₁₀₇

#2858

107

Subgroups — #28344817

Conjugacy classes of subgroups

Complements

Maximal subgroups