Group #28363857 — subgroups

Subgroups — #28363857

Order 13952.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16812620	C₂	1	4	yes	maximal, complemented
2	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294892	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
7	#16812620	—	2	8	no	—
8	#16812610	—	2	2	no	—
9	#16812606	—	2	2	no	—
10	C₂²×D₄×C₁₀₉	—	2	2	no	—
11	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
12	#127×C₁₀₉	—	2	2	no	—
13	#127×C₁₀₉	C₄	1	4	yes	complemented
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	—	2	2	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₂×#11724784	D₄	1	4	yes	complemented
19	C₄⋊Dic₁₀₉	—	2	4	no	—
20	C₂⁴×C₁₀₉	—	2	2	no	—
21	C₂×D₄×C₁₀₉	—	2	8	no	—
22	C₂×D₄×C₁₀₉	—	2	4	no	—
23	C₂×D₄×C₁₀₉	—	2	2	no	—
24	C₂×D₄×C₁₀₉	—	2	4	no	—
25	C₂×D₄×C₁₀₉	—	4	8	no	—
26	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
28	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₉	D₄	1	2	yes	—
31	C₂³×C₁₀₉	—	2	4	no	—
32	C₂³×C₁₀₉	—	4	2	no	—
33	C₂³×C₁₀₉	—	4	2	no	—
34	C₂³×C₁₀₉	—	4	8	no	—
35	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	D₄×C₁₀₉	—	2	8	no	—
37	D₄×C₁₀₉	—	4	16	no	—
38	D₄×C₁₀₉	—	4	4	no	—
39	C₂×C₄×C₁₀₉	—	2	4	no	—
40	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	#41	1	2	yes	—
45	C₂×Dic₁₀₉	—	4	2	no	—
46	C₂×Dic₁₀₉	—	4	4	no	—
47	#11724784	—	2	4	no	—
48	C₂²×C₁₀₉	—	4	4	no	—
49	C₂²×C₁₀₉	—	4	8	no	—
50	C₂²×C₁₀₉	—	8	4	no	—
51	C₂²×C₁₀₉	—	8	4	no	—
52	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₀₉	#124	1	1	yes	characteristic
54	C₂²×C₁₀₉	#127	1	1	yes	characteristic
55	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
56	C₄×C₁₀₉	—	2	4	no	—
57	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
58	C₄×C₁₀₉	#121	1	2	yes	—
59	Dic₁₀₉	—	8	2	no	—
60	C₂×C₁₀₉	—	8	4	no	—
61	C₂×C₁₀₉	#390	1	1	yes	characteristic
62	C₂×C₁₀₉	#418	1	2	yes	—
63	C₂×C₁₀₉	#490	1	2	yes	—
64	C₂×C₁₀₉	#496	1	2	yes	—
65	#1961	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1961	1	1	yes	characteristic, complemented, Sylow
67	C₂×#127	D₁₀₉	1	1	yes	characteristic
68	C₂×#102	—	109	4	no	—
69	C₂×#97	—	109	1	no	—
70	#382	—	109	1	no	—
71	C₂²×D₄	—	2	2	no	—
72	C₂²×D₄	C₂×D₁₀₉	1	2	yes	—
73	C₂²×C₈	—	109	2	no	—
74	#127	—	2	2	no	—
75	#127	Dic₁₀₉	1	4	yes	complemented
76	C₂×C₄⋊C₄	—	109	2	no	—
77	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
78	#102	—	218	8	no	—
79	#97	—	218	2	no	—
80	#95	—	218	2	no	—
81	C₂⁴	—	2	2	no	—
82	C₂×D₄	—	2	8	no	—
83	C₂×D₄	—	2	4	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	4	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂×Dic₁₀₉	1	4	yes	—
88	C₂²×C₄	—	218	2	no	—
89	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₂×C₈	—	109	4	no	—
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	—	218	2	no	—
94	C₄⋊C₄	—	218	4	no	—
95	C₄²	C₂×Dic₁₀₉	1	2	yes	—
96	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
97	C₂³	—	2	4	no	—
98	C₂³	—	4	2	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	8	no	—
101	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	436	2	no	—
107	C₂×C₄	—	436	4	no	—
108	C₂×C₄	C₂²⋊Dic₁₀₉	1	4	yes	—
109	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
110	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
112	C₂×C₄	#13182028	1	2	yes	—
113	C₈	—	218	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	—	4	8	no	—
116	C₂²	—	8	4	no	—
117	C₂²	—	8	4	no	—
118	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
119	C₂²	#16812686	1	1	yes	characteristic
120	C₂²	#16812695	1	1	yes	characteristic
121	C₂²	#16812725	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	872	2	no	—
124	C₄	D₄×Dic₁₀₉	1	2	yes	—
125	C₄	#16812670	1	2	yes	—
126	C₂	—	8	4	no	—
127	C₂	#20294929	1	1	yes	characteristic
128	C₂	#20295031	1	2	yes	—
129	C₂	#20295390	1	2	yes	—
130	C₂	#20295413	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812620

C₂

yes

maximal, complemented

C₂×#16812610

C₂

yes

maximal, characteristic, complemented

#20294892

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812620

—

#16812610

—

#16812606

—

C₂²×D₄×C₁₀₉

—

C₂²×D₄×C₁₀₉

C₂²

yes

—

#127×C₁₀₉

—

#127×C₁₀₉

C₄

yes

complemented

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

D₄

yes

complemented

C₄⋊Dic₁₀₉

—

C₂⁴×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

#121

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

#418

yes

—

C₂×C₁₀₉

#490

yes

—

C₂×C₁₀₉

#496

yes

—

#1961

—

109

maximal, Sylow

C₁₀₉

#1961

yes

characteristic, complemented, Sylow

C₂×#127

D₁₀₉

yes

characteristic

C₂×#102

—

109

—

C₂×#97

—

109

—

#382

—

109

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

#127

—

#127

Dic₁₀₉

yes

complemented

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#102

—

218

—

#97

—

218

—

#95

—

218

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₄²

C₂×Dic₁₀₉

yes

—

C₄²

C₂²⋊D₁₀₉

yes

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

436

—

107

C₂×C₄

—

436

—

108

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

109

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

110

C₂×C₄

D₄×D₁₀₉

yes

—

111

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

112

C₂×C₄

#13182028

yes

—

113

C₈

—

218

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

119

C₂²

#16812686

yes

characteristic

120

C₂²

#16812695

yes

characteristic

121

C₂²

#16812725

yes

—

122

C₄

—

123

C₄

—

872

—

124

C₄

D₄×Dic₁₀₉

yes

—

125

C₄

#16812670

yes

—

126

C₂

—

127

C₂

#20294929

yes

characteristic

128

C₂

#20295031

yes

—

129

C₂

#20295390

yes

—

130

C₂

#20295413

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting, prime radical	C₂×#127×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#127	D₁₀₉
hypocenter	C₁₀₉	#1961
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting, prime radical

C₂×#127×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#16812620	C₂	semidirect
2. C₂×#16812610	C₂	semidirect
3. #20294892	C₂	semidirect
13. #127×C₁₀₉	C₄	semidirect
18. C₂×#11724784	D₄	semidirect
66. C₁₀₉	#1961	semidirect
75. #127	Dic₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#16812620

C₂

semidirect

2. C₂×#16812610

C₂

semidirect

3. #20294892

C₂

semidirect

13. #127×C₁₀₉

C₄

semidirect

18. C₂×#11724784

D₄

semidirect

66. C₁₀₉

#1961

semidirect

75. #127

Dic₁₀₉

semidirect

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

subgroup	index
C₂×#16812620	2
C₂×#16812610	2
#20294892	2
C₂×#127×C₁₀₉	2
#1961	109

subgroup

index

C₂×#127×C₁₀₉

109

Subgroups — #28363857

Conjugacy classes of subgroups

Complements

Maximal subgroups