Group #28364585 — subgroups

Subgroups — #28364585

Order 13952.

210 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20295075	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295052	C₂	1	2	yes	maximal, complemented
3	#20295043	C₂	1	2	yes	maximal
4	#20294783	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	#16812658	C₂²	1	2	yes	—
8	#16812635	—	2	2	no	—
9	#16812628	—	2	4	no	—
10	#16812626	—	2	4	no	—
11	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
15	#13182013	—	2	2	no	—
16	#13182013	—	2	2	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₄⋊Dic₁₀₉	—	2	2	no	—
19	C₂²⋊Dic₁₀₉	—	2	2	no	—
20	C₄×Dic₁₀₉	D₄	1	4	yes	—
21	#41×C₁₀₉	—	2	2	no	—
22	#41×C₁₀₉	—	2	2	no	—
23	C₂×Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
24	C₂×D₄×C₁₀₉	—	2	1	no	—
25	C₂×D₄×C₁₀₉	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₉	—	2	1	no	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	—	2	1	no	—
30	C₂×C₈×C₁₀₉	—	2	2	no	—
31	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
32	C₂³×C₁₀₉	—	2	1	no	—
33	C₂³×C₁₀₉	#41	1	1	yes	characteristic
34	Q₈×C₁₀₉	—	2	2	no	—
35	D₄×C₁₀₉	—	2	2	no	—
36	D₄×C₁₀₉	—	4	2	no	—
37	C₂×C₄×C₁₀₉	—	2	1	no	—
38	C₂×C₄×C₁₀₉	—	4	1	no	—
39	C₂×C₄×C₁₀₉	—	4	1	no	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	#41	1	2	yes	—
44	C₂×Dic₁₀₉	—	2	4	no	—
45	C₈×C₁₀₉	—	2	2	no	—
46	C₈×C₁₀₉	—	4	1	no	—
47	#11724784	—	4	2	no	—
48	C₂²×C₁₀₉	—	4	1	no	—
49	C₂²×C₁₀₉	—	4	2	no	—
50	C₂²×C₁₀₉	—	4	1	no	—
51	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
52	C₂²×C₁₀₉	#123	1	2	yes	—
53	C₄×C₁₀₉	—	4	1	no	—
54	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₀₉	#123	1	1	yes	characteristic
56	C₄×C₁₀₉	#127	1	1	yes	characteristic
57	Dic₁₀₉	—	4	4	no	—
58	C₂×C₁₀₉	—	2	2	no	—
59	C₂×C₁₀₉	—	2	1	no	—
60	C₂×C₁₀₉	—	8	1	no	—
61	C₂×C₁₀₉	#399	1	1	yes	characteristic
62	#2098	—	109	1	no	maximal, Sylow
63	C₁₀₉	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	—	109	1	no	—
65	#424	—	109	2	no	—
66	#421	—	109	2	no	—
67	#408	D₁₀₉	1	1	yes	characteristic
68	H(3,ℤ/4)	—	109	1	no	—
69	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
70	C₂×M₁₆	—	109	2	no	—
71	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
73	#117	—	109	2	no	—
74	#108	—	218	2	no	—
75	C₄⋊C₈	—	218	4	no	—
76	#104	—	218	4	no	—
77	C₂²⋊C₈	—	2	2	no	—
78	#41	—	2	2	no	—
79	#41	—	2	2	no	—
80	C₂×Q₈	Dic₂₁₈	1	1	yes	characteristic
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	Dic₂₁₈	1	1	yes	characteristic
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	218	2	no	—
87	M₁₆	—	218	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	218	2	no	—
91	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
92	C₄⋊C₄	—	218	2	no	—
93	C₂²⋊C₄	—	218	2	no	—
94	C₄²	—	109	4	no	—
95	C₂³	—	2	1	no	—
96	C₂³	#13182029	1	1	yes	characteristic
97	Q₈	—	2	2	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	218	4	no	—
104	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
107	C₂×C₄	#13182028	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	4	1	no	—
110	C₈	—	436	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	#16812678	1	1	yes	characteristic
115	C₂²	#16812681	1	2	yes	—
116	C₄	—	4	1	no	—
117	C₄	—	436	4	no	—
118	C₄	#16812676	1	2	yes	—
119	C₄	#16812680	1	1	yes	characteristic
120	C₄	#16812695	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	8	1	no	—
124	C₂	#20294962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295075

C₂

yes

maximal, characteristic, complemented

#20295052

C₂

yes

maximal, complemented

#20295043

C₂

yes

maximal

#20294783

C₂

yes

maximal, characteristic, complemented

#408×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

#16812658

C₂²

yes

—

#16812635

—

#16812628

—

#16812626

—

C₂×#41×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²⋊C₈×C₁₀₉

—

#13182013

—

#13182013

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

Q₈

yes

characteristic

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#123

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#399

yes

characteristic

#2098

—

109

maximal, Sylow

C₁₀₉

#2098

yes

characteristic, complemented, Sylow

#430

—

109

—

#424

—

109

—

#421

—

109

—

#408

D₁₀₉

yes

characteristic

H(3,ℤ/4)

—

109

—

C₂×#41

C₂×D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

#108

—

218

—

C₄⋊C₈

—

218

—

#104

—

218

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₂₁₈

yes

characteristic

C₂×D₄

—

C₂×D₄

Dic₂₁₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₂³

—

C₂³

#13182029

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

218

—

104

C₂×C₄

D₄×D₁₀₉

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

106

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

107

C₂×C₄

#13182028

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

436

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#16812678

yes

characteristic

115

C₂²

#16812681

yes

—

116

C₄

—

117

C₄

—

436

—

118

C₄

#16812676

yes

—

119

C₄

#16812680

yes

characteristic

120

C₄

#16812695

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

#20294962

yes

characteristic

	subgroup	G/H
center	C₄	#16812695
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#408×C₁₀₉	C₂
socle	C₂×C₁₀₉	#399
prime radical	C₂×#41×C₁₀₉	C₂²
hypercenter	#408	D₁₀₉
hypocenter	C₁₀₉	#2098
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#16812695

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#408×C₁₀₉

C₂

socle

C₂×C₁₀₉

#399

prime radical

C₂×#41×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. #20295075	C₂	semidirect
2. #20295052	C₂	semidirect
4. #20294783	C₂	semidirect
63. C₁₀₉	#2098	semidirect

H ≅ G/N

type

1. #20295075

C₂

semidirect

2. #20295052

C₂

semidirect

4. #20294783

C₂

semidirect

63. C₁₀₉

#2098

semidirect

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

subgroup	index
#20295075	2
#20295052	2
#20295043	2
#20294783	2
#408×C₁₀₉	2
#2098	109

subgroup

index

#408×C₁₀₉

109

Subgroups — #28364585

Conjugacy classes of subgroups

Complements

Maximal subgroups