Group #28373534 — subgroups

Subgroups — C₂×#20295426 #28373534

Order 13952.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812690	C₂	1	2	yes	maximal
2	#20295426	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812632	C₂	1	2	yes	maximal
4	C₂×#16812631	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	#16812690	—	2	4	no	—
9	#16812690	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₁₀₉	—	2	2	no	—
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
12	C₂×#13182008	—	2	2	no	—
13	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#16812632	C₂²	1	8	yes	—
15	#16812631	C₂²	1	4	yes	—
16	C₈×Dic₁₀₉	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	#106×C₁₀₉	C₂²	1	4	yes	—
20	C₂²×Dic₁₀₉	—	2	2	no	—
21	C₂²×Dic₁₀₉	D₄	1	2	yes	—
22	C₂×#11724784	Q₈	1	4	yes	—
23	C₂×#11724784	C₂³	1	2	yes	—
24	C₄⋊Dic₁₀₉	—	2	2	no	—
25	C₄⋊Dic₁₀₉	—	2	8	no	—
26	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
27	#13182008	—	2	8	no	—
28	C₄×Dic₁₀₉	D₄	1	2	yes	—
29	C₄×Dic₁₀₉	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	Q₈	1	4	yes	—
33	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
34	C₄⋊C₄×C₁₀₉	—	2	2	no	—
35	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
36	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	—	2	4	no	—
38	C₂×C₄×C₁₀₉	—	4	4	no	—
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₁₀₉	—	2	4	no	—
44	C₂×Dic₁₀₉	—	2	4	no	—
45	C₂×Dic₁₀₉	—	4	4	no	—
46	C₂×Dic₁₀₉	C₂×D₄	1	4	yes	—
47	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
48	#11724784	C₂×Q₈	1	4	yes	—
49	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
50	C₂²×C₁₀₉	#135	1	4	yes	—
51	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₁₀₉	—	4	4	no	—
53	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
54	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
55	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
56	Dic₁₀₉	—	2	4	no	—
57	Dic₁₀₉	—	4	4	no	—
58	C₂×C₁₀₉	#499	1	4	yes	complemented
59	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂×#135	1	2	yes	—
61	C₂×#499	—	109	1	no	maximal, Sylow
62	C₁₀₉	C₂×#499	1	1	yes	characteristic, complemented, Sylow
63	C₂×#125	—	109	2	no	—
64	#499	—	109	8	no	—
65	C₂×#107	—	109	2	no	—
66	C₂×#106	—	109	1	no	—
67	C₂×#106	D₁₀₉	1	1	yes	characteristic
68	C₂×C₄×C₈	—	109	1	no	—
69	C₂²×C₈	—	109	1	no	—
70	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
71	#125	—	109	8	no	—
72	#125	—	218	4	no	—
73	C₂×C₄⋊C₄	—	109	2	no	—
74	C₂×C₄⋊C₄	—	218	2	no	—
75	C₂×C₄⋊C₄	—	218	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
77	C₂×C₄²	—	109	1	no	—
78	#107	—	109	8	no	—
79	#106	—	109	4	no	—
80	#106	C₂×D₁₀₉	1	4	yes	—
81	C₄×C₈	—	109	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	109	2	no	—
84	C₂²×C₄	—	218	2	no	—
85	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
86	C₂×C₈	—	109	4	no	—
87	C₂×C₈	—	109	2	no	—
88	C₂×C₈	Dic₂₁₈	1	4	yes	—
89	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	109	4	no	—
92	C₄⋊C₄	—	218	8	no	—
93	C₄⋊C₄	—	218	8	no	—
94	C₄⋊C₄	—	218	2	no	—
95	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
96	C₄²	—	109	2	no	—
97	C₄²	—	109	2	no	—
98	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	109	4	no	—
102	C₂×C₄	—	436	4	no	—
103	C₂×C₄	—	218	4	no	—
104	C₂×C₄	—	218	4	no	—
105	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
107	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
108	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
109	C₈	—	109	4	no	—
110	C₈	C₂×Dic₂₁₈	1	4	yes	—
111	C₂²	#16812698	1	2	yes	—
112	C₂²	#16812720	1	4	yes	—
113	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
114	C₄	—	4	4	no	—
115	C₄	—	436	4	no	—
116	C₄	—	218	4	no	—
117	C₄	#16812698	1	2	yes	—
118	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
119	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
120	C₂	#20295426	1	4	yes	direct product factor
121	C₂	C₂×#16812698	1	1	yes	characteristic
122	C₂	C₂×#16812720	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812690

C₂

yes

maximal

#20295426

C₂

yes

maximal, direct product factor

C₂×#16812632

C₂

yes

maximal

C₂×#16812631

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#106×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812690

—

#16812690

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812632

C₂²

yes

—

#16812631

C₂²

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

—

C₂×#11724784

Q₈

yes

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₈×C₁₀₉

C₂×Q₈

yes

—

#11724784

C₂×Q₈

yes

—

C₂²×C₁₀₉

C₄⋊Q₈

yes

—

C₂²×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#499

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#135

yes

—

C₂×#499

—

109

maximal, Sylow

C₁₀₉

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#125

—

109

—

#499

—

109

—

C₂×#107

—

109

—

C₂×#106

—

109

—

C₂×#106

D₁₀₉

yes

characteristic

C₂×C₄×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#125

—

109

—

#125

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

#107

—

109

—

#106

—

109

—

#106

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂²×D₁₀₉

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

D₄×D₁₀₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

436

—

103

C₂×C₄

—

218

—

104

C₂×C₄

—

218

—

105

C₂×C₄

D₄×D₁₀₉

yes

characteristic

106

C₂×C₄

Q₈×D₁₀₉

yes

—

107

C₂×C₄

C₂×Dic₂₁₈

yes

—

108

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

109

C₈

—

109

—

110

C₈

C₂×Dic₂₁₈

yes

—

111

C₂²

#16812698

yes

—

112

C₂²

#16812720

yes

—

113

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

114

C₄

—

115

C₄

—

436

—

116

C₄

—

218

—

117

C₄

#16812698

yes

—

118

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

119

C₄

C₂²×Dic₂₁₈

yes

characteristic

120

C₂

#20295426

yes

direct product factor

121

C₂

C₂×#16812698

yes

characteristic

122

C₂

C₂×#16812720

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#106×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×#106	D₁₀₉
hypocenter	C₁₀₉	C₂×#499
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#106×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #20295426	C₂	direct (N × H)
58. C₂×C₁₀₉	#499	semidirect
62. C₁₀₉	C₂×#499	semidirect
120. C₂	#20295426	direct (N × H)

H ≅ G/N

type

2. #20295426

C₂

direct (N × H)

58. C₂×C₁₀₉

#499

semidirect

62. C₁₀₉

C₂×#499

semidirect

120. C₂

#20295426

direct (N × H)

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

subgroup	index
C₂×#16812690	2
#20295426	2
C₂×#16812632	2
C₂×#16812631	2
C₂×C₈×Dic₁₀₉	2
C₂×#106×C₁₀₉	2
C₂×#499	109

subgroup

index

C₂×C₈×Dic₁₀₉

C₂×#106×C₁₀₉

C₂×#499

109

Subgroups — C₂×#20295426 #28373534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295426 #28373534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295426 #28373534