Group #28373535 — subgroups

Subgroups — C₂×#20295427 #28373535

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	#20295427	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812633	C₂	1	1	yes	maximal, characteristic
4	C₂×#16812630	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×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	#16812633	C₂²	1	4	yes	—
15	#16812630	C₂²	1	8	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	#107×C₁₀₉	C₂²	1	4	yes	—
20	C₂²×Dic₁₀₉	—	2	2	no	—
21	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
22	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
23	C₂×#11724784	Q₈	1	4	yes	—
24	C₂×#11724784	C₂³	1	2	yes	—
25	C₄⋊Dic₁₀₉	—	2	2	no	—
26	C₄⋊Dic₁₀₉	—	2	8	no	—
27	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
28	#13182008	—	2	8	no	—
29	C₄×Dic₁₀₉	D₄	1	2	yes	—
30	C₄×Dic₁₀₉	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₀₉	—	2	2	no	—
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	Q₈	1	4	yes	—
34	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₀₉	—	2	2	no	—
36	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
37	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	—	2	4	no	—
39	C₂×C₄×C₁₀₉	—	4	4	no	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
44	C₂×Dic₁₀₉	—	2	2	no	—
45	C₂×Dic₁₀₉	—	2	2	no	—
46	C₂×Dic₁₀₉	—	2	4	no	—
47	C₂×Dic₁₀₉	—	4	4	no	—
48	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
49	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
50	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
51	#11724784	C₂×Q₈	1	4	yes	—
52	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
53	C₂²×C₁₀₉	#135	1	2	yes	—
54	C₂²×C₁₀₉	#135	1	2	yes	—
55	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₀₉	—	4	4	no	—
57	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
58	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
60	Dic₁₀₉	—	2	2	no	—
61	Dic₁₀₉	—	2	2	no	—
62	Dic₁₀₉	—	4	4	no	—
63	C₂×C₁₀₉	#499	1	4	yes	complemented
64	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
66	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
67	C₂×#499	—	109	1	no	maximal, Sylow
68	C₁₀₉	C₂×#499	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	—	109	2	no	—
70	#499	—	109	8	no	—
71	C₂×#107	—	109	1	no	—
72	C₂×#107	D₁₀₉	1	1	yes	characteristic
73	C₂×#106	—	109	2	no	—
74	C₂×C₄×C₈	—	109	1	no	—
75	C₂²×C₈	—	109	1	no	—
76	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
77	#125	—	109	8	no	—
78	#125	—	218	4	no	—
79	C₂×C₄⋊C₄	—	109	2	no	—
80	C₂×C₄⋊C₄	—	218	2	no	—
81	C₂×C₄⋊C₄	—	218	2	no	—
82	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
83	C₂×C₄²	—	109	1	no	—
84	#107	—	109	4	no	—
85	#107	C₂×D₁₀₉	1	4	yes	—
86	#106	—	109	8	no	—
87	C₄×C₈	—	109	4	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	109	1	no	—
90	C₂²×C₄	—	109	1	no	—
91	C₂²×C₄	—	218	2	no	—
92	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
93	C₂×C₈	—	109	4	no	—
94	C₂×C₈	—	109	2	no	—
95	C₂×C₈	Dic₂₁₈	1	4	yes	—
96	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	109	4	no	—
99	C₄⋊C₄	—	218	8	no	—
100	C₄⋊C₄	—	218	8	no	—
101	C₄⋊C₄	—	218	2	no	—
102	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
103	C₄²	—	109	2	no	—
104	C₄²	—	109	2	no	—
105	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	109	2	no	—
109	C₂×C₄	—	109	2	no	—
110	C₂×C₄	—	436	4	no	—
111	C₂×C₄	—	218	4	no	—
112	C₂×C₄	—	218	2	no	—
113	C₂×C₄	—	218	2	no	—
114	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
116	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
117	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
118	C₈	—	109	4	no	—
119	C₈	C₂×Dic₂₁₈	1	4	yes	—
120	C₂²	#16812698	1	2	yes	—
121	C₂²	#16812718	1	2	yes	—
122	C₂²	#16812722	1	2	yes	—
123	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
124	C₄	—	4	4	no	—
125	C₄	—	436	4	no	—
126	C₄	—	218	2	no	—
127	C₄	—	218	2	no	—
128	C₄	#16812698	1	2	yes	—
129	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
130	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
131	C₂	#20295427	1	4	yes	direct product factor
132	C₂	C₂×#16812698	1	1	yes	characteristic
133	C₂	C₂×#16812718	1	1	yes	characteristic
134	C₂	C₂×#16812722	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812690

C₂

yes

maximal

#20295427

C₂

yes

maximal, direct product factor

C₂×#16812633

C₂

yes

maximal, characteristic

C₂×#16812630

C₂

yes

maximal

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#107×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

#16812633

C₂²

yes

—

#16812630

C₂²

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

D₄

yes

characteristic

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×Dic₁₀₉

C₂×D₄

yes

—

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

#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

—

Dic₁₀₉

—

C₂×C₁₀₉

#499

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#125

—

109

—

#499

—

109

—

C₂×#107

—

109

—

C₂×#107

D₁₀₉

yes

characteristic

C₂×#106

—

109

—

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

—

#107

C₂×D₁₀₉

yes

—

#106

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

C₂²×C₄

—

109

—

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

—

100

C₄⋊C₄

—

218

—

101

C₄⋊C₄

—

218

—

102

C₄⋊C₄

C₂²×D₁₀₉

yes

—

103

C₄²

—

109

—

104

C₄²

—

109

—

105

C₂³

D₄×D₁₀₉

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

436

—

111

C₂×C₄

—

218

—

112

C₂×C₄

—

218

—

113

C₂×C₄

—

218

—

114

C₂×C₄

D₄×D₁₀₉

yes

characteristic

115

C₂×C₄

Q₈×D₁₀₉

yes

—

116

C₂×C₄

C₂×Dic₂₁₈

yes

—

117

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

118

C₈

—

109

—

119

C₈

C₂×Dic₂₁₈

yes

—

120

C₂²

#16812698

yes

—

121

C₂²

#16812718

yes

—

122

C₂²

#16812722

yes

—

123

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

124

C₄

—

125

C₄

—

436

—

126

C₄

—

218

—

127

C₄

—

218

—

128

C₄

#16812698

yes

—

129

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

130

C₄

C₂²×Dic₂₁₈

yes

characteristic

131

C₂

#20295427

yes

direct product factor

132

C₂

C₂×#16812698

yes

characteristic

133

C₂

C₂×#16812718

yes

characteristic

134

C₂

C₂×#16812722

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#107×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×#107	D₁₀₉
hypocenter	C₁₀₉	C₂×#499
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#107×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #20295427	C₂	direct (N × H)
63. C₂×C₁₀₉	#499	semidirect
68. C₁₀₉	C₂×#499	semidirect
131. C₂	#20295427	direct (N × H)

H ≅ G/N

type

2. #20295427

C₂

direct (N × H)

63. C₂×C₁₀₉

#499

semidirect

68. C₁₀₉

C₂×#499

semidirect

131. C₂

#20295427

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
#20295427	2
C₂×#16812633	2
C₂×#16812630	2
C₂×C₈×Dic₁₀₉	2
C₂×#107×C₁₀₉	2
C₂×#499	109

subgroup

index

C₂×C₈×Dic₁₀₉

C₂×#107×C₁₀₉

C₂×#499

109

Subgroups — C₂×#20295427 #28373535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295427 #28373535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295427 #28373535