Group #28373533 — subgroups

Subgroups — C₂×#20295425 #28373533

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×#16812691	C₂	1	1	yes	maximal, characteristic
2	#20295425	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812632	C₂	1	2	yes	maximal
4	C₂×#16812630	C₂	1	2	yes	maximal
5	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	#16812691	—	2	2	no	—
10	#16812691	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
12	C₂×#13182008	—	2	2	no	—
13	#16812632	C₂²	1	8	yes	—
14	#16812630	C₂²	1	8	yes	—
15	C₄×#11724784	C₂²	1	4	yes	—
16	#125×C₁₀₉	—	2	2	no	—
17	#125×C₁₀₉	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×Dic₁₀₉	—	2	2	no	—
22	C₂×#11724784	Q₈	1	4	yes	—
23	C₂×#11724784	Q₈	1	4	yes	—
24	C₂×#11724784	C₂³	1	2	yes	—
25	C₂×#11724784	C₂³	1	2	yes	—
26	C₄⋊Dic₁₀₉	—	2	2	no	—
27	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
28	#13182008	—	2	8	no	—
29	C₂²×C₄×C₁₀₉	—	2	2	no	—
30	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
31	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₉	—	2	8	no	—
33	C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
35	C₄²×C₁₀₉	D₄	1	2	yes	—
36	C₄²×C₁₀₉	C₂³	1	2	yes	—
37	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	—	2	4	no	—
39	C₂×C₄×C₁₀₉	—	2	4	no	—
40	C₂×C₄×C₁₀₉	—	4	4	no	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₁₀₉	—	2	4	no	—
47	C₂×Dic₁₀₉	—	4	4	no	—
48	#11724784	C₂×Q₈	1	4	yes	—
49	#11724784	C₂×Q₈	1	4	yes	—
50	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
51	C₂²×C₁₀₉	#135	1	4	yes	—
52	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
53	C₄×C₁₀₉	—	2	4	no	—
54	C₄×C₁₀₉	—	4	4	no	—
55	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
56	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
57	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
58	Dic₁₀₉	—	4	4	no	—
59	C₂×C₁₀₉	#499	1	4	yes	complemented
60	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂×#135	1	2	yes	—
62	C₂×#499	—	109	1	no	maximal, Sylow
63	C₁₀₉	C₂×#499	1	1	yes	characteristic, complemented, Sylow
64	C₂×#125	—	109	1	no	—
65	C₂×#125	D₁₀₉	1	1	yes	characteristic
66	#499	—	109	8	no	—
67	C₂×#107	—	109	2	no	—
68	C₂×#106	—	109	2	no	—
69	C₂×C₄×C₈	—	109	1	no	—
70	C₂²×C₈	—	109	1	no	—
71	C₂²×C₈	—	109	1	no	—
72	#125	—	2	2	no	—
73	#125	—	109	4	no	—
74	#125	—	218	2	no	—
75	#125	C₂×D₁₀₉	1	4	yes	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	109	2	no	—
78	C₂×C₄⋊C₄	—	218	2	no	—
79	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
80	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
81	#107	—	109	8	no	—
82	#106	—	109	8	no	—
83	C₄×C₈	—	109	4	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	218	2	no	—
86	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
88	C₂×C₈	—	109	4	no	—
89	C₂×C₈	—	109	4	no	—
90	C₂×C₈	—	109	2	no	—
91	C₂×C₈	—	109	2	no	—
92	C₄⋊C₄	—	2	8	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	109	4	no	—
95	C₄⋊C₄	—	218	8	no	—
96	C₄⋊C₄	—	218	2	no	—
97	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
98	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
99	C₄²	C₂²×D₁₀₉	1	2	yes	—
100	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	436	4	no	—
105	C₂×C₄	—	218	4	no	—
106	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	4	yes	—
107	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
109	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
110	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
111	C₈	—	109	4	no	—
112	C₈	—	109	4	no	—
113	C₂²	#16812697	1	2	yes	—
114	C₂²	#16812719	1	4	yes	—
115	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	4	4	no	—
118	C₄	—	436	4	no	—
119	C₄	#16812697	1	2	yes	—
120	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
121	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
122	C₂	#20295425	1	4	yes	direct product factor
123	C₂	C₂×#16812697	1	1	yes	characteristic
124	C₂	C₂×#16812719	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812691

C₂

yes

maximal, characteristic

#20295425

C₂

yes

maximal, direct product factor

C₂×#16812632

C₂

yes

maximal

C₂×#16812630

C₂

yes

maximal

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

C₂×#125×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812691

—

#16812691

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

—

#16812632

C₂²

yes

—

#16812630

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

#125×C₁₀₉

—

#125×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

C₂×#11724784

Q₈

yes

—

C₂×#11724784

Q₈

yes

—

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

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

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

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×C₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

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

—

C₂×#125

D₁₀₉

yes

characteristic

#499

—

109

—

C₂×#107

—

109

—

C₂×#106

—

109

—

C₂×C₄×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

#125

—

#125

—

109

—

#125

—

218

—

#125

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

#106

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂²×D₁₀₉

yes

—

C₄²

C₂²⋊D₁₀₉

yes

—

C₄²

C₂²×D₁₀₉

yes

—

100

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

436

—

105

C₂×C₄

—

218

—

106

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

107

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

108

C₂×C₄

Q₈×D₁₀₉

yes

—

109

C₂×C₄

Q₈×D₁₀₉

yes

—

110

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

111

C₈

—

109

—

112

C₈

—

109

—

113

C₂²

#16812697

yes

—

114

C₂²

#16812719

yes

—

115

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

436

—

119

C₄

#16812697

yes

—

120

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

121

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

122

C₂

#20295425

yes

direct product factor

123

C₂

C₂×#16812697

yes

characteristic

124

C₂

C₂×#16812719

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#125×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×#125	D₁₀₉
hypocenter	C₁₀₉	C₂×#499
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#125×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #20295425	C₂	direct (N × H)
59. C₂×C₁₀₉	#499	semidirect
63. C₁₀₉	C₂×#499	semidirect
122. C₂	#20295425	direct (N × H)

H ≅ G/N

type

2. #20295425

C₂

direct (N × H)

59. C₂×C₁₀₉

#499

semidirect

63. C₁₀₉

C₂×#499

semidirect

122. C₂

#20295425

direct (N × H)

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

subgroup	index
C₂×#16812691	2
#20295425	2
C₂×#16812632	2
C₂×#16812630	2
C₂×C₄×#11724784	2
C₂×#125×C₁₀₉	2
C₂×#499	109

subgroup

index

C₂×#125×C₁₀₉

109

Subgroups — C₂×#20295425 #28373533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295425 #28373533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295425 #28373533