Group #28363193 — subgroups

Subgroups — #28363193

Order 13952.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16812632	C₂	1	2	yes	maximal
2	C₂×#16812631	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294891	C₂	1	2	yes	maximal
5	C₂×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812632	—	2	4	no	—
10	#16812631	—	2	2	no	—
11	C₈×Dic₁₀₉	C₄	1	4	yes	complemented
12	#16812606	—	2	4	no	—
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
15	#106×C₁₀₉	—	2	2	no	—
16	C₂²×Dic₁₀₉	—	2	2	no	—
17	C₂²×Dic₁₀₉	D₄	1	2	yes	—
18	C₂×#11724784	C₂×C₄	1	2	yes	—
19	C₂×#11724784	D₄	1	2	yes	—
20	C₂×#11724784	Q₈	1	2	yes	—
21	C₄⋊Dic₁₀₉	—	2	2	no	—
22	C₄⋊Dic₁₀₉	—	2	2	no	—
23	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₀₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₀₉	—	2	2	no	—
31	C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	4	2	no	—
34	C₂×C₄×C₁₀₉	—	4	4	no	—
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	2	4	no	—
42	C₂×Dic₁₀₉	—	4	2	no	—
43	C₂×Dic₁₀₉	—	4	4	no	—
44	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
45	C₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
46	#11724784	C₄⋊C₄	1	4	yes	—
47	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₀₉	#127	1	1	yes	characteristic
49	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₀₉	#135	1	4	yes	—
51	C₄×C₁₀₉	—	8	2	no	—
52	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
55	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₀₉	—	2	4	no	—
57	Dic₁₀₉	—	8	2	no	—
58	C₂×C₁₀₉	#384	1	1	yes	characteristic
59	C₂×C₁₀₉	#427	1	2	yes	—
60	C₂×C₁₀₉	#495	1	2	yes	—
61	C₂×C₁₀₉	#499	1	2	yes	—
62	#1846	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1846	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	109	2	no	—
65	C₂×#106	—	109	1	no	—
66	C₂×#106	D₁₀₉	1	1	yes	characteristic
67	C₂×C₄×C₈	—	109	1	no	—
68	#382	—	109	2	no	—
69	C₂²×C₈	—	109	1	no	—
70	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂×C₄⋊C₄	—	109	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
73	C₂×C₄²	—	109	1	no	—
74	#107	—	218	4	no	—
75	#106	—	2	2	no	—
76	#106	—	218	2	no	—
77	C₄×C₈	—	109	4	no	—
78	#95	—	218	4	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	109	2	no	—
81	C₂²×C₄	—	218	2	no	—
82	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
83	C₂×C₈	—	109	2	no	—
84	C₂×C₈	—	109	2	no	—
85	C₂×C₈	—	109	2	no	—
86	C₂×C₈	Dic₂₁₈	1	2	yes	—
87	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
88	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	218	2	no	—
92	C₄⋊C₄	—	218	2	no	—
93	C₄²	—	109	2	no	—
94	C₄²	—	109	2	no	—
95	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	109	4	no	—
99	C₂×C₄	—	436	2	no	—
100	C₂×C₄	—	436	4	no	—
101	C₂×C₄	—	218	4	no	—
102	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
103	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
108	C₈	—	109	4	no	—
109	C₈	#13182008	1	4	yes	—
110	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
111	C₂²	#16812695	1	1	yes	characteristic
112	C₂²	#16812698	1	1	yes	characteristic
113	C₂²	#16812720	1	4	yes	—
114	C₄	—	8	2	no	—
115	C₄	—	872	2	no	—
116	C₄	—	218	4	no	—
117	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
118	C₄	C₂×#13182008	1	1	yes	characteristic
119	C₄	#16812697	1	1	yes	characteristic
120	C₄	#16812698	1	1	yes	characteristic
121	C₂	#20294895	1	1	yes	characteristic
122	C₂	#20295061	1	2	yes	—
123	C₂	#20295407	1	2	yes	—
124	C₂	#20295426	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812632

C₂

yes

maximal

C₂×#16812631

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294891

C₂

yes

maximal

C₂×#106×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812632

—

#16812631

—

C₈×Dic₁₀₉

C₄

yes

complemented

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

D₄

yes

—

C₂×#11724784

Q₈

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

#11724784

C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#127

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#135

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#384

yes

characteristic

C₂×C₁₀₉

#427

yes

—

C₂×C₁₀₉

#495

yes

—

C₂×C₁₀₉

#499

yes

—

#1846

—

109

maximal, Sylow

C₁₀₉

#1846

yes

characteristic, complemented, Sylow

C₂×#107

—

109

—

C₂×#106

—

109

—

C₂×#106

D₁₀₉

yes

characteristic

C₂×C₄×C₈

—

109

—

#382

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

#107

—

218

—

#106

—

#106

—

218

—

C₄×C₈

—

109

—

#95

—

218

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

D₄×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

100

C₂×C₄

—

436

—

101

C₂×C₄

—

218

—

102

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

103

C₂×C₄

D₄×D₁₀₉

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

105

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

106

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

107

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

108

C₈

—

109

—

109

C₈

#13182008

yes

—

110

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

111

C₂²

#16812695

yes

characteristic

112

C₂²

#16812698

yes

characteristic

113

C₂²

#16812720

yes

—

114

C₄

—

115

C₄

—

872

—

116

C₄

—

218

—

117

C₄

C₄⋊C₄×D₁₀₉

yes

characteristic

118

C₄

C₂×#13182008

yes

characteristic

119

C₄

#16812697

yes

characteristic

120

C₄

#16812698

yes

characteristic

121

C₂

#20294895

yes

characteristic

122

C₂

#20295061

yes

—

123

C₂

#20295407

yes

—

124

C₂

#20295426

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#106×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#106	D₁₀₉
hypocenter	C₁₀₉	#1846
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#106×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
11. C₈×Dic₁₀₉	C₄	semidirect
63. C₁₀₉	#1846	semidirect

H ≅ G/N

type

11. C₈×Dic₁₀₉

C₄

semidirect

63. C₁₀₉

#1846

semidirect

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

subgroup	index
C₂×#16812632	2
C₂×#16812631	2
C₂×C₈×Dic₁₀₉	2
#20294891	2
C₂×#106×C₁₀₉	2
#1846	109

subgroup

index

C₂×C₈×Dic₁₀₉

C₂×#106×C₁₀₉

109

Subgroups — #28363193

Conjugacy classes of subgroups

Complements

Maximal subgroups