Group #28363144 — subgroups

Subgroups — #28363144

Order 13952.

236 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 81 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	#20295504	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812618	C₂	1	1	yes	maximal, characteristic
3	#20294798	C₂	1	2	yes	maximal
4	#20294797	C₂	1	2	yes	maximal
5	C₂×#101×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	Q₈×Dic₁₀₉	—	2	2	no	—
8	Q₈×Dic₁₀₉	—	2	2	no	—
9	#16812658	C₄	1	4	yes	—
10	#16812658	C₂²	1	1	yes	characteristic
11	#16812658	C₂²	1	1	yes	characteristic
12	#16812618	—	2	2	no	—
13	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
15	#101×C₁₀₉	—	2	2	no	—
16	#13182013	—	2	2	no	—
17	#13182013	—	2	2	no	—
18	C₂×#11724784	—	2	2	no	—
19	C₄⋊Dic₁₀₉	—	2	4	no	—
20	C₄⋊Dic₁₀₉	—	2	1	no	—
21	C₄⋊Dic₁₀₉	—	2	1	no	—
22	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₀₉	—	2	4	no	—
25	C₄×Dic₁₀₉	—	2	1	no	—
26	C₄×Dic₁₀₉	—	2	1	no	—
27	C₂×Q₈×C₁₀₉	—	2	2	no	—
28	C₂×Q₈×C₁₀₉	—	2	2	no	—
29	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
30	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
31	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₀₉	—	2	2	no	—
34	M₁₆×C₁₀₉	—	2	2	no	—
35	C₂×C₈×C₁₀₉	—	2	2	no	—
36	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₀₉	—	2	4	no	—
38	Q₈×C₁₀₉	—	2	4	no	—
39	C₂×C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	#41	1	2	yes	—
45	C₂×C₄×C₁₀₉	#41	1	2	yes	—
46	C₂×Dic₁₀₉	—	2	2	no	—
47	C₂×Dic₁₀₉	—	2	2	no	—
48	C₈×C₁₀₉	—	4	2	no	—
49	#11724784	—	4	2	no	—
50	C₂²×C₁₀₉	—	4	1	no	—
51	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₀₉	#117	1	1	yes	characteristic
53	C₂²×C₁₀₉	#117	1	1	yes	characteristic
54	C₄×C₁₀₉	—	2	4	no	—
55	C₄×C₁₀₉	#123	1	2	yes	—
56	C₄×C₁₀₉	#123	1	2	yes	—
57	Dic₁₀₉	—	4	2	no	—
58	Dic₁₀₉	—	4	2	no	—
59	C₂×C₁₀₉	—	2	1	no	—
60	C₂×C₁₀₉	—	2	1	no	—
61	C₂×C₁₀₉	—	2	1	no	—
62	C₂×C₁₀₉	#381	1	1	yes	characteristic
63	#1830	—	109	1	no	maximal, Sylow
64	C₁₀₉	#1830	1	1	yes	characteristic, complemented, Sylow
65	#519	—	109	1	no	—
66	C₂×#101	—	109	1	no	—
67	C₂×#101	D₁₀₉	1	1	yes	characteristic
68	#344	—	109	2	no	—
69	#344	—	109	2	no	—
70	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂×M₁₆	—	109	2	no	—
72	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
73	C₄×Q₈	—	218	2	no	—
74	C₄×Q₈	—	218	2	no	—
75	#117	—	109	4	no	—
76	#117	—	109	1	no	—
77	#117	—	109	1	no	—
78	#101	—	2	2	no	—
79	#101	—	218	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	D₄₃₆	1	2	yes	—
83	C₂×Q₈	C₂²⋊D₁₀₉	1	2	yes	—
84	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	218	2	no	—
89	M₁₆	—	218	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	218	2	no	—
92	C₄⋊C₄	—	218	4	no	—
93	C₄⋊C₄	—	218	1	no	—
94	C₄⋊C₄	—	218	1	no	—
95	C₂²⋊C₄	—	109	2	no	—
96	C₂²⋊C₄	—	109	2	no	—
97	C₄²	—	218	4	no	—
98	C₄²	—	218	1	no	—
99	C₄²	—	218	1	no	—
100	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	218	2	no	—
106	C₂×C₄	—	218	2	no	—
107	C₂×C₄	#13182006	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
109	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
110	C₂×C₄	#13182028	1	2	yes	—
111	C₂×C₄	#13182028	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	436	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₂×#13182006	1	1	yes	characteristic
116	C₂²	#16812655	1	1	yes	characteristic
117	C₂²	#16812655	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	436	2	no	—
120	C₄	—	436	2	no	—
121	C₄	#16812680	1	2	yes	—
122	C₄	#16812683	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#20294887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295504

C₂

yes

maximal, characteristic

C₂×#16812618

C₂

yes

maximal, characteristic

#20294798

C₂

yes

maximal

#20294797

C₂

yes

maximal

C₂×#101×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

Q₈×Dic₁₀₉

—

Q₈×Dic₁₀₉

—

#16812658

C₄

yes

—

#16812658

C₂²

yes

characteristic

#16812658

C₂²

yes

characteristic

#16812618

—

C₂²×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

—

#101×C₁₀₉

—

#13182013

—

#13182013

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

C₂×C₄

yes

—

C₂²⋊Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

#123

yes

—

C₄×C₁₀₉

#123

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#381

yes

characteristic

#1830

—

109

maximal, Sylow

C₁₀₉

#1830

yes

characteristic, complemented, Sylow

#519

—

109

—

C₂×#101

—

109

—

C₂×#101

D₁₀₉

yes

characteristic

#344

—

109

—

#344

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₄×Q₈

—

218

—

C₄×Q₈

—

218

—

#117

—

109

—

#117

—

109

—

#117

—

109

—

#101

—

#101

—

218

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄₃₆

yes

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₄²

—

218

—

100

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

218

—

106

C₂×C₄

—

218

—

107

C₂×C₄

#13182006

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

109

C₂×C₄

C₂×D₄₃₆

yes

characteristic

110

C₂×C₄

#13182028

yes

—

111

C₂×C₄

#13182028

yes

—

112

C₈

—

113

C₈

—

436

—

114

C₂²

—

115

C₂²

C₂×#13182006

yes

characteristic

116

C₂²

#16812655

yes

characteristic

117

C₂²

#16812655

yes

characteristic

118

C₄

—

119

C₄

—

436

—

120

C₄

—

436

—

121

C₄

#16812680

yes

—

122

C₄

#16812683

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#20294887

yes

characteristic

	subgroup	G/H
center	C₂	#20294887
commutator, prime radical	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#101×C₁₀₉	C₂
socle	C₂×C₁₀₉	#381
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#101	D₁₀₉
hypocenter	C₁₀₉	#1830
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#20294887

commutator, prime radical

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#101×C₁₀₉

C₂

socle

C₂×C₁₀₉

#381

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
64. C₁₀₉	#1830	semidirect

H ≅ G/N

type

64. C₁₀₉

#1830

semidirect

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

subgroup	index
#20295504	2
C₂×#16812618	2
#20294798	2
#20294797	2
C₂×#101×C₁₀₉	2
#1830	109

subgroup

index

C₂×#101×C₁₀₉

109

Subgroups — #28363144

Conjugacy classes of subgroups

Complements

Maximal subgroups