Group #28360928 — subgroups

Subgroups — #28360928

Order 13952.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#20295050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182013	—	2	1	no	—
5	C₂²×#11724784	—	2	1	no	—
6	C₄⋊#11724784	—	2	1	no	—
7	C₄⋊#11724784	—	2	1	no	—
8	C₄⋊#11724784	C₄	1	2	yes	—
9	C₄⋊#11724784	C₄	1	2	yes	—
10	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₀₉	—	2	1	no	—
12	C₂²⋊C₈×C₁₀₉	C₄	1	2	yes	—
13	C₄×C₈×C₁₀₉	—	2	1	no	—
14	#13182013	—	4	1	no	—
15	#13182013	—	4	1	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₂×#11724784	—	4	1	no	—
19	C₂×#11724784	—	4	1	no	—
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	—	2	2	no	—
24	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
27	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	—	4	1	no	—
31	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	Q₁₆	1	1	yes	characteristic
41	C₈×C₁₀₉	—	4	2	no	—
42	#11724784	—	4	2	no	—
43	#11724784	—	4	2	no	—
44	C₂²×C₁₀₉	—	2	1	no	—
45	C₂²×C₁₀₉	—	2	1	no	—
46	C₂²×C₁₀₉	#95	1	1	yes	characteristic
47	C₂²×C₁₀₉	#106	1	1	yes	characteristic
48	C₂²×C₁₀₉	#107	1	1	yes	characteristic
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	—	2	1	no	—
51	C₄×C₁₀₉	—	2	1	no	—
52	C₄×C₁₀₉	#102	1	1	yes	characteristic
53	C₄×C₁₀₉	#102	1	1	yes	characteristic
54	C₄×C₁₀₉	#103	1	1	yes	characteristic
55	C₄×C₁₀₉	#103	1	1	yes	characteristic
56	C₄×C₁₀₉	#108	1	2	yes	—
57	C₂×C₁₀₉	—	2	1	no	—
58	C₂×C₁₀₉	—	2	1	no	—
59	C₂×C₁₀₉	#340	1	1	yes	characteristic
60	C₂×C₁₀₉	#341	1	1	yes	characteristic
61	C₂×C₁₀₉	#344	1	1	yes	characteristic
62	#1301	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1301	1	1	yes	characteristic, complemented, Sylow
64	#431	D₁₀₉	1	1	yes	characteristic
65	#423	—	109	1	no	—
66	C₂×C₄⋊C₈	—	109	1	no	—
67	C₂×M₁₆	—	218	1	no	—
68	C₂²×C₈	—	218	1	no	—
69	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	109	2	no	—
72	C₄⋊C₈	—	109	2	no	—
73	C₄⋊C₈	—	218	1	no	—
74	C₄⋊C₈	—	218	1	no	—
75	C₂²⋊C₈	Dic₁₀₉	1	2	yes	—
76	C₄×C₈	—	2	1	no	—
77	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
78	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
79	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
80	M₁₆	—	436	1	no	—
81	M₁₆	—	436	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	436	1	no	—
84	C₂×C₈	—	436	1	no	—
85	C₂×C₈	—	218	2	no	—
86	C₂×C₈	—	218	2	no	—
87	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
88	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
89	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
90	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
91	C₂³	C₄⋊Dic₁₀₉	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
95	C₂×C₄	#13182006	1	1	yes	characteristic
96	C₂×C₄	#13182006	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
98	C₂×C₄	#13182008	1	1	yes	characteristic
99	C₂×C₄	#13182008	1	1	yes	characteristic
100	C₂×C₄	D₈₇₂	1	1	yes	characteristic
101	C₂×C₄	#13182018	1	1	yes	characteristic
102	C₂×C₄	#13182018	1	1	yes	characteristic
103	C₂×C₄	Dic₄₃₆	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	436	2	no	—
106	C₈	—	436	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#16812606	1	1	yes	characteristic
110	C₂²	#16812631	1	1	yes	characteristic
111	C₂²	#16812633	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#16812621	1	1	yes	characteristic
116	C₄	#16812621	1	1	yes	characteristic
117	C₄	#16812624	1	1	yes	characteristic
118	C₄	#16812624	1	1	yes	characteristic
119	C₄	#16812634	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#20294789	1	1	yes	characteristic
123	C₂	#20294790	1	1	yes	characteristic
124	C₂	#20294796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295050

C₂

yes

maximal, characteristic

C₂×C₄⋊#11724784

C₂

yes

maximal, characteristic

#431×C₁₀₉

C₂

yes

maximal, characteristic

—

—

—

—

C₄

yes

—

C₄⋊#11724784

C₄

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

C₄

yes

—

C₄×C₈×C₁₀₉

—

—

—

—

—

—

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

D₈

yes

characteristic

C₂×C₄×C₁₀₉

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₉

Q₁₆

yes

characteristic

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1301

yes

characteristic, complemented, Sylow

#431

D₁₀₉

yes

characteristic

#423

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

Dic₁₀₉

yes

—

C₄×C₈

—

C₂²×C₄

Dic₂₁₈

yes

characteristic

C₂²×C₄

D₄₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

M₁₆

—

436

—

M₁₆

—

436

—

C₂×C₈

—

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

C₄×D₁₀₉

yes

characteristic

C₄²

C₄×D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₂³

C₄⋊Dic₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₉

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₀₉

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

D₈₇₂

yes

characteristic

101

C₂×C₄

#13182018

yes

characteristic

102

C₂×C₄

#13182018

yes

characteristic

103

C₂×C₄

Dic₄₃₆

yes

characteristic

104

C₈

—

105

C₈

—

436

—

106

C₈

—

436

—

107

C₂²

—

108

C₂²

—

109

C₂²

#16812606

yes

characteristic

110

C₂²

#16812631

yes

characteristic

111

C₂²

#16812633

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#16812621

yes

characteristic

116

C₄

#16812621

yes

characteristic

117

C₄

#16812624

yes

characteristic

118

C₄

#16812624

yes

characteristic

119

C₄

#16812634

yes

—

120

C₂

—

121

C₂

—

122

C₂

#20294789

yes

characteristic

123

C₂

#20294790

yes

characteristic

124

C₂

#20294796

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13182006
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₂×C₄²	C₂×D₁₀₉
Fitting	#431×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#95
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₀₉	C₂²
hypercenter	#431	D₁₀₉
hypocenter	C₁₀₉	#1301
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

#13182006

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₂×C₄²

C₂×D₁₀₉

Fitting

#431×C₁₀₉

C₂

socle

C₂²×C₁₀₉

#95

prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₂×C₄²×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
63. C₁₀₉	#1301	semidirect

H ≅ G/N

type

63. C₁₀₉

#1301

semidirect

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

subgroup	index
#20295050	2
C₂×C₄⋊#11724784	2
#431×C₁₀₉	2
#1301	109

subgroup

index

#20295050

C₂×C₄⋊#11724784

#431×C₁₀₉

#1301

109

Subgroups — #28360928

Conjugacy classes of subgroups

Complements

Maximal subgroups