Group #28361782 — subgroups

Subgroups — #28361782

Order 13952.

180 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20295079	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
3	#20294778	C₂	1	2	yes	maximal, complemented
4	#20294776	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₄	1	1	yes	characteristic
7	C₂×#13182013	C₄	1	1	yes	characteristic
8	C₄⋊#11724784	C₂²	1	1	yes	characteristic
9	C₄⋊#11724784	C₂²	1	1	yes	characteristic
10	C₂²⋊#11724784	—	2	1	no	—
11	#16812610	C₄	1	2	yes	—
12	C₄×#11724784	—	2	1	no	—
13	C₄×#11724784	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₉	—	2	1	no	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₉	—	2	1	no	—
17	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
18	#13182013	—	2	2	no	—
19	#13182013	C₈	1	4	yes	complemented
20	C₂×#11724784	—	2	1	no	—
21	C₂×#11724784	—	2	1	no	—
22	C₂×#11724784	C₂×C₄	1	2	yes	—
23	C₂×#11724784	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	—	2	2	no	—
28	C₂×C₈×C₁₀₉	—	4	2	no	—
29	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
31	C₄²×C₁₀₉	Q₈	1	1	yes	characteristic
32	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄×C₁₀₉	—	2	1	no	—
36	C₂×C₄×C₁₀₉	—	4	1	no	—
37	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
41	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₁₀₉	—	4	2	no	—
46	#11724784	—	2	2	no	—
47	#11724784	—	4	1	no	—
48	#11724784	—	4	1	no	—
49	#11724784	C₂×C₈	1	4	yes	complemented
50	C₂²×C₁₀₉	—	2	1	no	—
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	C₄⋊C₈	1	2	yes	—
53	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₀₉	—	2	1	no	—
55	C₄×C₁₀₉	—	2	1	no	—
56	C₄×C₁₀₉	—	2	1	no	—
57	C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
58	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
59	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₁₀₉	#137	1	1	yes	characteristic
62	C₂×C₁₀₉	—	2	2	no	—
63	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₁₀₉	#428	1	1	yes	characteristic
65	C₂×C₁₀₉	#430	1	1	yes	characteristic
66	#1565	—	109	1	no	maximal, Sylow
67	C₁₀₉	#1565	1	1	yes	characteristic, complemented, Sylow
68	#431	—	109	1	no	—
69	C₂×C₄⋊C₈	D₁₀₉	1	1	yes	characteristic
70	C₄×M₁₆	—	109	1	no	—
71	#335	—	109	2	no	—
72	#334	—	109	2	no	—
73	C₂×M₁₆	—	109	1	no	—
74	C₂×M₁₆	—	109	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
77	C₄⋊C₈	—	2	1	no	—
78	C₄⋊C₈	—	109	1	no	—
79	C₄⋊C₈	—	109	1	no	—
80	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
81	C₂²⋊C₈	—	218	1	no	—
82	#97	—	109	2	no	—
83	C₄×C₈	—	109	2	no	—
84	C₄×C₈	—	218	1	no	—
85	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
88	M₁₆	—	109	4	no	—
89	M₁₆	—	218	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	109	2	no	—
93	C₂×C₈	—	109	2	no	—
94	C₂×C₈	—	218	1	no	—
95	C₂×C₈	—	218	1	no	—
96	C₄²	Dic₂₁₈	1	1	yes	characteristic
97	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
98	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
99	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
100	C₂³	#13182008	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	#13182008	1	1	yes	characteristic
105	C₂×C₄	#13182008	1	1	yes	characteristic
106	C₂×C₄	#13182008	1	1	yes	characteristic
107	C₂×C₄	C₈×D₁₀₉	1	2	yes	—
108	C₂×C₄	#13182012	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	109	4	no	—
114	C₈	—	436	1	no	—
115	C₈	—	436	1	no	—
116	C₈	—	218	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#16812628	1	2	yes	—
120	C₂²	C₂×#13182008	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	#16812628	1	2	yes	—
125	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
126	C₄	C₂×#13182012	1	1	yes	characteristic
127	C₄	#16812725	1	1	yes	characteristic
128	C₄	#16812730	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×#16812628	1	1	yes	characteristic
131	C₂	#20295066	1	1	yes	characteristic
132	C₂	#20295074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295079

C₂

yes

maximal, characteristic

C₄×#13182013

C₂

yes

maximal, characteristic

#20294778

C₂

yes

maximal, complemented

#20294776

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₄

yes

characteristic

C₂×#13182013

C₄

yes

characteristic

C₄⋊#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

characteristic

C₂²⋊#11724784

—

#16812610

C₄

yes

—

C₄×#11724784

—

C₄×#11724784

C₂²

yes

—

C₂²×C₈×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

C₂²

yes

—

#13182013

—

#13182013

C₈

yes

—

—

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

Q₈

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₈×C₁₀₉

—

—

—

—

C₂×C₈

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄⋊C₈

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊C₈

yes

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1565

yes

characteristic, complemented, Sylow

#431

—

109

—

C₂×C₄⋊C₈

D₁₀₉

yes

characteristic

C₄×M₁₆

—

109

—

#335

—

109

—

#334

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

C₂²⋊C₈

—

218

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂²×C₄

Dic₂₁₈

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

Dic₂₁₈

yes

characteristic

C₄²

C₄×D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₄²

C₂²×D₁₀₉

yes

characteristic

100

C₂³

#13182008

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#13182008

yes

characteristic

105

C₂×C₄

#13182008

yes

characteristic

106

C₂×C₄

#13182008

yes

characteristic

107

C₂×C₄

C₈×D₁₀₉

yes

—

108

C₂×C₄

#13182012

yes

—

109

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

111

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

112

C₈

—

113

C₈

—

109

—

114

C₈

—

436

—

115

C₈

—

436

—

116

C₈

—

218

—

117

C₂²

—

118

C₂²

—

119

C₂²

#16812628

yes

—

120

C₂²

C₂×#13182008

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#16812628

yes

—

125

C₄

C₂×C₈×D₁₀₉

yes

characteristic

126

C₄

C₂×#13182012

yes

characteristic

127

C₄

#16812725

yes

characteristic

128

C₄

#16812730

yes

characteristic

129

C₂

—

130

C₂

C₂×#16812628

yes

characteristic

131

C₂

#20295066

yes

characteristic

132

C₂

#20295074

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₄²	C₂²×D₁₀₉
Fitting	C₂×C₄⋊C₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#1565
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₄²

C₂²×D₁₀₉

Fitting

C₂×C₄⋊C₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₉

hypocenter

C₁₀₉

#1565

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #20294778	C₂	semidirect
4. #20294776	C₂	semidirect
19. #13182013	C₈	semidirect
49. #11724784	C₂×C₈	semidirect
67. C₁₀₉	#1565	semidirect

H ≅ G/N

type

3. #20294778

C₂

semidirect

4. #20294776

C₂

semidirect

19. #13182013

C₈

semidirect

49. #11724784

C₂×C₈

semidirect

67. C₁₀₉

#1565

semidirect

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

subgroup	index
#20295079	2
C₄×#13182013	2
#20294778	2
#20294776	2
C₂×C₄⋊C₈×C₁₀₉	2
#1565	109

subgroup

index

C₂×C₄⋊C₈×C₁₀₉

109

Subgroups — #28361782

Conjugacy classes of subgroups

Complements

Maximal subgroups