Group #28363317 — subgroups

Subgroups — #28363317

Order 13952.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20295793	C₂	1	1	yes	maximal, characteristic
2	#20295052	C₂	1	2	yes	maximal, complemented
3	#20295000	C₂	1	1	yes	maximal, characteristic
4	#20294798	C₂	1	2	yes	maximal, complemented
5	#408×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812706	—	2	4	no	—
7	C₂×#13182013	—	2	2	no	—
8	C₂×#13182013	C₂²	1	2	yes	—
9	#16812658	C₂²	1	2	yes	—
10	#16812628	—	2	4	no	—
11	#16812614	C₄	1	4	yes	—
12	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
16	#13182013	—	2	4	no	—
17	#13182013	—	2	2	no	—
18	#13182013	—	2	2	no	—
19	C₂×#11724784	—	2	4	no	—
20	C₂×#11724784	D₄	1	4	yes	—
21	C₄⋊Dic₁₀₉	—	2	2	no	—
22	C₂²⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	#41×C₁₀₉	—	2	2	no	—
25	#41×C₁₀₉	—	2	2	no	—
26	C₂×Q₈×C₁₀₉	Q₈	1	1	yes	characteristic
27	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₁₀₉	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	—	2	1	no	—
33	C₂×C₈×C₁₀₉	—	2	1	no	—
34	C₂×C₈×C₁₀₉	—	2	2	no	—
35	C₂³×C₁₀₉	C₄⋊C₄	1	2	yes	—
36	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₀₉	—	2	2	no	—
38	D₄×C₁₀₉	—	2	4	no	—
39	D₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	#41	1	2	yes	—
47	C₂×Dic₁₀₉	—	2	4	no	—
48	C₈×C₁₀₉	—	4	1	no	—
49	C₈×C₁₀₉	—	4	1	no	—
50	#11724784	—	2	4	no	—
51	C₂²×C₁₀₉	—	2	2	no	—
52	C₂²×C₁₀₉	—	4	1	no	—
53	C₂²×C₁₀₉	—	4	2	no	—
54	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
56	C₄×C₁₀₉	—	2	2	no	—
57	C₄×C₁₀₉	#120	1	1	yes	characteristic
58	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₀₉	#123	1	1	yes	characteristic
60	Dic₁₀₉	—	4	4	no	—
61	C₂×C₁₀₉	—	2	2	no	—
62	C₂×C₁₀₉	—	2	1	no	—
63	C₂×C₁₀₉	—	4	2	no	—
64	C₂×C₁₀₉	#385	1	1	yes	characteristic
65	#1868	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1868	1	1	yes	characteristic, complemented, Sylow
67	#568	—	109	1	no	—
68	#424	—	109	2	no	—
69	#410	—	109	1	no	—
70	#408	D₁₀₉	1	1	yes	characteristic
71	#344	—	109	2	no	—
72	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
73	#131	—	218	4	no	—
74	C₂×M₁₆	—	109	2	no	—
75	C₂×M₁₆	—	218	2	no	—
76	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
78	#117	—	109	2	no	—
79	C₄⋊C₈	—	218	4	no	—
80	#99	—	109	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₂₁₈	1	1	yes	characteristic
85	C₂×D₄	Dic₂₁₈	1	1	yes	characteristic
86	C₂×D₄	C₄×D₁₀₉	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	218	4	no	—
91	M₁₆	—	218	2	no	—
92	M₁₆	—	218	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	109	4	no	—
97	C₂×C₈	—	218	4	no	—
98	C₄⋊C₄	—	218	2	no	—
99	C₂²⋊C₄	—	109	4	no	—
100	C₄²	—	218	2	no	—
101	C₂³	#13182008	1	2	yes	—
102	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	218	4	no	—
109	C₂×C₄	#13182008	1	2	yes	—
110	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
113	C₂×C₄	#13182028	1	2	yes	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	218	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×#13182008	1	1	yes	characteristic
121	C₂²	#16812659	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	436	4	no	—
124	C₄	#16812667	1	1	yes	characteristic
125	C₄	#16812676	1	2	yes	—
126	C₄	#16812680	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#20294898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295793

C₂

yes

maximal, characteristic

#20295052

C₂

yes

maximal, complemented

#20295000

C₂

yes

maximal, characteristic

#20294798

C₂

yes

maximal, complemented

#408×C₁₀₉

C₂

yes

maximal, characteristic

#16812706

—

C₂×#13182013

—

C₂×#13182013

C₂²

yes

—

#16812658

C₂²

yes

—

#16812628

—

#16812614

C₄

yes

—

C₂×#41×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²⋊C₈×C₁₀₉

—

—

—

—

—

D₄

yes

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

Q₈

yes

characteristic

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂×D₄×C₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂³×C₁₀₉

C₄⋊C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#123

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#385

yes

characteristic

#1868

—

109

maximal, Sylow

C₁₀₉

#1868

yes

characteristic, complemented, Sylow

#568

—

109

—

#424

—

109

—

#410

—

109

—

#408

D₁₀₉

yes

characteristic

#344

—

109

—

C₂×#41

C₂×D₁₀₉

yes

characteristic

#131

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

218

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

C₄⋊C₈

—

218

—

#99

—

109

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₂₁₈

yes

characteristic

C₂×D₄

Dic₂₁₈

yes

characteristic

C₂×D₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

109

—

100

C₄²

—

218

—

101

C₂³

#13182008

yes

—

102

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

218

—

109

C₂×C₄

#13182008

yes

—

110

C₂×C₄

D₄×D₁₀₉

yes

—

111

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

112

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

113

C₂×C₄

#13182028

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

218

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#13182008

yes

characteristic

121

C₂²

#16812659

yes

—

122

C₄

—

123

C₄

—

436

—

124

C₄

#16812667

yes

characteristic

125

C₄

#16812676

yes

—

126

C₄

#16812680

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#20294898

yes

characteristic

	subgroup	G/H
center	C₄	#16812667
commutator	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#408×C₁₀₉	C₂
socle	C₂×C₁₀₉	#385
prime radical	C₂×#41×C₁₀₉	C₂²
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#408	D₁₀₉
hypocenter	C₁₀₉	#1868
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#16812667

commutator

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#408×C₁₀₉

C₂

socle

C₂×C₁₀₉

#385

prime radical

C₂×#41×C₁₀₉

C₂²

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #20295052	C₂	semidirect
4. #20294798	C₂	semidirect
66. C₁₀₉	#1868	semidirect

H ≅ G/N

type

2. #20295052

C₂

semidirect

4. #20294798

C₂

semidirect

66. C₁₀₉

#1868

semidirect

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

subgroup	index
#20295793	2
#20295052	2
#20295000	2
#20294798	2
#408×C₁₀₉	2
#1868	109

subgroup

index

#408×C₁₀₉

109

Subgroups — #28363317

Conjugacy classes of subgroups

Complements

Maximal subgroups