Group #28361786 — subgroups

Subgroups — #28361786

Order 13952.

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

normal subgroups only

This group has 184 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	#20295048	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294777	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294776	C₂	1	2	yes	maximal, complemented
5	#334×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182012	—	2	2	no	—
7	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
9	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
10	C₄⋊#11724784	C₂²	1	2	yes	—
11	#16812628	—	2	2	no	—
12	C₄×#11724784	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
20	#13182012	—	4	4	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
23	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
24	C₈×D₁₀₉	—	2	2	no	—
25	C₈×D₁₀₉	—	2	2	no	—
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄×D₁₀₉	—	2	1	no	—
41	C₄×D₁₀₉	—	2	1	no	—
42	C₄×D₁₀₉	—	2	2	no	—
43	C₄×D₁₀₉	SD₁₆	1	4	yes	—
44	C₈×C₁₀₉	—	2	1	no	—
45	C₈×C₁₀₉	—	4	2	no	—
46	C₈×C₁₀₉	M₁₆	1	2	yes	complemented
47	#11724784	—	2	1	no	—
48	#11724784	—	4	2	no	—
49	#11724784	M₁₆	1	2	yes	complemented
50	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₁₀₉	—	2	1	no	—
52	C₂×D₁₀₉	—	2	1	no	—
53	C₂×D₁₀₉	#106	1	2	yes	—
54	C₄×C₁₀₉	—	2	1	no	—
55	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
56	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
57	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
58	Dic₁₀₉	—	2	1	no	—
59	Dic₁₀₉	#106	1	2	yes	—
60	Dic₁₀₉	#108	1	2	yes	—
61	C₂×C₁₀₉	#423	1	1	yes	characteristic
62	C₂×C₁₀₉	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
64	D₁₀₉	—	2	2	no	—
65	#1566	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1566	1	1	yes	characteristic, complemented, Sylow
67	#423	—	109	2	no	—
68	C₂×C₄×C₈	—	109	1	no	—
69	#334	—	109	2	no	—
70	#334	—	109	1	no	—
71	#334	D₁₀₉	1	1	yes	characteristic, complemented
72	C₂×M₁₆	—	218	2	no	—
73	C₂²×C₈	—	109	1	no	—
74	C₂²×C₈	—	109	1	no	—
75	C₂×C₄²	—	109	1	no	—
76	C₄⋊C₈	—	109	2	no	—
77	C₄⋊C₈	—	218	2	no	—
78	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
79	C₄×C₈	—	109	1	no	—
80	C₄×C₈	—	109	1	no	—
81	C₄×C₈	—	109	1	no	—
82	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
83	C₂²×C₄	—	109	1	no	—
84	C₂²×C₄	—	109	1	no	—
85	C₂²×C₄	—	109	1	no	—
86	M₁₆	—	436	4	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	109	1	no	—
89	C₂×C₈	—	109	1	no	—
90	C₂×C₈	—	218	2	no	—
91	C₂×C₈	—	218	2	no	—
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
94	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
95	C₄²	—	109	1	no	—
96	C₄²	—	109	1	no	—
97	C₄²	—	109	1	no	—
98	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
99	C₂³	—	109	1	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	109	1	no	—
102	C₂×C₄	—	109	4	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	218	2	no	—
105	C₂×C₄	—	218	1	no	—
106	C₂×C₄	—	218	1	no	—
107	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
108	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	109	2	no	—
113	C₈	—	436	2	no	—
114	C₈	—	218	1	no	—
115	C₈	#13182012	1	2	yes	complemented
116	C₂²	—	109	2	no	—
117	C₂²	—	218	1	no	—
118	C₂²	—	218	1	no	—
119	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	109	2	no	—
122	C₄	—	109	2	no	—
123	C₄	—	218	1	no	—
124	C₄	M₁₆×D₁₀₉	1	1	yes	characteristic
125	C₄	C₂×#13182012	1	1	yes	characteristic
126	C₄	SD₁₆×D₁₀₉	1	2	yes	—
127	C₂	—	218	2	no	—
128	C₂	#20295048	1	1	yes	characteristic
129	C₂	#106×D₁₀₉	1	1	yes	characteristic
130	C₂	#108×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295048

C₂

yes

maximal

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#20294777

C₂

yes

maximal, characteristic, complemented

#20294776

C₂

yes

maximal, complemented

#334×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182012

—

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

#16812628

—

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

—

—

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

SD₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

M₁₆

yes

—

—

M₁₆

yes

complemented

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#106

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

C₂×SD₁₆

yes

—

Dic₁₀₉

—

Dic₁₀₉

#106

yes

—

Dic₁₀₉

#108

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1566

—

109

maximal, Sylow

C₁₀₉

#1566

yes

characteristic, complemented, Sylow

#423

—

109

—

C₂×C₄×C₈

—

109

—

#334

—

109

—

#334

—

109

—

#334

D₁₀₉

yes

characteristic, complemented

C₂×M₁₆

—

218

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

436

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

218

—

105

C₂×C₄

—

218

—

106

C₂×C₄

—

218

—

107

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

108

C₂×C₄

D₄×D₁₀₉

yes

characteristic

109

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

109

—

113

C₈

—

436

—

114

C₈

—

218

—

115

C₈

#13182012

yes

complemented

116

C₂²

—

109

—

117

C₂²

—

218

—

118

C₂²

—

218

—

119

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

120

C₄

—

121

C₄

—

109

—

122

C₄

—

109

—

123

C₄

—

218

—

124

C₄

M₁₆×D₁₀₉

yes

characteristic

125

C₄

C₂×#13182012

yes

characteristic

126

C₄

SD₁₆×D₁₀₉

yes

—

127

C₂

—

218

—

128

C₂

#20295048

yes

characteristic

129

C₂

#106×D₁₀₉

yes

characteristic

130

C₂

#108×D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#334×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#334	D₁₀₉
hypocenter	C₁₀₉	#1566
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#334×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
3. #20294777	C₂	semidirect
4. #20294776	C₂	semidirect
5. #334×C₁₀₉	C₂	semidirect
46. C₈×C₁₀₉	M₁₆	semidirect
49. #11724784	M₁₆	semidirect
66. C₁₀₉	#1566	semidirect
71. #334	D₁₀₉	semidirect
115. C₈	#13182012	semidirect

H ≅ G/N

type

3. #20294777

C₂

semidirect

4. #20294776

C₂

semidirect

5. #334×C₁₀₉

C₂

semidirect

46. C₈×C₁₀₉

M₁₆

semidirect

49. #11724784

M₁₆

semidirect

66. C₁₀₉

#1566

semidirect

71. #334

D₁₀₉

semidirect

115. C₈

#13182012

semidirect

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

subgroup	index
#20295048	2
C₄×C₈×D₁₀₉	2
#20294777	2
#20294776	2
#334×C₁₀₉	2
#1566	109

subgroup

index

C₄×C₈×D₁₀₉

#334×C₁₀₉

109

Subgroups — #28361786

Conjugacy classes of subgroups

Complements

Maximal subgroups