Group #28362776 — subgroups

Subgroups — #28362776

Order 13952.

296 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×C₈×Dic₁₀₉	C₂	1	2	yes	maximal
2	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294781	C₂	1	1	yes	maximal, characteristic
4	#20294780	C₂	1	2	yes	maximal
5	#336×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₀₉	C₄	1	8	yes	complemented
12	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
15	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×#11724784	—	2	2	no	—
18	C₂×#11724784	—	2	2	no	—
19	C₂×#11724784	C₂×C₄	1	4	yes	—
20	C₄×Dic₁₀₉	—	2	1	no	—
21	C₄×Dic₁₀₉	—	2	1	no	—
22	C₄×Dic₁₀₉	—	2	4	no	—
23	C₄×Dic₁₀₉	C₈	1	8	yes	complemented
24	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	—	2	2	no	—
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
30	C₄²×C₁₀₉	—	2	2	no	—
31	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₁₀₉	#41	1	2	yes	—
37	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
39	C₂×Dic₁₀₉	—	2	2	no	—
40	C₂×Dic₁₀₉	—	2	2	no	—
41	C₂×Dic₁₀₉	C₄²	1	2	yes	—
42	C₂×Dic₁₀₉	C₄²	1	2	yes	—
43	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
44	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
45	C₂×Dic₁₀₉	M₁₆	1	4	yes	—
46	C₈×C₁₀₉	—	2	4	no	—
47	#11724784	—	2	4	no	—
48	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
49	C₂²×C₁₀₉	#117	1	1	yes	characteristic
50	C₂²×C₁₀₉	#117	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₂²×C₈	1	2	yes	—
52	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
53	C₄×C₁₀₉	—	2	4	no	—
54	C₄×C₁₀₉	#117	1	2	yes	—
55	C₄×C₁₀₉	#117	1	2	yes	—
56	Dic₁₀₉	—	2	4	no	—
57	Dic₁₀₉	C₄×C₈	1	4	yes	—
58	Dic₁₀₉	#97	1	4	yes	—
59	C₂×C₁₀₉	#376	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₀₉	#431	1	2	yes	—
63	C₂×C₁₀₉	#431	1	2	yes	—
64	#1744	—	109	1	no	maximal, Sylow
65	C₁₀₉	#1744	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄×C₈	—	109	2	no	—
67	C₄³	—	109	1	no	—
68	#336	—	109	2	no	—
69	#336	—	109	1	no	—
70	#336	D₁₀₉	1	1	yes	characteristic
71	C₂²×C₈	—	109	2	no	—
72	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
73	C₂×C₄²	—	109	2	no	—
74	C₂×C₄²	—	109	1	no	—
75	C₂×C₄²	—	109	2	no	—
76	C₂×C₄²	—	109	1	no	—
77	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
78	C₄×C₈	—	109	8	no	—
79	C₂²×C₄	—	109	1	no	—
80	C₂²×C₄	—	109	1	no	—
81	C₂²×C₄	—	109	2	no	—
82	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	109	4	no	—
87	C₂×C₈	—	218	2	no	—
88	C₂×C₈	—	218	2	no	—
89	C₂×C₈	C₄×D₁₀₉	1	4	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	109	8	no	—
92	C₄²	—	109	4	no	—
93	C₄²	—	218	4	no	—
94	C₄²	—	218	1	no	—
95	C₄²	—	218	1	no	—
96	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	109	2	no	—
100	C₂×C₄	—	109	2	no	—
101	C₂×C₄	—	109	4	no	—
102	C₂×C₄	—	109	4	no	—
103	C₂×C₄	—	109	4	no	—
104	C₂×C₄	—	218	2	no	—
105	C₂×C₄	—	218	2	no	—
106	C₂×C₄	C₈×D₁₀₉	1	4	yes	—
107	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
108	C₂×C₄	#13182028	1	1	yes	characteristic
109	C₂×C₄	#13182028	1	2	yes	—
110	C₂×C₄	#13182030	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	218	4	no	—
113	C₂²	C₄²×D₁₀₉	1	1	yes	characteristic
114	C₂²	#16812655	1	1	yes	characteristic
115	C₂²	#16812657	1	1	yes	characteristic
116	C₂²	C₂×C₈×D₁₀₉	1	2	yes	—
117	C₂²	M₁₆×D₁₀₉	1	2	yes	—
118	C₄	—	2	4	no	—
119	C₄	—	109	4	no	—
120	C₄	—	109	4	no	—
121	C₄	—	218	4	no	—
122	C₄	#16812655	1	2	yes	—
123	C₄	#16812657	1	2	yes	—
124	C₂	#20294871	1	1	yes	characteristic
125	C₂	C₄×C₈×D₁₀₉	1	1	yes	characteristic
126	C₂	#97×D₁₀₉	1	1	yes	characteristic
127	C₂	#20295076	1	2	yes	—
128	C₂	#20295078	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₀₉

C₂

yes

maximal

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294781

C₂

yes

maximal, characteristic

#20294780

C₂

yes

maximal

#336×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₈

yes

complemented

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

C₂²×C₈

yes

—

C₂²×C₁₀₉

C₂×M₁₆

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#117

yes

—

C₄×C₁₀₉

#117

yes

—

Dic₁₀₉

—

Dic₁₀₉

C₄×C₈

yes

—

Dic₁₀₉

#97

yes

—

C₂×C₁₀₉

#376

yes

characteristic

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

C₂×#97

yes

characteristic

C₂×C₁₀₉

#431

yes

—

C₂×C₁₀₉

#431

yes

—

#1744

—

109

maximal, Sylow

C₁₀₉

#1744

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

109

—

C₄³

—

109

—

#336

—

109

—

#336

—

109

—

#336

D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₄²

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₄²

—

218

—

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

C₈×D₁₀₉

yes

—

107

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

108

C₂×C₄

#13182028

yes

characteristic

109

C₂×C₄

#13182028

yes

—

110

C₂×C₄

#13182030

yes

characteristic

111

C₈

—

112

C₈

—

218

—

113

C₂²

C₄²×D₁₀₉

yes

characteristic

114

C₂²

#16812655

yes

characteristic

115

C₂²

#16812657

yes

characteristic

116

C₂²

C₂×C₈×D₁₀₉

yes

—

117

C₂²

M₁₆×D₁₀₉

yes

—

118

C₄

—

119

C₄

—

109

—

120

C₄

—

109

—

121

C₄

—

218

—

122

C₄

#16812655

yes

—

123

C₄

#16812657

yes

—

124

C₂

#20294871

yes

characteristic

125

C₂

C₄×C₈×D₁₀₉

yes

characteristic

126

C₂

#97×D₁₀₉

yes

characteristic

127

C₂

#20295076

yes

—

128

C₂

#20295078

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Fitting	#336×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#336	D₁₀₉
hypocenter	C₁₀₉	#1744
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₉

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Fitting

#336×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. C₈×Dic₁₀₉	C₄	semidirect
23. C₄×Dic₁₀₉	C₈	semidirect
65. C₁₀₉	#1744	semidirect

H ≅ G/N

type

11. C₈×Dic₁₀₉

C₄

semidirect

23. C₄×Dic₁₀₉

C₈

semidirect

65. C₁₀₉

#1744

semidirect

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

subgroup	index
C₂×C₈×Dic₁₀₉	2
C₄²×Dic₁₀₉	2
#20294781	2
#20294780	2
#336×C₁₀₉	2
#1744	109

subgroup

index

C₂×C₈×Dic₁₀₉

C₄²×Dic₁₀₉

#20294781

#20294780

#336×C₁₀₉

#1744

109

Subgroups — #28362776

Conjugacy classes of subgroups

Complements

Maximal subgroups