Group #28362884 — subgroups

Subgroups — #28362884

Order 13952.

292 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16812628	C₂	1	2	yes	maximal
2	C₂×#16812609	C₂	1	2	yes	maximal
3	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294781	C₂	1	1	yes	maximal, characteristic
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	#16812628	C₄	1	8	yes	complemented
12	#16812609	—	2	4	no	—
13	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	—	2	2	no	—
19	C₂×#11724784	—	2	2	no	—
20	C₂×#11724784	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₀₉	—	2	4	no	—
22	C₄×Dic₁₀₉	—	2	4	no	—
23	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
24	C₄×Dic₁₀₉	D₄	1	2	yes	—
25	C₄×Dic₁₀₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
31	C₄²×C₁₀₉	—	2	2	no	—
32	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	M₁₆	1	4	yes	—
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	#41	1	2	yes	—
40	C₂×Dic₁₀₉	—	2	2	no	—
41	C₂×Dic₁₀₉	—	2	2	no	—
42	C₂×Dic₁₀₉	—	2	2	no	—
43	C₂×Dic₁₀₉	C₄²	1	2	yes	—
44	C₂×Dic₁₀₉	C₄²	1	2	yes	—
45	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
46	C₂×Dic₁₀₉	M₁₆	1	4	yes	—
47	C₈×C₁₀₉	—	2	4	no	—
48	#11724784	—	2	4	no	—
49	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
50	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₀₉	#117	1	1	yes	characteristic
52	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
53	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
54	C₄×C₁₀₉	—	2	4	no	—
55	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
56	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
57	Dic₁₀₉	—	2	2	no	—
58	Dic₁₀₉	—	2	4	no	—
59	Dic₁₀₉	#97	1	4	yes	complemented
60	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₁₀₉	#423	1	2	yes	—
64	C₂×C₁₀₉	#432	1	2	yes	—
65	#1767	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1767	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄⋊C₈	—	109	2	no	—
68	C₂×#97	—	109	2	no	—
69	C₄³	—	109	1	no	—
70	#336	—	109	1	no	—
71	#336	D₁₀₉	1	1	yes	characteristic
72	C₂²×C₈	—	109	2	no	—
73	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
74	C₂×C₄²	—	109	1	no	—
75	C₂×C₄²	—	109	1	no	—
76	C₂×C₄²	—	109	2	no	—
77	C₂×C₄²	—	109	2	no	—
78	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
79	C₄⋊C₈	—	109	8	no	—
80	#97	—	218	4	no	—
81	C₂²×C₄	—	109	1	no	—
82	C₂²×C₄	—	109	1	no	—
83	C₂²×C₄	—	109	2	no	—
84	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	109	4	no	—
89	C₂×C₈	—	218	2	no	—
90	C₂×C₈	—	218	2	no	—
91	C₂×C₈	C₄×D₁₀₉	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	109	4	no	—
94	C₄²	—	109	2	no	—
95	C₄²	—	109	2	no	—
96	C₄²	—	218	4	no	—
97	C₄²	—	218	4	no	—
98	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	109	2	no	—
102	C₂×C₄	—	109	4	no	—
103	C₂×C₄	—	109	2	no	—
104	C₂×C₄	—	109	4	no	—
105	C₂×C₄	—	218	2	no	—
106	C₂×C₄	—	218	2	no	—
107	C₂×C₄	—	218	2	no	—
108	C₂×C₄	#13182012	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
110	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
112	C₂×C₄	#13182028	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	218	4	no	—
115	C₂²	C₄²×D₁₀₉	1	1	yes	characteristic
116	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
117	C₂²	#16812655	1	1	yes	characteristic
118	C₂²	M₁₆×D₁₀₉	1	2	yes	—
119	C₂²	C₂×#13182012	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	109	4	no	—
122	C₄	—	218	2	no	—
123	C₄	—	218	4	no	—
124	C₄	#16812659	1	2	yes	—
125	C₄	#16812664	1	2	yes	—
126	C₂	#20294877	1	1	yes	characteristic
127	C₂	#97×D₁₀₉	1	1	yes	characteristic
128	C₂	C₄×#13182012	1	1	yes	characteristic
129	C₂	#20295048	1	2	yes	—
130	C₂	#20295080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812628

C₂

yes

maximal

C₂×#16812609

C₂

yes

maximal

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294781

C₂

yes

maximal, characteristic

#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

#16812628

C₄

yes

complemented

#16812609

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

Q₈

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

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#117

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

—

C₂²×C₁₀₉

C₂×M₁₆

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

C₄×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#97

yes

complemented

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×#97

yes

characteristic

C₂×C₁₀₉

C₄×M₁₆

yes

characteristic

C₂×C₁₀₉

#423

yes

—

C₂×C₁₀₉

#432

yes

—

#1767

—

109

maximal, Sylow

C₁₀₉

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×#97

—

109

—

C₄³

—

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

—

#97

—

218

—

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

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

218

—

106

C₂×C₄

—

218

—

107

C₂×C₄

—

218

—

108

C₂×C₄

#13182012

yes

—

109

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

110

C₂×C₄

D₄×D₁₀₉

yes

characteristic

111

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

112

C₂×C₄

#13182028

yes

—

113

C₈

—

114

C₈

—

218

—

115

C₂²

C₄²×D₁₀₉

yes

characteristic

116

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

117

C₂²

#16812655

yes

characteristic

118

C₂²

M₁₆×D₁₀₉

yes

—

119

C₂²

C₂×#13182012

yes

—

120

C₄

—

121

C₄

—

109

—

122

C₄

—

218

—

123

C₄

—

218

—

124

C₄

#16812659

yes

—

125

C₄

#16812664

yes

—

126

C₂

#20294877

yes

characteristic

127

C₂

#97×D₁₀₉

yes

characteristic

128

C₂

C₄×#13182012

yes

characteristic

129

C₂

#20295048

yes

—

130

C₂

#20295080

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂×C₄²
Fitting	#336×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#336	D₁₀₉
hypocenter	C₁₀₉	#1767
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂×C₄²

Fitting

#336×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. #16812628	C₄	semidirect
59. Dic₁₀₉	#97	semidirect
66. C₁₀₉	#1767	semidirect

H ≅ G/N

type

11. #16812628

C₄

semidirect

59. Dic₁₀₉

#97

semidirect

66. C₁₀₉

#1767

semidirect

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

subgroup	index
C₂×#16812628	2
C₂×#16812609	2
C₄²×Dic₁₀₉	2
#20294781	2
#336×C₁₀₉	2
#1767	109

subgroup

index

C₄²×Dic₁₀₉

#336×C₁₀₉

109

Subgroups — #28362884

Conjugacy classes of subgroups

Complements

Maximal subgroups