Group #28361333 — subgroups

Subgroups — #28361333

Order 13952.

178 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20295077	C₂	1	1	yes	maximal, characteristic
2	#20295077	C₂	1	1	yes	maximal, characteristic
3	#20295077	C₂	1	1	yes	maximal, characteristic
4	#20294748	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294747	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
7	#322×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
9	#16812628	C₄	1	2	yes	—
10	#16812628	C₄	1	2	yes	—
11	#16812628	C₄	1	2	yes	—
12	#16812611	C₄	1	2	yes	—
13	#16812611	C₄	1	2	yes	—
14	#16812611	C₄	1	2	yes	—
15	C₄×#11724784	C₂²	1	1	yes	characteristic
16	C₄×#11724784	C₂²	1	1	yes	characteristic
17	C₄×#11724784	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×#11724784	C₂×C₄	1	2	yes	—
25	C₂×#11724784	C₂×C₄	1	2	yes	—
26	C₂×#11724784	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
41	C₄×D₁₀₉	—	2	2	no	—
42	C₄×D₁₀₉	—	2	2	no	—
43	C₄×D₁₀₉	—	2	2	no	—
44	C₈×C₁₀₉	—	2	2	no	—
45	C₈×C₁₀₉	—	2	2	no	—
46	C₈×C₁₀₉	—	2	2	no	—
47	#11724784	—	2	2	no	—
48	#11724784	—	2	2	no	—
49	#11724784	—	2	2	no	—
50	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
51	C₂×D₁₀₉	—	2	1	no	—
52	C₂×D₁₀₉	—	2	1	no	—
53	C₂×D₁₀₉	—	2	1	no	—
54	C₄×C₁₀₉	#131	1	2	yes	—
55	C₄×C₁₀₉	#131	1	2	yes	—
56	C₄×C₁₀₉	#131	1	2	yes	—
57	Dic₁₀₉	—	2	1	no	—
58	Dic₁₀₉	—	2	1	no	—
59	Dic₁₀₉	—	2	1	no	—
60	C₂×C₁₀₉	#405	1	1	yes	characteristic
61	C₂×C₁₀₉	#405	1	1	yes	characteristic
62	C₂×C₁₀₉	#405	1	1	yes	characteristic
63	D₁₀₉	—	4	1	no	—
64	#1454	—	109	1	no	maximal, Sylow
65	C₁₀₉	#1454	1	1	yes	characteristic, complemented, Sylow
66	#431	—	109	1	no	—
67	#431	—	109	1	no	—
68	#431	—	109	1	no	—
69	#322	—	109	1	no	—
70	#322	—	109	1	no	—
71	#322	D₁₀₉	1	1	yes	characteristic, complemented
72	C₈²	—	109	1	no	—
73	C₂×C₄²	—	109	1	no	—
74	C₄⋊C₈	—	109	2	no	—
75	C₄⋊C₈	—	109	2	no	—
76	C₄⋊C₈	—	109	2	no	—
77	C₂²⋊C₈	—	109	2	no	—
78	C₂²⋊C₈	—	109	2	no	—
79	C₂²⋊C₈	—	109	2	no	—
80	C₄×C₈	—	109	1	no	—
81	C₄×C₈	—	109	1	no	—
82	C₄×C₈	—	109	1	no	—
83	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
84	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
85	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
86	C₂²×C₄	—	109	1	no	—
87	C₂²×C₄	—	109	1	no	—
88	C₂²×C₄	—	109	1	no	—
89	C₂×C₈	—	109	2	no	—
90	C₂×C₈	—	109	2	no	—
91	C₂×C₈	—	109	2	no	—
92	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
93	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
94	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
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	1	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	—	218	2	no	—
105	C₂×C₄	—	218	2	no	—
106	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	218	2	no	—
113	C₈	—	218	2	no	—
114	C₈	—	218	2	no	—
115	C₂²	—	218	1	no	—
116	C₂²	—	218	1	no	—
117	C₂²	—	218	1	no	—
118	C₂²	C₄²×D₁₀₉	1	1	yes	characteristic
119	C₄	—	218	1	no	—
120	C₄	—	218	1	no	—
121	C₄	—	218	1	no	—
122	C₄	#16812705	1	2	yes	—
123	C₄	#16812707	1	2	yes	—
124	C₄	#16812707	1	2	yes	—
125	C₂	—	436	1	no	—
126	C₂	#20294980	1	1	yes	characteristic
127	C₂	#20294983	1	1	yes	characteristic
128	C₂	#20294983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295077

C₂

yes

maximal, characteristic

#20295077

C₂

yes

maximal, characteristic

#20295077

C₂

yes

maximal, characteristic

#20294748

C₂

yes

maximal, characteristic, complemented

#20294747

C₂

yes

maximal, characteristic, complemented

C₈×#11724784

C₂

yes

maximal, characteristic, complemented

#322×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812628

C₄

yes

—

#16812628

C₄

yes

—

#16812628

C₄

yes

—

#16812611

C₄

yes

—

#16812611

C₄

yes

—

#16812611

C₄

yes

—

C₄×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₄²

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

#131

yes

—

C₄×C₁₀₉

#131

yes

—

C₄×C₁₀₉

#131

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1454

—

109

maximal, Sylow

C₁₀₉

#1454

yes

characteristic, complemented, Sylow

#431

—

109

—

#431

—

109

—

#431

—

109

—

#322

—

109

—

#322

—

109

—

#322

D₁₀₉

yes

characteristic, complemented

C₈²

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

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

—

218

—

104

C₂×C₄

—

218

—

105

C₂×C₄

—

218

—

106

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

107

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

108

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

218

—

113

C₈

—

218

—

114

C₈

—

218

—

115

C₂²

—

218

—

116

C₂²

—

218

—

117

C₂²

—

218

—

118

C₂²

C₄²×D₁₀₉

yes

characteristic

119

C₄

—

218

—

120

C₄

—

218

—

121

C₄

—

218

—

122

C₄

#16812705

yes

—

123

C₄

#16812707

yes

—

124

C₄

#16812707

yes

—

125

C₂

—

436

—

126

C₂

#20294980

yes

characteristic

127

C₂

#20294983

yes

characteristic

128

C₂

#20294983

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂×C₄²
Fitting	#322×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₄²
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#322	D₁₀₉
hypocenter	C₁₀₉	#1454
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂×C₄²

Fitting

#322×C₁₀₉

C₂

prime radical

C₂²×D₁₀₉

C₄²

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #20294748	C₂	semidirect
5. #20294747	C₂	semidirect
6. C₈×#11724784	C₂	semidirect
7. #322×C₁₀₉	C₂	semidirect
65. C₁₀₉	#1454	semidirect
71. #322	D₁₀₉	semidirect

H ≅ G/N

type

4. #20294748

C₂

semidirect

5. #20294747

C₂

semidirect

6. C₈×#11724784

C₂

semidirect

7. #322×C₁₀₉

C₂

semidirect

65. C₁₀₉

#1454

semidirect

71. #322

D₁₀₉

semidirect

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

subgroup	index
#20295077	2
#20294748	2
#20294747	2
C₈×#11724784	2
#322×C₁₀₉	2
#1454	109

subgroup

index

#322×C₁₀₉

109

Subgroups — #28361333

Conjugacy classes of subgroups

Complements

Maximal subgroups