Group #28366335 — subgroups

Subgroups — #28366335

Order 13952.

432 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20294967	C₂	1	3	yes	maximal, complemented
2	#20294955	C₂	1	3	yes	maximal
3	#20294950	C₂	1	1	yes	maximal, characteristic
4	#20294877	C₂	1	3	yes	maximal, complemented
5	#20294873	C₂	1	3	yes	maximal
6	#20294871	C₂	1	1	yes	maximal, characteristic, complemented
7	#401×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#13182008	C₂²	1	3	yes	—
10	C₂×#13182008	C₂²	1	3	yes	—
11	C₂×#13182006	C₂²	1	3	yes	—
12	C₂×#13182006	C₂²	1	3	yes	—
13	C₂×C₄×Dic₁₀₉	C₂²	1	3	yes	—
14	C₂×C₄×Dic₁₀₉	C₂²	1	3	yes	—
15	#16812606	C₂²	1	3	yes	—
16	#16812606	C₂²	1	3	yes	—
17	#16812606	C₂²	1	3	yes	—
18	#95×C₁₀₉	C₂²	1	1	yes	characteristic
19	#95×C₁₀₉	C₂²	1	3	yes	—
20	#95×C₁₀₉	C₂²	1	3	yes	—
21	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
23	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
24	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×C₄×D₁₀₉	—	2	6	no	—
26	#13182008	—	2	6	no	—
27	#13182008	—	2	6	no	—
28	#13182006	—	2	6	no	—
29	#13182006	—	2	6	no	—
30	C₄×Dic₁₀₉	—	2	6	no	—
31	C₄×Dic₁₀₉	—	2	6	no	—
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
34	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
35	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
36	C₂²×D₁₀₉	—	2	3	no	—
37	C₂²×D₁₀₉	—	2	3	no	—
38	C₂²×D₁₀₉	#41	1	2	yes	—
39	C₂×C₄×C₁₀₉	—	2	3	no	—
40	C₂×C₄×C₁₀₉	—	2	3	no	—
41	C₂×C₄×C₁₀₉	—	2	3	no	—
42	C₂×C₄×C₁₀₉	—	2	3	no	—
43	C₂×C₄×C₁₀₉	—	2	3	no	—
44	C₂×C₄×C₁₀₉	—	2	3	no	—
45	C₂×C₄×C₁₀₉	—	2	3	no	—
46	C₂×Dic₁₀₉	—	2	3	no	—
47	C₂×Dic₁₀₉	—	2	3	no	—
48	C₂×Dic₁₀₉	—	2	3	no	—
49	C₂×Dic₁₀₉	—	2	3	no	—
50	C₂×Dic₁₀₉	—	2	3	no	—
51	C₂×Dic₁₀₉	#41	1	6	yes	—
52	C₂×Dic₁₀₉	#41	1	6	yes	—
53	C₄×D₁₀₉	—	4	4	no	—
54	C₂²×C₁₀₉	C₂×#41	1	3	yes	—
55	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
56	C₂²×C₁₀₉	C₂×#41	1	3	yes	—
57	C₂×D₁₀₉	—	2	6	no	—
58	C₂×D₁₀₉	—	4	1	no	—
59	C₂×D₁₀₉	—	4	3	no	—
60	C₄×C₁₀₉	—	4	1	no	—
61	C₄×C₁₀₉	—	4	3	no	—
62	C₄×C₁₀₉	—	4	3	no	—
63	Dic₁₀₉	—	2	6	no	—
64	Dic₁₀₉	—	4	3	no	—
65	Dic₁₀₉	#126	1	4	yes	—
66	C₂×C₁₀₉	C₂×#126	1	1	yes	characteristic
67	C₂×C₁₀₉	#529	1	3	yes	—
68	C₂×C₁₀₉	#529	1	3	yes	—
69	D₁₀₉	—	4	2	no	—
70	#2401	—	109	1	no	maximal, Sylow
71	C₁₀₉	#2401	1	1	yes	characteristic, complemented, Sylow
72	#401	D₁₀₉	1	1	yes	characteristic, complemented
73	#400	—	109	3	no	—
74	#397	—	109	3	no	—
75	#396	—	109	1	no	—
76	C₄×C₄⋊C₄	—	109	3	no	—
77	C₄×C₂²⋊C₄	—	109	3	no	—
78	#376	—	109	1	no	—
79	C₂³×C₄	—	109	1	no	—
80	C₂×C₄⋊C₄	—	109	3	no	—
81	C₂×C₄⋊C₄	—	109	3	no	—
82	C₂×C₂²⋊C₄	—	109	3	no	—
83	C₂×C₂²⋊C₄	—	109	3	no	—
84	C₂×C₄²	—	109	3	no	—
85	C₂×C₄²	—	109	3	no	—
86	#95	—	109	3	no	—
87	#95	—	109	3	no	—
88	#95	—	109	3	no	—
89	#95	C₂×D₁₀₉	1	1	yes	characteristic
90	#95	C₂×D₁₀₉	1	3	yes	—
91	#95	C₂×D₁₀₉	1	3	yes	—
92	C₂⁴	—	109	1	no	—
93	C₂²×C₄	—	109	1	no	—
94	C₂²×C₄	—	109	3	no	—
95	C₂²×C₄	—	109	3	no	—
96	C₂²×C₄	—	218	6	no	—
97	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
99	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
100	C₄⋊C₄	—	218	6	no	—
101	C₄⋊C₄	—	218	6	no	—
102	C₂²⋊C₄	—	218	6	no	—
103	C₂²⋊C₄	—	218	6	no	—
104	C₄²	—	218	6	no	—
105	C₄²	—	218	6	no	—
106	C₂³	—	109	2	no	—
107	C₂³	—	218	3	no	—
108	C₂³	—	218	3	no	—
109	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
110	C₂×C₄	—	2	3	no	—
111	C₂×C₄	—	2	3	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	2	3	no	—
116	C₂×C₄	—	2	3	no	—
117	C₂×C₄	—	109	6	no	—
118	C₂×C₄	—	109	6	no	—
119	C₂×C₄	—	436	4	no	—
120	C₂×C₄	—	218	3	no	—
121	C₂×C₄	—	218	3	no	—
122	C₂×C₄	—	218	3	no	—
123	C₂×C₄	—	218	3	no	—
124	C₂×C₄	—	218	3	no	—
125	C₂²	—	436	1	no	—
126	C₂²	—	436	3	no	—
127	C₂²	—	218	6	no	—
128	C₂²	#41×D₁₀₉	1	3	yes	—
129	C₂²	#41×D₁₀₉	1	3	yes	—
130	C₂²	#41×D₁₀₉	1	1	yes	characteristic
131	C₄	—	4	1	no	—
132	C₄	—	4	3	no	—
133	C₄	—	4	3	no	—
134	C₄	—	109	4	no	—
135	C₄	—	436	3	no	—
136	C₄	—	218	6	no	—
137	C₂	—	436	2	no	—
138	C₂	#126×D₁₀₉	1	1	yes	characteristic
139	C₂	#20295559	1	3	yes	—
140	C₂	#20295559	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20294967

C₂

yes

maximal, complemented

#20294955

C₂

yes

maximal

#20294950

C₂

yes

maximal, characteristic

#20294877

C₂

yes

maximal, complemented

#20294873

C₂

yes

maximal

#20294871

C₂

yes

maximal, characteristic, complemented

#401×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×#13182008

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

#16812606

C₂²

yes

—

#16812606

C₂²

yes

—

#16812606

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

characteristic

#95×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

—

C₂³×D₁₀₉

C₂³

yes

characteristic

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₄×D₁₀₉

—

—

—

—

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

#41

yes

—

C₂×Dic₁₀₉

#41

yes

—

C₄×D₁₀₉

—

C₂²×C₁₀₉

C₂×#41

yes

—

C₂²×C₁₀₉

C₂×#41

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#126

yes

—

C₂×C₁₀₉

C₂×#126

yes

characteristic

C₂×C₁₀₉

#529

yes

—

C₂×C₁₀₉

#529

yes

—

D₁₀₉

—

#2401

—

109

maximal, Sylow

C₁₀₉

#2401

yes

characteristic, complemented, Sylow

#401

D₁₀₉

yes

characteristic, complemented

#400

—

109

—

#397

—

109

—

#396

—

109

—

C₄×C₄⋊C₄

—

109

—

C₄×C₂²⋊C₄

—

109

—

#376

—

109

—

C₂³×C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

#95

—

109

—

#95

—

109

—

#95

—

109

—

#95

C₂×D₁₀₉

yes

characteristic

#95

C₂×D₁₀₉

yes

—

#95

C₂×D₁₀₉

yes

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

100

C₄⋊C₄

—

218

—

101

C₄⋊C₄

—

218

—

102

C₂²⋊C₄

—

218

—

103

C₂²⋊C₄

—

218

—

104

C₄²

—

218

—

105

C₄²

—

218

—

106

C₂³

—

109

—

107

C₂³

—

218

—

108

C₂³

—

218

—

109

C₂³

C₂³×D₁₀₉

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

109

—

118

C₂×C₄

—

109

—

119

C₂×C₄

—

436

—

120

C₂×C₄

—

218

—

121

C₂×C₄

—

218

—

122

C₂×C₄

—

218

—

123

C₂×C₄

—

218

—

124

C₂×C₄

—

218

—

125

C₂²

—

436

—

126

C₂²

—

436

—

127

C₂²

—

218

—

128

C₂²

#41×D₁₀₉

yes

—

129

C₂²

#41×D₁₀₉

yes

—

130

C₂²

#41×D₁₀₉

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

109

—

135

C₄

—

436

—

136

C₄

—

218

—

137

C₂

—

436

—

138

C₂

#126×D₁₀₉

yes

characteristic

139

C₂

#20295559

yes

—

140

C₂

#20295559

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	#401×C₁₀₉	C₂
prime radical	C₂³×D₁₀₉	C₂³
hypercenter	#401	D₁₀₉
hypocenter	C₁₀₉	#2401
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #401

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

#401×C₁₀₉

C₂

prime radical

C₂³×D₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ #401

N	H ≅ G/N	type
1. #20294967	C₂	semidirect
4. #20294877	C₂	semidirect
6. #20294871	C₂	semidirect
7. #401×C₁₀₉	C₂	semidirect
71. C₁₀₉	#2401	semidirect
72. #401	D₁₀₉	semidirect

H ≅ G/N

type

1. #20294967

C₂

semidirect

4. #20294877

C₂

semidirect

6. #20294871

C₂

semidirect

7. #401×C₁₀₉

C₂

semidirect

71. C₁₀₉

#2401

semidirect

72. #401

D₁₀₉

semidirect

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

subgroup	index
#20294967	2
#20294955	2
#20294950	2
#20294877	2
#20294873	2
#20294871	2
#401×C₁₀₉	2
#2401	109

subgroup

index

#401×C₁₀₉

109

Subgroups — #28366335

Conjugacy classes of subgroups

Complements

Maximal subgroups