Group #28371284 — subgroups

Subgroups — #28371284

Order 13952.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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	#20294939	C₂	1	6	yes	maximal, complemented
2	#20294938	C₂	1	3	yes	maximal, complemented
3	#20294937	C₂	1	2	yes	maximal, complemented
4	#20294881	C₂	1	3	yes	maximal
5	#393×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₁₈	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊Dic₁₀₉	C₂²	1	3	yes	—
9	C₂×#13182006	C₂²	1	6	yes	—
10	C₂×#13182006	C₂²	1	6	yes	—
11	#16812606	C₂²	1	6	yes	complemented
12	#16812606	C₂²	1	3	yes	complemented
13	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	3	yes	—
15	#95×C₁₀₉	C₂²	1	3	yes	complemented
16	C₂³×D₁₀₉	—	4	3	no	—
17	C₂³×D₁₀₉	—	4	3	no	—
18	C₂³×D₁₀₉	Q₈	1	4	yes	complemented
19	C₂³×D₁₀₉	C₂³	1	2	yes	—
20	C₂×Dic₂₁₈	—	4	3	no	—
21	C₂×Dic₂₁₈	—	4	6	no	—
22	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
23	C₂²⋊Dic₁₀₉	—	4	3	no	—
24	#13182006	—	4	6	no	—
25	#13182006	—	4	6	no	—
26	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₁₀₉	—	4	3	no	—
28	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
29	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
30	C₂²⋊C₄×C₁₀₉	—	4	3	no	—
31	C₂³×C₁₀₉	—	2	3	no	—
32	C₂³×C₁₀₉	—	2	3	no	—
33	C₂³×C₁₀₉	C₂×Q₈	1	2	yes	—
34	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
35	C₂²×D₁₀₉	—	2	6	no	—
36	C₂²×D₁₀₉	—	2	6	no	—
37	C₂²×D₁₀₉	—	4	12	no	—
38	C₂²×D₁₀₉	—	4	12	no	—
39	C₂²×D₁₀₉	—	4	4	no	—
40	C₂²×D₁₀₉	C₂×Q₈	1	4	yes	—
41	Q₈×C₁₀₉	—	4	4	no	—
42	C₂×C₄×C₁₀₉	—	2	3	no	—
43	C₂×C₄×C₁₀₉	—	2	3	no	—
44	C₂×C₄×C₁₀₉	—	2	6	no	—
45	C₂×C₄×C₁₀₉	—	2	6	no	—
46	C₂×Dic₁₀₉	—	2	6	no	—
47	C₂×Dic₁₀₉	—	2	6	no	—
48	C₂×Dic₁₀₉	—	2	6	no	—
49	Dic₂₁₈	—	4	12	no	—
50	C₂²×C₁₀₉	—	2	6	no	—
51	C₂²×C₁₀₉	—	4	3	no	—
52	C₂²×C₁₀₉	—	4	1	no	—
53	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
54	C₂²×C₁₀₉	H(5,𝔽₂)	1	3	yes	—
55	C₂²×C₁₀₉	H(5,𝔽₂)	1	3	yes	—
56	C₂×D₁₀₉	—	2	12	no	—
57	C₂×D₁₀₉	—	4	16	no	—
58	C₂×D₁₀₉	—	4	6	no	—
59	C₂×D₁₀₉	—	4	2	no	—
60	C₄×C₁₀₉	—	4	3	no	—
61	C₄×C₁₀₉	—	4	3	no	—
62	Dic₁₀₉	—	4	6	no	—
63	C₂×C₁₀₉	—	4	2	no	—
64	C₂×C₁₀₉	#543	1	3	yes	—
65	C₂×C₁₀₉	H(3,𝔽₄)	1	3	yes	—
66	C₂×C₁₀₉	H(3,𝔽₄)	1	1	yes	characteristic
67	D₁₀₉	—	4	4	no	—
68	#2840	—	109	1	no	maximal, Sylow
69	C₁₀₉	#2840	1	1	yes	characteristic, complemented, Sylow
70	#393	—	109	2	no	—
71	#393	—	109	3	no	—
72	#393	—	109	6	no	—
73	#393	D₁₀₉	1	1	yes	characteristic, complemented
74	#379	—	109	3	no	—
75	C₂⁵	—	109	1	no	—
76	C₂²×Q₈	—	109	3	no	—
77	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊C₄	—	109	3	no	—
79	C₂×C₂²⋊C₄	—	109	6	no	—
80	C₂×C₂²⋊C₄	—	109	6	no	—
81	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	3	yes	—
82	#95	—	109	6	no	—
83	#95	—	109	3	no	—
84	#95	C₂×D₁₀₉	1	3	yes	complemented
85	C₂⁴	—	109	4	no	—
86	C₂⁴	—	109	2	no	—
87	C₂⁴	—	436	3	no	—
88	C₂⁴	—	436	3	no	—
89	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
90	C₂×Q₈	—	4	3	no	—
91	C₂×Q₈	—	436	3	no	—
92	C₂×Q₈	—	436	6	no	—
93	C₂²×C₄	—	109	6	no	—
94	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
95	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
96	C₂²⋊C₄	—	4	3	no	—
97	C₂²⋊C₄	—	436	6	no	—
98	C₂²⋊C₄	—	436	6	no	—
99	C₂²⋊C₄	—	436	3	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	2	3	no	—
102	C₂³	—	109	4	no	—
103	C₂³	—	436	12	no	—
104	C₂³	—	436	12	no	—
105	C₂³	—	436	4	no	—
106	C₂³	—	218	6	no	—
107	C₂³	—	218	6	no	—
108	C₂³	Q₈×D₁₀₉	1	2	yes	—
109	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	436	12	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	218	6	no	—
117	C₂×C₄	—	218	6	no	—
118	C₂×C₄	—	218	6	no	—
119	C₂²	—	2	6	no	—
120	C₂²	—	4	3	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	436	16	no	—
123	C₂²	—	436	6	no	—
124	C₂²	—	436	2	no	—
125	C₂²	—	218	12	no	—
126	C₂²	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
127	C₂²	#16812744	1	3	yes	—
128	C₂²	#16812745	1	3	yes	—
129	C₄	—	4	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	436	6	no	—
132	C₂	—	4	2	no	—
133	C₂	—	436	4	no	—
134	C₂	#20295651	1	3	yes	—
135	C₂	#20295768	1	3	yes	—
136	C₂	#20295769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20294939

C₂

yes

maximal, complemented

#20294938

C₂

yes

maximal, complemented

#20294937

C₂

yes

maximal, complemented

#20294881

C₂

yes

maximal

#393×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₀₉

C₂²

yes

characteristic

C₂²×Dic₂₁₈

C₂²

yes

complemented

C₂×C₂²⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

#16812606

C₂²

yes

complemented

#16812606

C₂²

yes

complemented

C₂²×Q₈×C₁₀₉

C₂²

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

complemented

C₂³×D₁₀₉

—

C₂³×D₁₀₉

—

C₂³×D₁₀₉

Q₈

yes

complemented

C₂³×D₁₀₉

C₂³

yes

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²⋊Dic₁₀₉

—

#13182006

—

#13182006

—

C₂⁴×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×Q₈

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂×Q₈

yes

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₉

H(5,𝔽₂)

yes

—

C₂²×C₁₀₉

H(5,𝔽₂)

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#543

yes

—

C₂×C₁₀₉

H(3,𝔽₄)

yes

—

C₂×C₁₀₉

H(3,𝔽₄)

yes

characteristic

D₁₀₉

—

#2840

—

109

maximal, Sylow

C₁₀₉

#2840

yes

characteristic, complemented, Sylow

#393

—

109

—

#393

—

109

—

#393

—

109

—

#393

D₁₀₉

yes

characteristic, complemented

#379

—

109

—

C₂⁵

—

109

—

C₂²×Q₈

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

C₂×D₁₀₉

yes

—

#95

—

109

—

#95

—

109

—

#95

C₂×D₁₀₉

yes

complemented

C₂⁴

—

109

—

C₂⁴

—

109

—

C₂⁴

—

436

—

C₂⁴

—

436

—

C₂⁴

C₂²×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

436

—

C₂×Q₈

—

436

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

436

—

C₂²⋊C₄

—

436

—

C₂²⋊C₄

—

436

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

109

—

103

C₂³

—

436

—

104

C₂³

—

436

—

105

C₂³

—

436

—

106

C₂³

—

218

—

107

C₂³

—

218

—

108

C₂³

Q₈×D₁₀₉

yes

—

109

C₂³

C₂³×D₁₀₉

yes

characteristic

110

Q₈

—

111

Q₈

—

436

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

218

—

117

C₂×C₄

—

218

—

118

C₂×C₄

—

218

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

436

—

123

C₂²

—

436

—

124

C₂²

—

436

—

125

C₂²

—

218

—

126

C₂²

C₂×Q₈×D₁₀₉

yes

characteristic

127

C₂²

#16812744

yes

—

128

C₂²

#16812745

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

436

—

132

C₂

—

133

C₂

—

436

—

134

C₂

#20295651

yes

—

135

C₂

#20295768

yes

—

136

C₂

#20295769

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	#393×C₁₀₉	C₂
prime radical	C₂⁴×D₁₀₉	C₂²
hypercenter	#393	D₁₀₉
hypocenter	C₁₀₉	#2840
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

#393×C₁₀₉

C₂

prime radical

C₂⁴×D₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
1. #20294939	C₂	semidirect
2. #20294938	C₂	semidirect
3. #20294937	C₂	semidirect
5. #393×C₁₀₉	C₂	semidirect
7. C₂²×Dic₂₁₈	C₂²	semidirect
11. #16812606	C₂²	semidirect
12. #16812606	C₂²	semidirect
13. C₂²×Q₈×C₁₀₉	C₂²	semidirect
15. #95×C₁₀₉	C₂²	semidirect
18. C₂³×D₁₀₉	Q₈	semidirect
69. C₁₀₉	#2840	semidirect
73. #393	D₁₀₉	semidirect
77. C₂²×Q₈	C₂×D₁₀₉	semidirect
84. #95	C₂×D₁₀₉	semidirect

H ≅ G/N

type

1. #20294939

C₂

semidirect

2. #20294938

C₂

semidirect

3. #20294937

C₂

semidirect

5. #393×C₁₀₉

C₂

semidirect

7. C₂²×Dic₂₁₈

C₂²

semidirect

11. #16812606

C₂²

semidirect

12. #16812606

C₂²

semidirect

13. C₂²×Q₈×C₁₀₉

C₂²

semidirect

15. #95×C₁₀₉

C₂²

semidirect

18. C₂³×D₁₀₉

Q₈

semidirect

69. C₁₀₉

#2840

semidirect

73. #393

D₁₀₉

semidirect

77. C₂²×Q₈

C₂×D₁₀₉

semidirect

84. #95

C₂×D₁₀₉

semidirect

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

subgroup	index
#20294939	2
#20294938	2
#20294937	2
#20294881	2
#393×C₁₀₉	2
#2840	109

subgroup

index

#393×C₁₀₉

109

Subgroups — #28371284

Conjugacy classes of subgroups

Complements

Maximal subgroups