Group #28371297 — subgroups

Subgroups — #28371297

Order 13952.

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

normal subgroups only

This group has 548 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	#20294962	C₂	1	3	yes	maximal, complemented
2	#20294950	C₂	1	2	yes	maximal, complemented
3	#20294948	C₂	1	6	yes	maximal, complemented
4	#20294881	C₂	1	3	yes	maximal
5	#399×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	complemented
8	C₂×#13182008	C₂²	1	6	yes	complemented
9	C₂×C₂²⋊Dic₁₀₉	C₂²	1	3	yes	—
10	C₂×#13182006	C₂²	1	6	yes	—
11	C₂×#13182006	C₂²	1	6	yes	—
12	#16812606	C₂²	1	3	yes	complemented
13	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	3	yes	complemented
14	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	3	yes	—
15	#95×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
16	C₂³×D₁₀₉	—	4	3	no	—
17	C₂³×D₁₀₉	—	4	3	no	—
18	C₂³×D₁₀₉	Q₈	1	4	yes	—
19	C₂³×D₁₀₉	C₂³	1	2	yes	—
20	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
21	C₄⋊Dic₁₀₉	—	4	3	no	—
22	#13182008	—	4	6	no	—
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	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
28	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
29	C₄⋊C₄×C₁₀₉	—	4	3	no	—
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	4	no	—
39	C₂²×D₁₀₉	—	4	12	no	—
40	C₂²×D₁₀₉	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₀₉	—	2	3	no	—
42	C₂×C₄×C₁₀₉	—	2	3	no	—
43	C₂×C₄×C₁₀₉	—	2	6	no	—
44	C₂×C₄×C₁₀₉	—	2	6	no	—
45	C₂×Dic₁₀₉	—	2	6	no	—
46	C₂×Dic₁₀₉	—	2	6	no	—
47	C₂×Dic₁₀₉	—	2	6	no	—
48	C₂²×C₁₀₉	—	2	6	no	—
49	C₂²×C₁₀₉	—	4	1	no	—
50	C₂²×C₁₀₉	—	4	3	no	—
51	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₁₀₉	H(5,𝔽₂)	1	3	yes	—
53	C₂²×C₁₀₉	H(5,𝔽₂)	1	3	yes	—
54	C₂×D₁₀₉	—	2	12	no	—
55	C₂×D₁₀₉	—	4	16	no	—
56	C₂×D₁₀₉	—	4	2	no	—
57	C₂×D₁₀₉	—	4	6	no	—
58	C₄×C₁₀₉	—	4	3	no	—
59	C₄×C₁₀₉	—	4	3	no	—
60	Dic₁₀₉	—	4	6	no	—
61	C₂×C₁₀₉	—	4	2	no	—
62	C₂×C₁₀₉	#543	1	3	yes	—
63	C₂×C₁₀₉	#560	1	1	yes	characteristic
64	C₂×C₁₀₉	#560	1	3	yes	—
65	D₁₀₉	—	4	4	no	—
66	#2842	—	109	1	no	maximal, Sylow
67	C₁₀₉	#2842	1	1	yes	characteristic, complemented, Sylow
68	#399	—	109	3	no	—
69	#399	D₁₀₉	1	1	yes	characteristic, complemented
70	#396	—	109	6	no	—
71	#396	—	109	2	no	—
72	#379	—	109	3	no	—
73	C₂⁵	—	109	1	no	—
74	C₂×C₄⋊C₄	—	109	6	no	—
75	C₂×C₄⋊C₄	—	109	3	no	—
76	C₂×C₄⋊C₄	C₂×D₁₀₉	1	3	yes	complemented
77	C₂×C₂²⋊C₄	—	109	3	no	—
78	C₂×C₂²⋊C₄	—	109	6	no	—
79	C₂×C₂²⋊C₄	—	109	6	no	—
80	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	3	yes	—
81	#95	—	109	3	no	—
82	#95	C₂×D₁₀₉	1	1	yes	characteristic, complemented
83	C₂⁴	—	109	4	no	—
84	C₂⁴	—	109	2	no	—
85	C₂⁴	—	436	3	no	—
86	C₂⁴	—	436	3	no	—
87	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
88	C₂²×C₄	—	109	6	no	—
89	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
90	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
91	C₄⋊C₄	—	4	3	no	—
92	C₄⋊C₄	—	436	6	no	—
93	C₄⋊C₄	—	436	3	no	—
94	C₂²⋊C₄	—	4	3	no	—
95	C₂²⋊C₄	—	436	6	no	—
96	C₂²⋊C₄	—	436	6	no	—
97	C₂²⋊C₄	—	436	3	no	—
98	C₂³	—	2	3	no	—
99	C₂³	—	2	3	no	—
100	C₂³	—	109	4	no	—
101	C₂³	—	436	12	no	—
102	C₂³	—	436	4	no	—
103	C₂³	—	436	12	no	—
104	C₂³	—	218	6	no	—
105	C₂³	—	218	6	no	—
106	C₂³	Q₈×D₁₀₉	1	2	yes	—
107	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	3	no	—
110	C₂×C₄	—	2	6	no	—
111	C₂×C₄	—	2	6	no	—
112	C₂×C₄	—	218	6	no	—
113	C₂×C₄	—	218	6	no	—
114	C₂×C₄	—	218	6	no	—
115	C₂²	—	2	6	no	—
116	C₂²	—	4	1	no	—
117	C₂²	—	4	3	no	—
118	C₂²	—	436	16	no	—
119	C₂²	—	436	2	no	—
120	C₂²	—	436	6	no	—
121	C₂²	—	218	12	no	—
122	C₂²	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
123	C₂²	#16812744	1	3	yes	—
124	C₂²	#16812745	1	3	yes	—
125	C₄	—	4	3	no	—
126	C₄	—	4	3	no	—
127	C₄	—	436	6	no	—
128	C₂	—	4	2	no	—
129	C₂	—	436	4	no	—
130	C₂	#20295651	1	3	yes	—
131	C₂	#20295763	1	1	yes	characteristic
132	C₂	#20295766	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20294962

C₂

yes

maximal, complemented

#20294950

C₂

yes

maximal, complemented

#20294948

C₂

yes

maximal, complemented

#20294881

C₂

yes

maximal

#399×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

complemented

C₂×#13182008

C₂²

yes

complemented

C₂×C₂²⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

#16812606

C₂²

yes

complemented

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

complemented

C₂×C₂²⋊C₄×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

characteristic, complemented

C₂³×D₁₀₉

—

C₂³×D₁₀₉

—

C₂³×D₁₀₉

Q₈

yes

—

C₂³×D₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

—

#13182008

—

C₂²⋊Dic₁₀₉

—

#13182006

—

#13182006

—

C₂⁴×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

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

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

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

#560

yes

characteristic

C₂×C₁₀₉

#560

yes

—

D₁₀₉

—

#2842

—

109

maximal, Sylow

C₁₀₉

#2842

yes

characteristic, complemented, Sylow

#399

—

109

—

#399

D₁₀₉

yes

characteristic, complemented

#396

—

109

—

#396

—

109

—

#379

—

109

—

C₂⁵

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

complemented

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

C₂×D₁₀₉

yes

—

#95

—

109

—

#95

C₂×D₁₀₉

yes

characteristic, complemented

C₂⁴

—

109

—

C₂⁴

—

109

—

C₂⁴

—

436

—

C₂⁴

—

436

—

C₂⁴

C₂²×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₂²⋊C₄

—

C₂²⋊C₄

—

436

—

C₂²⋊C₄

—

436

—

C₂²⋊C₄

—

436

—

C₂³

—

C₂³

—

100

C₂³

—

109

—

101

C₂³

—

436

—

102

C₂³

—

436

—

103

C₂³

—

436

—

104

C₂³

—

218

—

105

C₂³

—

218

—

106

C₂³

Q₈×D₁₀₉

yes

—

107

C₂³

C₂³×D₁₀₉

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

218

—

113

C₂×C₄

—

218

—

114

C₂×C₄

—

218

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

436

—

119

C₂²

—

436

—

120

C₂²

—

436

—

121

C₂²

—

218

—

122

C₂²

C₂×Q₈×D₁₀₉

yes

characteristic

123

C₂²

#16812744

yes

—

124

C₂²

#16812745

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

436

—

128

C₂

—

129

C₂

—

436

—

130

C₂

#20295651

yes

—

131

C₂

#20295763

yes

characteristic

132

C₂

#20295766

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	#399×C₁₀₉	C₂
prime radical	C₂⁴×D₁₀₉	C₂²
hypercenter	#399	D₁₀₉
hypocenter	C₁₀₉	#2842
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

#399×C₁₀₉

C₂

prime radical

C₂⁴×D₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
1. #20294962	C₂	semidirect
2. #20294950	C₂	semidirect
3. #20294948	C₂	semidirect
5. #399×C₁₀₉	C₂	semidirect
7. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
8. C₂×#13182008	C₂²	semidirect
12. #16812606	C₂²	semidirect
13. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
15. #95×C₁₀₉	C₂²	semidirect
67. C₁₀₉	#2842	semidirect
69. #399	D₁₀₉	semidirect
76. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect
82. #95	C₂×D₁₀₉	semidirect

H ≅ G/N

type

1. #20294962

C₂

semidirect

2. #20294950

C₂

semidirect

3. #20294948

C₂

semidirect

5. #399×C₁₀₉

C₂

semidirect

7. C₂×C₄⋊Dic₁₀₉

C₂²

semidirect

8. C₂×#13182008

C₂²

semidirect

12. #16812606

C₂²

semidirect

13. C₂×C₄⋊C₄×C₁₀₉

C₂²

semidirect

15. #95×C₁₀₉

C₂²

semidirect

67. C₁₀₉

#2842

semidirect

69. #399

D₁₀₉

semidirect

76. C₂×C₄⋊C₄

C₂×D₁₀₉

semidirect

82. #95

C₂×D₁₀₉

semidirect

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

subgroup	index
#20294962	2
#20294950	2
#20294948	2
#20294881	2
#399×C₁₀₉	2
#2842	109

subgroup

index

#399×C₁₀₉

109

Subgroups — #28371297

Conjugacy classes of subgroups

Complements

Maximal subgroups