Group #28371399 — subgroups

Subgroups — #28371399

Order 13952.

516 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812693	C₂	1	2	yes	maximal, complemented
2	#20294928	C₂	1	3	yes	maximal
3	#20294924	C₂	1	3	yes	maximal, complemented
4	#20294917	C₂	1	6	yes	maximal, complemented
5	C₄³×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
7	#16812693	—	2	8	no	—
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	—
9	C₂×#13182008	C₂²	1	6	yes	—
10	C₂×#13182006	C₂²	1	6	yes	—
11	C₂×#13182006	C₂²	1	6	yes	—
12	#16812606	C₂²	1	6	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂³×D₁₀₉	C₂³	1	2	yes	—
17	C₂×D₄₃₆	—	2	6	no	—
18	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
19	C₄⋊Dic₁₀₉	—	2	6	no	—
20	#13182008	—	2	12	no	—
21	#13182006	—	2	12	no	—
22	#13182006	—	2	12	no	—
23	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
24	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₄²×C₁₀₉	—	2	6	no	—
27	C₄²×C₁₀₉	—	2	6	no	—
28	C₄²×C₁₀₉	D₄	1	4	yes	complemented
29	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
30	C₂²×D₁₀₉	—	2	2	no	—
31	C₂²×D₁₀₉	—	2	6	no	—
32	C₂²×D₁₀₉	—	2	6	no	—
33	C₂×C₄×C₁₀₉	—	2	3	no	—
34	C₂×C₄×C₁₀₉	—	2	3	no	—
35	C₂×C₄×C₁₀₉	—	2	3	no	—
36	C₂×C₄×C₁₀₉	—	2	3	no	—
37	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
38	C₂×C₄×C₁₀₉	#41	1	6	yes	—
39	C₂×C₄×C₁₀₉	#41	1	6	yes	—
40	C₂×Dic₁₀₉	—	2	6	no	—
41	C₂×Dic₁₀₉	—	2	6	no	—
42	C₂×Dic₁₀₉	—	2	6	no	—
43	D₄₃₆	—	4	4	no	—
44	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
45	C₂²×C₁₀₉	C₂×#41	1	3	yes	—
46	C₂²×C₁₀₉	C₂×#41	1	3	yes	—
47	C₂×D₁₀₉	—	4	2	no	—
48	C₂×D₁₀₉	—	4	6	no	—
49	C₂×D₁₀₉	—	4	6	no	—
50	C₄×C₁₀₉	—	2	6	no	—
51	C₄×C₁₀₉	—	2	6	no	—
52	C₄×C₁₀₉	#126	1	4	yes	complemented
53	Dic₁₀₉	—	4	6	no	—
54	C₂×C₁₀₉	C₂×#126	1	1	yes	characteristic
55	C₂×C₁₀₉	#529	1	3	yes	—
56	C₂×C₁₀₉	#532	1	3	yes	—
57	D₁₀₉	—	8	2	no	—
58	#2860	—	109	1	no	maximal, Sylow
59	C₁₀₉	#2860	1	1	yes	characteristic, complemented, Sylow
60	C₂×#126	—	109	2	no	—
61	#390	—	109	3	no	—
62	#389	—	109	3	no	—
63	#388	—	109	6	no	—
64	C₄³	D₁₀₉	1	1	yes	characteristic, complemented
65	C₂²×D₄	—	109	1	no	—
66	#126	—	218	8	no	—
67	C₂×C₄⋊C₄	—	109	6	no	—
68	C₂×C₄⋊C₄	—	109	3	no	—
69	C₂×C₂²⋊C₄	—	109	6	no	—
70	C₂×C₂²⋊C₄	—	109	6	no	—
71	C₂×C₄²	C₂×D₁₀₉	1	3	yes	—
72	C₂×C₄²	C₂×D₁₀₉	1	3	yes	—
73	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
74	#95	—	109	6	no	—
75	C₂⁴	—	109	2	no	—
76	C₂×D₄	—	218	6	no	—
77	C₂²×C₄	—	109	6	no	—
78	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
79	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
80	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
81	C₄⋊C₄	—	218	12	no	—
82	C₄⋊C₄	—	218	6	no	—
83	C₂²⋊C₄	—	218	12	no	—
84	C₂²⋊C₄	—	218	12	no	—
85	C₄²	—	2	6	no	—
86	C₄²	—	2	6	no	—
87	C₄²	D₄₃₆	1	4	yes	complemented
88	C₂³	—	218	2	no	—
89	C₂³	—	218	6	no	—
90	C₂³	—	218	6	no	—
91	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
92	D₄	—	436	4	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	218	6	no	—
98	C₂×C₄	—	218	6	no	—
99	C₂×C₄	—	218	6	no	—
100	C₂×C₄	C₂×D₄₃₆	1	6	yes	—
101	C₂×C₄	#13182029	1	6	yes	—
102	C₂×C₄	#13182029	1	6	yes	—
103	C₂²	—	436	2	no	—
104	C₂²	—	436	6	no	—
105	C₂²	—	436	6	no	—
106	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
107	C₂²	C₂×#13182029	1	3	yes	—
108	C₂²	C₂×#13182029	1	3	yes	—
109	C₄	—	2	6	no	—
110	C₄	—	2	6	no	—
111	C₄	—	436	6	no	—
112	C₄	#16812693	1	4	yes	complemented
113	C₂	—	872	2	no	—
114	C₂	C₂×#16812693	1	1	yes	characteristic
115	C₂	#20295560	1	3	yes	—
116	C₂	#20295576	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812693

C₂

yes

maximal, complemented

#20294928

C₂

yes

maximal

#20294924

C₂

yes

maximal, complemented

#20294917

C₂

yes

maximal, complemented

C₄³×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

C₂²

yes

characteristic

#16812693

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

C₂×#13182006

C₂²

yes

—

#16812606

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂³×D₁₀₉

C₂³

yes

—

C₂×D₄₃₆

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

—

#13182008

—

#13182006

—

#13182006

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

complemented

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₂²×C₁₀₉

C₂×#41

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#126

yes

complemented

Dic₁₀₉

—

C₂×C₁₀₉

C₂×#126

yes

characteristic

C₂×C₁₀₉

#529

yes

—

C₂×C₁₀₉

#532

yes

—

D₁₀₉

—

#2860

—

109

maximal, Sylow

C₁₀₉

#2860

yes

characteristic, complemented, Sylow

C₂×#126

—

109

—

#390

—

109

—

#389

—

109

—

#388

—

109

—

C₄³

D₁₀₉

yes

characteristic, complemented

C₂²×D₄

—

109

—

#126

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#95

—

109

—

C₂⁴

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

C₄²

—

C₄²

D₄₃₆

yes

complemented

C₂³

—

218

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₂³×D₁₀₉

yes

characteristic

D₄

—

436

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

100

C₂×C₄

C₂×D₄₃₆

yes

—

101

C₂×C₄

#13182029

yes

—

102

C₂×C₄

#13182029

yes

—

103

C₂²

—

436

—

104

C₂²

—

436

—

105

C₂²

—

436

—

106

C₂²

C₂²×D₄₃₆

yes

characteristic

107

C₂²

C₂×#13182029

yes

—

108

C₂²

C₂×#13182029

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

436

—

112

C₄

#16812693

yes

complemented

113

C₂

—

872

—

114

C₂

C₂×#16812693

yes

characteristic

115

C₂

#20295560

yes

—

116

C₂

#20295576

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	C₄³×C₁₀₉	C₂
prime radical	C₂²×D₄₃₆	C₂²
hypercenter	C₄³	D₁₀₉
hypocenter	C₁₀₉	#2860
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

C₄³×C₁₀₉

C₂

prime radical

C₂²×D₄₃₆

C₂²

hypercenter

C₄³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
1. C₂×#16812693	C₂	semidirect
3. #20294924	C₂	semidirect
4. #20294917	C₂	semidirect
5. C₄³×C₁₀₉	C₂	semidirect
28. C₄²×C₁₀₉	D₄	semidirect
52. C₄×C₁₀₉	#126	semidirect
59. C₁₀₉	#2860	semidirect
64. C₄³	D₁₀₉	semidirect
87. C₄²	D₄₃₆	semidirect
112. C₄	#16812693	semidirect

H ≅ G/N

type

1. C₂×#16812693

C₂

semidirect

3. #20294924

C₂

semidirect

4. #20294917

C₂

semidirect

5. C₄³×C₁₀₉

C₂

semidirect

28. C₄²×C₁₀₉

D₄

semidirect

52. C₄×C₁₀₉

#126

semidirect

59. C₁₀₉

#2860

semidirect

64. C₄³

D₁₀₉

semidirect

87. C₄²

D₄₃₆

semidirect

112. C₄

#16812693

semidirect

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

subgroup	index
C₂×#16812693	2
#20294928	2
#20294924	2
#20294917	2
C₄³×C₁₀₉	2
#2860	109

subgroup

index

C₄³×C₁₀₉

109

Subgroups — #28371399

Conjugacy classes of subgroups

Complements

Maximal subgroups