Group #28371632 — subgroups

Subgroups — #28371632

Order 13952.

476 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 293 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	#20295509	C₂	1	2	yes	maximal
2	#20295068	C₂	1	6	yes	maximal, complemented
3	#20295064	C₂	1	3	yes	maximal, complemented
4	C₂×#16812633	C₂	1	3	yes	maximal, complemented
5	C₂×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₀₉	—	2	2	no	—
7	D₄×Dic₁₀₉	—	2	6	no	—
8	#16812658	C₂²	1	6	yes	—
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	6	yes	—
10	#16812633	C₂²	1	3	yes	—
11	#16812633	C₂²	1	6	yes	—
12	#16812633	C₂²	1	1	yes	characteristic
13	#16812631	C₂²	1	6	yes	—
14	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
15	#131×C₁₀₉	C₄	1	8	yes	complemented
16	C₂×M₁₆×C₁₀₉	C₂²	1	3	yes	—
17	C₂²×C₈×C₁₀₉	C₂²	1	3	yes	—
18	C₂²×Dic₁₀₉	—	2	6	no	—
19	C₄⋊Dic₁₀₉	—	2	6	no	—
20	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
21	C₄⋊Dic₁₀₉	C₂³	1	6	yes	—
22	C₂²⋊Dic₁₀₉	—	2	6	no	—
23	C₄×Dic₁₀₉	—	2	6	no	—
24	#41×C₁₀₉	C₂×C₄	1	4	yes	—
25	#41×C₁₀₉	Q₈	1	4	yes	—
26	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂×D₄×C₁₀₉	D₄	1	3	yes	—
28	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
29	M₁₆×C₁₀₉	C₂×C₄	1	12	yes	complemented
30	C₂×C₈×C₁₀₉	C₂×C₄	1	12	yes	complemented
31	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	C₂³	1	3	yes	—
33	C₂³×C₁₀₉	C₂×D₄	1	3	yes	—
34	Q₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
35	D₄×C₁₀₉	C₄⋊C₄	1	12	yes	—
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
38	C₂×C₄×C₁₀₉	C₂×Q₈	1	6	yes	—
39	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₁₀₉	—	2	2	no	—
41	C₂×Dic₁₀₉	—	2	6	no	—
42	C₂×Dic₁₀₉	—	8	6	no	—
43	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
44	C₈×C₁₀₉	C₂²×C₄	1	6	yes	—
45	C₂²×C₁₀₉	—	2	3	no	—
46	C₂²×C₁₀₉	—	2	3	no	—
47	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
48	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
50	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
51	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
52	Dic₁₀₉	—	4	6	no	—
53	Dic₁₀₉	—	4	2	no	—
54	C₂×C₁₀₉	—	2	6	no	—
55	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₁₀₉	#576	1	1	yes	characteristic
57	C₂×C₁₀₉	#578	1	1	yes	characteristic
58	#2913	—	109	1	no	maximal, Sylow
59	C₁₀₉	#2913	1	1	yes	characteristic, complemented, Sylow
60	C₂×#131	D₁₀₉	1	1	yes	characteristic
61	#520	—	109	2	no	—
62	#428	—	109	6	no	—
63	#427	—	109	3	no	—
64	C₂×#107	—	109	3	no	—
65	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
66	#131	Dic₁₀₉	1	8	yes	complemented
67	C₂×M₁₆	C₂×D₁₀₉	1	3	yes	—
68	C₂²×C₈	C₂×D₁₀₉	1	3	yes	—
69	C₄×Q₈	—	218	2	no	—
70	C₄×D₄	—	218	6	no	—
71	#117	—	109	6	no	—
72	C₂×C₄⋊C₄	—	109	6	no	—
73	#107	—	109	3	no	—
74	#107	—	109	6	no	—
75	#107	—	109	1	no	—
76	#106	—	109	6	no	—
77	#41	Dic₂₁₈	1	4	yes	—
78	#41	C₂×Dic₁₀₉	1	4	yes	—
79	C₂×Q₈	D₄₃₆	1	1	yes	characteristic
80	C₂×D₄	D₄₃₆	1	3	yes	—
81	C₂²×C₄	—	218	6	no	—
82	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
83	M₁₆	C₂×Dic₁₀₉	1	12	yes	complemented
84	C₂×C₈	C₂×Dic₁₀₉	1	12	yes	complemented
85	C₂×C₈	C₂²×D₁₀₉	1	3	yes	—
86	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
87	C₄⋊C₄	—	109	2	no	—
88	C₄⋊C₄	—	109	6	no	—
89	C₄⋊C₄	—	218	6	no	—
90	C₂²⋊C₄	—	218	6	no	—
91	C₄²	—	218	6	no	—
92	C₂³	C₂×D₄₃₆	1	3	yes	—
93	Q₈	C₄⋊Dic₁₀₉	1	4	yes	—
94	D₄	C₄⋊Dic₁₀₉	1	12	yes	—
95	C₂×C₄	—	872	6	no	—
96	C₂×C₄	—	218	2	no	—
97	C₂×C₄	—	218	6	no	—
98	C₂×C₄	C₂²×Dic₁₀₉	1	6	yes	—
99	C₂×C₄	C₂×D₄₃₆	1	3	yes	—
100	C₂×C₄	C₂×Dic₂₁₈	1	6	yes	—
101	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
102	C₈	C₂²×Dic₁₀₉	1	2	yes	—
103	C₈	C₂²×Dic₁₀₉	1	6	yes	—
104	C₂²	—	2	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	C₂×C₄⋊Dic₁₀₉	1	6	yes	—
107	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
108	C₄	—	436	6	no	—
109	C₄	—	436	2	no	—
110	C₄	C₂×C₄⋊Dic₁₀₉	1	6	yes	—
111	C₄	C₂³×Dic₁₀₉	1	1	yes	characteristic
112	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
113	C₂	—	2	6	no	—
114	C₂	C₂²×C₄⋊Dic₁₀₉	1	1	yes	characteristic
115	C₂	#20295831	1	1	yes	characteristic
116	C₂	#20295843	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295509

C₂

yes

maximal

#20295068

C₂

yes

maximal, complemented

#20295064

C₂

yes

maximal, complemented

C₂×#16812633

C₂

yes

maximal, complemented

C₂×#131×C₁₀₉

C₂

yes

maximal, characteristic

Q₈×Dic₁₀₉

—

D₄×Dic₁₀₉

—

#16812658

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812633

C₂²

yes

—

#16812633

C₂²

yes

—

#16812633

C₂²

yes

characteristic

#16812631

C₂²

yes

—

C₂×#41×C₁₀₉

C₂²

yes

characteristic

#131×C₁₀₉

C₄

yes

complemented

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

#41×C₁₀₉

C₂×C₄

yes

—

#41×C₁₀₉

Q₈

yes

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×D₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

M₁₆×C₁₀₉

C₂×C₄

yes

complemented

C₂×C₈×C₁₀₉

C₂×C₄

yes

complemented

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

—

Q₈×C₁₀₉

C₄⋊C₄

yes

—

D₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₂²×C₄

yes

—

C₈×C₁₀₉

C₂²×C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×C₄⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2913

yes

characteristic, complemented, Sylow

C₂×#131

D₁₀₉

yes

characteristic

#520

—

109

—

#428

—

109

—

#427

—

109

—

C₂×#107

—

109

—

C₂×#41

C₂×D₁₀₉

yes

characteristic

#131

Dic₁₀₉

yes

complemented

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₄×Q₈

—

218

—

C₄×D₄

—

218

—

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

#107

—

109

—

#107

—

109

—

#107

—

109

—

#106

—

109

—

#41

Dic₂₁₈

yes

—

#41

C₂×Dic₁₀₉

yes

—

C₂×Q₈

D₄₃₆

yes

characteristic

C₂×D₄

D₄₃₆

yes

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

M₁₆

C₂×Dic₁₀₉

yes

complemented

C₂×C₈

C₂×Dic₁₀₉

yes

complemented

C₂×C₈

C₂²×D₁₀₉

yes

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₂³

C₂×D₄₃₆

yes

—

Q₈

C₄⋊Dic₁₀₉

yes

—

D₄

C₄⋊Dic₁₀₉

yes

—

C₂×C₄

—

872

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×Dic₁₀₉

yes

—

C₂×C₄

C₂×D₄₃₆

yes

—

100

C₂×C₄

C₂×Dic₂₁₈

yes

—

101

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

102

C₈

C₂²×Dic₁₀₉

yes

—

103

C₈

C₂²×Dic₁₀₉

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂×C₄⋊Dic₁₀₉

yes

—

107

C₂²

C₂²×D₄₃₆

yes

characteristic

108

C₄

—

436

—

109

C₄

—

436

—

110

C₄

C₂×C₄⋊Dic₁₀₉

yes

—

111

C₄

C₂³×Dic₁₀₉

yes

characteristic

112

C₄

C₂²×Dic₂₁₈

yes

characteristic

113

C₂

—

114

C₂

C₂²×C₄⋊Dic₁₀₉

yes

characteristic

115

C₂

#20295831

yes

characteristic

116

C₂

#20295843

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#131×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#41×C₁₀₉	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×#131	D₁₀₉
hypocenter	C₁₀₉	#2913
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₄₃₆

commutator

C₄×C₁₀₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#131×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂×#41×C₁₀₉

C₂²

cosocle

C₂×C₄×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
2. #20295068	C₂	semidirect
3. #20295064	C₂	semidirect
4. C₂×#16812633	C₂	semidirect
15. #131×C₁₀₉	C₄	semidirect
29. M₁₆×C₁₀₉	C₂×C₄	semidirect
30. C₂×C₈×C₁₀₉	C₂×C₄	semidirect
59. C₁₀₉	#2913	semidirect
66. #131	Dic₁₀₉	semidirect
83. M₁₆	C₂×Dic₁₀₉	semidirect
84. C₂×C₈	C₂×Dic₁₀₉	semidirect

H ≅ G/N

type

2. #20295068

C₂

semidirect

3. #20295064

C₂

semidirect

4. C₂×#16812633

C₂

semidirect

15. #131×C₁₀₉

C₄

semidirect

29. M₁₆×C₁₀₉

C₂×C₄

semidirect

30. C₂×C₈×C₁₀₉

C₂×C₄

semidirect

59. C₁₀₉

#2913

semidirect

66. #131

Dic₁₀₉

semidirect

83. M₁₆

C₂×Dic₁₀₉

semidirect

84. C₂×C₈

C₂×Dic₁₀₉

semidirect

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

subgroup	index
#20295509	2
#20295068	2
#20295064	2
C₂×#16812633	2
C₂×#131×C₁₀₉	2
#2913	109

subgroup

index

C₂×#131×C₁₀₉

109

Subgroups — #28371632

Conjugacy classes of subgroups

Complements

Maximal subgroups