Group #28364001 — subgroups

Subgroups — #28364001

Order 13952.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#16812691	C₂	1	1	yes	maximal, characteristic
2	#20294896	C₂	1	1	yes	maximal, characteristic
3	#20294789	C₂	1	4	yes	maximal
4	C₂×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812691	—	2	2	no	—
6	#16812691	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₁₀₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
10	C₂×#13182008	—	2	2	no	—
11	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
13	#97×C₁₀₉	—	2	2	no	—
14	C₂²×Dic₁₀₉	—	2	2	no	—
15	C₂²×Dic₁₀₉	—	2	2	no	—
16	C₄⋊Dic₁₀₉	—	2	2	no	—
17	C₄⋊Dic₁₀₉	—	2	4	no	—
18	C₄⋊Dic₁₀₉	—	4	4	no	—
19	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
20	#13182008	—	2	8	no	—
21	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
22	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	—	2	2	no	—
24	C₂×C₈×C₁₀₉	—	2	2	no	—
25	C₂×C₈×C₁₀₉	Q₈	1	4	yes	—
26	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₄²×C₁₀₉	D₄	1	2	yes	—
28	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₀₉	#41	1	2	yes	—
36	C₂×Dic₁₀₉	—	2	4	no	—
37	C₂×Dic₁₀₉	—	4	2	no	—
38	C₂×Dic₁₀₉	—	4	2	no	—
39	C₂×Dic₁₀₉	—	4	2	no	—
40	C₂×Dic₁₀₉	—	4	4	no	—
41	C₈×C₁₀₉	—	2	4	no	—
42	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₀₉	#124	1	1	yes	characteristic
44	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₀₉	#137	1	2	yes	—
47	C₄×C₁₀₉	—	2	4	no	—
48	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
49	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
50	Dic₁₀₉	—	4	4	no	—
51	Dic₁₀₉	—	8	2	no	—
52	C₂×C₁₀₉	#391	1	1	yes	characteristic
53	C₂×C₁₀₉	#417	1	2	yes	—
54	C₂×C₁₀₉	#490	1	1	yes	characteristic
55	C₂×C₁₀₉	#491	1	1	yes	characteristic
56	C₂×C₁₀₉	#501	1	2	yes	—
57	#1992	—	109	1	no	maximal, Sylow
58	C₁₀₉	#1992	1	1	yes	characteristic, complemented, Sylow
59	C₂×#125	—	109	1	no	—
60	C₂×#97	D₁₀₉	1	1	yes	characteristic
61	#384	—	109	1	no	—
62	#340	—	109	4	no	—
63	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
64	#125	—	109	4	no	—
65	#125	—	218	2	no	—
66	C₂×C₄⋊C₄	—	109	2	no	—
67	C₂×C₄⋊C₄	—	109	2	no	—
68	C₂×C₄⋊C₄	—	218	2	no	—
69	C₂×C₄⋊C₄	—	218	2	no	—
70	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
71	#97	—	2	2	no	—
72	C₂²×C₄	—	218	2	no	—
73	C₂²×C₄	—	218	2	no	—
74	C₂²×C₄	D₄₃₆	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Dic₂₁₈	1	4	yes	—
79	C₄⋊C₄	—	109	4	no	—
80	C₄⋊C₄	—	436	4	no	—
81	C₄⋊C₄	—	218	4	no	—
82	C₄⋊C₄	—	218	8	no	—
83	C₄⋊C₄	—	218	2	no	—
84	C₄²	C₄×D₁₀₉	1	2	yes	—
85	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
86	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	436	2	no	—
90	C₂×C₄	—	436	4	no	—
91	C₂×C₄	—	436	2	no	—
92	C₂×C₄	—	436	2	no	—
93	C₂×C₄	—	218	4	no	—
94	C₂×C₄	#13182006	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
98	C₂×C₄	#13182029	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×#13182006	1	1	yes	characteristic
101	C₂²	#16812687	1	1	yes	characteristic
102	C₂²	#16812699	1	1	yes	characteristic
103	C₂²	#16812727	1	2	yes	—
104	C₂²	#16812732	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	872	2	no	—
107	C₄	—	436	4	no	—
108	C₄	C₄×Dic₂₁₈	1	2	yes	—
109	C₄	#16812678	1	2	yes	—
110	C₂	#20294932	1	1	yes	characteristic
111	C₂	#20295028	1	2	yes	—
112	C₂	#20295391	1	1	yes	characteristic
113	C₂	#20295395	1	1	yes	characteristic
114	C₂	#20295435	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812691

C₂

yes

maximal, characteristic

#20294896

C₂

yes

maximal, characteristic

#20294789

C₂

yes

maximal

C₂×#97×C₁₀₉

C₂

yes

maximal, characteristic

#16812691

—

#16812691

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

#124

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

#137

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#391

yes

characteristic

C₂×C₁₀₉

#417

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

#1992

—

109

maximal, Sylow

C₁₀₉

#1992

yes

characteristic, complemented, Sylow

C₂×#125

—

109

—

C₂×#97

D₁₀₉

yes

characteristic

#384

—

109

—

#340

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

—

#125

—

109

—

#125

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#97

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

D₄₃₆

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₁₈

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

C₄×D₁₀₉

yes

—

C₄²

C₂²⋊D₁₀₉

yes

—

C₂³

C₂×D₄₃₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

#13182006

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

—

C₂×C₄

#13182029

yes

—

C₈

—

100

C₂²

C₂×#13182006

yes

characteristic

101

C₂²

#16812687

yes

characteristic

102

C₂²

#16812699

yes

characteristic

103

C₂²

#16812727

yes

—

104

C₂²

#16812732

yes

—

105

C₄

—

106

C₄

—

872

—

107

C₄

—

436

—

108

C₄

C₄×Dic₂₁₈

yes

—

109

C₄

#16812678

yes

—

110

C₂

#20294932

yes

characteristic

111

C₂

#20295028

yes

—

112

C₂

#20295391

yes

characteristic

113

C₂

#20295395

yes

characteristic

114

C₂

#20295435

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#97×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#97	D₁₀₉
hypocenter	C₁₀₉	#1992
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₄₃₆

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#97×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂×#97

D₁₀₉

hypocenter

C₁₀₉

#1992

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
6. #16812691	C₄	semidirect
58. C₁₀₉	#1992	semidirect

H ≅ G/N

type

6. #16812691

C₄

semidirect

58. C₁₀₉

#1992

semidirect

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

subgroup	index
C₂×#16812691	2
#20294896	2
#20294789	2
C₂×#97×C₁₀₉	2
#1992	109

subgroup

index

C₂×#97×C₁₀₉

109

Subgroups — #28364001

Conjugacy classes of subgroups

Complements

Maximal subgroups