Group #28364000 — subgroups

Subgroups — #28364000

Order 13952.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
3	#20294788	C₂	1	4	yes	maximal
4	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₂²	1	2	yes	—
6	#16812691	—	2	2	no	—
7	#16812691	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
9	C₂×#13182008	—	2	2	no	—
10	#16812610	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₀₉	—	2	2	no	—
15	C₂×#11724784	—	2	2	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₂×#11724784	Q₈	1	4	yes	—
18	C₄⋊Dic₁₀₉	—	2	2	no	—
19	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
20	#13182008	—	2	8	no	—
21	C₂²×C₄×C₁₀₉	—	2	2	no	—
22	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
23	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₀₉	—	2	4	no	—
25	C₄⋊C₄×C₁₀₉	—	4	4	no	—
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	—	4	2	no	—
32	C₂×C₄×C₁₀₉	—	4	4	no	—
33	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₀₉	#41	1	2	yes	—
38	C₂×Dic₁₀₉	—	2	4	no	—
39	C₂×Dic₁₀₉	—	4	2	no	—
40	C₂×Dic₁₀₉	—	4	2	no	—
41	#11724784	—	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	—	8	2	no	—
49	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
50	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
51	Dic₁₀₉	—	4	4	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	—	109	1	no	—
61	#384	D₁₀₉	1	1	yes	characteristic
62	#340	—	109	4	no	—
63	C₂²×C₈	—	109	2	no	—
64	#125	—	109	4	no	—
65	#125	—	218	2	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	109	2	no	—
68	C₂×C₄⋊C₄	—	218	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
70	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
71	#97	—	218	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	218	2	no	—
74	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
76	C₂×C₈	—	109	4	no	—
77	C₂×C₈	—	218	2	no	—
78	C₂×C₈	—	218	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	4	4	no	—
81	C₄⋊C₄	—	109	4	no	—
82	C₄⋊C₄	—	218	8	no	—
83	C₄⋊C₄	—	218	2	no	—
84	C₄²	C₄×D₁₀₉	1	2	yes	—
85	C₄²	D₄₃₆	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	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×D₄₃₆	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
98	C₂×C₄	#13182028	1	2	yes	—
99	C₈	—	218	4	no	—
100	C₂²	C₂×#13182006	1	1	yes	characteristic
101	C₂²	#16812686	1	1	yes	characteristic
102	C₂²	#16812697	1	1	yes	characteristic
103	C₂²	#16812725	1	2	yes	—
104	C₂²	#16812730	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	8	2	no	—
107	C₄	—	436	4	no	—
108	C₄	#16812664	1	2	yes	—
109	C₄	#16812679	1	2	yes	—
110	C₂	#20294931	1	1	yes	characteristic
111	C₂	#20295025	1	2	yes	—
112	C₂	#20295390	1	1	yes	characteristic
113	C₂	#20295394	1	1	yes	characteristic
114	C₂	#20295432	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812691

C₂

yes

maximal, characteristic

C₂×#16812610

C₂

yes

maximal, characteristic

#20294788

C₂

yes

maximal

#384×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

#16812691

—

#16812691

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

—

#16812610

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

Q₈

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

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

—

#11724784

—

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×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

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

—

109

—

#384

D₁₀₉

yes

characteristic

#340

—

109

—

C₂²×C₈

—

109

—

#125

—

109

—

#125

—

218

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#97

—

218

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

C₄×D₁₀₉

yes

—

C₄²

D₄₃₆

yes

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

#13182006

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

Q₈×D₁₀₉

yes

—

C₂×C₄

#13182028

yes

—

C₈

—

218

—

100

C₂²

C₂×#13182006

yes

characteristic

101

C₂²

#16812686

yes

characteristic

102

C₂²

#16812697

yes

characteristic

103

C₂²

#16812725

yes

—

104

C₂²

#16812730

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

436

—

108

C₄

#16812664

yes

—

109

C₄

#16812679

yes

—

110

C₂

#20294931

yes

characteristic

111

C₂

#20295025

yes

—

112

C₂

#20295390

yes

characteristic

113

C₂

#20295394

yes

characteristic

114

C₂

#20295432

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#384×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#384	D₁₀₉
hypocenter	C₁₀₉	#1992
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#384×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #16812691	C₄	semidirect
58. C₁₀₉	#1992	semidirect

H ≅ G/N

type

7. #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
C₂×#16812610	2
#20294788	2
#384×C₁₀₉	2
#1992	109

subgroup

index

#384×C₁₀₉

109

Subgroups — #28364000

Conjugacy classes of subgroups

Complements

Maximal subgroups