Group #28364025 — subgroups

Subgroups — #28364025

Order 13952.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20295050	C₂	1	1	yes	maximal, characteristic
3	#20294794	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₂²	1	2	yes	—
6	#16812691	—	2	2	no	—
7	#16812691	—	2	1	no	—
8	#16812691	—	2	1	no	—
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
10	C₂×#13182008	C₄	1	4	yes	—
11	C₄⋊#11724784	—	2	2	no	—
12	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	#97×C₁₀₉	—	2	1	no	—
15	C₄×C₈×C₁₀₉	—	2	1	no	—
16	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
17	#13182013	—	2	4	no	—
18	C₂×#11724784	—	2	2	no	—
19	C₄⋊Dic₁₀₉	—	2	2	no	—
20	C₄⋊Dic₁₀₉	—	2	2	no	—
21	#13182008	—	2	4	no	—
22	#13182008	—	2	4	no	—
23	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₀₉	—	2	2	no	—
26	M₁₆×C₁₀₉	Q₈	1	4	yes	—
27	C₂×C₈×C₁₀₉	—	2	2	no	—
28	C₄²×C₁₀₉	D₄	1	2	yes	—
29	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
30	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	4	2	no	—
33	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₀₉	#41	1	2	yes	—
39	C₂×Dic₁₀₉	—	2	4	no	—
40	C₂×Dic₁₀₉	—	4	4	no	—
41	C₈×C₁₀₉	—	2	4	no	—
42	#11724784	—	4	2	no	—
43	C₂²×C₁₀₉	—	2	2	no	—
44	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
46	C₄×C₁₀₉	—	2	2	no	—
47	C₄×C₁₀₉	—	2	2	no	—
48	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₀₉	#124	1	1	yes	characteristic
50	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₁₀₉	—	4	4	no	—
52	C₂×C₁₀₉	—	2	2	no	—
53	C₂×C₁₀₉	#391	1	1	yes	characteristic
54	C₂×C₁₀₉	#413	1	2	yes	—
55	#1998	—	109	1	no	maximal, Sylow
56	C₁₀₉	#1998	1	1	yes	characteristic, complemented, Sylow
57	C₂×#125	—	109	1	no	—
58	#423	—	109	1	no	—
59	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic
60	#343	—	109	4	no	—
61	C₂×M₁₆	—	109	2	no	—
62	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
63	#125	—	218	2	no	—
64	#125	—	218	1	no	—
65	#125	—	218	1	no	—
66	C₂×C₄⋊C₄	—	109	4	no	—
67	C₂×C₄⋊C₄	—	109	2	no	—
68	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
69	C₄⋊C₈	—	218	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	—	109	4	no	—
73	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	218	4	no	—
77	M₁₆	Dic₂₁₈	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	218	2	no	—
80	C₄⋊C₄	—	218	4	no	—
81	C₄⋊C₄	—	218	4	no	—
82	C₄⋊C₄	—	218	2	no	—
83	C₄⋊C₄	—	218	2	no	—
84	C₄²	D₄₃₆	1	2	yes	—
85	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
86	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
87	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	436	4	no	—
90	C₂×C₄	—	218	4	no	—
91	C₂×C₄	#13182006	1	2	yes	—
92	C₂×C₄	#13182006	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
94	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
95	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
96	C₂×C₄	#13182029	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	436	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×#13182006	1	1	yes	characteristic
101	C₂²	C₄×Dic₂₁₈	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	436	4	no	—
105	C₄	#16812678	1	2	yes	—
106	C₄	#16812687	1	1	yes	characteristic
107	C₄	#16812699	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#20294932	1	1	yes	characteristic
110	C₂	#20295008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812691

C₂

yes

maximal, characteristic

#20295050

C₂

yes

maximal, characteristic

#20294794

C₂

yes

maximal

C₄×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

#16812691

—

#16812691

—

#16812691

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

C₄

yes

—

C₄⋊#11724784

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

#13182013

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

#13182008

—

#13182008

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#391

yes

characteristic

C₂×C₁₀₉

#413

yes

—

#1998

—

109

maximal, Sylow

C₁₀₉

#1998

yes

characteristic, complemented, Sylow

C₂×#125

—

109

—

#423

—

109

—

C₄×M₁₆

D₁₀₉

yes

characteristic

#343

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

#125

—

218

—

#125

—

218

—

#125

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

218

—

#97

—

C₄×C₈

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

M₁₆

Dic₂₁₈

yes

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

D₄₃₆

yes

—

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

#13182006

yes

—

C₂×C₄

#13182006

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

—

C₂×C₄

#13182029

yes

—

C₈

—

C₈

—

436

—

C₂²

—

100

C₂²

C₂×#13182006

yes

characteristic

101

C₂²

C₄×Dic₂₁₈

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

436

—

105

C₄

#16812678

yes

—

106

C₄

#16812687

yes

characteristic

107

C₄

#16812699

yes

characteristic

108

C₂

—

109

C₂

#20294932

yes

characteristic

110

C₂

#20295008

yes

—

	subgroup	G/H
center	C₂²	C₂×#13182006
commutator, prime radical	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₄×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#1998
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#13182006

commutator, prime radical

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₄×M₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₄×M₁₆

D₁₀₉

hypocenter

C₁₀₉

#1998

LCS

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
56. C₁₀₉	#1998	semidirect

H ≅ G/N

type

56. C₁₀₉

#1998

semidirect

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

subgroup	index
C₂×#16812691	2
#20295050	2
#20294794	2
C₄×M₁₆×C₁₀₉	2
#1998	109

subgroup

index

C₄×M₁₆×C₁₀₉

109

Subgroups — #28364025

Conjugacy classes of subgroups

Complements

Maximal subgroups