Group #28362939 — subgroups

Subgroups — #28362939

Order 13952.

220 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16812634	C₂	1	2	yes	maximal
2	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
4	#20294790	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	#16812634	C₄	1	8	yes	complemented
10	#16812610	—	2	1	no	—
11	#16812610	C₄	1	4	yes	—
12	C₄×#11724784	—	2	1	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	#97×C₁₀₉	—	2	1	no	—
16	C₄×C₈×C₁₀₉	—	2	1	no	—
17	#13182013	—	2	2	no	—
18	#13182013	C₂×C₄	1	4	yes	—
19	C₂×#11724784	—	2	2	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	C₂×C₄	1	2	yes	—
22	C₂×#11724784	C₂×C₄	1	2	yes	—
23	C₂×#11724784	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₉	—	2	2	no	—
27	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
31	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	4	2	no	—
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	#41	1	2	yes	—
40	C₈×C₁₀₉	—	2	4	no	—
41	#11724784	—	2	2	no	—
42	#11724784	—	2	4	no	—
43	#11724784	C₄²	1	4	yes	—
44	C₂²×C₁₀₉	—	2	2	no	—
45	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
47	C₄×C₁₀₉	—	2	2	no	—
48	C₄×C₁₀₉	—	2	2	no	—
49	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₁₀₉	#117	1	1	yes	characteristic
51	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
52	C₂×C₁₀₉	—	2	2	no	—
53	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₁₀₉	#430	1	2	yes	—
55	#1778	—	109	1	no	maximal, Sylow
56	C₁₀₉	#1778	1	1	yes	characteristic, complemented, Sylow
57	C₂×#108	—	109	2	no	—
58	C₄×M₁₆	—	109	1	no	—
59	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic
60	C₂×#97	—	109	1	no	—
61	#341	—	109	2	no	—
62	C₂×M₁₆	—	109	2	no	—
63	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
64	C₂²×C₈	—	109	1	no	—
65	C₂²×C₈	—	109	1	no	—
66	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
67	#108	—	109	8	no	—
68	#97	—	2	1	no	—
69	#97	—	109	4	no	—
70	#97	—	218	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	218	1	no	—
73	C₂²×C₄	Dic₂₁₈	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	109	4	no	—
77	M₁₆	—	218	2	no	—
78	M₁₆	C₄×D₁₀₉	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	109	2	no	—
81	C₂×C₈	—	109	4	no	—
82	C₂×C₈	—	109	2	no	—
83	C₂×C₈	—	218	2	no	—
84	C₂×C₈	—	218	2	no	—
85	C₄²	C₄×D₁₀₉	1	2	yes	—
86	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
87	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
88	C₂³	C₂×Dic₂₁₈	1	1	yes	characteristic
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	#13182008	1	2	yes	—
91	C₂×C₄	#13182008	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
95	C₂×C₄	#13182029	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	109	4	no	—
98	C₈	—	218	2	no	—
99	C₈	—	218	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#13182008	1	1	yes	characteristic
102	C₂²	C₄×Dic₂₁₈	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	C₄²×D₁₀₉	1	1	yes	characteristic
106	C₄	#16812656	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#13182008	1	1	yes	characteristic
110	C₂	#20295074	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812634

C₂

yes

maximal

C₄×#13182013

C₂

yes

maximal, characteristic

C₂×#16812610

C₂

yes

maximal, characteristic

#20294790

C₂

yes

maximal

C₄×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812634

C₄

yes

complemented

#16812610

—

#16812610

C₄

yes

—

C₄×#11724784

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

#13182013

—

#13182013

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

#11724784

C₄²

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#430

yes

—

#1778

—

109

maximal, Sylow

C₁₀₉

#1778

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

C₄×M₁₆

—

109

—

C₄×M₁₆

D₁₀₉

yes

characteristic

C₂×#97

—

109

—

#341

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#108

—

109

—

#97

—

#97

—

109

—

#97

—

218

—

C₄×C₈

—

C₄×C₈

—

218

—

C₂²×C₄

Dic₂₁₈

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

C₄×D₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

C₄×D₁₀₉

yes

—

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₂³

C₂×Dic₂₁₈

yes

characteristic

C₂×C₄

—

C₂×C₄

#13182008

yes

—

C₂×C₄

#13182008

yes

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

#13182029

yes

—

C₈

—

C₈

—

109

—

C₈

—

218

—

C₈

—

218

—

100

C₂²

—

101

C₂²

C₂×#13182008

yes

characteristic

102

C₂²

C₄×Dic₂₁₈

yes

—

103

C₄

—

104

C₄

—

105

C₄

C₄²×D₁₀₉

yes

characteristic

106

C₄

#16812656

yes

characteristic

107

C₄

C₄×C₂²⋊D₁₀₉

yes

—

108

C₂

—

109

C₂

C₄×#13182008

yes

characteristic

110

C₂

#20295074

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₄×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#1778
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₄×M₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₄×M₁₆

D₁₀₉

hypocenter

C₁₀₉

#1778

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #16812634	C₄	semidirect
56. C₁₀₉	#1778	semidirect

H ≅ G/N

type

9. #16812634

C₄

semidirect

56. C₁₀₉

#1778

semidirect

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

subgroup	index
C₂×#16812634	2
C₄×#13182013	2
C₂×#16812610	2
#20294790	2
C₄×M₁₆×C₁₀₉	2
#1778	109

subgroup

index

C₄×M₁₆×C₁₀₉

109

Subgroups — #28362939

Conjugacy classes of subgroups

Complements

Maximal subgroups