Group #28377902 — subgroups

Subgroups — #28377902

Order 13952.

152 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 56 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	#20296029	C₂	1	2	yes	maximal
2	C₂×#16812628	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182038	—	2	2	no	—
4	#16812766	C₄	1	4	yes	—
5	C₄⋊#11724786	—	2	2	no	—
6	C₂²×#11724784	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
8	#16812628	—	2	2	no	—
9	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
10	#13182038	—	4	2	no	—
11	#13182038	—	4	2	no	—
12	C₂×#11724786	—	2	4	no	—
13	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
14	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
15	C₂×#11724784	—	2	1	no	—
16	C₂×#11724784	—	4	1	no	—
17	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₀₉	D₄	1	2	yes	—
19	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	—	2	1	no	—
21	C₂×C₈×C₁₀₉	—	4	1	no	—
22	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₁₀₉	—	4	1	no	—
28	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
29	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
30	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
31	C₂×Dic₁₀₉	SD₁₆	1	2	yes	—
32	C₂×Dic₁₀₉	Q₁₆	1	1	yes	characteristic
33	#11724786	—	4	4	no	—
34	C₈×C₁₀₉	—	4	1	no	—
35	#11724784	—	4	1	no	—
36	C₂²×C₁₀₉	—	2	1	no	—
37	C₂²×C₁₀₉	—	2	1	no	—
38	C₂²×C₁₀₉	#95	1	1	yes	characteristic
39	C₂²×C₁₀₉	#106	1	1	yes	characteristic
40	C₂²×C₁₀₉	#107	1	1	yes	characteristic
41	C₄×C₁₀₉	—	2	1	no	—
42	C₄×C₁₀₉	#100	1	1	yes	characteristic
43	C₄×C₁₀₉	#101	1	1	yes	characteristic
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	#102	1	2	yes	—
46	Dic₁₀₉	#103	1	2	yes	—
47	C₂×C₁₀₉	—	2	1	no	—
48	C₂×C₁₀₉	—	2	1	no	—
49	C₂×C₁₀₉	#338	1	1	yes	characteristic
50	C₂×C₁₀₉	#340	1	1	yes	characteristic
51	C₂×C₁₀₉	#343	1	1	yes	characteristic
52	#1292	—	109	1	no	maximal, Sylow
53	C₁₀₉	#1292	1	1	yes	characteristic, complemented, Sylow
54	#423	—	109	2	no	—
55	C₂×C₄⋊C₈	—	109	1	no	—
56	C₂×M₁₆	—	218	2	no	—
57	C₂²×C₈	—	109	1	no	—
58	C₂²×C₈	#89137	1	1	yes	characteristic
59	C₂×C₄²	—	109	1	no	—
60	C₄⋊C₈	—	109	4	no	—
61	C₄⋊C₈	—	218	2	no	—
62	C₄⋊C₈	—	218	2	no	—
63	C₂²×C₄	—	109	1	no	—
64	C₂²×C₄	—	109	1	no	—
65	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
66	M₁₆	—	436	2	no	—
67	M₁₆	—	436	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	4	1	no	—
70	C₂×C₈	—	436	1	no	—
71	C₂×C₈	—	218	4	no	—
72	C₂×C₈	—	218	1	no	—
73	C₄²	—	109	2	no	—
74	C₄²	—	109	2	no	—
75	C₂³	C₄⋊#89137	1	1	yes	characteristic
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	109	2	no	—
79	C₂×C₄	—	109	2	no	—
80	C₂×C₄	—	109	1	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	109	2	no	—
83	C₂×C₄	—	436	1	no	—
84	C₂×C₄	C₄×#89137	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
86	C₈	—	4	1	no	—
87	C₈	—	436	4	no	—
88	C₈	—	436	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
92	C₂²	#16812767	1	1	yes	characteristic
93	C₂²	#16812768	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	109	2	no	—
96	C₄	—	109	2	no	—
97	C₄	—	218	2	no	—
98	C₄	#16812759	1	1	yes	characteristic
99	C₄	#16812760	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	1	no	—
102	C₂	#20295895	1	1	yes	characteristic
103	C₂	#20295900	1	1	yes	characteristic
104	C₂	#20295906	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296029

C₂

yes

maximal

C₂×#16812628

C₂

yes

maximal, characteristic

C₂×#13182038

—

#16812766

C₄

yes

—

C₄⋊#11724786

—

C₂²×#11724784

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812628

—

C₂²×C₈×C₁₀₉

C₄

yes

—

—

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

Q₈

yes

characteristic

C₂×#11724784

—

C₂×#11724784

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

D₈

yes

characteristic

C₂×Dic₁₀₉

SD₁₆

yes

—

C₂×Dic₁₀₉

Q₁₆

yes

characteristic

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

#102

yes

—

Dic₁₀₉

#103

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1292

yes

characteristic, complemented, Sylow

#423

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

109

—

C₂²×C₈

#89137

yes

characteristic

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

M₁₆

—

436

—

M₁₆

—

436

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₄⋊#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

436

—

C₈

—

436

—

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

109

—

C₄

—

109

—

C₄

—

218

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

—

102

C₂

#20295895

yes

characteristic

103

C₂

#20295900

yes

characteristic

104

C₂

#20295906

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#89137)⋊C₄
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₂²×C₄	C₂×#89137
Fitting	C₂²×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₂×C₄×Dic₁₀₉	C₂²
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	#1292
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂²

(C₂×#89137)⋊C₄

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₂²×C₄

C₂×#89137

Fitting

C₂²×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₂×C₄×Dic₁₀₉

C₂²

C₂²×C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
53. C₁₀₉	#1292	semidirect

H ≅ G/N

type

53. C₁₀₉

#1292

semidirect

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

subgroup	index
#20296029	2
C₂×#16812628	2
#1292	109

subgroup

index

#20296029

C₂×#16812628

#1292

109

Subgroups — #28377902

Conjugacy classes of subgroups

Complements

Maximal subgroups