Group #28378958 — subgroups

Subgroups — #28378958

Order 13952.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20296031	C₂	1	1	yes	maximal, characteristic
2	#20296030	C₂	1	1	yes	maximal, characteristic
3	#20295902	C₂	1	4	yes	maximal
4	#97×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	C₂²	1	2	yes	—
6	C₂×#13182039	C₂²	1	2	yes	—
7	#16812766	—	2	2	no	—
8	C₄⋊#11724786	—	2	2	no	—
9	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
10	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
11	#16812610	C₄	1	1	yes	characteristic
12	#16812609	C₄	1	2	yes	—
13	#97×C₁₀₉	C₄	1	1	yes	characteristic
14	#13182039	—	2	4	no	—
15	#13182039	—	2	4	no	—
16	C₂×#11724786	—	2	2	no	—
17	C₂×#11724786	—	2	2	no	—
18	C₂×C₄×D₁₀₉	Q₈	1	2	yes	—
19	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×#11724784	C₂×C₄	1	2	yes	—
21	C₈×D₁₀₉	—	2	2	no	—
22	C₈×D₁₀₉	—	2	2	no	—
23	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₀₉	D₄	1	2	yes	—
25	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₄×D₁₀₉	—	4	2	no	—
33	C₄×D₁₀₉	#41	1	2	yes	—
34	C₄×D₁₀₉	#41	1	2	yes	—
35	#11724786	—	4	2	no	—
36	#11724786	—	4	2	no	—
37	C₈×C₁₀₉	—	2	2	no	—
38	#11724784	—	2	2	no	—
39	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×D₁₀₉	—	2	2	no	—
41	C₂×D₁₀₉	#125	1	2	yes	—
42	C₄×C₁₀₉	—	2	2	no	—
43	C₄×C₁₀₉	#117	1	2	yes	—
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	#124	1	1	yes	characteristic
46	Dic₁₀₉	#124	1	1	yes	characteristic
47	C₂×C₁₀₉	#382	1	1	yes	characteristic
48	C₂×C₁₀₉	#430	1	2	yes	—
49	D₁₀₉	—	2	2	no	—
50	#1838	—	109	1	no	maximal, Sylow
51	C₁₀₉	#1838	1	1	yes	characteristic, complemented, Sylow
52	#423	—	109	1	no	—
53	#423	—	109	1	no	—
54	C₂×#97	—	109	1	no	—
55	#341	—	109	4	no	—
56	C₂×M₁₆	—	109	2	no	—
57	C₂×M₁₆	—	109	2	no	—
58	C₂²×C₈	—	109	2	no	—
59	C₂×C₄²	—	109	1	no	—
60	C₄⋊C₈	—	218	2	no	—
61	C₄⋊C₈	—	218	2	no	—
62	#97	—	109	2	no	—
63	#97	—	109	1	no	—
64	#97	#89137	1	1	yes	characteristic
65	C₂²×C₄	—	109	1	no	—
66	C₂²×C₄	—	109	2	no	—
67	M₁₆	—	218	4	no	—
68	M₁₆	—	218	4	no	—
69	C₂×C₈	—	109	2	no	—
70	C₂×C₈	—	218	2	no	—
71	C₂×C₈	—	218	2	no	—
72	C₂×C₈	—	218	2	no	—
73	C₂×C₈	—	218	2	no	—
74	C₂×C₈	C₂×#89137	1	2	yes	—
75	C₄²	—	109	2	no	—
76	C₄²	—	109	1	no	—
77	C₄²	C₂×#89137	1	1	yes	characteristic
78	C₂³	—	109	1	no	—
79	C₂×C₄	—	109	1	no	—
80	C₂×C₄	—	109	2	no	—
81	C₂×C₄	—	109	2	no	—
82	C₂×C₄	—	109	2	no	—
83	C₂×C₄	—	436	2	no	—
84	C₂×C₄	C₄⋊#89137	1	2	yes	—
85	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	436	2	no	—
88	C₈	—	436	2	no	—
89	C₈	—	218	2	no	—
90	C₂²	—	109	2	no	—
91	C₂²	—	218	2	no	—
92	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	109	1	no	—
95	C₄	—	109	1	no	—
96	C₄	—	218	2	no	—
97	C₄	#16812778	1	2	yes	—
98	C₂	—	218	2	no	—
99	C₂	#20295970	1	1	yes	characteristic
100	C₂	#20296041	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20296031

C₂

yes

maximal, characteristic

#20296030

C₂

yes

maximal, characteristic

#20295902

C₂

yes

maximal

#97×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×#13182039

C₂²

yes

—

#16812766

—

C₄⋊#11724786

—

C₂×C₈×D₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812610

C₄

yes

characteristic

#16812609

C₄

yes

—

#97×C₁₀₉

C₄

yes

—

—

—

—

C₂×C₄×D₁₀₉

Q₈

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

#41

yes

—

C₄×D₁₀₉

#41

yes

—

#11724786

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

#125

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#117

yes

—

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

D₁₀₉

—

#1838

—

109

maximal, Sylow

C₁₀₉

#1838

yes

characteristic, complemented, Sylow

#423

—

109

—

#423

—

109

—

C₂×#97

—

109

—

#341

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

#97

—

109

—

#97

—

109

—

#97

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

C₄⋊#89137

yes

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

436

—

C₈

—

436

—

C₈

—

218

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

C₂×C₄⋊#89137

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

109

—

C₄

—

218

—

C₄

#16812778

yes

—

C₂

—

218

—

C₂

#20295970

yes

characteristic

100

C₂

#20296041

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#97×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₂×Q₈
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#97	#89137
hypocenter	C₁₀₉	#1838
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

#97×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₂×Q₈

cosocle

C₂×C₄×D₁₀₉

C₂³

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
51. C₁₀₉	#1838	semidirect

H ≅ G/N

type

51. C₁₀₉

#1838

semidirect

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

subgroup	index
#20296031	2
#20296030	2
#20295902	2
#97×D₁₀₉	2
#1838	109

subgroup

index

#97×D₁₀₉

109

Subgroups — #28378958

Conjugacy classes of subgroups

Complements

Maximal subgroups