Group #28378591 — subgroups

Subgroups — #28378591

Order 13952.

184 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20296030	C₂	1	2	yes	maximal
2	#20295890	C₂	1	4	yes	maximal, complemented
3	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182039	—	2	2	no	—
5	C₄⋊#11724786	—	2	2	no	—
6	C₄⋊#11724786	C₂²	1	4	yes	—
7	C₂×C₈×D₁₀₉	C₄	1	2	yes	—
8	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₄×#11724784	C₄	1	1	yes	characteristic
10	C₈×Dic₁₀₉	C₂²	1	2	yes	—
11	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
12	#13182039	—	4	4	no	—
13	C₂×#11724786	—	2	4	no	—
14	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
17	C₂×#11724784	C₂×C₄	1	2	yes	—
18	C₈×D₁₀₉	—	2	4	no	—
19	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
20	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
21	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
31	C₄×D₁₀₉	—	2	2	no	—
32	C₄×D₁₀₉	—	2	2	no	—
33	C₄×D₁₀₉	D₈	1	2	yes	—
34	C₄×D₁₀₉	Q₁₆	1	2	yes	—
35	#11724786	—	4	4	no	—
36	C₈×C₁₀₉	—	2	2	no	—
37	#11724784	M₁₆	1	4	yes	complemented
38	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×D₁₀₉	—	2	1	no	—
40	C₂×D₁₀₉	—	2	1	no	—
41	C₂×D₁₀₉	#107	1	2	yes	—
42	C₄×C₁₀₉	#107	1	2	yes	—
43	C₄×C₁₀₉	#108	1	2	yes	—
44	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
45	Dic₁₀₉	—	2	1	no	—
46	Dic₁₀₉	—	2	1	no	—
47	Dic₁₀₉	C₂×D₈	1	1	yes	characteristic
48	Dic₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₁₀₉	#423	1	1	yes	characteristic
50	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
51	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
52	D₁₀₉	—	2	2	no	—
53	#1567	—	109	1	no	maximal, Sylow
54	C₁₀₉	#1567	1	1	yes	characteristic, complemented, Sylow
55	#423	—	109	2	no	—
56	C₂×C₄×C₈	—	109	1	no	—
57	#335	—	109	4	no	—
58	C₂×M₁₆	—	218	2	no	—
59	C₂²×C₈	—	109	2	no	—
60	C₂×C₄²	—	109	1	no	—
61	C₄⋊C₈	—	109	4	no	—
62	C₄⋊C₈	—	218	2	no	—
63	C₄×C₈	—	109	2	no	—
64	C₄×C₈	—	109	1	no	—
65	C₄×C₈	#89137	1	1	yes	characteristic
66	C₂²×C₄	—	109	1	no	—
67	C₂²×C₄	—	109	1	no	—
68	C₂²×C₄	—	109	1	no	—
69	M₁₆	—	436	4	no	—
70	C₂×C₈	—	109	2	no	—
71	C₂×C₈	—	218	4	no	—
72	C₂×C₈	—	218	4	no	—
73	C₂×C₈	C₂×#89137	1	2	yes	—
74	C₄²	—	109	1	no	—
75	C₄²	—	109	1	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	1	no	—
83	C₂×C₄	—	109	1	no	—
84	C₂×C₄	—	218	2	no	—
85	C₂×C₄	—	218	2	no	—
86	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
87	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
88	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	109	4	no	—
91	C₈	—	436	4	no	—
92	C₂²	—	109	2	no	—
93	C₂²	—	218	1	no	—
94	C₂²	—	218	1	no	—
95	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
96	C₄	—	109	1	no	—
97	C₄	—	109	1	no	—
98	C₄	—	218	1	no	—
99	C₄	—	218	1	no	—
100	C₄	#16812768	1	2	yes	—
101	C₄	#16812769	1	2	yes	—
102	C₄	D₁₀₉⋊M₁₆	1	2	yes	—
103	C₂	—	218	2	no	—
104	C₂	#20296031	1	1	yes	characteristic
105	C₂	C₂×#16812768	1	1	yes	characteristic
106	C₂	C₂×#16812769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296030

C₂

yes

maximal

#20295890

C₂

yes

maximal, complemented

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

—

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

—

C₂×C₈×D₁₀₉

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

C₂×#11724786

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₈

yes

—

C₄×D₁₀₉

Q₁₆

yes

—

#11724786

—

C₈×C₁₀₉

—

#11724784

M₁₆

yes

complemented

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#107

yes

—

C₄×C₁₀₉

#107

yes

—

C₄×C₁₀₉

#108

yes

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×D₈

yes

characteristic

Dic₁₀₉

C₂×Q₁₆

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1567

—

109

maximal, Sylow

C₁₀₉

#1567

yes

characteristic, complemented, Sylow

#423

—

109

—

C₂×C₄×C₈

—

109

—

#335

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

436

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

—

C₄²

—

109

—

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

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

436

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×C₄⋊#89137

yes

characteristic

C₄

—

109

—

C₄

—

109

—

C₄

—

218

—

C₄

—

218

—

100

C₄

#16812768

yes

—

101

C₄

#16812769

yes

—

102

C₄

D₁₀₉⋊M₁₆

yes

—

103

C₂

—

218

—

104

C₂

#20296031

yes

characteristic

105

C₂

C₂×#16812768

yes

characteristic

106

C₂

C₂×#16812769

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1567
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. #20295890	C₂	semidirect
37. #11724784	M₁₆	semidirect
54. C₁₀₉	#1567	semidirect

H ≅ G/N

type

2. #20295890

C₂

semidirect

37. #11724784

M₁₆

semidirect

54. C₁₀₉

#1567

semidirect

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

subgroup	index
#20296030	2
#20295890	2
C₄×C₈×D₁₀₉	2
#1567	109

subgroup

index

#20296030

#20295890

C₄×C₈×D₁₀₉

#1567

109

Subgroups — #28378591

Conjugacy classes of subgroups

Complements

Maximal subgroups