Group #28336087 — subgroups

Subgroups — #28336087

Order 13696.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20293025	C₂	1	1	yes	maximal, characteristic, complemented
2	#20292999	C₂	1	2	yes	maximal, complemented
3	#20292973	C₂	1	2	yes	maximal, complemented
4	#20292348	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133806	—	2	4	no	—
7	#16812109	C₂²	1	2	yes	—
8	#16812041	—	2	2	no	—
9	#16812030	—	2	4	no	—
10	C₄×C₁₀₇⋊C₈	C₂²	1	2	yes	—
11	C₂×D₈×C₁₀₇	—	2	2	no	—
12	#127×C₁₀₇	C₂²	1	1	yes	characteristic
13	#127×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₄	—	2	2	no	—
16	#13133806	—	4	8	no	—
17	C₂×C₁₀₇⋊C₈	D₄	1	4	yes	—
18	C₄⋊Dic₁₀₇	—	2	2	no	—
19	C₄⋊Dic₁₀₇	—	2	2	no	—
20	C₂×D₄×C₁₀₇	—	2	2	no	—
21	C₂×D₄×C₁₀₇	—	2	1	no	—
22	C₂×D₄×C₁₀₇	—	2	1	no	—
23	C₂×D₄×C₁₀₇	—	2	2	no	—
24	D₈×C₁₀₇	—	4	4	no	—
25	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
26	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
27	C₂³×C₁₀₇	—	4	1	no	—
28	C₂³×C₁₀₇	—	4	1	no	—
29	D₄×C₁₀₇	—	4	2	no	—
30	D₄×C₁₀₇	—	4	2	no	—
31	D₄×C₁₀₇	—	4	4	no	—
32	D₄×C₁₀₇	—	4	4	no	—
33	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₇	—	4	2	no	—
36	Dic₂₁₄	—	4	4	no	—
37	C₈×C₁₀₇	SD₁₆	1	4	yes	complemented
38	C₁₀₇⋊C₈	—	2	4	no	—
39	C₂²×C₁₀₇	—	8	1	no	—
40	C₂²×C₁₀₇	—	8	1	no	—
41	C₂²×C₁₀₇	—	8	2	no	—
42	C₂²×C₁₀₇	—	8	2	no	—
43	C₂²×C₁₀₇	#127	1	1	yes	characteristic
44	C₄×C₁₀₇	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₀₇	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₀₇	—	8	2	no	—
47	C₂×C₁₀₇	—	16	1	no	—
48	C₂×C₁₀₇	—	16	1	no	—
49	C₂×C₁₀₇	#492	1	1	yes	characteristic
50	C₂×C₁₀₇	#496	1	2	yes	—
51	#1715	—	107	1	no	maximal, Sylow
52	C₁₀₇	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	—	107	1	no	—
54	#493	D₁₀₇	1	1	yes	characteristic
55	#492	—	107	2	no	—
56	#486	—	107	2	no	—
57	#322	—	107	1	no	—
58	C₂×SD₁₆	—	214	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	107	2	no	—
61	#127	C₂×D₁₀₇	1	1	yes	characteristic
62	#127	C₂×D₁₀₇	1	1	yes	characteristic
63	#106	—	214	2	no	—
64	#102	—	214	4	no	—
65	C₄×C₈	—	107	2	no	—
66	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
67	C₂×Q₈	—	214	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	SD₁₆	—	428	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	—	107	4	no	—
75	C₂×C₈	C₂²⋊D₁₀₇	1	2	yes	—
76	C₄⋊C₄	—	214	2	no	—
77	C₄⋊C₄	—	214	2	no	—
78	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	428	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	428	2	no	—
87	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
89	C₈	—	214	4	no	—
90	C₈	#13133806	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	#16812105	1	1	yes	characteristic
96	C₄	—	856	2	no	—
97	C₄	C₂×#13133806	1	2	yes	—
98	C₄	#16812133	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#20292999	1	1	yes	characteristic
102	C₂	#20293015	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20293025

C₂

yes

maximal, characteristic, complemented

#20292999

C₂

yes

maximal, complemented

#20292973

C₂

yes

maximal, complemented

#20292348

C₂

yes

maximal, characteristic, complemented

#493×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133806

—

#16812109

C₂²

yes

—

#16812041

—

#16812030

—

C₄×C₁₀₇⋊C₈

C₂²

yes

—

C₂×D₈×C₁₀₇

—

#127×C₁₀₇

C₂²

yes

characteristic

#127×C₁₀₇

C₂²

yes

characteristic

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

#13133806

—

C₂×C₁₀₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

D₈×C₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×Dic₁₀₇

—

Dic₂₁₄

—

C₈×C₁₀₇

SD₁₆

yes

complemented

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#127

yes

characteristic

C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₄×C₁₀₇

AGL(1,ℤ/8)

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#492

yes

characteristic

C₂×C₁₀₇

#496

yes

—

#1715

—

107

maximal, Sylow

C₁₀₇

#1715

yes

characteristic, complemented, Sylow

#498

—

107

—

#493

D₁₀₇

yes

characteristic

#492

—

107

—

#486

—

107

—

#322

—

107

—

C₂×SD₁₆

—

214

—

C₂×D₈

—

C₄⋊Q₈

—

107

—

#127

C₂×D₁₀₇

yes

characteristic

#127

C₂×D₁₀₇

yes

characteristic

#106

—

214

—

#102

—

214

—

C₄×C₈

—

107

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×Q₈

—

214

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

428

—

D₈

—

C₂×C₈

—

107

—

C₂×C₈

C₂²⋊D₁₀₇

yes

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

428

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

428

—

C₂×C₄

D₄×D₁₀₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

C₈

—

214

—

C₈

#13133806

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#16812105

yes

characteristic

C₄

—

856

—

C₄

C₂×#13133806

yes

—

C₄

#16812133

yes

—

C₂

—

100

C₂

—

101

C₂

#20292999

yes

characteristic

102

C₂

#20293015

yes

—

	subgroup	G/H
center	C₂²	#16812105
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting, prime radical	#493×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#127
hypercenter	#493	D₁₀₇
hypocenter	C₁₀₇	#1715
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#16812105

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting, prime radical

#493×C₁₀₇

C₂

C₂²×C₁₀₇

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #20293025	C₂	semidirect
2. #20292999	C₂	semidirect
3. #20292973	C₂	semidirect
4. #20292348	C₂	semidirect
37. C₈×C₁₀₇	SD₁₆	semidirect
52. C₁₀₇	#1715	semidirect
90. C₈	#13133806	semidirect

H ≅ G/N

type

1. #20293025

C₂

semidirect

2. #20292999

C₂

semidirect

3. #20292973

C₂

semidirect

4. #20292348

C₂

semidirect

37. C₈×C₁₀₇

SD₁₆

semidirect

52. C₁₀₇

#1715

semidirect

90. C₈

#13133806

semidirect

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

subgroup	index
#20293025	2
#20292999	2
#20292973	2
#20292348	2
#493×C₁₀₇	2
#1715	107

subgroup

index

#493×C₁₀₇

107

Subgroups — #28336087

Conjugacy classes of subgroups

Complements

Maximal subgroups