Group #28336089 — subgroups

Subgroups — #28336089

Order 13696.

220 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20293004	C₂	1	1	yes	maximal, characteristic
3	#20293000	C₂	1	2	yes	maximal, complemented
4	#20292974	C₂	1	2	yes	maximal, complemented
5	#322×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133807	—	2	4	no	—
7	C₂×D₈₅₆	—	2	2	no	—
8	#16812109	C₂²	1	2	yes	—
9	#16812106	C₂²	1	1	yes	characteristic
10	#16812106	C₂²	1	1	yes	characteristic
11	#16812041	—	2	2	no	—
12	#16812031	—	2	4	no	—
13	C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₄	—	2	2	no	—
16	C₂×D₄₂₈	—	2	1	no	—
17	C₂×D₄₂₈	—	2	1	no	—
18	C₂×D₄₂₈	—	2	2	no	—
19	C₂×D₄₂₈	—	2	2	no	—
20	#13133807	—	4	8	no	—
21	D₈₅₆	—	4	4	no	—
22	C₄⋊Dic₁₀₇	—	2	2	no	—
23	C₄⋊Dic₁₀₇	—	2	2	no	—
24	C₂×C₈×C₁₀₇	D₄	1	4	yes	—
25	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
26	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
27	C₂²×D₁₀₇	—	4	1	no	—
28	C₂²×D₁₀₇	—	4	1	no	—
29	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
31	C₂×Dic₁₀₇	—	4	2	no	—
32	D₄₂₈	—	4	2	no	—
33	D₄₂₈	—	4	2	no	—
34	D₄₂₈	—	4	4	no	—
35	D₄₂₈	—	4	4	no	—
36	Dic₂₁₄	—	4	4	no	—
37	C₈×C₁₀₇	—	2	4	no	—
38	C₈×C₁₀₇	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₀₇	#127	1	1	yes	characteristic
40	C₂×D₁₀₇	—	8	1	no	—
41	C₂×D₁₀₇	—	8	1	no	—
42	C₂×D₁₀₇	—	8	2	no	—
43	C₂×D₁₀₇	—	8	2	no	—
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	#492	1	1	yes	characteristic
48	C₂×C₁₀₇	#496	1	2	yes	—
49	D₁₀₇	—	16	1	no	—
50	D₁₀₇	—	16	1	no	—
51	#1715	—	107	1	no	maximal, Sylow
52	C₁₀₇	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	—	107	1	no	—
54	#493	—	107	1	no	—
55	#492	—	107	2	no	—
56	#486	—	107	2	no	—
57	#322	D₁₀₇	1	1	yes	characteristic, complemented
58	C₂×SD₁₆	—	214	4	no	—
59	C₂×D₈	—	214	2	no	—
60	C₄⋊Q₈	—	107	2	no	—
61	#127	—	107	1	no	—
62	#127	—	107	1	no	—
63	#106	—	214	2	no	—
64	#102	—	214	4	no	—
65	C₄×C₈	C₂×D₁₀₇	1	2	yes	—
66	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
67	C₂×Q₈	—	214	2	no	—
68	C₂×D₄	—	214	2	no	—
69	C₂×D₄	—	214	1	no	—
70	C₂×D₄	—	214	1	no	—
71	C₂×D₄	—	214	2	no	—
72	SD₁₆	—	428	8	no	—
73	D₈	—	428	4	no	—
74	C₂×C₈	D₄₂₈	1	4	yes	—
75	C₂×C₈	D₄₂₈	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	—	428	1	no	—
80	C₂³	—	428	1	no	—
81	Q₈	—	428	4	no	—
82	D₄	—	428	2	no	—
83	D₄	—	428	2	no	—
84	D₄	—	428	4	no	—
85	D₄	—	428	4	no	—
86	C₂×C₄	—	428	2	no	—
87	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄₂₈	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#13133807	1	4	yes	complemented
91	C₂²	—	856	1	no	—
92	C₂²	—	856	1	no	—
93	C₂²	—	856	2	no	—
94	C₂²	—	856	2	no	—
95	C₂²	#16812106	1	1	yes	characteristic
96	C₄	—	856	2	no	—
97	C₄	C₂×#13133807	1	2	yes	—
98	C₄	#16812137	1	4	yes	—
99	C₂	—	1712	1	no	—
100	C₂	—	1712	1	no	—
101	C₂	#20293000	1	1	yes	characteristic
102	C₂	#20293016	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20293025

C₂

yes

maximal, characteristic, complemented

#20293004

C₂

yes

maximal, characteristic

#20293000

C₂

yes

maximal, complemented

#20292974

C₂

yes

maximal, complemented

#322×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×#13133807

—

C₂×D₈₅₆

—

#16812109

C₂²

yes

—

#16812106

C₂²

yes

characteristic

#16812106

C₂²

yes

characteristic

#16812041

—

#16812031

—

C₄×C₈×C₁₀₇

C₂²

yes

—

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

C₂×D₄₂₈

—

C₂×D₄₂₈

—

C₂×D₄₂₈

—

C₂×D₄₂₈

—

#13133807

—

D₈₅₆

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

—

C₂²×D₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×Dic₁₀₇

—

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₈×C₁₀₇

SD₁₆

yes

complemented

C₂²×C₁₀₇

#127

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₄×C₁₀₇

AGL(1,ℤ/8)

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

#492

yes

characteristic

C₂×C₁₀₇

#496

yes

—

D₁₀₇

—

D₁₀₇

—

#1715

—

107

maximal, Sylow

C₁₀₇

#1715

yes

characteristic, complemented, Sylow

#498

—

107

—

#493

—

107

—

#492

—

107

—

#486

—

107

—

#322

D₁₀₇

yes

characteristic, complemented

C₂×SD₁₆

—

214

—

C₂×D₈

—

214

—

C₄⋊Q₈

—

107

—

#127

—

107

—

#127

—

107

—

#106

—

214

—

#102

—

214

—

C₄×C₈

C₂×D₁₀₇

yes

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×Q₈

—

214

—

C₂×D₄

—

214

—

C₂×D₄

—

214

—

C₂×D₄

—

214

—

C₂×D₄

—

214

—

SD₁₆

—

428

—

D₈

—

428

—

C₂×C₈

D₄₂₈

yes

—

C₂×C₈

D₄₂₈

yes

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

—

428

—

C₂³

—

428

—

Q₈

—

428

—

D₄

—

428

—

D₄

—

428

—

D₄

—

428

—

D₄

—

428

—

C₂×C₄

—

428

—

C₂×C₄

C₂×D₄₂₈

yes

characteristic

C₂×C₄

C₂×D₄₂₈

yes

—

C₈

—

C₈

#13133807

yes

complemented

C₂²

—

856

—

C₂²

—

856

—

C₂²

—

856

—

C₂²

—

856

—

C₂²

#16812106

yes

characteristic

C₄

—

856

—

C₄

C₂×#13133807

yes

—

C₄

#16812137

yes

—

C₂

—

1712

—

100

C₂

—

1712

—

101

C₂

#20293000

yes

characteristic

102

C₂

#20293016

yes

—

	subgroup	G/H
center	C₂²	#16812106
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#322×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#127
prime radical	#20293004	C₂
hypercenter	#322	D₁₀₇
hypocenter	C₁₀₇	#1715
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#16812106

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#322×C₁₀₇

C₂

C₂²×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #20293025	C₂	semidirect
3. #20293000	C₂	semidirect
4. #20292974	C₂	semidirect
5. #322×C₁₀₇	C₂	semidirect
38. C₈×C₁₀₇	SD₁₆	semidirect
52. C₁₀₇	#1715	semidirect
57. #322	D₁₀₇	semidirect
90. C₈	#13133807	semidirect

H ≅ G/N

type

1. #20293025

C₂

semidirect

3. #20293000

C₂

semidirect

4. #20292974

C₂

semidirect

5. #322×C₁₀₇

C₂

semidirect

38. C₈×C₁₀₇

SD₁₆

semidirect

52. C₁₀₇

#1715

semidirect

57. #322

D₁₀₇

semidirect

90. C₈

#13133807

semidirect

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

subgroup	index
#20293025	2
#20293004	2
#20293000	2
#20292974	2
#322×C₁₀₇	2
#1715	107

subgroup

index

#322×C₁₀₇

107

Subgroups — #28336089

Conjugacy classes of subgroups

Complements

Maximal subgroups