Group #28335199 — subgroups

Subgroups — #28335199

Order 13696.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20292651	C₂	1	2	yes	maximal
2	#20292373	C₂	1	4	yes	maximal, complemented
3	#533×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133802	—	2	2	no	—
5	C₄⋊(C₁₀₇⋊C₈)	—	2	2	no	—
6	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	4	yes	—
7	C₄⋊Q₈×C₁₀₇	C₂²	1	2	yes	—
8	#125×C₁₀₇	C₄	1	2	yes	—
9	C₂²⋊Q₈×C₁₀₇	—	2	2	no	—
10	C₄×Q₈×C₁₀₇	—	2	2	no	—
11	#117×C₁₀₇	C₄	1	2	yes	—
12	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
13	#13133802	—	4	4	no	—
14	C₂×C₁₀₇⋊C₈	—	2	4	no	—
15	C₂×Q₈×C₁₀₇	—	2	2	no	—
16	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
18	C₄⋊C₄×C₁₀₇	—	2	4	no	—
19	C₄⋊C₄×C₁₀₇	—	2	2	no	—
20	C₄⋊C₄×C₁₀₇	C₂×C₄	1	4	yes	—
21	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
22	C₄²×C₁₀₇	—	2	2	no	—
23	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₀₇	D₄	1	2	yes	—
25	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
26	C₂³×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₁₀₇	—	4	2	no	—
28	Q₈×C₁₀₇	—	4	2	no	—
29	C₂×C₄×C₁₀₇	—	2	2	no	—
30	C₂×C₄×C₁₀₇	—	2	2	no	—
31	C₂×C₄×C₁₀₇	—	2	2	no	—
32	C₂×C₄×C₁₀₇	—	2	4	no	—
33	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
37	C₁₀₇⋊C₈	—	4	4	no	—
38	C₂²×C₁₀₇	—	2	1	no	—
39	C₂²×C₁₀₇	—	2	2	no	—
40	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₁₀₇	—	2	1	no	—
42	C₄×C₁₀₇	—	2	2	no	—
43	C₄×C₁₀₇	—	4	4	no	—
44	C₄×C₁₀₇	#101	1	2	yes	—
45	C₄×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
46	C₄×C₁₀₇	#137	1	2	yes	—
47	C₂×C₁₀₇	—	4	1	no	—
48	C₂×C₁₀₇	C₂×#101	1	1	yes	characteristic
49	C₂×C₁₀₇	#417	1	2	yes	—
50	#1561	—	107	1	no	maximal, Sylow
51	C₁₀₇	#1561	1	1	yes	characteristic, complemented, Sylow
52	#533	D₁₀₇	1	1	yes	characteristic
53	#423	—	107	2	no	—
54	#332	—	107	4	no	—
55	C₂×M₁₆	—	214	2	no	—
56	C₄⋊Q₈	C₂×D₁₀₇	1	2	yes	—
57	#125	Dic₁₀₇	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	Dic₁₀₇	1	2	yes	—
61	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
62	C₄⋊C₈	—	107	4	no	—
63	C₄⋊C₈	—	214	2	no	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
67	M₁₆	—	428	4	no	—
68	C₂×C₈	—	214	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	C₂×Dic₁₀₇	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
75	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
76	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
77	C₂³	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
85	C₂×C₄	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
86	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
88	C₈	—	428	4	no	—
89	C₂²	—	2	2	no	—
90	C₂²	—	2	1	no	—
91	C₂²	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	1	no	—
94	C₄	—	4	4	no	—
95	C₄	#16812028	1	2	yes	—
96	C₄	#16812134	1	2	yes	—
97	C₄	#16812139	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#16812028	1	1	yes	characteristic
100	C₂	#20292628	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20292651

C₂

yes

maximal

#20292373

C₂

yes

maximal, complemented

#533×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

—

C₄⋊(C₁₀₇⋊C₈)

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

—

#125×C₁₀₇

C₄

yes

—

C₂²⋊Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

—

#117×C₁₀₇

C₄

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂×Q₈×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

D₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#101

yes

—

C₄×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₇

#137

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×#101

yes

characteristic

C₂×C₁₀₇

#417

yes

—

#1561

—

107

maximal, Sylow

C₁₀₇

#1561

yes

characteristic, complemented, Sylow

#533

D₁₀₇

yes

characteristic

#423

—

107

—

#332

—

107

—

C₂×M₁₆

—

214

—

C₄⋊Q₈

C₂×D₁₀₇

yes

—

#125

Dic₁₀₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

107

—

C₄⋊C₈

—

214

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

M₁₆

—

428

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₀₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₁₀₇

yes

characteristic

C₄²

C₂²⋊D₁₀₇

yes

—

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

C₂²⋊Dic₁₀₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

C₂×C₄

C₂²⋊Dic₁₀₇

yes

characteristic

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

C₈

—

428

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#16812028

yes

—

C₄

#16812134

yes

—

C₄

#16812139

yes

—

C₂

—

C₂

C₂×#16812028

yes

characteristic

100

C₂

#20292628

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#533×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₇	C₂²⋊C₄
cosocle	C₄²×C₁₀₇	C₂³
hypercenter	#533	D₁₀₇
hypocenter	C₁₀₇	#1561
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₇

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#533×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₇

C₂²⋊C₄

cosocle

C₄²×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20292373	C₂	semidirect
51. C₁₀₇	#1561	semidirect

H ≅ G/N

type

2. #20292373

C₂

semidirect

51. C₁₀₇

#1561

semidirect

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

subgroup	index
#20292651	2
#20292373	2
#533×C₁₀₇	2
#1561	107

subgroup

index

#20292651

#20292373

#533×C₁₀₇

#1561

107

Subgroups — #28335199

Conjugacy classes of subgroups

Complements

Maximal subgroups