Group #28338478 — subgroups

Subgroups — #28338478

Order 13696.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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	#20292799	C₂	1	3	yes	maximal, complemented
2	#20292424	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812157	C₂²	1	1	yes	characteristic, complemented
5	#16812135	—	2	3	no	—
6	#16812037	—	2	3	no	—
7	#16812028	C₂²	1	3	yes	complemented
8	#124×C₁₀₇	—	2	3	no	—
9	#124×C₁₀₇	C₂²	1	3	yes	—
10	#120×C₁₀₇	—	2	3	no	—
11	#13133819	—	4	1	no	—
12	#13133818	—	2	3	no	—
13	Q₈×D₁₀₇	—	4	1	no	—
14	#13133806	—	4	3	no	—
15	#13133803	—	4	3	no	—
16	#13133802	—	2	3	no	—
17	C₂⁴×C₁₀₇	—	2	1	no	—
18	C₂×Q₈×C₁₀₇	—	2	1	no	—
19	C₂×Q₈×C₁₀₇	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₁₀₇	—	2	3	no	—
21	C₂×D₄×C₁₀₇	D₄	1	3	yes	—
22	C₂²⋊C₄×C₁₀₇	—	2	3	no	—
23	C₂²⋊C₄×C₁₀₇	—	2	6	no	—
24	C₄²×C₁₀₇	D₄	1	3	yes	—
25	C₂³×C₁₀₇	—	2	3	no	—
26	C₂³×C₁₀₇	—	8	2	no	—
27	C₂³×C₁₀₇	—	8	1	no	—
28	Q₈×C₁₀₇	—	4	1	no	—
29	Q₈×C₁₀₇	—	8	1	no	—
30	D₄×C₁₀₇	—	4	3	no	—
31	D₄×C₁₀₇	—	8	3	no	—
32	C₂×C₄×C₁₀₇	—	2	3	no	—
33	C₂×C₄×C₁₀₇	C₂×D₄	1	3	yes	—
34	C₂²⋊D₁₀₇	—	4	1	no	—
35	D₄₂₈	—	2	3	no	—
36	C₄×D₁₀₇	—	4	3	no	—
37	Dic₂₁₄	—	2	3	no	—
38	C₁₀₇⋊C₈	—	4	3	no	—
39	C₂²×C₁₀₇	—	4	6	no	—
40	C₂²×C₁₀₇	—	4	3	no	—
41	C₂²×C₁₀₇	—	8	4	no	—
42	C₂²×C₁₀₇	#120	1	1	yes	characteristic
43	C₂×D₁₀₇	—	4	1	no	—
44	C₄×C₁₀₇	—	2	3	no	—
45	C₄×C₁₀₇	—	4	3	no	—
46	Dic₁₀₇	—	4	1	no	—
47	C₂×C₁₀₇	—	2	1	no	—
48	C₂×C₁₀₇	—	8	3	no	—
49	C₂×C₁₀₇	B(4,𝔽₂)	1	1	yes	characteristic
50	D₁₀₇	—	8	1	no	—
51	#2202	—	107	1	no	maximal, Sylow
52	C₁₀₇	#2202	1	1	yes	characteristic, complemented, Sylow
53	H(3,𝔽₄)	D₁₀₇	1	1	yes	characteristic, complemented
54	#454	—	107	3	no	—
55	#355	—	107	3	no	—
56	2₋¹⁺⁴	—	107	1	no	—
57	AGL(1,ℤ/8)	—	214	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₁₀₇	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	214	3	no	—
62	#101	—	107	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	428	1	no	—
65	#41	—	214	3	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	428	1	no	—
68	C₂×Q₈	C₂²×D₁₀₇	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₁₀₇	1	3	yes	—
71	SD₁₆	—	428	3	no	—
72	D₈	—	428	3	no	—
73	M₁₆	—	214	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₁₀₇	1	3	yes	—
77	C₂³	—	2	3	no	—
78	C₂³	—	8	2	no	—
79	C₂³	—	8	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	8	1	no	—
82	Q₈	—	214	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	428	1	no	—
86	D₄	—	214	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	428	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	3	yes	—
90	C₈	—	428	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	428	1	no	—
95	C₂²	#16812078	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	428	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	856	1	no	—
102	C₂	#20292814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20292799

C₂

yes

maximal, complemented

#20292424

C₂

yes

maximal, complemented

H(3,𝔽₄)×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#16812157

C₂²

yes

characteristic, complemented

#16812135

—

#16812037

—

#16812028

C₂²

yes

complemented

#124×C₁₀₇

—

#124×C₁₀₇

C₂²

yes

—

#120×C₁₀₇

—

#13133819

—

#13133818

—

Q₈×D₁₀₇

—

#13133806

—

#13133803

—

#13133802

—

C₂⁴×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

C₂³

yes

characteristic

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

D₄

yes

—

C₂²⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂²⋊D₁₀₇

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#120

yes

characteristic

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

B(4,𝔽₂)

yes

characteristic

D₁₀₇

—

#2202

—

107

maximal, Sylow

C₁₀₇

#2202

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

D₁₀₇

yes

characteristic, complemented

#454

—

107

—

#355

—

107

—

2₋¹⁺⁴

—

107

—

AGL(1,ℤ/8)

—

214

—

#124

—

#124

C₂×D₁₀₇

yes

—

#120

—

#104

—

214

—

#101

—

107

—

C₂⁴

—

#41

—

428

—

#41

—

214

—

C₂×Q₈

—

C₂×Q₈

—

428

—

C₂×Q₈

C₂²×D₁₀₇

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₀₇

yes

—

SD₁₆

—

428

—

D₈

—

428

—

M₁₆

—

214

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₁₀₇

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

214

—

D₄

—

D₄

—

D₄

—

428

—

D₄

—

214

—

C₂×C₄

—

C₂×C₄

—

428

—

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

C₈

—

428

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

428

—

C₂²

#16812078

yes

characteristic

C₄

—

C₄

—

C₄

—

428

—

C₂

—

100

C₂

—

101

C₂

—

856

—

102

C₂

#20292814

yes

characteristic

	subgroup	G/H
center	C₂	#20292814
commutator, cosocle	C₂×Q₈×C₁₀₇	C₂³
Frattini	C₂×Q₈	C₂²×D₁₀₇
Fitting	H(3,𝔽₄)×C₁₀₇	C₂
socle	C₂×C₁₀₇	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₁₀₇
hypocenter	C₁₀₇	#2202
LCS	G ▷ C₂×Q₈×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#20292814

commutator, cosocle

C₂×Q₈×C₁₀₇

C₂³

Frattini

C₂×Q₈

C₂²×D₁₀₇

Fitting

H(3,𝔽₄)×C₁₀₇

C₂

C₂×C₁₀₇

D₁₀₇

C₁₀₇

G ▷ C₂×Q₈×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #20292799	C₂	semidirect
2. #20292424	C₂	semidirect
3. H(3,𝔽₄)×C₁₀₇	C₂	semidirect
4. #16812157	C₂²	semidirect
7. #16812028	C₂²	semidirect
52. C₁₀₇	#2202	semidirect
53. H(3,𝔽₄)	D₁₀₇	semidirect

H ≅ G/N

type

1. #20292799

C₂

semidirect

2. #20292424

C₂

semidirect

3. H(3,𝔽₄)×C₁₀₇

C₂

semidirect

4. #16812157

C₂²

semidirect

7. #16812028

C₂²

semidirect

52. C₁₀₇

#2202

semidirect

53. H(3,𝔽₄)

D₁₀₇

semidirect

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

subgroup	index
#20292799	2
#20292424	2
H(3,𝔽₄)×C₁₀₇	2
#2202	107

subgroup

index

#20292799

#20292424

H(3,𝔽₄)×C₁₀₇

#2202

107

Subgroups — #28338478

Conjugacy classes of subgroups

Complements

Maximal subgroups