Group #23765417 — subgroups

Subgroups — #23765417

Order 10112.

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	#19098735	C₂	1	3	yes	maximal, complemented
2	#19098360	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#14579209	C₂²	1	1	yes	characteristic, complemented
5	#14579187	—	2	3	no	—
6	#14579089	—	2	3	no	—
7	#14579080	C₂²	1	3	yes	complemented
8	#124×C₇₉	—	2	3	no	—
9	#124×C₇₉	C₂²	1	3	yes	—
10	#120×C₇₉	—	2	3	no	—
11	#11929625	—	4	1	no	—
12	#11929624	—	2	3	no	—
13	Q₈×D₇₉	—	4	1	no	—
14	#11929612	—	4	3	no	—
15	#11929609	—	4	3	no	—
16	#11929608	—	2	3	no	—
17	C₂⁴×C₇₉	—	2	1	no	—
18	C₂×Q₈×C₇₉	—	2	1	no	—
19	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₇₉	—	2	3	no	—
21	C₂×D₄×C₇₉	D₄	1	3	yes	—
22	C₂²⋊C₄×C₇₉	—	2	3	no	—
23	C₂²⋊C₄×C₇₉	—	2	6	no	—
24	C₄²×C₇₉	D₄	1	3	yes	—
25	C₂³×C₇₉	—	2	3	no	—
26	C₂³×C₇₉	—	8	2	no	—
27	C₂³×C₇₉	—	8	1	no	—
28	Q₈×C₇₉	—	4	1	no	—
29	Q₈×C₇₉	—	8	1	no	—
30	D₄×C₇₉	—	4	3	no	—
31	D₄×C₇₉	—	8	3	no	—
32	C₂×C₄×C₇₉	—	2	3	no	—
33	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
34	C₂²⋊D₇₉	—	4	1	no	—
35	D₃₁₆	—	2	3	no	—
36	C₄×D₇₉	—	4	3	no	—
37	Dic₁₅₈	—	2	3	no	—
38	C₇₉⋊C₈	—	4	3	no	—
39	C₂²×C₇₉	—	4	6	no	—
40	C₂²×C₇₉	—	4	3	no	—
41	C₂²×C₇₉	—	8	4	no	—
42	C₂²×C₇₉	#120	1	1	yes	characteristic
43	C₂×D₇₉	—	4	1	no	—
44	C₄×C₇₉	—	2	3	no	—
45	C₄×C₇₉	—	4	3	no	—
46	Dic₇₉	—	4	1	no	—
47	C₂×C₇₉	—	2	1	no	—
48	C₂×C₇₉	—	8	3	no	—
49	C₂×C₇₉	B(4,𝔽₂)	1	1	yes	characteristic
50	D₇₉	—	8	1	no	—
51	#2202	—	79	1	no	maximal, Sylow
52	C₇₉	#2202	1	1	yes	characteristic, complemented, Sylow
53	H(3,𝔽₄)	D₇₉	1	1	yes	characteristic, complemented
54	#454	—	79	3	no	—
55	#355	—	79	3	no	—
56	2₋¹⁺⁴	—	79	1	no	—
57	AGL(1,ℤ/8)	—	158	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₇₉	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	158	3	no	—
62	#101	—	79	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	316	1	no	—
65	#41	—	158	3	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	316	1	no	—
68	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₇₉	1	3	yes	—
71	SD₁₆	—	316	3	no	—
72	D₈	—	316	3	no	—
73	M₁₆	—	158	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₇₉	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	—	158	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	316	1	no	—
86	D₄	—	158	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	316	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₇₉	1	3	yes	—
90	C₈	—	316	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	316	1	no	—
95	C₂²	#14579130	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	316	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	632	1	no	—
102	C₂	#19098750	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098735

C₂

yes

maximal, complemented

#19098360

C₂

yes

maximal, complemented

H(3,𝔽₄)×C₇₉

C₂

yes

maximal, characteristic, complemented

#14579209

C₂²

yes

characteristic, complemented

#14579187

—

#14579089

—

#14579080

C₂²

yes

complemented

#124×C₇₉

—

#124×C₇₉

C₂²

yes

—

#120×C₇₉

—

#11929625

—

#11929624

—

Q₈×D₇₉

—

#11929612

—

#11929609

—

#11929608

—

C₂⁴×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂³

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

D₄

yes

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂²⋊D₇₉

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

B(4,𝔽₂)

yes

characteristic

D₇₉

—

#2202

—

maximal, Sylow

C₇₉

#2202

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

D₇₉

yes

characteristic, complemented

#454

—

#355

—

2₋¹⁺⁴

—

AGL(1,ℤ/8)

—

158

—

#124

—

#124

C₂×D₇₉

yes

—

#120

—

#104

—

158

—

#101

—

C₂⁴

—

#41

—

316

—

#41

—

158

—

C₂×Q₈

—

C₂×Q₈

—

316

—

C₂×Q₈

C₂²×D₇₉

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₇₉

yes

—

SD₁₆

—

316

—

D₈

—

316

—

M₁₆

—

158

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₇₉

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

158

—

D₄

—

D₄

—

D₄

—

316

—

D₄

—

158

—

C₂×C₄

—

C₂×C₄

—

316

—

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

C₈

—

316

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

316

—

C₂²

#14579130

yes

characteristic

C₄

—

C₄

—

C₄

—

316

—

C₂

—

100

C₂

—

101

C₂

—

632

—

102

C₂

#19098750

yes

characteristic

	subgroup	G/H
center	C₂	#19098750
commutator, cosocle	C₂×Q₈×C₇₉	C₂³
Frattini	C₂×Q₈	C₂²×D₇₉
Fitting	H(3,𝔽₄)×C₇₉	C₂
socle	C₂×C₇₉	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₇₉
hypocenter	C₇₉	#2202
LCS	G ▷ C₂×Q₈×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#19098750

commutator, cosocle

C₂×Q₈×C₇₉

C₂³

Frattini

C₂×Q₈

C₂²×D₇₉

Fitting

H(3,𝔽₄)×C₇₉

C₂

C₂×C₇₉

D₇₉

C₇₉

G ▷ C₂×Q₈×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #19098735	C₂	semidirect
2. #19098360	C₂	semidirect
3. H(3,𝔽₄)×C₇₉	C₂	semidirect
4. #14579209	C₂²	semidirect
7. #14579080	C₂²	semidirect
52. C₇₉	#2202	semidirect
53. H(3,𝔽₄)	D₇₉	semidirect

H ≅ G/N

type

1. #19098735

C₂

semidirect

2. #19098360

C₂

semidirect

3. H(3,𝔽₄)×C₇₉

C₂

semidirect

4. #14579209

C₂²

semidirect

7. #14579080

C₂²

semidirect

52. C₇₉

#2202

semidirect

53. H(3,𝔽₄)

D₇₉

semidirect

Subgroups — #23765417

Conjugacy classes of subgroups

Complements