Group #23776406 — subgroups

Subgroups — #23776406

Order 10112.

348 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 141 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	#19099292	C₂	1	2	yes	maximal
2	#19098943	C₂	1	3	yes	maximal
3	#19098774	C₂	1	6	yes	maximal
4	C₄×Dic₃₁₆	C₂	1	3	yes	maximal
5	C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₁₆	C₂²	1	3	yes	—
7	C₂×Dic₃₁₆	C₂²	1	6	yes	—
8	#14579161	C₂²	1	6	yes	—
9	#14579152	—	2	6	no	—
10	Q₈×Dic₇₉	—	2	2	no	—
11	C₄×Dic₁₅₈	C₂²	1	6	yes	—
12	#14579095	C₂²	1	1	yes	characteristic
13	#14579086	C₂²	1	6	yes	—
14	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₇₉	C₂²	1	3	yes	—
16	C₄×C₈×C₇₉	C₂²	1	3	yes	—
17	C₂×Dic₁₅₈	C₂³	1	6	yes	—
18	Dic₃₁₆	—	2	6	no	—
19	Dic₃₁₆	—	4	6	no	—
20	C₄⋊Dic₇₉	—	2	6	no	—
21	C₄⋊Dic₇₉	—	2	6	no	—
22	C₄⋊Dic₇₉	C₂³	1	2	yes	—
23	Dic₇₉⋊C₄	—	2	6	no	—
24	C₄×Dic₇₉	—	2	6	no	—
25	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₉	C₂³	1	3	yes	—
27	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₇₉	D₄	1	3	yes	—
29	C₄²×C₇₉	C₂³	1	3	yes	—
30	Q₈×C₇₉	Q₁₆	1	4	yes	complemented
31	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
32	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
34	C₂×Dic₇₉	—	2	2	no	—
35	C₂×Dic₇₉	—	2	6	no	—
36	Dic₁₅₈	—	2	6	no	—
37	Dic₁₅₈	—	2	6	no	—
38	C₈×C₇₉	—	2	3	no	—
39	C₈×C₇₉	#41	1	2	yes	—
40	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₇₉	—	2	3	no	—
42	C₄×C₇₉	C₂×Q₁₆	1	6	yes	—
43	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
44	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
45	Dic₇₉	—	4	6	no	—
46	Dic₇₉	—	4	2	no	—
47	C₂×C₇₉	#550	1	1	yes	characteristic
48	C₂×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
49	C₂×C₇₉	#578	1	1	yes	characteristic
50	#3395	—	79	1	no	maximal, Sylow
51	C₇₉	#3395	1	1	yes	characteristic, complemented, Sylow
52	#557	—	79	2	no	—
53	#494	—	79	3	no	—
54	#462	—	79	6	no	—
55	C₈×Q₈	D₇₉	1	1	yes	characteristic
56	C₄×Q₁₆	—	79	3	no	—
57	C₂×Q₁₆	—	79	6	no	—
58	C₂×Q₁₆	—	79	3	no	—
59	C₄⋊Q₈	—	79	6	no	—
60	#125	—	158	6	no	—
61	C₄×Q₈	—	79	6	no	—
62	C₄×Q₈	—	158	2	no	—
63	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
64	#107	—	79	1	no	—
65	C₄⋊C₈	C₂×D₇₉	1	3	yes	—
66	#103	—	79	6	no	—
67	C₄×C₈	C₂×D₇₉	1	3	yes	—
68	C₂×Q₈	—	79	6	no	—
69	C₂×Q₈	D₃₁₆	1	1	yes	characteristic
70	Q₁₆	—	316	6	no	—
71	Q₁₆	—	158	6	no	—
72	C₂×C₈	C₂²×D₇₉	1	3	yes	—
73	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
74	C₄⋊C₄	—	79	2	no	—
75	C₄⋊C₄	—	158	6	no	—
76	C₄⋊C₄	—	158	6	no	—
77	C₄⋊C₄	—	158	6	no	—
78	C₄⋊C₄	D₃₁₆	1	3	yes	—
79	C₄²	—	158	6	no	—
80	C₄²	C₂²×D₇₉	1	3	yes	—
81	Q₈	—	158	6	no	—
82	Q₈	—	158	6	no	—
83	Q₈	Dic₃₁₆	1	4	yes	complemented
84	C₂×C₄	—	158	2	no	—
85	C₂×C₄	—	158	6	no	—
86	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
87	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
88	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	#11929625	1	2	yes	—
91	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
92	C₄	—	2	3	no	—
93	C₄	—	316	6	no	—
94	C₄	—	316	2	no	—
95	C₄	C₂×Dic₃₁₆	1	6	yes	—
96	C₄	C₂×#11929625	1	1	yes	characteristic
97	C₄	#14579207	1	1	yes	characteristic
98	C₂	#19099235	1	1	yes	characteristic
99	C₂	C₂²×Dic₃₁₆	1	1	yes	characteristic
100	C₂	#19099380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099292

C₂

yes

maximal

#19098943

C₂

yes

maximal

#19098774

C₂

yes

maximal

C₄×Dic₃₁₆

C₂

yes

maximal

C₈×Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

—

C₂×Dic₃₁₆

C₂²

yes

—

#14579161

C₂²

yes

—

#14579152

—

Q₈×Dic₇₉

—

C₄×Dic₁₅₈

C₂²

yes

—

#14579095

C₂²

yes

characteristic

#14579086

C₂²

yes

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

C₂³

yes

—

Dic₃₁₆

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

Q₈×C₇₉

Q₁₆

yes

complemented

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

#41

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂×Q₁₆

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#550

yes

characteristic

C₂×C₇₉

C₂²×Q₁₆

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#557

—

#494

—

#462

—

C₈×Q₈

D₇₉

yes

characteristic

C₄×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#125

—

158

—

C₄×Q₈

—

C₄×Q₈

—

158

—

C₄×Q₈

C₂×D₇₉

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₇₉

yes

—

#103

—

C₄×C₈

C₂×D₇₉

yes

—

C₂×Q₈

—

C₂×Q₈

D₃₁₆

yes

characteristic

Q₁₆

—

316

—

Q₁₆

—

158

—

C₂×C₈

C₂²×D₇₉

yes

—

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄⋊C₄

—

158

—

C₄⋊C₄

—

158

—

C₄⋊C₄

D₃₁₆

yes

—

C₄²

—

158

—

C₄²

C₂²×D₇₉

yes

—

Q₈

—

158

—

Q₈

—

158

—

Q₈

Dic₃₁₆

yes

complemented

C₂×C₄

—

158

—

C₂×C₄

—

158

—

C₂×C₄

C₂×D₃₁₆

yes

—

C₂×C₄

C₂×D₃₁₆

yes

—

C₂×C₄

C₂³×D₇₉

yes

characteristic

C₈

—

C₈

#11929625

yes

—

C₂²

C₂²×D₃₁₆

yes

characteristic

C₄

—

C₄

—

316

—

C₄

—

316

—

C₄

C₂×Dic₃₁₆

yes

—

C₄

yes

C₄

yes

C₂

yes

C₂

C₂²×Dic₃₁₆

yes

characteristic

100

C₂

#19099380

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₈×Q₈×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₂²×D₄
hypercenter	C₈×Q₈	D₇₉
hypocenter	C₇₉	#3395
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₈×Q₈×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

C₂²×D₄

hypercenter

C₈×Q₈

D₇₉

hypocenter

C₇₉

#3395

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
30. Q₈×C₇₉	Q₁₆	semidirect
51. C₇₉	#3395	semidirect
83. Q₈	Dic₃₁₆	semidirect

H ≅ G/N

type

30. Q₈×C₇₉

Q₁₆

semidirect

51. C₇₉

#3395

semidirect

83. Q₈

Dic₃₁₆

semidirect

Subgroups — #23776406

Conjugacy classes of subgroups

Complements