Group #23776674 — subgroups

Subgroups — C₂²×#14579159 #23776674

Order 10112.

1624 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 725 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	C₂³×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579159	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Dic₇₉⋊C₄	C₂	1	4	yes	maximal
4	C₂²×C₄×Dic₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	—	2	8	no	—
7	C₂²×Dic₁₅₈	C₂²	1	12	yes	—
8	C₂³×Dic₇₉	C₂²	1	2	yes	—
9	C₂³×Dic₇₉	C₂²	1	2	yes	—
10	#14579159	C₂²	1	64	yes	direct product factor
11	C₂×Dic₇₉⋊C₄	C₂²	1	48	yes	—
12	C₂×C₄×Dic₇₉	C₂²	1	12	yes	—
13	C₂²×Q₈×C₇₉	—	2	8	no	—
14	C₂²×Q₈×C₇₉	C₂²	1	12	yes	—
15	C₂³×C₄×C₇₉	C₂²	1	2	yes	—
16	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₅₈	—	2	48	no	—
18	C₂×Dic₁₅₈	C₂³	1	16	yes	—
19	C₂²×Dic₇₉	—	2	8	no	—
20	C₂²×Dic₇₉	Q₈	1	16	yes	complemented
21	C₂²×Dic₇₉	C₂³	1	12	yes	—
22	C₂²×Dic₇₉	C₂³	1	12	yes	—
23	Dic₇₉⋊C₄	C₂³	1	64	yes	—
24	C₄×Dic₇₉	C₂³	1	16	yes	—
25	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₇₉	—	2	48	no	—
27	C₂×Q₈×C₇₉	C₂³	1	16	yes	—
28	C₂²×C₄×C₇₉	—	2	8	no	—
29	C₂²×C₄×C₇₉	D₄	1	8	yes	—
30	C₂²×C₄×C₇₉	C₂³	1	12	yes	—
31	C₂²×C₄×C₇₉	C₂³	1	6	yes	—
32	C₂³×C₇₉	C₂×D₄	1	4	yes	—
33	C₂³×C₇₉	C₂×Q₈	1	8	yes	—
34	C₂³×C₇₉	C₂⁴	1	3	yes	—
35	Q₈×C₇₉	—	2	32	no	—
36	C₂×C₄×C₇₉	—	2	24	no	—
37	C₂×C₄×C₇₉	C₂×D₄	1	24	yes	—
38	C₂×C₄×C₇₉	C₂⁴	1	8	yes	—
39	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
40	C₂×Dic₇₉	—	2	24	no	—
41	C₂×Dic₇₉	C₂×Q₈	1	48	yes	complemented
42	C₂×Dic₇₉	C₂⁴	1	8	yes	—
43	C₂×Dic₇₉	C₂⁴	1	8	yes	—
44	Dic₁₅₈	—	2	32	no	—
45	C₂²×C₇₉	C₄⋊Q₈	1	16	yes	complemented
46	C₂²×C₇₉	C₂²×D₄	1	6	yes	—
47	C₂²×C₇₉	C₂²×Q₈	1	12	yes	—
48	C₂²×C₇₉	C₂⁵	1	1	yes	characteristic
49	C₄×C₇₉	—	2	8	no	—
50	C₄×C₇₉	C₂²×D₄	1	8	yes	—
51	Dic₇₉	—	2	8	no	—
52	Dic₇₉	C₂²×Q₈	1	16	yes	complemented
53	C₂×C₇₉	C₂×C₄⋊Q₈	1	12	yes	complemented
54	C₂×C₇₉	C₂³×D₄	1	1	yes	characteristic
55	C₂×C₇₉	C₂³×Q₈	1	2	yes	—
56	C₂²×C₄⋊Q₈	—	79	1	no	maximal, Sylow
57	C₇₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
58	C₂³×Q₈	—	79	1	no	—
59	C₂³×Q₈	D₇₉	1	1	yes	characteristic
60	C₂×C₄⋊Q₈	—	79	24	no	—
61	C₂²×C₄⋊C₄	—	79	4	no	—
62	C₂²×C₄²	—	79	1	no	—
63	C₂²×Q₈	—	2	8	no	—
64	C₂²×Q₈	—	79	12	no	—
65	C₂²×Q₈	—	158	8	no	—
66	C₂²×Q₈	C₂×D₇₉	1	12	yes	—
67	C₂³×C₄	—	79	2	no	—
68	C₂³×C₄	—	79	2	no	—
69	C₂³×C₄	C₂×D₇₉	1	2	yes	—
70	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
71	C₄⋊Q₈	—	79	64	no	—
72	C₂×C₄⋊C₄	—	79	48	no	—
73	C₂×C₄²	—	79	12	no	—
74	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
75	C₂×Q₈	—	2	48	no	—
76	C₂×Q₈	—	79	16	no	—
77	C₂×Q₈	—	158	48	no	—
78	C₂×Q₈	C₂²×D₇₉	1	16	yes	—
79	C₂²×C₄	—	2	8	no	—
80	C₂²×C₄	—	79	16	no	—
81	C₂²×C₄	—	79	12	no	—
82	C₂²×C₄	—	79	12	no	—
83	C₂²×C₄	—	158	8	no	—
84	C₂²×C₄	C₂²⋊D₇₉	1	8	yes	—
85	C₂²×C₄	C₂²×D₇₉	1	12	yes	—
86	C₂²×C₄	C₂²×D₇₉	1	6	yes	—
87	C₄⋊C₄	—	79	64	no	—
88	C₄²	—	79	16	no	—
89	C₂³	C₂×C₂²⋊D₇₉	1	4	yes	—
90	C₂³	Q₈×D₇₉	1	8	yes	—
91	C₂³	C₂³×D₇₉	1	3	yes	—
92	Q₈	—	2	32	no	—
93	Q₈	—	158	32	no	—
94	C₂×C₄	—	2	24	no	—
95	C₂×C₄	—	79	48	no	—
96	C₂×C₄	—	79	8	no	—
97	C₂×C₄	—	79	8	no	—
98	C₂×C₄	—	158	24	no	—
99	C₂×C₄	C₂×C₂²⋊D₇₉	1	24	yes	—
100	C₂×C₄	C₂³×D₇₉	1	8	yes	—
101	C₂×C₄	C₂³×D₇₉	1	4	yes	—
102	C₂²	#14579159	1	16	yes	direct product factor
103	C₂²	C₂²×C₂²⋊D₇₉	1	6	yes	—
104	C₂²	C₂×Q₈×D₇₉	1	12	yes	—
105	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
106	C₄	—	2	8	no	—
107	C₄	—	79	16	no	—
108	C₄	—	158	8	no	—
109	C₄	C₂²×C₂²⋊D₇₉	1	8	yes	—
110	C₂	C₂×#14579159	1	12	yes	direct product factor
111	C₂	C₂³×C₂²⋊D₇₉	1	1	yes	characteristic
112	C₂	C₂²×Q₈×D₇₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₅₈

C₂

yes

maximal, characteristic

C₂×#14579159

C₂

yes

maximal, direct product factor

C₂²×Dic₇₉⋊C₄

C₂

yes

maximal

C₂²×C₄×Dic₇₉

C₂

yes

maximal, characteristic

C₂³×Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

—

C₂²×Dic₁₅₈

C₂²

yes

—

C₂³×Dic₇₉

C₂²

yes

—

C₂³×Dic₇₉

C₂²

yes

—

#14579159

C₂²

yes

direct product factor

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₂²×Q₈×C₇₉

—

C₂²×Q₈×C₇₉

C₂²

yes

—

C₂³×C₄×C₇₉

C₂²

yes

—

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

Q₈

yes

complemented

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

C₂³

yes

—

C₄×Dic₇₉

C₂³

yes

—

C₂⁴×C₇₉

C₂³

yes

characteristic

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

C₂⁴

yes

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×Q₈

yes

complemented

C₂×Dic₇₉

C₂⁴

yes

—

C₂×Dic₇₉

C₂⁴

yes

—

Dic₁₅₈

—

C₂²×C₇₉

C₄⋊Q₈

yes

complemented

C₂²×C₇₉

C₂²×D₄

yes

—

C₂²×C₇₉

C₂²×Q₈

yes

—

C₂²×C₇₉

C₂⁵

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂²×D₄

yes

—

Dic₇₉

—

Dic₇₉

C₂²×Q₈

yes

complemented

C₂×C₇₉

C₂×C₄⋊Q₈

yes

complemented

C₂×C₇₉

C₂³×D₄

yes

characteristic

C₂×C₇₉

C₂³×Q₈

yes

—

C₂²×C₄⋊Q₈

—

maximal, Sylow

C₇₉

C₂²×C₄⋊Q₈

yes

characteristic, complemented, Sylow

C₂³×Q₈

—

C₂³×Q₈

D₇₉

yes

characteristic

C₂×C₄⋊Q₈

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

158

—

C₂²×Q₈

C₂×D₇₉

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₇₉

yes

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂⁴

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂×Q₈

C₂²×D₇₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂×C₂²⋊D₇₉

yes

—

C₂³

Q₈×D₇₉

yes

—

C₂³

C₂³×D₇₉

yes

—

Q₈

—

Q₈

—

158

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

158

—

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

100

C₂×C₄

C₂³×D₇₉

yes

—

101

C₂×C₄

C₂³×D₇₉

yes

—

102

C₂²

#14579159

yes

direct product factor

103

C₂²

C₂²×C₂²⋊D₇₉

yes

—

104

C₂²

C₂×Q₈×D₇₉

yes

—

105

C₂²

C₂⁴×D₇₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

158

—

109

C₄

C₂²×C₂²⋊D₇₉

yes

—

110

C₂

C₂×#14579159

yes

direct product factor

111

C₂

C₂³×C₂²⋊D₇₉

yes

characteristic

112

C₂

C₂²×Q₈×D₇₉

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁵
Frattini	C₂²	C₂⁴×D₇₉
Fitting	C₂³×Q₈×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂³
hypercenter	C₂³×Q₈	D₇₉
hypocenter	C₇₉	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

subgroup

G/H

center

C₂⁴

C₂²×D₇₉

commutator, cosocle

C₂²×C₇₉

C₂⁵

Frattini

C₂²

C₂⁴×D₇₉

Fitting

C₂³×Q₈×C₇₉

C₂

socle, prime radical

C₂⁴×C₇₉

C₂³

hypercenter

C₂³×Q₈

D₇₉

hypocenter

C₇₉

C₂²×C₄⋊Q₈

LCS

G ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂⁴ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. C₂×#14579159	C₂	direct (N × H)
10. #14579159	C₂²	direct (N × H)
20. C₂²×Dic₇₉	Q₈	semidirect
41. C₂×Dic₇₉	C₂×Q₈	semidirect
45. C₂²×C₇₉	C₄⋊Q₈	semidirect
52. Dic₇₉	C₂²×Q₈	semidirect
53. C₂×C₇₉	C₂×C₄⋊Q₈	semidirect
57. C₇₉	C₂²×C₄⋊Q₈	semidirect
102. C₂²	#14579159	direct (N × H)
110. C₂	C₂×#14579159	direct (N × H)

H ≅ G/N

type

2. C₂×#14579159

C₂

direct (N × H)

10. #14579159

C₂²

direct (N × H)

20. C₂²×Dic₇₉

Q₈

semidirect

41. C₂×Dic₇₉

C₂×Q₈

semidirect

45. C₂²×C₇₉

C₄⋊Q₈

semidirect

52. Dic₇₉

C₂²×Q₈

semidirect

53. C₂×C₇₉

C₂×C₄⋊Q₈

semidirect

57. C₇₉

C₂²×C₄⋊Q₈

semidirect

102. C₂²

#14579159

direct (N × H)

110. C₂

C₂×#14579159

direct (N × H)

Subgroups — C₂²×#14579159 #23776674

Conjugacy classes of subgroups

Complements

Subgroups — C22×#14579159 #23776674

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14579159 #23776674