Group #23759830 — subgroups

Subgroups — C₇₉×#2208 #23759830

Order 10112.

166 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 68 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	C₂×SD₃₂×C₇₉	C₂	1	2	yes	maximal
2	#494×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#358×C₇₉	C₂	1	2	yes	maximal, complemented
6	C₂×Q₁₆×C₇₉	—	2	1	no	—
7	C₂×Q₁₆×C₇₉	C₂²	1	2	yes	—
8	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₇₉	—	2	1	no	—
10	C₄×D₄×C₇₉	—	2	1	no	—
11	SD₃₂×C₇₉	—	2	4	no	—
12	C₂×C₁₆×C₇₉	C₂²	1	2	yes	—
13	#107×C₇₉	C₂²	1	1	yes	characteristic
14	#102×C₇₉	—	2	1	no	—
15	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₇₉	—	2	2	no	—
17	C₂×D₄×C₇₉	—	2	1	no	—
18	C₂²×C₄×C₇₉	—	4	1	no	—
19	Q₁₆×C₇₉	—	2	2	no	—
20	Q₁₆×C₇₉	—	2	2	no	—
21	Q₁₆×C₇₉	—	4	2	no	—
22	D₈×C₇₉	—	2	1	no	—
23	D₈×C₇₉	D₄	1	2	yes	—
24	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₇₉	—	2	1	no	—
27	C₄⋊C₄×C₇₉	—	4	1	no	—
28	C₄⋊C₄×C₇₉	—	4	1	no	—
29	C₂²⋊C₄×C₇₉	—	4	1	no	—
30	C₄²×C₇₉	D₄	1	1	yes	characteristic
31	C₁₆×C₇₉	—	2	2	no	—
32	C₂³×C₇₉	—	4	1	no	—
33	Q₈×C₇₉	—	4	2	no	—
34	Q₈×C₇₉	—	4	2	no	—
35	D₄×C₇₉	—	2	2	no	—
36	D₄×C₇₉	—	4	1	no	—
37	C₂×C₄×C₇₉	—	4	1	no	—
38	C₂×C₄×C₇₉	—	4	2	no	—
39	C₂×C₄×C₇₉	—	8	2	no	—
40	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
41	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₈×C₇₉	—	2	1	no	—
44	C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₈×C₇₉	#41	1	1	yes	characteristic
46	C₂²×C₇₉	—	4	2	no	—
47	C₂²×C₇₉	—	8	1	no	—
48	C₂²×C₇₉	—	8	1	no	—
49	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
50	C₄×C₇₉	—	2	1	no	—
51	C₄×C₇₉	—	8	2	no	—
52	C₄×C₇₉	—	8	1	no	—
53	C₄×C₇₉	SD₃₂	1	2	yes	complemented
54	C₄×C₇₉	#121	1	1	yes	characteristic
55	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
56	C₂×C₇₉	—	8	2	no	—
57	C₂×C₇₉	#459	1	1	yes	characteristic
58	C₂×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
59	C₂×C₇₉	#510	1	1	yes	characteristic
60	#2208	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₇₉	#2208	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×SD₃₂	C₂×C₇₉	1	2	yes	—
63	#494	C₂×C₇₉	1	1	yes	characteristic, complemented
64	C₄×D₈	C₂×C₇₉	1	1	yes	characteristic
65	C₄⋊C₁₆	C₂×C₇₉	1	1	yes	characteristic, complemented
66	#358	C₂×C₇₉	1	2	yes	complemented
67	C₂×Q₁₆	—	2	1	no	—
68	C₂×Q₁₆	C₂²×C₇₉	1	2	yes	—
69	C₂×D₈	C₂²×C₇₉	1	1	yes	characteristic
70	C₄⋊Q₈	—	2	1	no	—
71	C₄×D₄	—	2	1	no	—
72	SD₃₂	—	2	4	no	—
73	C₂×C₁₆	C₂²×C₇₉	1	2	yes	—
74	#107	C₂²×C₇₉	1	1	yes	characteristic
75	#102	—	2	1	no	—
76	C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic
77	C₂×Q₈	—	2	2	no	—
78	C₂×D₄	—	2	1	no	—
79	C₂²×C₄	—	4	1	no	—
80	Q₁₆	—	2	2	no	—
81	Q₁₆	—	2	2	no	—
82	Q₁₆	—	4	2	no	—
83	D₈	—	2	1	no	—
84	D₈	D₄×C₇₉	1	2	yes	—
85	C₂×C₈	D₄×C₇₉	1	1	yes	characteristic
86	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	4	1	no	—
90	C₂²⋊C₄	—	4	1	no	—
91	C₄²	D₄×C₇₉	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	C₂³	—	4	1	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	D₈×C₇₉	1	1	yes	characteristic
102	C₂×C₄	D₈×C₇₉	1	1	yes	characteristic
103	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
106	C₈	#41×C₇₉	1	1	yes	characteristic
107	C₂²	—	4	2	no	—
108	C₂²	—	8	1	no	—
109	C₂²	—	8	1	no	—
110	C₂²	C₂×D₈×C₇₉	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	8	2	no	—
113	C₄	—	8	1	no	—
114	C₄	SD₃₂×C₇₉	1	2	yes	complemented
115	C₄	#121×C₇₉	1	1	yes	characteristic
116	C₄	AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
117	C₂	—	8	2	no	—
118	C₂	#459×C₇₉	1	1	yes	characteristic
119	C₂	C₂×SD₃₂×C₇₉	1	1	yes	characteristic
120	C₂	#510×C₇₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×SD₃₂×C₇₉

C₂

yes

maximal

#494×C₇₉

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₇₉

C₂

yes

maximal, characteristic

C₄⋊C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

#358×C₇₉

C₂

yes

maximal, complemented

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

—

C₄×D₄×C₇₉

—

SD₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

characteristic

#102×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

D₄

yes

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

D₄

yes

characteristic

C₁₆×C₇₉

—

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

SD₃₂

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

#459

yes

characteristic

C₂×C₇₉

C₂×SD₃₂

yes

C₂×C₇₉

yes

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₇₉

#2208

yes

characteristic, direct product factor, Sylow

C₂×SD₃₂

C₂×C₇₉

yes

—

#494

C₂×C₇₉

yes

characteristic, complemented

C₄×D₈

C₂×C₇₉

yes

characteristic

C₄⋊C₁₆

C₂×C₇₉

yes

characteristic, complemented

#358

C₂×C₇₉

yes

complemented

C₂×Q₁₆

—

C₂×Q₁₆

C₂²×C₇₉

yes

—

C₂×D₈

C₂²×C₇₉

yes

characteristic

C₄⋊Q₈

—

C₄×D₄

—

SD₃₂

—

C₂×C₁₆

C₂²×C₇₉

yes

—

#107

C₂²×C₇₉

yes

characteristic

#102

—

C₄×C₈

C₂²×C₇₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

D₈

D₄×C₇₉

yes

—

C₂×C₈

D₄×C₇₉

yes

characteristic

C₂×C₈

C₂³×C₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄×C₇₉

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

D₈×C₇₉

yes

characteristic

102

C₂×C₄

D₈×C₇₉

yes

characteristic

103

C₂×C₄

C₂×D₄×C₇₉

yes

characteristic

104

C₈

—

105

C₈

C₂×D₄×C₇₉

yes

characteristic

106

C₈

#41×C₇₉

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×D₈×C₇₉

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

SD₃₂×C₇₉

yes

complemented

115

C₄

#121×C₇₉

yes

characteristic

116

C₄

AGL(1,ℤ/8)×C₇₉

yes

characteristic

117

C₂

—

118

C₂

#459×C₇₉

yes

characteristic

119

C₂

C₂×SD₃₂×C₇₉

yes

characteristic

120

C₂

#510×C₇₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇₉
prime radical	C₂×D₈×C₇₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₄×C₈×C₇₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₉

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₇₉

prime radical

C₂×D₈×C₇₉

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₄×C₈×C₇₉ ◁ G

N	H ≅ G/N	type
2. #494×C₇₉	C₂	semidirect
4. C₄⋊C₁₆×C₇₉	C₂	semidirect
5. #358×C₇₉	C₂	semidirect
53. C₄×C₇₉	SD₃₂	semidirect
60. #2208	C₇₉	direct (N × H)
61. C₇₉	#2208	direct (N × H)
63. #494	C₂×C₇₉	semidirect
65. C₄⋊C₁₆	C₂×C₇₉	semidirect
66. #358	C₂×C₇₉	semidirect
114. C₄	SD₃₂×C₇₉	semidirect

H ≅ G/N

type

2. #494×C₇₉

C₂

semidirect

4. C₄⋊C₁₆×C₇₉

C₂

semidirect

5. #358×C₇₉

C₂

semidirect

53. C₄×C₇₉

SD₃₂

semidirect

60. #2208

C₇₉

direct (N × H)

61. C₇₉

#2208

direct (N × H)

63. #494

C₂×C₇₉

semidirect

65. C₄⋊C₁₆

C₂×C₇₉

semidirect

66. #358

C₂×C₇₉

semidirect

114. C₄

SD₃₂×C₇₉

semidirect

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

subgroup	index
C₂×SD₃₂×C₇₉	2
#494×C₇₉	2
C₄×D₈×C₇₉	2
C₄⋊C₁₆×C₇₉	2
#358×C₇₉	2
#2208	79

subgroup

index

C₂×SD₃₂×C₇₉

#494×C₇₉

C₄×D₈×C₇₉

C₄⋊C₁₆×C₇₉

#358×C₇₉

#2208

Subgroups — C₇₉×#2208 #23759830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#2208 #23759830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#2208 #23759830