Group #23761013 — subgroups

Subgroups — C₇₉×#3391 #23761013

Order 10112.

460 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 192 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	#550×C₇₉	C₂	1	2	yes	maximal, complemented
2	#493×C₇₉	C₂	1	3	yes	maximal, complemented
3	#459×C₇₉	C₂	1	6	yes	maximal, complemented
4	C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₇₉	C₂	1	3	yes	maximal, complemented
6	C₂×#41×C₇₉	—	2	2	no	—
7	C₂×D₈×C₇₉	C₂²	1	3	yes	—
8	C₂×D₈×C₇₉	C₂²	1	6	yes	—
9	#127×C₇₉	C₂²	1	6	yes	—
10	#121×C₇₉	—	2	6	no	—
11	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₇₉	C₂²	1	6	yes	—
13	#107×C₇₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₇₉	C₂²	1	3	yes	—
15	#102×C₇₉	C₂²	1	6	yes	—
16	C₄×C₈×C₇₉	C₂²	1	3	yes	—
17	#41×C₇₉	—	4	8	no	—
18	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
19	C₂×D₄×C₇₉	—	2	6	no	—
20	C₂×D₄×C₇₉	—	2	6	no	—
21	C₂×D₄×C₇₉	C₂³	1	6	yes	—
22	C₂²×C₄×C₇₉	—	2	6	no	—
23	D₈×C₇₉	—	2	6	no	—
24	D₈×C₇₉	—	4	6	no	—
25	C₂×C₈×C₇₉	C₂³	1	3	yes	—
26	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₇₉	D₄	1	3	yes	—
28	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
29	C₂²⋊C₄×C₇₉	—	2	6	no	—
30	C₄²×C₇₉	C₂³	1	3	yes	—
31	C₂³×C₇₉	—	2	6	no	—
32	Q₈×C₇₉	D₈	1	4	yes	complemented
33	D₄×C₇₉	—	2	6	no	—
34	D₄×C₇₉	—	2	6	no	—
35	D₄×C₇₉	—	4	12	no	—
36	D₄×C₇₉	—	8	6	no	—
37	C₂×C₄×C₇₉	—	2	2	no	—
38	C₂×C₄×C₇₉	—	4	12	no	—
39	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
40	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
41	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
42	C₈×C₇₉	—	2	3	no	—
43	C₈×C₇₉	#41	1	2	yes	complemented
44	C₂²×C₇₉	—	4	6	no	—
45	C₂²×C₇₉	—	4	6	no	—
46	C₂²×C₇₉	—	4	6	no	—
47	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
48	C₄×C₇₉	—	2	3	no	—
49	C₄×C₇₉	—	4	2	no	—
50	C₄×C₇₉	C₂×D₈	1	6	yes	—
51	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
52	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂×C₇₉	—	8	6	no	—
54	C₂×C₇₉	#550	1	1	yes	characteristic
55	C₂×C₇₉	C₂²×D₈	1	1	yes	characteristic
56	C₂×C₇₉	#576	1	1	yes	characteristic
57	#3391	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₇₉	#3391	1	1	yes	characteristic, direct product factor, Sylow
59	#550	C₂×C₇₉	1	2	yes	complemented
60	#493	C₂×C₇₉	1	3	yes	complemented
61	#459	C₂×C₇₉	1	6	yes	complemented
62	C₈×Q₈	C₂×C₇₉	1	1	yes	characteristic, complemented
63	C₄×D₈	C₂×C₇₉	1	3	yes	complemented
64	C₂×#41	—	2	2	no	—
65	C₂×D₈	C₂²×C₇₉	1	3	yes	—
66	C₂×D₈	C₂²×C₇₉	1	6	yes	—
67	#127	C₂²×C₇₉	1	6	yes	—
68	#121	—	2	6	no	—
69	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
70	C₄×D₄	C₂²×C₇₉	1	6	yes	—
71	#107	C₂²×C₇₉	1	1	yes	characteristic
72	C₄⋊C₈	C₂²×C₇₉	1	3	yes	—
73	#102	C₂²×C₇₉	1	6	yes	—
74	C₄×C₈	C₂²×C₇₉	1	3	yes	—
75	#41	—	4	8	no	—
76	C₂×Q₈	D₄×C₇₉	1	1	yes	characteristic
77	C₂×D₄	—	2	6	no	—
78	C₂×D₄	—	2	6	no	—
79	C₂×D₄	C₂³×C₇₉	1	6	yes	—
80	C₂²×C₄	—	2	6	no	—
81	D₈	—	2	6	no	—
82	D₈	—	4	6	no	—
83	C₂×C₈	C₂³×C₇₉	1	3	yes	—
84	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
85	C₄⋊C₄	D₄×C₇₉	1	3	yes	—
86	C₄⋊C₄	C₂³×C₇₉	1	2	yes	—
87	C₂²⋊C₄	—	2	6	no	—
88	C₄²	C₂³×C₇₉	1	3	yes	—
89	C₂³	—	2	6	no	—
90	Q₈	D₈×C₇₉	1	4	yes	complemented
91	D₄	—	2	6	no	—
92	D₄	—	2	6	no	—
93	D₄	—	4	12	no	—
94	D₄	—	8	6	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	12	no	—
97	C₂×C₄	C₂×D₄×C₇₉	1	3	yes	—
98	C₂×C₄	C₂×D₄×C₇₉	1	3	yes	—
99	C₂×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
100	C₈	—	2	3	no	—
101	C₈	#41×C₇₉	1	2	yes	complemented
102	C₂²	—	4	6	no	—
103	C₂²	—	4	6	no	—
104	C₂²	—	4	6	no	—
105	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
106	C₄	—	2	3	no	—
107	C₄	—	4	2	no	—
108	C₄	C₂×D₈×C₇₉	1	6	yes	—
109	C₄	C₂×#41×C₇₉	1	1	yes	characteristic
110	C₄	H(5,𝔽₂)×C₇₉	1	1	yes	characteristic
111	C₂	—	8	6	no	—
112	C₂	#550×C₇₉	1	1	yes	characteristic
113	C₂	C₂²×D₈×C₇₉	1	1	yes	characteristic
114	C₂	#576×C₇₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#550×C₇₉

C₂

yes

maximal, complemented

#493×C₇₉

C₂

yes

maximal, complemented

#459×C₇₉

C₂

yes

maximal, complemented

C₈×Q₈×C₇₉

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₇₉

C₂

yes

maximal, complemented

C₂×#41×C₇₉

—

C₂×D₈×C₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

—

#127×C₇₉

C₂²

yes

—

#121×C₇₉

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄×D₄×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

#102×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

—

#41×C₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

—

Q₈×C₇₉

D₈

yes

complemented

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

#41

yes

complemented

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

C₂×C₇₉

—

C₂×C₇₉

#550

yes

characteristic

C₂×C₇₉

C₂²×D₈

yes

C₂×C₇₉

yes

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₇₉

#3391

yes

characteristic, direct product factor, Sylow

#550

C₂×C₇₉

yes

complemented

#493

C₂×C₇₉

yes

complemented

#459

C₂×C₇₉

yes

complemented

C₈×Q₈

C₂×C₇₉

yes

characteristic, complemented

C₄×D₈

C₂×C₇₉

yes

complemented

C₂×#41

—

C₂×D₈

C₂²×C₇₉

yes

—

C₂×D₈

C₂²×C₇₉

yes

—

#127

C₂²×C₇₉

yes

—

#121

—

C₄×Q₈

C₂²×C₇₉

yes

characteristic

C₄×D₄

C₂²×C₇₉

yes

—

#107

C₂²×C₇₉

yes

characteristic

C₄⋊C₈

C₂²×C₇₉

yes

—

#102

C₂²×C₇₉

yes

—

C₄×C₈

C₂²×C₇₉

yes

—

#41

—

C₂×Q₈

D₄×C₇₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂³×C₇₉

yes

—

C₂²×C₄

—

D₈

—

D₈

—

C₂×C₈

C₂³×C₇₉

yes

—

C₂×C₈

C₂³×C₇₉

yes

characteristic

C₄⋊C₄

D₄×C₇₉

yes

—

C₄⋊C₄

C₂³×C₇₉

yes

—

C₂²⋊C₄

—

C₄²

C₂³×C₇₉

yes

—

C₂³

—

Q₈

D₈×C₇₉

yes

complemented

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₇₉

yes

—

C₂×C₄

C₂×D₄×C₇₉

yes

—

C₂×C₄

C₂⁴×C₇₉

yes

characteristic

100

C₈

—

101

C₈

#41×C₇₉

yes

complemented

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₂²×D₄×C₇₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×D₈×C₇₉

yes

—

109

C₄

C₂×#41×C₇₉

yes

characteristic

110

C₄

H(5,𝔽₂)×C₇₉

yes

characteristic

111

C₂

—

112

C₂

#550×C₇₉

yes

characteristic

113

C₂

C₂²×D₈×C₇₉

yes

characteristic

114

C₂

#576×C₇₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄×Q₈×C₇₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₉

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₇₉

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₉ ◁ C₄×Q₈×C₇₉ ◁ G

N	H ≅ G/N	type
1. #550×C₇₉	C₂	semidirect
2. #493×C₇₉	C₂	semidirect
3. #459×C₇₉	C₂	semidirect
4. C₈×Q₈×C₇₉	C₂	semidirect
5. C₄×D₈×C₇₉	C₂	semidirect
32. Q₈×C₇₉	D₈	semidirect
43. C₈×C₇₉	#41	semidirect
57. #3391	C₇₉	direct (N × H)
58. C₇₉	#3391	direct (N × H)
59. #550	C₂×C₇₉	semidirect
60. #493	C₂×C₇₉	semidirect
61. #459	C₂×C₇₉	semidirect
62. C₈×Q₈	C₂×C₇₉	semidirect
63. C₄×D₈	C₂×C₇₉	semidirect
90. Q₈	D₈×C₇₉	semidirect
101. C₈	#41×C₇₉	semidirect

H ≅ G/N

type

1. #550×C₇₉

C₂

semidirect

2. #493×C₇₉

C₂

semidirect

3. #459×C₇₉

C₂

semidirect

4. C₈×Q₈×C₇₉

C₂

semidirect

5. C₄×D₈×C₇₉

C₂

semidirect

32. Q₈×C₇₉

D₈

semidirect

43. C₈×C₇₉

#41

semidirect

57. #3391

C₇₉

direct (N × H)

58. C₇₉

#3391

direct (N × H)

59. #550

C₂×C₇₉

semidirect

60. #493

C₂×C₇₉

semidirect

61. #459

C₂×C₇₉

semidirect

62. C₈×Q₈

C₂×C₇₉

semidirect

63. C₄×D₈

C₂×C₇₉

semidirect

90. Q₈

D₈×C₇₉

semidirect

101. C₈

#41×C₇₉

semidirect

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

subgroup	index
#550×C₇₉	2
#493×C₇₉	2
#459×C₇₉	2
C₈×Q₈×C₇₉	2
C₄×D₈×C₇₉	2
#3391	79

subgroup

index

#550×C₇₉

#493×C₇₉

#459×C₇₉

C₈×Q₈×C₇₉

C₄×D₈×C₇₉

#3391

Subgroups — C₇₉×#3391 #23761013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#3391 #23761013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#3391 #23761013