Group #23762786 — subgroups

Subgroups — #23762786

Order 10112.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19098765	C₂	1	2	yes	maximal
2	C₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic
3	#19098316	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098309	C₂	1	2	yes	maximal, complemented
5	#500×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11929611	—	2	2	no	—
7	C₂×C₈×D₇₉	—	2	1	no	—
8	C₄×D₃₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
11	#14579085	—	2	2	no	—
12	(C₂×D₇₉)⋊C₈	—	2	1	no	—
13	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
14	#107×C₇₉	—	2	1	no	—
15	#107×C₇₉	—	2	1	no	—
16	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×D₃₁₆	D₄	1	1	yes	characteristic
18	C₂×C₄×D₇₉	—	2	1	no	—
19	#11929611	—	4	4	no	—
20	C₂×C₇₉⋊C₈	—	2	1	no	—
21	C₂×C₇₉⋊C₈	—	2	2	no	—
22	C₈×D₇₉	—	4	2	no	—
23	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
24	(C₂×D₇₉)⋊C₄	—	2	1	no	—
25	C₂×Q₈×C₇₉	—	2	2	no	—
26	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₄²×C₇₉	C₂³	1	1	yes	characteristic
31	C₂²×D₇₉	—	2	1	no	—
32	Q₈×C₇₉	—	4	2	no	—
33	Q₈×C₇₉	—	4	2	no	—
34	C₂×C₄×C₇₉	—	4	2	no	—
35	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
38	C₂×Dic₇₉	—	2	1	no	—
39	D₃₁₆	—	2	1	no	—
40	D₃₁₆	D₈	1	2	yes	—
41	C₄×D₇₉	—	4	1	no	—
42	C₄×D₇₉	—	4	1	no	—
43	C₈×C₇₉	—	2	1	no	—
44	C₈×C₇₉	SD₁₆	1	2	yes	complemented
45	C₇₉⋊C₈	—	4	2	no	—
46	C₇₉⋊C₈	—	4	1	no	—
47	C₂²×C₇₉	#121	1	1	yes	characteristic
48	C₂×D₇₉	—	2	2	no	—
49	C₂×D₇₉	—	4	1	no	—
50	C₂×D₇₉	—	4	1	no	—
51	C₄×C₇₉	—	2	1	no	—
52	C₄×C₇₉	—	8	2	no	—
53	C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
54	C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
55	C₄×C₇₉	#135	1	1	yes	characteristic
56	C₄×C₇₉	#137	1	1	yes	characteristic
57	Dic₇₉	—	4	1	no	—
58	C₂×C₇₉	#461	1	1	yes	characteristic
59	C₂×C₇₉	#466	1	1	yes	characteristic
60	C₂×C₇₉	#473	1	1	yes	characteristic
61	D₇₉	—	4	2	no	—
62	#1673	—	79	1	no	maximal, Sylow
63	C₇₉	#1673	1	1	yes	characteristic, complemented, Sylow
64	#500	D₇₉	1	1	yes	characteristic, complemented
65	#461	—	79	2	no	—
66	C₈×D₄	—	79	1	no	—
67	#335	—	79	1	no	—
68	#332	—	79	2	no	—
69	C₂×SD₁₆	—	158	2	no	—
70	C₂²×C₈	—	158	1	no	—
71	C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
72	C₄×D₄	—	79	1	no	—
73	#107	—	2	1	no	—
74	#107	—	2	1	no	—
75	C₄⋊C₈	—	79	1	no	—
76	C₄⋊C₈	—	79	2	no	—
77	#103	—	158	2	no	—
78	C₂²⋊C₈	—	158	1	no	—
79	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
80	C₂×Q₈	—	2	2	no	—
81	C₂×D₄	—	79	1	no	—
82	C₂²×C₄	—	158	1	no	—
83	SD₁₆	—	316	4	no	—
84	C₂×C₈	—	316	2	no	—
85	C₂×C₈	—	158	2	no	—
86	C₂×C₈	—	158	1	no	—
87	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	79	1	no	—
92	C₂²⋊C₄	—	158	1	no	—
93	C₄²	C₂²×D₇₉	1	1	yes	characteristic
94	C₂³	—	158	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	4	2	no	—
97	D₄	—	79	2	no	—
98	D₄	—	158	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	316	1	no	—
101	C₂×C₄	—	316	1	no	—
102	C₂×C₄	—	158	1	no	—
103	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
105	C₂×C₄	#11929623	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	316	2	no	—
108	C₈	—	316	1	no	—
109	C₈	#11929611	1	2	yes	complemented
110	C₂²	—	316	1	no	—
111	C₂²	—	316	1	no	—
112	C₂²	—	158	2	no	—
113	C₂²	#14579132	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	8	2	no	—
116	C₄	—	316	1	no	—
117	C₄	D₈×D₇₉	1	1	yes	characteristic
118	C₄	C₂×#11929611	1	1	yes	characteristic
119	C₄	#14579180	1	1	yes	characteristic
120	C₄	#14579192	1	1	yes	characteristic
121	C₂	—	316	2	no	—
122	C₂	#19098765	1	1	yes	characteristic
123	C₂	#19098796	1	1	yes	characteristic
124	C₂	#19098835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098765

C₂

yes

maximal

C₈×D₃₁₆

C₂

yes

maximal, characteristic

#19098316

C₂

yes

maximal, characteristic, complemented

#19098309

C₂

yes

maximal, complemented

#500×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929611

—

C₂×C₈×D₇₉

—

C₄×D₃₁₆

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579085

—

(C₂×D₇₉)⋊C₈

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×D₃₁₆

D₄

yes

characteristic

C₂×C₄×D₇₉

—

#11929611

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₈×D₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

D₈

yes

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

SD₁₆

yes

complemented

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#121

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

C₂×SD₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

—

#1673

—

maximal, Sylow

C₇₉

#1673

yes

characteristic, complemented, Sylow

#500

D₇₉

yes

characteristic, complemented

#461

—

C₈×D₄

—

#335

—

#332

—

C₂×SD₁₆

—

158

—

C₂²×C₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

—

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

158

—

C₂²⋊C₈

—

158

—

C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

158

—

SD₁₆

—

316

—

C₂×C₈

—

316

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

158

—

C₄²

C₂²×D₇₉

yes

characteristic

C₂³

—

158

—

Q₈

—

Q₈

—

D₄

—

D₄

—

158

—

C₂×C₄

—

100

C₂×C₄

—

316

—

101

C₂×C₄

—

316

—

102

C₂×C₄

—

158

—

103

C₂×C₄

D₄×D₇₉

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

105

C₂×C₄

#11929623

yes

characteristic

106

C₈

—

107

C₈

—

316

—

108

C₈

—

316

—

109

C₈

#11929611

yes

complemented

110

C₂²

—

316

—

111

C₂²

—

316

—

112

C₂²

—

158

—

113

C₂²

#14579132

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

316

—

117

C₄

D₈×D₇₉

yes

characteristic

118

C₄

C₂×#11929611

yes

characteristic

119

C₄

#14579180

yes

characteristic

120

C₄

#14579192

yes

characteristic

121

C₂

—

316

—

122

C₂

#19098765

yes

characteristic

123

C₂

#19098796

yes

characteristic

124

C₂

#19098835

yes

characteristic

	subgroup	G/H
center	C₂²	#14579132
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#500×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	C₂×D₃₁₆	D₄
hypercenter	#500	D₇₉
hypocenter	C₇₉	#1673
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14579132

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#500×C₇₉

C₂

socle

C₂²×C₇₉

#121

prime radical

C₂×D₃₁₆

D₄

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
3. #19098316	C₂	semidirect
4. #19098309	C₂	semidirect
5. #500×C₇₉	C₂	semidirect
44. C₈×C₇₉	SD₁₆	semidirect
63. C₇₉	#1673	semidirect
64. #500	D₇₉	semidirect
109. C₈	#11929611	semidirect

H ≅ G/N

type

3. #19098316

C₂

semidirect

4. #19098309

C₂

semidirect

5. #500×C₇₉

C₂

semidirect

44. C₈×C₇₉

SD₁₆

semidirect

63. C₇₉

#1673

semidirect

64. #500

D₇₉

semidirect

109. C₈

#11929611

semidirect

Subgroups — #23762786

Conjugacy classes of subgroups

Complements