Group #23764245 — subgroups

Subgroups — #23764245

Order 10112.

292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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	#19098587	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579080	C₂	1	4	yes	maximal
3	C₄×#11929608	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#11929608	C₂²	1	2	yes	—
6	C₂×#11929608	C₂²	1	2	yes	—
7	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
8	#14579080	—	2	8	no	—
9	#14579072	—	2	1	no	—
10	C₄×C₇₉⋊C₈	—	2	1	no	—
11	C₂²×Q₈×C₇₉	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₇₉	—	2	1	no	—
13	C₄⋊Q₈×C₇₉	—	2	1	no	—
14	C₄⋊Q₈×C₇₉	—	2	2	no	—
15	C₂×C₄⋊C₄×C₇₉	C₄	1	4	yes	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	#11929608	—	2	4	no	—
18	#11929608	—	2	2	no	—
19	#11929608	D₄	1	4	yes	—
20	C₂×C₇₉⋊C₈	—	2	2	no	—
21	C₂×C₇₉⋊C₈	—	2	2	no	—
22	C₂×Q₈×C₇₉	—	2	2	no	—
23	C₂×Q₈×C₇₉	—	2	2	no	—
24	C₂×Q₈×C₇₉	—	2	4	no	—
25	C₂×Q₈×C₇₉	—	2	4	no	—
26	C₂²×C₄×C₇₉	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₇₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₉	—	2	4	no	—
30	C₄⋊C₄×C₇₉	—	2	4	no	—
31	C₄²×C₇₉	D₄	1	1	yes	characteristic
32	C₄²×C₇₉	D₄	1	1	yes	characteristic
33	C₄²×C₇₉	D₄	1	2	yes	—
34	C₂³×C₇₉	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₇₉	—	4	4	no	—
36	Q₈×C₇₉	—	4	2	no	—
37	Q₈×C₇₉	—	4	2	no	—
38	C₂×C₄×C₇₉	—	2	4	no	—
39	C₂×C₄×C₇₉	—	2	4	no	—
40	C₂×C₄×C₇₉	—	2	8	no	—
41	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	#41	1	2	yes	—
47	C₇₉⋊C₈	—	2	4	no	—
48	C₇₉⋊C₈	—	4	2	no	—
49	C₂²×C₇₉	—	2	2	no	—
50	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₇₉	C₄×D₄	1	2	yes	—
52	C₄×C₇₉	—	2	2	no	—
53	C₄×C₇₉	—	4	4	no	—
54	C₄×C₇₉	#101	1	4	yes	complemented
55	C₄×C₇₉	#121	1	2	yes	—
56	C₄×C₇₉	#124	1	1	yes	characteristic
57	C₄×C₇₉	#127	1	1	yes	characteristic
58	C₂×C₇₉	—	2	2	no	—
59	C₂×C₇₉	#390	1	1	yes	characteristic
60	C₂×C₇₉	C₂×#101	1	2	yes	—
61	#1966	—	79	1	no	maximal, Sylow
62	C₇₉	#1966	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
64	#423	—	79	1	no	—
65	C₂×#101	—	79	4	no	—
66	C₄×M₁₆	—	79	1	no	—
67	C₂²×Q₈	C₂×D₇₉	1	2	yes	—
68	C₂×M₁₆	—	79	2	no	—
69	C₂×M₁₆	—	79	2	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄⋊Q₈	—	2	1	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₂×C₄⋊C₄	Dic₇₉	1	4	yes	—
74	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
75	C₄⋊C₈	—	158	2	no	—
76	#101	—	158	8	no	—
77	#97	—	158	1	no	—
78	C₄×C₈	—	158	1	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂²×C₄	C₂×Dic₇₉	1	4	yes	—
84	C₂²×C₄	C₂×Dic₇₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
86	M₁₆	—	79	4	no	—
87	M₁₆	—	158	4	no	—
88	M₁₆	—	158	2	no	—
89	C₂×C₈	—	158	2	no	—
90	C₂×C₈	—	158	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
94	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
95	C₄²	C₂²⋊D₇₉	1	2	yes	—
96	C₂³	C₂²×Dic₇₉	1	1	yes	characteristic
97	Q₈	—	4	4	no	—
98	Q₈	—	4	2	no	—
99	Q₈	—	4	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	C₂²⋊Dic₇₉	1	2	yes	—
104	C₂×C₄	C₂²⋊Dic₇₉	1	2	yes	—
105	C₂×C₄	D₄×D₇₉	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
108	C₂×C₄	#11929623	1	2	yes	—
109	C₈	—	316	2	no	—
110	C₈	—	158	4	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
113	C₂²	D₄×Dic₇₉	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	#14579080	1	4	yes	complemented
117	C₄	#14579132	1	2	yes	—
118	C₄	#14579148	1	1	yes	characteristic
119	C₄	#14579157	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#19098466	1	1	yes	characteristic
122	C₂	C₂×#14579080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19098587

C₂

yes

maximal, characteristic

C₂×#14579080

C₂

yes

maximal

C₄×#11929608

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₂²

yes

—

C₂×#11929608

C₂²

yes

—

C₄⋊(C₇₉⋊C₈)

—

#14579080

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₂²×Q₈×C₇₉

C₂²

yes

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₄

yes

—

C₂×C₄²×C₇₉

C₂²

yes

—

—

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

C₂²×C₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₉

C₄×D₄

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#101

yes

complemented

C₄×C₇₉

#121

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

#390

yes

characteristic

C₂×C₇₉

C₂×#101

yes

—

#1966

—

maximal, Sylow

C₇₉

#1966

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₇₉

yes

characteristic

#423

—

C₂×#101

—

C₄×M₁₆

—

C₂²×Q₈

C₂×D₇₉

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₇₉

yes

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

158

—

#101

—

158

—

#97

—

158

—

C₄×C₈

—

158

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

M₁₆

—

M₁₆

—

158

—

M₁₆

—

158

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₇₉

yes

characteristic

C₄²

C₂²⋊D₇₉

yes

characteristic

C₄²

C₂²⋊D₇₉

yes

—

C₂³

C₂²×Dic₇₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²⋊Dic₇₉

yes

—

104

C₂×C₄

C₂²⋊Dic₇₉

yes

—

105

C₂×C₄

D₄×D₇₉

yes

—

106

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

108

C₂×C₄

#11929623

yes

—

109

C₈

—

316

—

110

C₈

—

158

—

111

C₂²

—

112

C₂²

C₂×C₂²⋊Dic₇₉

yes

characteristic

113

C₂²

D₄×Dic₇₉

yes

—

114

C₄

—

115

C₄

—

116

C₄

#14579080

yes

complemented

117

C₄

#14579132

yes

—

118

C₄

#14579148

yes

characteristic

119

C₄

#14579157

yes

characteristic

120

C₂

—

121

C₂

#19098466

yes

characteristic

122

C₂

C₂×#14579080

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇₉
commutator, prime radical	C₂³×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×C₄⋊Q₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	#1966
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇₉

commutator, prime radical

C₂³×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×C₄⋊Q₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₇₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₇₉

hypocenter

C₇₉

#1966

LCS

G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
54. C₄×C₇₉	#101	semidirect
62. C₇₉	#1966	semidirect
116. C₄	#14579080	semidirect

H ≅ G/N

type

54. C₄×C₇₉

#101

semidirect

62. C₇₉

#1966

semidirect

116. C₄

#14579080

semidirect

Subgroups — #23764245

Conjugacy classes of subgroups

Complements