Group #23764221 — subgroups

Subgroups — #23764221

Order 10112.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14579085	C₂	1	4	yes	maximal
2	C₂×#14579072	C₂	1	1	yes	maximal, characteristic
3	#19098429	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
7	#14579085	—	2	8	no	—
8	#14579072	—	2	2	no	—
9	C₇₉⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₇₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₇₉	—	2	2	no	—
12	C₄⋊Q₈×C₇₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
14	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₇₉	—	2	2	no	—
16	C₂×C₇₉⋊C₈	—	2	2	no	—
17	C₂×C₇₉⋊C₈	—	2	2	no	—
18	C₂×C₇₉⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₇₉	—	2	4	no	—
20	C₂×Q₈×C₇₉	—	2	2	no	—
21	C₂×Q₈×C₇₉	—	2	4	no	—
22	C₂×Q₈×C₇₉	—	2	4	no	—
23	C₂×Q₈×C₇₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₇₉	—	2	2	no	—
25	C₂²×C₄×C₇₉	D₄	1	2	yes	—
26	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₇₉	—	2	8	no	—
28	C₄²×C₇₉	C₂×C₄	1	2	yes	—
29	C₄²×C₇₉	D₄	1	2	yes	—
30	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₇₉	—	2	8	no	—
32	Q₈×C₇₉	—	4	4	no	—
33	C₂×C₄×C₇₉	—	2	4	no	—
34	C₂×C₄×C₇₉	—	2	4	no	—
35	C₂×C₄×C₇₉	—	4	2	no	—
36	C₂×C₄×C₇₉	—	4	2	no	—
37	C₂×C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₉	#41	1	2	yes	—
42	C₂×Dic₇₉	—	4	2	no	—
43	C₂×Dic₇₉	—	4	4	no	—
44	C₇₉⋊C₈	—	2	4	no	—
45	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₇₉	#124	1	1	yes	characteristic
47	C₂²×C₇₉	#127	1	1	yes	characteristic
48	C₂²×C₇₉	#137	1	4	yes	—
49	C₄×C₇₉	—	2	4	no	—
50	C₄×C₇₉	—	4	4	no	—
51	C₄×C₇₉	C₄×D₄	1	2	yes	—
52	C₄×C₇₉	#121	1	2	yes	—
53	Dic₇₉	—	8	2	no	—
54	C₂×C₇₉	#390	1	1	yes	characteristic
55	C₂×C₇₉	#419	1	2	yes	—
56	C₂×C₇₉	#491	1	2	yes	—
57	C₂×C₇₉	#497	1	2	yes	—
58	#1960	—	79	1	no	maximal, Sylow
59	C₇₉	#1960	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
61	C₂×#103	—	79	4	no	—
62	C₂×#97	—	79	1	no	—
63	#382	—	79	1	no	—
64	C₂²×Q₈	C₂×D₇₉	1	2	yes	—
65	C₂²×C₈	—	79	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₇₉	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	79	2	no	—
70	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
71	#103	—	158	8	no	—
72	#97	—	158	2	no	—
73	#95	—	158	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₇₉	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	158	2	no	—
80	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
81	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
82	C₂×C₈	—	79	4	no	—
83	C₂×C₈	—	158	2	no	—
84	C₂×C₈	—	158	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	158	4	no	—
87	C₄²	C₂×Dic₇₉	1	2	yes	—
88	C₄²	C₂²⋊D₇₉	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	316	2	no	—
97	C₂×C₄	—	316	4	no	—
98	C₂×C₄	C₂²⋊Dic₇₉	1	4	yes	—
99	C₂×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₇₉	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
102	C₂×C₄	#11929623	1	2	yes	—
103	C₈	—	158	4	no	—
104	C₂²	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
105	C₂²	#14579148	1	1	yes	characteristic
106	C₂²	#14579157	1	1	yes	characteristic
107	C₂²	#14579192	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	632	2	no	—
111	C₄	D₄×Dic₇₉	1	2	yes	—
112	C₄	#14579132	1	2	yes	—
113	C₂	#19098466	1	1	yes	characteristic
114	C₂	#19098572	1	2	yes	—
115	C₂	#19098931	1	2	yes	—
116	C₂	#19098955	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579085

C₂

yes

maximal

C₂×#14579072

C₂

yes

maximal, characteristic

#19098429

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

#14579085

—

#14579072

—

C₇₉⋊#95

—

C₂²×Q₈×C₇₉

C₂²

yes

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

—

C₄×C₇₉

#121

yes

—

Dic₇₉

—

C₂×C₇₉

#390

yes

characteristic

C₂×C₇₉

#419

yes

—

C₂×C₇₉

#491

yes

—

C₂×C₇₉

#497

yes

—

#1960

—

maximal, Sylow

C₇₉

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₇₉

yes

characteristic

C₂×#103

—

C₂×#97

—

#382

—

C₂²×Q₈

C₂×D₇₉

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₇₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#103

—

158

—

#97

—

158

—

#95

—

158

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₇₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄²

C₂×Dic₇₉

yes

—

C₄²

C₂²⋊D₇₉

yes

—

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

316

—

C₂×C₄

—

316

—

C₂×C₄

C₂²⋊Dic₇₉

yes

—

C₂×C₄

C₂²×Dic₇₉

yes

characteristic

100

C₂×C₄

D₄×D₇₉

yes

—

101

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

102

C₂×C₄

#11929623

yes

—

103

C₈

—

158

—

104

C₂²

C₂×C₂²⋊Dic₇₉

yes

characteristic

105

C₂²

#14579148

yes

characteristic

106

C₂²

#14579157

yes

characteristic

107

C₂²

#14579192

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

632

—

111

C₄

D₄×Dic₇₉

yes

—

112

C₄

#14579132

yes

—

113

C₂

#19098466

yes

characteristic

114

C₂

#19098572

yes

—

115

C₂

#19098931

yes

—

116

C₂

#19098955

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×C₄⋊Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	#1960
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×C₄⋊Q₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₇₉

hypocenter

C₇₉

#1960

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₇₉	C₄	semidirect
59. C₇₉	#1960	semidirect
67. C₄⋊Q₈	Dic₇₉	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₇₉

C₄

semidirect

59. C₇₉

#1960

semidirect

67. C₄⋊Q₈

Dic₇₉

semidirect

Subgroups — #23764221

Conjugacy classes of subgroups

Complements