Group #34174940 — subgroups

Subgroups — #34174940

Order 20224.

134 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 51 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	#23762298	C₂	1	1	yes	maximal, characteristic
2	#23761464	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1311	C₂	1	1	yes	maximal, characteristic
4	#19098693	—	2	1	no	—
5	C₄×Dic₃₁₆	C₄	1	1	yes	characteristic
6	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
7	#19098316	C₄	1	1	yes	characteristic
8	#19098295	—	2	1	no	—
9	C₄×C₁₆×C₇₉	—	2	1	no	—
10	C₈²×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₁₆	C₈	1	1	yes	characteristic
12	C₄×Dic₁₅₈	—	2	1	no	—
13	C₂×C₇₉⋊C₁₆	—	4	1	no	—
14	#14579095	C₈	1	1	yes	characteristic
15	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
16	Dic₇₉⋊C₈	—	4	1	no	—
17	#14579086	—	2	1	no	—
18	#14579071	—	4	1	no	—
19	C₂×C₁₆×C₇₉	—	2	2	no	—
20	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
22	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₅₈	—	2	1	no	—
24	Dic₃₁₆	—	4	1	no	—
25	C₂×C₇₉⋊C₈	—	4	1	no	—
26	C₄⋊Dic₇₉	—	2	1	no	—
27	Dic₇₉⋊C₄	—	4	1	no	—
28	C₄×Dic₇₉	—	4	1	no	—
29	C₇₉⋊C₁₆	—	8	1	no	—
30	C₂×C₈×C₇₉	—	2	1	no	—
31	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
34	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
35	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₁₆×C₇₉	—	2	4	no	—
37	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
39	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
40	C₂×Dic₇₉	—	4	1	no	—
41	C₂×Dic₇₉	—	4	1	no	—
42	Dic₁₅₈	—	4	1	no	—
43	Dic₁₅₈	—	4	1	no	—
44	C₈×C₇₉	—	2	2	no	—
45	C₈×C₇₉	—	2	1	no	—
46	C₈×C₇₉	—	2	1	no	—
47	C₈×C₇₉	—	2	1	no	—
48	C₈×C₇₉	—	2	1	no	—
49	C₇₉⋊C₈	—	8	1	no	—
50	C₂²×C₇₉	D₄⋊C₈	1	1	yes	characteristic
51	C₄×C₇₉	—	2	1	no	—
52	C₄×C₇₉	#349	1	1	yes	characteristic
53	C₄×C₇₉	#350	1	1	yes	characteristic
54	C₄×C₇₉	#360	1	1	yes	characteristic
55	C₄×C₇₉	#362	1	1	yes	characteristic
56	Dic₇₉	—	8	1	no	—
57	Dic₇₉	—	8	1	no	—
58	#7075	—	79	1	no	maximal, Sylow
59	C₂×C₇₉	#1330	1	1	yes	characteristic
60	C₂×C₇₉	#1334	1	1	yes	characteristic
61	C₂×C₇₉	#1335	1	1	yes	characteristic
62	#1591	—	79	1	no	—
63	#1367	—	79	1	no	—
64	#1311	D₇₉	1	1	yes	characteristic
65	C₇₉	#7075	1	1	yes	characteristic, complemented, Sylow
66	#446	—	158	1	no	—
67	C₄×Q₁₆	—	79	1	no	—
68	C₄⋊C₁₆	—	158	1	no	—
69	C₄×C₁₆	—	2	1	no	—
70	#335	—	79	1	no	—
71	Q₈⋊C₈	—	158	1	no	—
72	C₈²	C₂×D₇₉	1	1	yes	characteristic
73	C₂×Q₁₆	—	79	1	no	—
74	C₄×Q₈	—	158	1	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	—	316	1	no	—
77	#107	—	79	1	no	—
78	C₄⋊C₈	—	316	1	no	—
79	C₄⋊C₈	—	158	1	no	—
80	#103	—	158	1	no	—
81	#97	—	316	1	no	—
82	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
83	C₄×C₈	D₃₁₆	1	1	yes	characteristic
84	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
85	C₂×Q₈	—	158	1	no	—
86	Q₁₆	—	316	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	316	1	no	—
89	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
90	C₂×C₈	#11929607	1	1	yes	characteristic
91	C₂×C₈	D₆₃₂	1	1	yes	characteristic
92	C₂×C₈	#11929613	1	1	yes	characteristic
93	C₄⋊C₄	—	316	1	no	—
94	C₄⋊C₄	—	158	1	no	—
95	C₄²	—	316	1	no	—
96	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
97	C₁₆	—	2	4	no	—
98	C₁₆	—	632	1	no	—
99	Q₈	—	316	1	no	—
100	Q₈	—	316	1	no	—
101	C₂×C₄	—	316	1	no	—
102	C₂×C₄	—	316	1	no	—
103	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
104	C₂×C₄	#14579083	1	1	yes	characteristic
105	C₂×C₄	#14579089	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	632	1	no	—
112	C₂²	#19098292	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	632	1	no	—
115	C₄	—	632	1	no	—
116	C₄	#19098344	1	1	yes	characteristic
117	C₄	#19098348	1	1	yes	characteristic
118	C₄	#19098374	1	1	yes	characteristic
119	C₄	#19098380	1	1	yes	characteristic
120	C₂	#23761359	1	1	yes	characteristic
121	C₂	#23761371	1	1	yes	characteristic
122	C₂	#23761374	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23762298

C₂

yes

maximal, characteristic

#23761464

C₂

yes

maximal, characteristic

C₇₉×#1311

C₂

yes

maximal, characteristic

#19098693

—

C₄×Dic₃₁₆

C₄

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

—

#19098316

C₄

yes

characteristic

#19098295

—

C₄×C₁₆×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

C₈

yes

characteristic

C₄×Dic₁₅₈

—

C₂×C₇₉⋊C₁₆

—

#14579095

C₈

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

#14579086

—

#14579071

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

M₁₆

yes

characteristic

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

D₄⋊C₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

C₇₉

#7075

yes

characteristic, complemented, Sylow

#446

—

158

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

158

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

158

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

158

—

C₂×C₁₆

—

C₂×C₁₆

—

316

—

#107

—

C₄⋊C₈

—

316

—

C₄⋊C₈

—

158

—

#103

—

158

—

#97

—

316

—

C₄×C₈

C₄×D₇₉

yes

characteristic

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×Q₈

—

158

—

Q₁₆

—

316

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

C₈×D₇₉

yes

characteristic

C₂×C₈

#11929607

yes

characteristic

C₂×C₈

D₆₃₂

yes

characteristic

C₂×C₈

#11929613

yes

characteristic

C₄⋊C₄

—

316

—

C₄⋊C₄

—

158

—

C₄²

—

316

—

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

C₁₆

—

C₁₆

—

632

—

Q₈

—

316

—

100

Q₈

—

316

—

101

C₂×C₄

—

316

—

102

C₂×C₄

—

316

—

103

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

characteristic

104

C₂×C₄

#14579083

yes

characteristic

105

C₂×C₄

#14579089

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

632

—

112

C₂²

#19098292

yes

characteristic

113

C₄

—

114

C₄

—

632

—

115

C₄

—

632

—

116

C₄

#19098344

yes

characteristic

117

C₄

#19098348

yes

characteristic

118

C₄

#19098374

yes

characteristic

119

C₄

#19098380

yes

characteristic

120

C₂

#23761359

yes

characteristic

121

C₂

#23761371

yes

characteristic

122

C₂

#23761374

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579083
commutator	C₂×C₈×C₇₉	C₂×C₈
Frattini	C₈²	C₂×D₇₉
Fitting	C₇₉×#1311	C₂
socle, prime radical	C₂²×C₇₉	D₄⋊C₈
cosocle	C₈²×C₇₉	C₂²
hypercenter	#1311	D₇₉
hypocenter	C₇₉	#7075
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

#14579083

commutator

C₂×C₈×C₇₉

C₂×C₈

Frattini

C₈²

C₂×D₇₉

Fitting

C₇₉×#1311

C₂

socle, prime radical

C₂²×C₇₉

D₄⋊C₈

cosocle

C₈²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

N	H ≅ G/N	type
65. C₇₉	#7075	semidirect

H ≅ G/N

type

65. C₇₉

#7075

semidirect

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

subgroup	index
#23762298	2
#23761464	2
C₇₉×#1311	2
#7075	79

subgroup

index

#23762298

#23761464

C₇₉×#1311

#7075

Subgroups — #34174940

Conjugacy classes of subgroups

Complements

Maximal subgroups