Group #34174933 — subgroups

Subgroups — #34174933

Order 20224.

132 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 59 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	#23761470	C₂	1	1	yes	maximal, characteristic
2	#23761318	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1591	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
5	C₄×C₇₉⋊C₁₆	—	2	1	no	—
6	#446×C₇₉	—	2	1	no	—
7	C₄×Q₁₆×C₇₉	C₄	1	1	yes	characteristic
8	#335×C₇₉	C₄	1	1	yes	characteristic
9	Q₈⋊C₈×C₇₉	—	2	1	no	—
10	C₈²×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₇₉⋊C₁₆	—	2	2	no	—
12	C₂×C₇₉⋊C₁₆	—	4	1	no	—
13	C₂×Q₁₆×C₇₉	C₈	1	1	yes	characteristic
14	C₄×Q₈×C₇₉	—	2	1	no	—
15	#107×C₇₉	C₈	1	1	yes	characteristic
16	C₄⋊C₈×C₇₉	—	2	1	no	—
17	C₄⋊C₈×C₇₉	—	4	1	no	—
18	#103×C₇₉	—	2	1	no	—
19	#97×C₇₉	—	4	1	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	C₇₉⋊C₁₆	—	4	2	no	—
24	C₇₉⋊C₁₆	—	8	1	no	—
25	C₂×Q₈×C₇₉	—	2	1	no	—
26	Q₁₆×C₇₉	—	4	1	no	—
27	C₂×C₈×C₇₉	—	2	1	no	—
28	C₂×C₈×C₇₉	—	4	1	no	—
29	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
32	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
33	C₄⋊C₄×C₇₉	—	2	1	no	—
34	C₄⋊C₄×C₇₉	—	4	1	no	—
35	C₄²×C₇₉	—	4	1	no	—
36	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
37	Q₈×C₇₉	—	4	1	no	—
38	Q₈×C₇₉	—	4	1	no	—
39	C₂×C₄×C₇₉	—	4	1	no	—
40	C₂×C₄×C₇₉	—	4	1	no	—
41	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
42	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
43	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
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	—	8	1	no	—
49	C₈×C₇₉	SD₃₂	1	1	yes	characteristic
50	C₈×C₇₉	Q₃₂	1	1	yes	characteristic
51	C₂²×C₇₉	D₄⋊C₈	1	1	yes	characteristic
52	C₄×C₇₉	—	2	1	no	—
53	C₄×C₇₉	—	8	1	no	—
54	C₄×C₇₉	—	8	1	no	—
55	C₄×C₇₉	#349	1	1	yes	characteristic
56	C₄×C₇₉	#350	1	1	yes	characteristic
57	C₄×C₇₉	#358	1	1	yes	characteristic
58	C₄×C₇₉	#359	1	1	yes	characteristic
59	#7073	—	79	1	no	maximal, Sylow
60	C₂×C₇₉	#1330	1	1	yes	characteristic
61	C₂×C₇₉	#1332	1	1	yes	characteristic
62	C₂×C₇₉	#1336	1	1	yes	characteristic
63	#1591	D₇₉	1	1	yes	characteristic
64	#1370	—	79	1	no	—
65	#1312	—	79	1	no	—
66	C₇₉	#7073	1	1	yes	characteristic, complemented, Sylow
67	#446	—	2	1	no	—
68	C₄×Q₁₆	Dic₇₉	1	1	yes	characteristic
69	C₄⋊C₁₆	—	158	1	no	—
70	C₄×C₁₆	—	158	1	no	—
71	#335	Dic₇₉	1	1	yes	characteristic
72	Q₈⋊C₈	—	2	1	no	—
73	C₈²	C₂×D₇₉	1	1	yes	characteristic
74	C₂×Q₁₆	C₇₉⋊C₈	1	1	yes	characteristic
75	C₄×Q₈	—	2	1	no	—
76	C₂×C₁₆	—	316	1	no	—
77	C₂×C₁₆	—	158	2	no	—
78	#107	C₇₉⋊C₈	1	1	yes	characteristic
79	C₄⋊C₈	—	2	1	no	—
80	C₄⋊C₈	—	4	1	no	—
81	#103	—	2	1	no	—
82	#97	—	4	1	no	—
83	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
84	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
85	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
86	C₂×Q₈	—	2	1	no	—
87	Q₁₆	—	4	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
91	C₂×C₈	#11929608	1	1	yes	characteristic
92	C₂×C₈	#11929609	1	1	yes	characteristic
93	C₂×C₈	#11929612	1	1	yes	characteristic
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄²	—	4	1	no	—
97	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
98	C₁₆	—	316	2	no	—
99	C₁₆	—	632	1	no	—
100	Q₈	—	4	1	no	—
101	Q₈	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
105	C₂×C₄	#14579082	1	1	yes	characteristic
106	C₂×C₄	#14579088	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	8	1	no	—
112	C₈	#14579104	1	1	yes	characteristic
113	C₈	#14579107	1	1	yes	characteristic
114	C₂²	D₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	#19098345	1	1	yes	characteristic
119	C₄	#19098347	1	1	yes	characteristic
120	C₄	#19098367	1	1	yes	characteristic
121	C₄	#19098370	1	1	yes	characteristic
122	C₂	#23761358	1	1	yes	characteristic
123	C₂	#23761364	1	1	yes	characteristic
124	C₂	#23761376	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23761470

C₂

yes

maximal, characteristic

#23761318

C₂

yes

maximal, characteristic

C₇₉×#1591

C₂

yes

maximal, characteristic

C₄⋊(C₇₉⋊C₁₆)

—

C₄×C₇₉⋊C₁₆

—

#446×C₇₉

—

C₄×Q₁₆×C₇₉

C₄

yes

characteristic

#335×C₇₉

C₄

yes

characteristic

Q₈⋊C₈×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×Q₁₆×C₇₉

C₈

yes

characteristic

C₄×Q₈×C₇₉

—

#107×C₇₉

C₈

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

#103×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

—

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

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

SD₃₂

yes

characteristic

C₈×C₇₉

Q₃₂

yes

characteristic

C₂²×C₇₉

D₄⋊C₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

#1370

—

#1312

—

C₇₉

#7073

yes

characteristic, complemented, Sylow

#446

—

C₄×Q₁₆

Dic₇₉

yes

characteristic

C₄⋊C₁₆

—

158

—

C₄×C₁₆

—

158

—

#335

Dic₇₉

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

C₇₉⋊C₈

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

316

—

C₂×C₁₆

—

158

—

#107

C₇₉⋊C₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₇₉⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²⋊Dic₇₉

yes

characteristic

C₁₆

—

316

—

C₁₆

—

632

—

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

characteristic

105

C₂×C₄

#14579082

yes

characteristic

106

C₂×C₄

#14579088

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

#14579104

yes

characteristic

113

C₈

#14579107

yes

characteristic

114

C₂²

D₄⋊(C₇₉⋊C₈)

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#19098345

yes

characteristic

119

C₄

#19098347

yes

characteristic

120

C₄

#19098367

yes

characteristic

121

C₄

#19098370

yes

characteristic

122

C₂

#23761358

yes

characteristic

123

C₂

#23761364

yes

characteristic

124

C₂

#23761376

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579082
commutator	C₂×C₈×C₇₉	C₂×C₈
Frattini	C₈²	C₂×D₇₉
Fitting	C₇₉×#1591	C₂
socle, prime radical	C₂²×C₇₉	D₄⋊C₈
cosocle	C₈²×C₇₉	C₂²
hypercenter	#1591	D₇₉
hypocenter	C₇₉	#7073
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1591

subgroup

G/H

center

C₂×C₄

#14579082

commutator

C₂×C₈×C₇₉

C₂×C₈

Frattini

C₈²

C₂×D₇₉

Fitting

C₇₉×#1591

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 ◁ #1591

N	H ≅ G/N	type
66. C₇₉	#7073	semidirect

H ≅ G/N

type

66. C₇₉

#7073

semidirect

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

subgroup	index
#23761470	2
#23761318	2
C₇₉×#1591	2
#7073	79

subgroup

index

#23761470

#23761318

C₇₉×#1591

#7073

Subgroups — #34174933

Conjugacy classes of subgroups

Complements

Maximal subgroups