Group #34203005 — subgroups

Subgroups — #34203005

Order 20224.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#23765675	C₂	1	1	yes	maximal, characteristic
2	#23765673	C₂	1	2	yes	maximal
3	#23763017	C₂	1	2	yes	maximal
4	C₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#2252	C₂	1	1	yes	maximal, characteristic
6	#19098968	C₂²	1	2	yes	—
7	#19098966	—	2	2	no	—
8	#19098942	—	2	2	no	—
9	#19098389	—	2	2	no	—
10	#19098388	—	2	4	no	—
11	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	#500×C₇₉	C₂²	1	2	yes	—
14	#366×C₇₉	—	2	2	no	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#11929614	—	4	4	no	—
17	#14579161	—	2	2	no	—
18	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
19	#14579095	—	2	2	no	—
20	#14579095	—	2	2	no	—
21	#14579094	—	4	4	no	—
22	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₇₉	—	2	2	no	—
25	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
26	#107×C₇₉	—	2	2	no	—
27	#107×C₇₉	—	2	2	no	—
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₅₈	—	4	2	no	—
30	#11929614	—	8	8	no	—
31	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
32	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
33	C₄⋊Dic₇₉	—	4	2	no	—
34	C₄⋊Dic₇₉	—	4	2	no	—
35	C₇₉⋊C₁₆	Q₁₆	1	4	yes	complemented
36	C₂×Q₈×C₇₉	—	4	2	no	—
37	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₇₉	—	4	2	no	—
40	C₄⋊C₄×C₇₉	—	4	2	no	—
41	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₉	Q₁₆	1	4	yes	complemented
43	Q₈×C₇₉	—	8	4	no	—
44	C₂×C₄×C₇₉	—	8	2	no	—
45	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₇₉	—	8	2	no	—
49	Dic₁₅₈	—	8	4	no	—
50	C₈×C₇₉	C₂×Q₁₆	1	2	yes	—
51	C₈×C₇₉	C₂×Q₁₆	1	2	yes	—
52	C₇₉⋊C₈	D₁₆	1	4	yes	—
53	C₇₉⋊C₈	Q₃₂	1	4	yes	—
54	C₂²×C₇₉	#493	1	1	yes	characteristic
55	C₄×C₇₉	—	16	2	no	—
56	C₄×C₇₉	#494	1	1	yes	characteristic
57	C₄×C₇₉	#494	1	1	yes	characteristic
58	C₄×C₇₉	C₂×D₁₆	1	2	yes	—
59	C₄×C₇₉	C₂×Q₃₂	1	2	yes	—
60	Dic₇₉	—	16	2	no	—
61	#12521	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	#1713	1	1	yes	characteristic
63	C₂×C₇₉	#2246	1	1	yes	characteristic
64	C₂×C₇₉	#2247	1	1	yes	characteristic
65	#2252	—	79	2	no	—
66	#2252	—	79	1	no	—
67	#2252	D₇₉	1	1	yes	characteristic
68	#1713	—	79	2	no	—
69	C₈×C₁₆	—	79	1	no	—
70	C₇₉	#12521	1	1	yes	characteristic, complemented, Sylow
71	#500	—	79	2	no	—
72	#500	—	158	2	no	—
73	#500	C₂×D₇₉	1	2	yes	—
74	#494	—	158	2	no	—
75	#366	—	2	2	no	—
76	#366	—	158	4	no	—
77	#366	—	158	2	no	—
78	C₄×C₁₆	—	79	1	no	—
79	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
80	C₈²	—	79	1	no	—
81	C₂×Q₁₆	—	316	4	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	158	2	no	—
84	C₂×C₁₆	—	79	2	no	—
85	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
86	#107	—	2	2	no	—
87	#107	—	2	2	no	—
88	#107	—	316	4	no	—
89	#107	—	158	2	no	—
90	#107	—	158	2	no	—
91	C₄×C₈	—	79	1	no	—
92	C₄×C₈	—	79	1	no	—
93	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	316	2	no	—
96	Q₁₆	—	632	8	no	—
97	C₂×C₈	—	79	2	no	—
98	C₂×C₈	—	79	2	no	—
99	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	316	2	no	—
104	C₄⋊C₄	—	316	2	no	—
105	C₄²	D₄×D₇₉	1	1	yes	characteristic
106	C₁₆	—	79	4	no	—
107	C₁₆	#11929614	1	4	yes	complemented
108	Q₈	—	8	4	no	—
109	Q₈	—	632	4	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	632	2	no	—
112	C₂×C₄	#14579157	1	1	yes	characteristic
113	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
114	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
115	C₈	—	79	4	no	—
116	C₈	—	79	4	no	—
117	C₈	Q₁₆×D₇₉	1	2	yes	—
118	C₈	C₂×#11929614	1	2	yes	—
119	C₂²	#19098938	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	1264	2	no	—
122	C₄	#19098941	1	1	yes	characteristic
123	C₄	#19098942	1	1	yes	characteristic
124	C₄	D₁₆×D₇₉	1	2	yes	—
125	C₄	Q₃₂×D₇₉	1	2	yes	—
126	C₂	#23763017	1	1	yes	characteristic
127	C₂	#23765648	1	1	yes	characteristic
128	C₂	#23765651	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765675

C₂

yes

maximal, characteristic

#23765673

C₂

yes

maximal

#23763017

C₂

yes

maximal

C₁₆×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₇₉×#2252

C₂

yes

maximal, characteristic

#19098968

C₂²

yes

—

—

—

—

—

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

#500×C₇₉

C₂²

yes

—

#366×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#11929614

—

#14579161

—

C₂×C₇₉⋊C₁₆

D₄

yes

—

#14579095

—

#14579095

—

#14579094

—

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₄⋊Q₈×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

#11929614

—

C₂×C₇₉⋊C₈

D₈

yes

—

C₂×C₇₉⋊C₈

D₈

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₇₉⋊C₁₆

Q₁₆

yes

complemented

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

Q₁₆

yes

complemented

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

C₂×Q₁₆

yes

—

C₈×C₇₉

C₂×Q₁₆

yes

—

C₇₉⋊C₈

D₁₆

yes

—

C₇₉⋊C₈

Q₃₂

yes

—

C₂²×C₇₉

#493

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×D₁₆

yes

—

C₄×C₇₉

C₂×Q₃₂

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

#1713

—

C₈×C₁₆

—

C₇₉

#12521

yes

characteristic, complemented, Sylow

#500

—

#500

—

158

—

#500

C₂×D₇₉

yes

—

#494

—

158

—

#366

—

#366

—

158

—

#366

—

158

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

—

C₂×Q₁₆

—

316

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₉

yes

—

#107

—

#107

—

#107

—

316

—

#107

—

158

—

#107

—

158

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

316

—

Q₁₆

—

632

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₇₉

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

316

—

104

C₄⋊C₄

—

316

—

105

C₄²

D₄×D₇₉

yes

characteristic

106

C₁₆

—

107

C₁₆

#11929614

yes

complemented

108

Q₈

—

109

Q₈

—

632

—

110

C₂×C₄

—

111

C₂×C₄

—

632

—

112

C₂×C₄

#14579157

yes

characteristic

113

C₂×C₄

D₈×D₇₉

yes

characteristic

114

C₂×C₄

D₈×D₇₉

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

Q₁₆×D₇₉

yes

—

118

C₈

C₂×#11929614

yes

—

119

C₂²

#19098938

yes

characteristic

120

C₄

—

121

C₄

—

1264

—

122

C₄

#19098941

yes

characteristic

123

C₄

#19098942

yes

characteristic

124

C₄

D₁₆×D₇₉

yes

—

125

C₄

Q₃₂×D₇₉

yes

—

126

C₂

#23763017

yes

characteristic

127

C₂

#23765648

yes

characteristic

128

C₂

#23765651

yes

characteristic

	subgroup	G/H
center	C₂²	#19098938
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2252	C₂
socle, prime radical	C₂²×C₇₉	#493
hypercenter	#2252	D₇₉
hypocenter	C₇₉	#12521
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#19098938

commutator, cosocle

C₄×C₈×C₇₉

C₂³

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#2252

C₂

C₂²×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
35. C₇₉⋊C₁₆	Q₁₆	semidirect
42. C₁₆×C₇₉	Q₁₆	semidirect
70. C₇₉	#12521	semidirect
107. C₁₆	#11929614	semidirect

H ≅ G/N

type

35. C₇₉⋊C₁₆

Q₁₆

semidirect

42. C₁₆×C₇₉

Q₁₆

semidirect

70. C₇₉

#12521

semidirect

107. C₁₆

#11929614

semidirect

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

subgroup	index
#23765675	2
#23765673	2
#23763017	2
C₁₆×C₇₉⋊C₈	2
C₇₉×#2252	2
#12521	79

subgroup

index

#23765675

#23765673

#23763017

C₁₆×C₇₉⋊C₈

C₇₉×#2252

#12521

Subgroups — #34203005

Conjugacy classes of subgroups

Complements

Maximal subgroups