Group #34200272 — subgroups

Subgroups — #34200272

Order 20224.

206 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 83 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	#23762297	C₂	1	2	yes	maximal
2	#23761474	C₂	1	1	yes	maximal, characteristic
3	#23761364	C₂	1	2	yes	maximal
4	#23761316	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098694	—	2	2	no	—
7	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#19098316	C₂²	1	2	yes	—
9	Q₈⋊(C₇₉⋊C₈)	—	2	2	no	—
10	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₇₉	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₇₉	C₂²	1	2	yes	—
13	#366×C₇₉	C₄	1	1	yes	characteristic
14	#358×C₇₉	C₄	1	2	yes	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₇₉⋊C₁₆	—	2	1	no	—
17	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
19	#14579072	—	4	2	no	—
20	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
21	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₇₉	C₂×C₄	1	2	yes	—
23	C₄×Q₈×C₇₉	—	2	2	no	—
24	Q₃₂×C₇₉	—	2	2	no	—
25	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
27	#107×C₇₉	C₂×C₄	1	2	yes	—
28	#103×C₇₉	—	2	2	no	—
29	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×C₇₉⋊C₈	—	2	1	no	—
31	C₂×C₇₉⋊C₈	—	4	2	no	—
32	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
33	C₇₉⋊C₁₆	—	2	2	no	—
34	C₂×Q₈×C₇₉	—	2	2	no	—
35	Q₁₆×C₇₉	—	2	2	no	—
36	Q₁₆×C₇₉	—	2	2	no	—
37	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₇₉	—	2	2	no	—
40	C₄⋊C₄×C₇₉	—	4	2	no	—
41	C₄²×C₇₉	—	4	2	no	—
42	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₉	—	2	1	no	—
44	C₁₆×C₇₉	M₁₆	1	2	yes	—
45	Q₈×C₇₉	—	4	2	no	—
46	Q₈×C₇₉	—	4	2	no	—
47	C₂×C₄×C₇₉	—	4	2	no	—
48	C₂×C₄×C₇₉	—	4	2	no	—
49	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
52	C₈×C₇₉	—	2	1	no	—
53	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
54	C₈×C₇₉	#131	1	1	yes	characteristic
55	C₇₉⋊C₈	—	2	2	no	—
56	C₇₉⋊C₈	—	4	1	no	—
57	C₇₉⋊C₈	—	8	2	no	—
58	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
59	C₄×C₇₉	—	2	1	no	—
60	C₄×C₇₉	—	8	2	no	—
61	C₄×C₇₉	—	8	2	no	—
62	C₄×C₇₉	#436	1	1	yes	characteristic
63	C₄×C₇₉	#443	1	1	yes	characteristic
64	C₄×C₇₉	#510	1	2	yes	—
65	#12085	—	79	1	no	maximal, Sylow
66	C₂×C₇₉	#1589	1	1	yes	characteristic
67	C₂×C₇₉	#2176	1	1	yes	characteristic
68	C₂×C₇₉	#2179	1	1	yes	characteristic
69	C₄×Q₃₂	D₇₉	1	1	yes	characteristic
70	#1591	—	79	2	no	—
71	#1372	—	79	1	no	—
72	#1332	—	79	2	no	—
73	#1311	—	79	1	no	—
74	C₇₉	#12085	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	Dic₇₉	1	1	yes	characteristic
76	#446	—	158	2	no	—
77	C₄×Q₁₆	C₂×D₇₉	1	2	yes	—
78	#366	Dic₇₉	1	1	yes	characteristic
79	#358	Dic₇₉	1	2	yes	—
80	C₄×C₁₆	—	79	1	no	—
81	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
82	#335	—	79	2	no	—
83	Q₈⋊C₈	—	158	2	no	—
84	C₈²	—	79	1	no	—
85	C₂×Q₁₆	C₂×Dic₇₉	1	2	yes	—
86	C₄×Q₈	—	2	2	no	—
87	Q₃₂	—	2	2	no	—
88	C₂×C₁₆	—	158	1	no	—
89	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
90	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
91	#107	C₂×Dic₇₉	1	2	yes	—
92	C₄⋊C₈	—	316	2	no	—
93	C₄⋊C₈	—	158	2	no	—
94	#103	—	2	2	no	—
95	#97	—	316	2	no	—
96	C₄×C₈	—	79	1	no	—
97	C₄×C₈	—	79	1	no	—
98	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	2	2	no	—
102	C₂×C₈	—	316	2	no	—
103	C₂×C₈	—	79	2	no	—
104	C₂×C₈	—	158	1	no	—
105	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
106	C₂×C₈	#11929623	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	—	4	2	no	—
110	C₄²	D₄×D₇₉	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	—	158	2	no	—
113	C₁₆	#11929608	1	2	yes	—
114	Q₈	—	4	2	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
119	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
120	C₂×C₄	#14579180	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	316	1	no	—
123	C₈	—	632	2	no	—
124	C₈	—	158	2	no	—
125	C₈	C₂×#11929608	1	1	yes	characteristic
126	C₈	#14579168	1	1	yes	characteristic
127	C₂²	D₈×Dic₇₉	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	8	2	no	—
130	C₄	—	8	2	no	—
131	C₄	#19098641	1	1	yes	characteristic
132	C₄	#19098680	1	1	yes	characteristic
133	C₄	#19099005	1	2	yes	—
134	C₂	#23762285	1	1	yes	characteristic
135	C₂	#23765271	1	1	yes	characteristic
136	C₂	#23765286	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23762297

C₂

yes

maximal

#23761474

C₂

yes

maximal, characteristic

#23761364

C₂

yes

maximal

#23761316

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₇₉

C₂

yes

maximal, characteristic

#19098694

—

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098316

C₂²

yes

—

Q₈⋊(C₇₉⋊C₈)

—

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₇₉

C₄

yes

characteristic

C₄×Q₁₆×C₇₉

C₂²

yes

—

#366×C₇₉

C₄

yes

characteristic

#358×C₇₉

C₄

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

#14579072

—

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₇₉

C₂×C₄

yes

—

C₄×Q₈×C₇₉

—

Q₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

#107×C₇₉

C₂×C₄

yes

—

#103×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₈

yes

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

M₁₆

yes

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

C₂×M₁₆

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₄×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₄×Q₃₂

D₇₉

yes

—

—

—

—

C₇₉

#12085

yes

characteristic, complemented, Sylow

C₂×Q₃₂

Dic₇₉

yes

characteristic

#446

—

158

—

C₄×Q₁₆

C₂×D₇₉

yes

—

#366

Dic₇₉

yes

characteristic

#358

Dic₇₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

—

Q₈⋊C₈

—

158

—

C₈²

—

C₂×Q₁₆

C₂×Dic₇₉

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

158

—

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

#107

C₂×Dic₇₉

yes

—

C₄⋊C₈

—

316

—

C₄⋊C₈

—

158

—

#103

—

#97

—

316

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

316

—

103

C₂×C₈

—

104

C₂×C₈

—

158

—

105

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

106

C₂×C₈

#11929623

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

D₄×D₇₉

yes

characteristic

111

C₁₆

—

112

C₁₆

—

158

—

113

C₁₆

#11929608

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

D₄×Dic₇₉

yes

characteristic

119

C₂×C₄

D₈×D₇₉

yes

characteristic

120

C₂×C₄

#14579180

yes

characteristic

121

C₈

—

122

C₈

—

316

—

123

C₈

—

632

—

124

C₈

—

158

—

125

C₈

C₂×#11929608

yes

characteristic

126

C₈

#14579168

yes

characteristic

127

C₂²

D₈×Dic₇₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#19098641

yes

characteristic

132

C₄

#19098680

yes

characteristic

133

C₄

#19099005

yes

—

134

C₂

#23762285

yes

characteristic

135

C₂

#23765271

yes

characteristic

136

C₂

#23765286

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₄×Q₃₂×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₄×D₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₄×Q₃₂	D₇₉
hypocenter	C₇₉	#12085
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂×C₄

D₈×D₇₉

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₄×Q₃₂×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

C₄×D₈

cosocle

C₄×C₈×C₇₉

C₂³

hypercenter

C₄×Q₃₂

D₇₉

hypocenter

C₇₉

#12085

LCS

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

N	H ≅ G/N	type
74. C₇₉	#12085	semidirect

H ≅ G/N

type

74. C₇₉

#12085

semidirect

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

subgroup	index
#23762297	2
#23761474	2
#23761364	2
#23761316	2
C₄×Q₃₂×C₇₉	2
#12085	79

subgroup

index

C₄×Q₃₂×C₇₉

Subgroups — #34200272

Conjugacy classes of subgroups

Complements

Maximal subgroups