Group #34200273 — subgroups

Subgroups — #34200273

Order 20224.

206 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₄×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic
2	#23762298	C₂	1	2	yes	maximal
3	#23761474	C₂	1	1	yes	maximal, characteristic
4	#23761365	C₂	1	2	yes	maximal
5	C₇₉×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₂	C₄	1	1	yes	characteristic
7	#19098693	—	2	2	no	—
8	C₄×Dic₃₁₆	C₂²	1	2	yes	—
9	#19098389	C₄	1	1	yes	characteristic
10	#19098368	C₄	1	2	yes	—
11	#19098316	C₂²	1	2	yes	—
12	#19098295	—	2	2	no	—
13	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₈²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₅₈	—	2	2	no	—
18	Dic₆₃₂	—	2	2	no	—
19	#14579095	C₂×C₄	1	2	yes	—
20	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
21	Dic₇₉⋊C₈	—	4	2	no	—
22	#14579086	—	2	2	no	—
23	#14579071	—	4	2	no	—
24	C₂×C₁₆×C₇₉	—	2	1	no	—
25	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₅₈	—	2	2	no	—
31	Dic₃₁₆	—	2	2	no	—
32	Dic₃₁₆	—	2	2	no	—
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₄⋊Dic₇₉	—	2	2	no	—
35	Dic₇₉⋊C₄	—	4	2	no	—
36	C₄×Dic₇₉	—	4	2	no	—
37	C₂×C₈×C₇₉	—	2	1	no	—
38	C₂×C₈×C₇₉	D₈	1	2	yes	—
39	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
41	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₉	—	2	2	no	—
43	C₁₆×C₇₉	—	2	1	no	—
44	C₁₆×C₇₉	M₁₆	1	2	yes	—
45	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
48	C₂×Dic₇₉	—	4	2	no	—
49	C₂×Dic₇₉	—	4	2	no	—
50	Dic₁₅₈	—	4	2	no	—
51	Dic₁₅₈	—	4	2	no	—
52	C₈×C₇₉	—	2	1	no	—
53	C₈×C₇₉	—	2	2	no	—
54	C₈×C₇₉	—	4	1	no	—
55	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₇₉	#131	1	1	yes	characteristic
57	C₇₉⋊C₈	—	8	2	no	—
58	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
59	C₄×C₇₉	—	2	1	no	—
60	C₄×C₇₉	#436	1	1	yes	characteristic
61	C₄×C₇₉	#443	1	1	yes	characteristic
62	C₄×C₇₉	#510	1	2	yes	—
63	Dic₇₉	—	8	2	no	—
64	Dic₇₉	—	8	2	no	—
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	—	79	1	no	—
70	#1591	—	79	2	no	—
71	#1372	—	79	1	no	—
72	#1332	—	79	2	no	—
73	#1311	D₇₉	1	1	yes	characteristic
74	C₇₉	#12085	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	—	79	1	no	—
76	#446	—	158	2	no	—
77	C₄×Q₁₆	—	79	2	no	—
78	#366	—	79	1	no	—
79	#358	—	79	2	no	—
80	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
82	#335	—	79	2	no	—
83	Q₈⋊C₈	—	158	2	no	—
84	C₈²	C₂×D₇₉	1	1	yes	characteristic
85	C₂×Q₁₆	—	79	2	no	—
86	C₄×Q₈	—	158	2	no	—
87	Q₃₂	—	158	2	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
90	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
91	#107	—	79	2	no	—
92	C₄⋊C₈	—	316	2	no	—
93	C₄⋊C₈	—	158	2	no	—
94	#103	—	158	2	no	—
95	#97	—	316	2	no	—
96	C₄×C₈	D₃₁₆	1	1	yes	characteristic
97	C₄×C₈	D₃₁₆	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
99	C₂×Q₈	—	158	2	no	—
100	Q₁₆	—	158	2	no	—
101	Q₁₆	—	158	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	316	2	no	—
104	C₂×C₈	D₆₃₂	1	2	yes	—
105	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
106	C₂×C₈	#11929624	1	1	yes	characteristic
107	C₄⋊C₄	—	316	2	no	—
108	C₄⋊C₄	—	158	2	no	—
109	C₄²	—	316	2	no	—
110	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	#11929607	1	2	yes	—
114	Q₈	—	316	2	no	—
115	Q₈	—	316	2	no	—
116	C₂×C₄	—	316	2	no	—
117	C₂×C₄	—	316	2	no	—
118	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
119	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
120	C₂×C₄	#14579183	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	2	no	—
123	C₈	—	4	1	no	—
124	C₈	—	632	2	no	—
125	C₈	C₂×#11929607	1	1	yes	characteristic
126	C₈	#14579167	1	1	yes	characteristic
127	C₂²	C₄×D₆₃₂	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	632	2	no	—
130	C₄	—	632	2	no	—
131	C₄	#19098642	1	1	yes	characteristic
132	C₄	#19098681	1	1	yes	characteristic
133	C₄	#19099006	1	2	yes	—
134	C₂	#23762286	1	1	yes	characteristic
135	C₂	#23765272	1	1	yes	characteristic
136	C₂	#23765287	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₆₃₂

C₂

yes

maximal, characteristic

#23762298

C₂

yes

maximal

#23761474

C₂

yes

maximal, characteristic

#23761365

C₂

yes

maximal

C₇₉×#1311

C₂

yes

maximal, characteristic

C₂×Dic₆₃₂

C₄

yes

characteristic

#19098693

—

C₄×Dic₃₁₆

C₂²

yes

—

#19098389

C₄

yes

characteristic

#19098368

C₄

yes

—

#19098316

C₂²

yes

—

#19098295

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

C₂×C₄

yes

—

C₄×Dic₁₅₈

—

Dic₆₃₂

—

#14579095

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

#14579086

—

#14579071

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

M₁₆

yes

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×M₁₆

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

C₄×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₄×Q₃₂

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#12085

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#446

—

158

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

—

Q₈⋊C₈

—

158

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

158

—

Q₃₂

—

158

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₉

yes

characteristic

C₂×C₁₆

C₄×D₇₉

yes

characteristic

#107

—

C₄⋊C₈

—

316

—

C₄⋊C₈

—

158

—

#103

—

158

—

#97

—

316

—

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂×Q₈

—

158

—

100

Q₁₆

—

158

—

101

Q₁₆

—

158

—

102

C₂×C₈

—

103

C₂×C₈

—

316

—

104

C₂×C₈

D₆₃₂

yes

—

105

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

106

C₂×C₈

#11929624

yes

characteristic

107

C₄⋊C₄

—

316

—

108

C₄⋊C₄

—

158

—

109

C₄²

—

316

—

110

C₄²

C₂×D₃₁₆

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

#11929607

yes

—

114

Q₈

—

316

—

115

Q₈

—

316

—

116

C₂×C₄

—

316

—

117

C₂×C₄

—

316

—

118

C₂×C₄

C₄×D₃₁₆

yes

characteristic

119

C₂×C₄

C₂×D₆₃₂

yes

characteristic

120

C₂×C₄

#14579183

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

632

—

125

C₈

C₂×#11929607

yes

characteristic

126

C₈

#14579167

yes

characteristic

127

C₂²

C₄×D₆₃₂

yes

characteristic

128

C₄

—

129

C₄

—

632

—

130

C₄

—

632

—

131

C₄

#19098642

yes

characteristic

132

C₄

#19098681

yes

characteristic

133

C₄

#19099006

yes

—

134

C₂

#23762286

yes

characteristic

135

C₂

#23765272

yes

characteristic

136

C₂

#23765287

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1311	C₂
socle, prime radical	C₂²×C₇₉	C₄×D₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1311	D₇₉
hypocenter	C₇₉	#12085
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

C₂×D₆₃₂

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1311

C₂

socle, prime radical

C₂²×C₇₉

C₄×D₈

cosocle

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

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
C₄×Dic₆₃₂	2
#23762298	2
#23761474	2
#23761365	2
C₇₉×#1311	2
#12085	79

subgroup

index

C₄×Dic₆₃₂

C₇₉×#1311

Subgroups — #34200273

Conjugacy classes of subgroups

Complements

Maximal subgroups