Group #34202625 — subgroups

Subgroups — #34202625

Order 20224.

204 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
2	#23762791	C₂	1	2	yes	maximal
3	#23761472	C₂	1	1	yes	maximal, characteristic
4	#23761455	C₂	1	2	yes	maximal
5	C₇₉×#2252	C₂	1	1	yes	maximal, characteristic
6	#19098770	—	2	2	no	—
7	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₇₉⋊C₁₆	—	2	1	no	—
10	C₁₆×Dic₇₉	—	2	1	no	—
11	#19098316	C₂²	1	2	yes	—
12	#19098309	—	2	2	no	—
13	#500×C₇₉	C₂²	1	2	yes	—
14	#366×C₇₉	—	2	1	no	—
15	#366×C₇₉	—	2	1	no	—
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929614	—	4	2	no	—
18	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
19	C₂×C₇₉⋊C₁₆	—	2	1	no	—
20	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₇₉⋊C₈	—	2	1	no	—
23	#14579085	—	4	2	no	—
24	C₈×Dic₇₉	—	2	1	no	—
25	C₄⋊Q₈×C₇₉	—	2	2	no	—
26	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
28	#107×C₇₉	—	2	2	no	—
29	#107×C₇₉	—	2	2	no	—
30	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
32	#11929614	—	8	4	no	—
33	C₂×C₇₉⋊C₈	—	2	1	no	—
34	C₂×C₇₉⋊C₈	—	4	2	no	—
35	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
36	Dic₇₉⋊C₄	—	2	1	no	—
37	C₄×Dic₇₉	—	2	1	no	—
38	C₇₉⋊C₁₆	—	4	1	no	—
39	C₂×Q₈×C₇₉	—	4	2	no	—
40	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₇₉	—	4	2	no	—
43	C₄⋊C₄×C₇₉	—	4	2	no	—
44	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₇₉	—	2	1	no	—
46	C₁₆×C₇₉	Q₁₆	1	2	yes	complemented
47	Q₈×C₇₉	—	8	2	no	—
48	Q₈×C₇₉	—	8	2	no	—
49	C₂×C₄×C₇₉	—	8	2	no	—
50	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
53	C₂×Dic₇₉	—	2	1	no	—
54	C₂×Dic₇₉	—	2	1	no	—
55	Dic₁₅₈	—	2	1	no	—
56	Dic₁₅₈	D₁₆	1	2	yes	—
57	C₈×C₇₉	—	2	1	no	—
58	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₇₉	#137	1	1	yes	characteristic
60	C₇₉⋊C₈	—	4	1	no	—
61	C₇₉⋊C₈	—	8	2	no	—
62	C₂²×C₇₉	#466	1	1	yes	characteristic
63	C₄×C₇₉	—	2	1	no	—
64	C₄×C₇₉	—	16	2	no	—
65	C₄×C₇₉	#462	1	1	yes	characteristic
66	C₄×C₇₉	#473	1	1	yes	characteristic
67	C₄×C₇₉	C₂×D₁₆	1	1	yes	characteristic
68	C₄×C₇₉	#508	1	1	yes	characteristic
69	Dic₇₉	—	2	2	no	—
70	Dic₇₉	—	4	1	no	—
71	#12457	—	79	1	no	maximal, Sylow
72	C₂×C₇₉	#1674	1	1	yes	characteristic
73	C₂×C₇₉	#2215	1	1	yes	characteristic
74	C₂×C₇₉	#2222	1	1	yes	characteristic
75	#2252	D₇₉	1	1	yes	characteristic
76	Q₈×C₁₆	—	79	1	no	—
77	#1674	—	79	2	no	—
78	#1371	—	79	1	no	—
79	#1364	—	79	2	no	—
80	C₇₉	#12457	1	1	yes	characteristic, complemented, Sylow
81	#500	C₂×D₇₉	1	2	yes	—
82	#462	—	158	2	no	—
83	C₈×Q₈	—	79	1	no	—
84	#366	—	2	1	no	—
85	#366	—	2	1	no	—
86	C₄⋊C₁₆	—	79	1	no	—
87	C₄⋊C₁₆	—	158	1	no	—
88	C₄×C₁₆	—	158	1	no	—
89	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
90	#335	—	79	2	no	—
91	#332	—	158	2	no	—
92	C₂×Q₁₆	—	316	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄×Q₈	—	79	1	no	—
95	C₂×C₁₆	—	158	1	no	—
96	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	C₄⋊C₈	—	79	1	no	—
101	C₄⋊C₈	—	158	1	no	—
102	C₄⋊C₈	—	158	2	no	—
103	#103	—	316	2	no	—
104	C₄×C₈	—	158	1	no	—
105	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	79	1	no	—
108	Q₁₆	—	632	4	no	—
109	C₂×C₈	—	316	2	no	—
110	C₂×C₈	—	158	1	no	—
111	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
112	C₂×C₈	#11929623	1	1	yes	characteristic
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	79	1	no	—
116	C₄⋊C₄	—	158	1	no	—
117	C₄²	—	158	1	no	—
118	C₄²	D₄×D₇₉	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	316	1	no	—
121	C₁₆	#11929614	1	2	yes	complemented
122	Q₈	—	8	2	no	—
123	Q₈	—	8	2	no	—
124	Q₈	—	79	2	no	—
125	Q₈	—	158	1	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	—	158	1	no	—
128	C₂×C₄	—	158	1	no	—
129	C₂×C₄	#14579132	1	1	yes	characteristic
130	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
131	C₂×C₄	#14579180	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	316	1	no	—
134	C₈	—	632	2	no	—
135	C₈	C₂×#11929614	1	1	yes	characteristic
136	C₈	#14579192	1	1	yes	characteristic
137	C₂²	#19098796	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	16	2	no	—
140	C₄	—	316	1	no	—
141	C₄	—	158	2	no	—
142	C₄	#19098770	1	1	yes	characteristic
143	C₄	#19098835	1	1	yes	characteristic
144	C₄	D₁₆×D₇₉	1	1	yes	characteristic
145	C₄	#19098992	1	1	yes	characteristic
146	C₂	#23762791	1	1	yes	characteristic
147	C₂	#23765483	1	1	yes	characteristic
148	C₂	#23765522	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₅₈

C₂

yes

maximal, characteristic

#23762791

C₂

yes

maximal

#23761472

C₂

yes

maximal, characteristic

#23761455

C₂

yes

maximal

C₇₉×#2252

C₂

yes

maximal, characteristic

#19098770

—

C₈×Dic₁₅₈

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

C₂²

yes

characteristic

Dic₇₉⋊C₁₆

—

C₁₆×Dic₇₉

—

#19098316

C₂²

yes

—

#19098309

—

#500×C₇₉

C₂²

yes

—

#366×C₇₉

—

#366×C₇₉

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₂×#11929614

—

C₄×Dic₁₅₈

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

characteristic

Dic₇₉⋊C₈

—

#14579085

—

C₈×Dic₇₉

—

C₄⋊Q₈×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

D₄

yes

characteristic

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

D₈

yes

characteristic

#11929614

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

D₈

yes

characteristic

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

Q₁₆

yes

complemented

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

D₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

C₂×Q₁₆

yes

characteristic

C₈×C₇₉

#137

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#466

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

#508

yes

characteristic

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

Q₈×C₁₆

—

#1674

—

#1371

—

#1364

—

C₇₉

#12457

yes

characteristic, complemented, Sylow

#500

C₂×D₇₉

yes

—

#462

—

158

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

158

—

C₄×C₁₆

—

158

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

#335

—

#332

—

158

—

C₂×Q₁₆

—

316

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

158

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

#107

—

#107

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

158

—

102

C₄⋊C₈

—

158

—

103

#103

—

316

—

104

C₄×C₈

—

158

—

105

C₄×C₈

C₂²×D₇₉

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

108

Q₁₆

—

632

—

109

C₂×C₈

—

316

—

110

C₂×C₈

—

158

—

111

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

112

C₂×C₈

#11929623

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

158

—

117

C₄²

—

158

—

118

C₄²

D₄×D₇₉

yes

characteristic

119

C₁₆

—

120

C₁₆

—

316

—

121

C₁₆

#11929614

yes

complemented

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

158

—

126

C₂×C₄

—

127

C₂×C₄

—

158

—

128

C₂×C₄

—

158

—

129

C₂×C₄

#14579132

yes

characteristic

130

C₂×C₄

D₈×D₇₉

yes

characteristic

131

C₂×C₄

#14579180

yes

characteristic

132

C₈

—

133

C₈

—

316

—

134

C₈

—

632

—

135

C₈

C₂×#11929614

yes

characteristic

136

C₈

#14579192

yes

characteristic

137

C₂²

#19098796

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

316

—

141

C₄

—

158

—

142

C₄

#19098770

yes

characteristic

143

C₄

#19098835

yes

characteristic

144

C₄

D₁₆×D₇₉

yes

characteristic

145

C₄

#19098992

yes

characteristic

146

C₂

#23762791

yes

characteristic

147

C₂

#23765483

yes

characteristic

148

C₂

#23765522

yes

characteristic

	subgroup	G/H
center	C₂²	#19098796
commutator, cosocle	C₄×C₈×C₇₉	C₂³
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2252	C₂
socle, prime radical	C₂²×C₇₉	#466
hypercenter	#2252	D₇₉
hypocenter	C₇₉	#12457
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#19098796

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
46. C₁₆×C₇₉	Q₁₆	semidirect
80. C₇₉	#12457	semidirect
121. C₁₆	#11929614	semidirect

H ≅ G/N

type

46. C₁₆×C₇₉

Q₁₆

semidirect

80. C₇₉

#12457

semidirect

121. C₁₆

#11929614

semidirect

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

subgroup	index
C₁₆×Dic₁₅₈	2
#23762791	2
#23761472	2
#23761455	2
C₇₉×#2252	2
#12457	79

subgroup

index

C₁₆×Dic₁₅₈

C₇₉×#2252

Subgroups — #34202625

Conjugacy classes of subgroups

Complements

Maximal subgroups