Group #34174866 — subgroups

Subgroups — #34174866

Order 20224.

304 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 107 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	#23763253	C₂	1	2	yes	maximal
2	C₇₉×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
4	C₂×#14579072	C₄	1	4	yes	—
5	#19098318	—	2	2	no	—
6	C₂×C₄⋊C₈×C₇₉	C₄	1	2	yes	—
7	C₄³×C₇₉	C₂²	1	1	yes	characteristic
8	#336×C₇₉	—	2	2	no	—
9	C₂²×C₇₉⋊C₈	—	2	4	no	—
10	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
11	#14579072	—	4	4	no	—
12	C₂²×C₈×C₇₉	—	2	2	no	—
13	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₇₉	D₄	1	2	yes	—
16	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
18	C₄⋊C₈×C₇₉	—	4	2	no	—
19	C₂×C₇₉⋊C₈	—	2	8	no	—
20	C₂×C₇₉⋊C₈	—	4	4	no	—
21	C₂×C₇₉⋊C₈	—	4	4	no	—
22	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
23	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
24	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
26	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₇₉	—	4	2	no	—
28	C₂×C₈×C₇₉	—	4	4	no	—
29	C₄²×C₇₉	—	2	1	no	—
30	C₄²×C₇₉	—	2	2	no	—
31	C₄²×C₇₉	—	2	1	no	—
32	C₄²×C₇₉	—	2	2	no	—
33	C₄²×C₇₉	—	2	4	no	—
34	C₄²×C₇₉	—	2	2	no	—
35	C₄²×C₇₉	—	4	1	no	—
36	C₂³×C₇₉	#95	1	1	yes	characteristic
37	C₂×C₄×C₇₉	—	2	2	no	—
38	C₂×C₄×C₇₉	—	2	1	no	—
39	C₂×C₄×C₇₉	—	2	2	no	—
40	C₂×C₄×C₇₉	—	2	1	no	—
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	—	2	2	no	—
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	#100	1	2	yes	—
47	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
48	C₂×C₄×C₇₉	#101	1	2	yes	—
49	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#104	1	4	yes	—
51	C₂×C₄×C₇₉	#108	1	2	yes	—
52	C₈×C₇₉	—	8	2	no	—
53	C₇₉⋊C₈	—	4	8	no	—
54	C₂²×C₇₉	H(3,ℤ/4)	1	1	yes	characteristic
55	C₂²×C₇₉	#339	1	2	yes	—
56	C₂²×C₇₉	#341	1	1	yes	characteristic
57	C₂²×C₇₉	#343	1	2	yes	—
58	C₂²×C₇₉	#343	1	1	yes	characteristic
59	C₄×C₇₉	—	2	2	no	—
60	C₄×C₇₉	—	2	2	no	—
61	C₄×C₇₉	—	2	2	no	—
62	C₄×C₇₉	—	4	2	no	—
63	C₄×C₇₉	—	4	1	no	—
64	C₄×C₇₉	#329	1	2	yes	—
65	C₄×C₇₉	#330	1	2	yes	—
66	#7042	—	79	1	no	maximal, Sylow
67	C₂×C₇₉	#1285	1	2	yes	—
68	C₂×C₇₉	#1290	1	1	yes	characteristic
69	C₂×C₇₉	#1295	1	1	yes	characteristic
70	C₂×C₇₉	#1299	1	2	yes	—
71	C₂×C₇₉	#1309	1	1	yes	characteristic
72	#1831	D₇₉	1	1	yes	characteristic
73	#1767	—	79	2	no	—
74	C₇₉	#7042	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄⋊C₈	—	158	2	no	—
76	C₂×C₄⋊C₈	Dic₇₉	1	2	yes	—
77	C₂×#97	—	79	4	no	—
78	C₄³	C₂×D₇₉	1	1	yes	characteristic
79	#336	—	2	2	no	—
80	#336	—	158	2	no	—
81	C₂²×C₈	—	2	2	no	—
82	C₂²×C₈	—	158	4	no	—
83	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
84	C₂×C₄²	C₄×D₇₉	1	2	yes	—
85	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
86	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
87	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	158	8	no	—
90	#97	—	316	4	no	—
91	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
92	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
93	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
94	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
95	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	—	316	4	no	—
99	C₂×C₈	—	316	4	no	—
100	C₂×C₈	—	158	8	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	4	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	4	1	no	—
108	C₂³	C₇₉⋊#95	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	#14579077	1	2	yes	—
119	C₂×C₄	#14579078	1	1	yes	characteristic
120	C₂×C₄	#14579079	1	2	yes	—
121	C₂×C₄	#14579080	1	1	yes	characteristic
122	C₂×C₄	#14579088	1	4	yes	—
123	C₂×C₄	#14579097	1	2	yes	—
124	C₈	—	8	2	no	—
125	C₈	—	316	8	no	—
126	C₂²	#19098320	1	1	yes	characteristic
127	C₂²	#19098322	1	2	yes	—
128	C₂²	#19098328	1	1	yes	characteristic
129	C₂²	#19098331	1	2	yes	—
130	C₂²	#19098332	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	4	1	no	—
136	C₄	#19098303	1	2	yes	—
137	C₄	#19098305	1	2	yes	—
138	C₂	#23761257	1	2	yes	—
139	C₂	#23761270	1	1	yes	characteristic
140	C₂	#23761281	1	1	yes	characteristic
141	C₂	#23761291	1	2	yes	—
142	C₂	C₇₉⋊#1309	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763253

C₂

yes

maximal

C₇₉×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

C₂×#14579072

C₄

yes

—

#19098318

—

C₂×C₄⋊C₈×C₇₉

C₄

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

#14579072

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

Q₈

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄²

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

#95

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#100

yes

—

C₂×C₄×C₇₉

#100

yes

characteristic

C₂×C₄×C₇₉

#101

yes

—

C₂×C₄×C₇₉

#101

yes

characteristic

C₂×C₄×C₇₉

#104

yes

—

C₂×C₄×C₇₉

#108

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

H(3,ℤ/4)

yes

characteristic

C₂²×C₇₉

#339

yes

—

C₂²×C₇₉

#341

yes

characteristic

C₂²×C₇₉

#343

yes

—

C₂²×C₇₉

#343

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#329

yes

—

C₄×C₇₉

#330

yes

—

#7042

—

maximal, Sylow

C₂×C₇₉

#1285

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#1309

yes

characteristic

#1831

D₇₉

yes

characteristic

#1767

—

C₇₉

#7042

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

158

—

C₂×C₄⋊C₈

Dic₇₉

yes

—

C₂×#97

—

C₄³

C₂×D₇₉

yes

characteristic

#336

—

#336

—

158

—

C₂²×C₈

—

C₂²×C₈

—

158

—

C₂×C₄²

Dic₁₅₈

yes

characteristic

C₂×C₄²

C₄×D₇₉

yes

—

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

C₂²⋊D₇₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

158

—

#97

—

316

—

C₂²×C₄

C₄×Dic₇₉

yes

characteristic

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

C₂²×C₄

Dic₇₉⋊C₄

yes

—

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

—

316

—

100

C₂×C₈

—

158

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂³

C₇₉⋊#95

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#14579077

yes

—

119

C₂×C₄

#14579078

yes

characteristic

120

C₂×C₄

#14579079

yes

—

121

C₂×C₄

#14579080

yes

characteristic

122

C₂×C₄

#14579088

yes

—

123

C₂×C₄

#14579097

yes

—

124

C₈

—

125

C₈

—

316

—

126

C₂²

#19098320

yes

characteristic

127

C₂²

#19098322

yes

—

128

C₂²

#19098328

yes

characteristic

129

C₂²

#19098331

yes

—

130

C₂²

#19098332

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#19098303

yes

—

137

C₄

#19098305

yes

—

138

C₂

#23761257

yes

—

139

C₂

#23761270

yes

characteristic

140

C₂

#23761281

yes

characteristic

141

C₂

#23761291

yes

—

142

C₂

C₇₉⋊#1309

yes

characteristic

	subgroup	G/H
center	C₂³	C₇₉⋊#95
commutator	C₂²×C₄×C₇₉	C₄²
Frattini	C₄³	C₂×D₇₉
Fitting	C₇₉×#1831	C₂
socle, prime radical	C₂³×C₇₉	#95
cosocle	C₄³×C₇₉	C₂²
hypercenter	#1831	D₇₉
hypocenter	C₇₉	#7042
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₇₉⋊#95

commutator

C₂²×C₄×C₇₉

C₄²

Frattini

C₄³

C₂×D₇₉

Fitting

C₇₉×#1831

C₂

socle, prime radical

C₂³×C₇₉

#95

cosocle

C₄³×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
74. C₇₉	#7042	semidirect

H ≅ G/N

type

74. C₇₉

#7042

semidirect

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

subgroup	index
#23763253	2
C₇₉×#1831	2
#7042	79

subgroup

index

#23763253

C₇₉×#1831

#7042

Subgroups — #34174866

Conjugacy classes of subgroups

Complements

Maximal subgroups