Group #34288655 — subgroups

Subgroups — #34288655

Order 20224.

1118 conjugacy classes of subgroups; only the 387 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 387 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	C₂×#19098585	C₂	1	6	yes	maximal
2	#376×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23763554	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763553	C₂	1	6	yes	maximal, complemented
5	C₇₉×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11929607	C₂²	1	4	yes	—
7	C₂×C₄²×D₇₉	C₂²	1	3	yes	—
8	#19098585	—	2	24	no	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	6	yes	—
10	C₂×Dic₇₉⋊C₈	C₂²	1	12	yes	—
11	#19098409	C₂²	1	3	yes	—
12	#19098408	C₄	1	4	yes	—
13	#95×D₇₉	C₄	1	4	yes	—
14	C₂×C₄⋊C₈×C₇₉	C₂²	1	6	yes	—
15	#376×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₇₉	C₂×C₄	1	6	yes	—
17	C₂²×C₄×D₇₉	C₂³	1	1	yes	characteristic
18	C₂×#11929607	—	2	24	no	—
19	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
20	C₂×C₄×Dic₇₉	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₇₉	C₂³	1	3	yes	—
22	C₄²×D₇₉	D₄	1	12	yes	—
23	C₄²×D₇₉	Q₈	1	12	yes	—
24	C₄⋊(C₇₉⋊C₈)	—	2	12	no	—
25	Dic₇₉⋊C₈	—	2	24	no	—
26	C₇₉⋊#95	C₂×C₄	1	12	yes	—
27	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
28	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
29	C₄⋊C₈×C₇₉	—	2	12	no	—
30	#95×C₇₉	C₂×C₄	1	4	yes	—
31	C₂³×D₇₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₇₉	C₂²×C₄	1	6	yes	—
33	C₂²×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×D₇₉	—	4	12	no	—
35	C₂×C₄×D₇₉	C₄⋊C₄	1	24	yes	—
36	C₂×C₄×D₇₉	C₂×D₄	1	6	yes	—
37	C₂×C₄×D₇₉	C₂×Q₈	1	6	yes	—
38	#11929607	—	4	16	no	—
39	C₂×C₇₉⋊C₈	—	2	12	no	—
40	C₄×Dic₇₉	C₄⋊C₄	1	24	yes	—
41	C₄×Dic₇₉	C₂×D₄	1	6	yes	—
42	C₄×Dic₇₉	C₂×Q₈	1	6	yes	—
43	C₂²×C₄×C₇₉	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₇₉	—	2	12	no	—
46	C₄²×C₇₉	C₂×D₄	1	6	yes	—
47	C₄²×C₇₉	C₂×Q₈	1	6	yes	—
48	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
49	C₂²×D₇₉	—	2	4	no	—
50	C₂²×D₇₉	C₂×C₄⋊C₄	1	6	yes	—
51	C₂×C₄×C₇₉	—	2	12	no	—
52	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	12	yes	—
53	C₂×C₄×C₇₉	C₂²×D₄	1	3	yes	—
54	C₂×C₄×C₇₉	C₂²×Q₈	1	3	yes	—
55	C₂×Dic₇₉	—	2	12	no	—
56	C₂×Dic₇₉	C₂×C₄⋊C₄	1	12	yes	—
57	C₂×Dic₇₉	C₂×C₄⋊C₄	1	6	yes	—
58	C₄×D₇₉	—	4	24	no	—
59	C₄×D₇₉	#127	1	4	yes	—
60	C₄×D₇₉	C₄⋊Q₈	1	12	yes	—
61	C₈×C₇₉	—	4	4	no	—
62	C₇₉⋊C₈	—	4	4	no	—
63	C₂²×C₇₉	C₂²×C₄⋊C₄	1	3	yes	—
64	C₂²×C₇₉	#568	1	4	yes	—
65	C₂×D₇₉	—	2	12	no	—
66	C₂×D₇₉	#384	1	4	yes	—
67	C₄×C₇₉	—	2	12	no	—
68	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
69	C₄×C₇₉	C₂×C₄⋊Q₈	1	3	yes	—
70	Dic₇₉	—	2	12	no	—
71	Dic₇₉	#384	1	4	yes	—
72	#20792	—	79	1	no	maximal, Sylow
73	C₂×C₇₉	C₂×#384	1	1	yes	characteristic
74	C₂×C₇₉	#2905	1	6	yes	—
75	D₇₉	—	2	4	no	—
76	C₂×#423	—	79	6	no	—
77	C₂×#376	—	79	1	no	—
78	#1843	—	79	6	no	—
79	#1843	—	79	1	no	—
80	#1843	D₇₉	1	1	yes	characteristic, complemented
81	C₇₉	#20792	1	1	yes	characteristic, complemented, Sylow
82	C₂²×M₁₆	—	79	4	no	—
83	C₂²×C₄²	—	79	3	no	—
84	#423	—	158	24	no	—
85	C₂×C₄⋊C₈	—	79	12	no	—
86	C₂×C₄⋊C₈	—	79	6	no	—
87	C₂×C₄⋊C₈	C₂×D₇₉	1	6	yes	—
88	#376	—	79	3	no	—
89	#376	—	79	4	no	—
90	#376	C₂×D₇₉	1	1	yes	characteristic
91	C₂×#95	—	79	4	no	—
92	C₂³×C₄	—	79	6	no	—
93	C₂³×C₄	—	79	1	no	—
94	C₂×M₁₆	—	158	24	no	—
95	C₂²×C₈	—	79	4	no	—
96	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
97	C₂×C₄²	—	79	6	no	—
98	C₂×C₄²	—	79	3	no	—
99	C₂×C₄²	—	79	12	no	—
100	C₂×C₄²	—	79	12	no	—
101	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
102	C₄⋊C₈	—	2	12	no	—
103	C₄⋊C₈	—	158	24	no	—
104	C₄⋊C₈	—	158	12	no	—
105	#95	—	79	12	no	—
106	#95	C₄×D₇₉	1	4	yes	—
107	C₂⁴	—	79	1	no	—
108	C₂²×C₄	—	316	12	no	—
109	C₂²×C₄	—	79	24	no	—
110	C₂²×C₄	—	79	1	no	—
111	C₂²×C₄	—	79	6	no	—
112	C₂²×C₄	—	79	6	no	—
113	C₂²×C₄	—	79	6	no	—
114	C₂²×C₄	C₂×C₄×D₇₉	1	6	yes	—
115	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
116	M₁₆	—	316	16	no	—
117	C₂×C₈	—	2	12	no	—
118	C₂×C₈	—	158	12	no	—
119	C₄²	—	79	6	no	—
120	C₄²	—	79	24	no	—
121	C₄²	—	79	6	no	—
122	C₄²	D₄×D₇₉	1	6	yes	—
123	C₄²	Q₈×D₇₉	1	6	yes	—
124	C₂³	—	79	6	no	—
125	C₂³	—	158	4	no	—
126	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	316	24	no	—
129	C₂×C₄	—	79	6	no	—
130	C₂×C₄	—	79	4	no	—
131	C₂×C₄	—	79	12	no	—
132	C₂×C₄	—	79	12	no	—
133	C₂×C₄	—	158	12	no	—
134	C₂×C₄	C₄⋊C₄×D₇₉	1	12	yes	—
135	C₂×C₄	C₂×D₄×D₇₉	1	3	yes	—
136	C₂×C₄	C₂×Q₈×D₇₉	1	3	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	316	4	no	—
139	C₂²	—	79	4	no	—
140	C₂²	—	158	12	no	—
141	C₂²	C₂×C₄⋊C₄×D₇₉	1	3	yes	—
142	C₂²	#19099328	1	4	yes	—
143	C₄	—	2	12	no	—
144	C₄	—	79	4	no	—
145	C₄	—	158	12	no	—
146	C₄	#127×D₇₉	1	1	yes	characteristic
147	C₄	C₄⋊Q₈×D₇₉	1	3	yes	—
148	C₂	—	158	4	no	—
149	C₂	#384×D₇₉	1	1	yes	characteristic
150	C₂	#23771961	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098585

C₂

yes

maximal

#376×D₇₉

C₂

yes

maximal, characteristic

#23763554

C₂

yes

maximal, characteristic, complemented

#23763553

C₂

yes

maximal, complemented

C₇₉×#1843

C₂

yes

maximal, characteristic, complemented

C₂²×#11929607

C₂²

yes

—

C₂×C₄²×D₇₉

C₂²

yes

—

#19098585

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂×Dic₇₉⋊C₈

C₂²

yes

—

#19098409

C₂²

yes

—

#19098408

C₄

yes

—

#95×D₇₉

C₄

yes

—

C₂×C₄⋊C₈×C₇₉

C₂²

yes

—

#376×C₇₉

C₂²

yes

characteristic

C₂²×C₄×D₇₉

C₂×C₄

yes

—

C₂²×C₄×D₇₉

C₂³

yes

characteristic

C₂×#11929607

—

C₂²×C₇₉⋊C₈

C₂³

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂³

yes

—

C₄²×D₇₉

D₄

yes

—

C₄²×D₇₉

Q₈

yes

—

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

C₇₉⋊#95

C₂×C₄

yes

—

C₂²×C₈×C₇₉

C₂³

yes

—

C₂×C₄²×C₇₉

C₂³

yes

—

C₄⋊C₈×C₇₉

—

#95×C₇₉

C₂×C₄

yes

—

C₂³×D₇₉

C₂²×C₄

yes

characteristic

C₂²×Dic₇₉

C₂²×C₄

yes

—

C₂²×Dic₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

C₄⋊C₄

yes

—

C₂×C₄×D₇₉

C₂×D₄

yes

—

C₂×C₄×D₇₉

C₂×Q₈

yes

—

#11929607

—

C₂×C₇₉⋊C₈

—

C₄×Dic₇₉

C₄⋊C₄

yes

—

C₄×Dic₇₉

C₂×D₄

yes

—

C₄×Dic₇₉

C₂×Q₈

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

C₂³×C₄

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²×D₄

yes

—

C₂×C₄×C₇₉

C₂²×Q₈

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₉

C₂×C₄⋊C₄

yes

—

C₄×D₇₉

—

C₄×D₇₉

#127

yes

—

C₄×D₇₉

C₄⋊Q₈

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂²×C₄⋊C₄

yes

—

C₂²×C₇₉

#568

yes

—

C₂×D₇₉

—

C₂×D₇₉

#384

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#127

yes

characteristic

C₄×C₇₉

C₂×C₄⋊Q₈

yes

—

Dic₇₉

—

Dic₇₉

#384

yes

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

—

—

—

—

D₇₉

yes

characteristic, complemented

C₇₉

#20792

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×C₄²

—

#423

—

158

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₇₉

yes

—

#376

—

#376

—

#376

C₂×D₇₉

yes

characteristic

C₂×#95

—

C₂³×C₄

—

C₂³×C₄

—

C₂×M₁₆

—

158

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂²×D₇₉

yes

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

158

—

104

C₄⋊C₈

—

158

—

105

#95

—

106

#95

C₄×D₇₉

yes

—

107

C₂⁴

—

108

C₂²×C₄

—

316

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×C₄×D₇₉

yes

—

115

C₂²×C₄

C₂³×D₇₉

yes

characteristic

116

M₁₆

—

316

—

117

C₂×C₈

—

118

C₂×C₈

—

158

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

D₄×D₇₉

yes

—

123

C₄²

Q₈×D₇₉

yes

—

124

C₂³

—

125

C₂³

—

158

—

126

C₂³

C₂²×C₄×D₇₉

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

316

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

158

—

134

C₂×C₄

C₄⋊C₄×D₇₉

yes

—

135

C₂×C₄

C₂×D₄×D₇₉

yes

—

136

C₂×C₄

C₂×Q₈×D₇₉

yes

—

137

C₈

—

138

C₈

—

316

—

139

C₂²

—

140

C₂²

—

158

—

141

C₂²

C₂×C₄⋊C₄×D₇₉

yes

—

142

C₂²

#19099328

yes

—

143

C₄

—

144

C₄

—

145

C₄

—

158

—

146

C₄

#127×D₇₉

yes

characteristic

147

C₄

C₄⋊Q₈×D₇₉

yes

—

148

C₂

—

158

—

149

C₂

#384×D₇₉

yes

characteristic

150

C₂

#23771961

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator, socle	C₂³×C₇₉	C₂³×C₄
Fitting	C₇₉×#1843	C₂
prime radical	C₂³×D₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1843	D₇₉
hypocenter	C₇₉	#20792
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1843

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₇₉

commutator, socle

C₂³×C₇₉

C₂³×C₄

Fitting

C₇₉×#1843

C₂

prime radical

C₂³×D₇₉

C₂²×C₄

cosocle

C₂²×C₄×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂²×C₄ ◁ #1843

N	H ≅ G/N	type
3. #23763554	C₂	semidirect
4. #23763553	C₂	semidirect
5. C₇₉×#1843	C₂	semidirect
80. #1843	D₇₉	semidirect
81. C₇₉	#20792	semidirect

H ≅ G/N

type

3. #23763554

C₂

semidirect

4. #23763553

C₂

semidirect

5. C₇₉×#1843

C₂

semidirect

80. #1843

D₇₉

semidirect

81. C₇₉

#20792

semidirect

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

subgroup	index
C₂×#19098585	2
#376×D₇₉	2
#23763554	2
#23763553	2
C₇₉×#1843	2
#20792	79

subgroup

index

#376×D₇₉

C₇₉×#1843

Subgroups — #34288655

Conjugacy classes of subgroups

Complements

Maximal subgroups