Group #34314814 — subgroups

Subgroups — #34314814

Order 20224.

984 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 255 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×#14579158	C₂	1	1	yes	maximal, characteristic
2	#23763691	C₂	1	12	yes	maximal, complemented
3	#23763543	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#1839	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₇₉	C₂²	1	4	yes	—
6	C₂×#14579158	C₄	1	1	yes	characteristic
7	C₂×C₄×D₃₁₆	C₂²	1	6	yes	—
8	C₂×(C₂×D₇₉)⋊C₈	C₂²	1	12	yes	—
9	#19098465	C₄	1	6	yes	—
10	#19098433	C₄	1	1	yes	characteristic
11	#19098318	C₂²	1	6	yes	—
12	C₄³×C₇₉	C₂²	1	1	yes	characteristic
13	#336×C₇₉	C₂²	1	6	yes	—
14	C₂²×D₃₁₆	C₂×C₄	1	6	yes	—
15	C₂²×C₄×D₇₉	C₂³	1	4	yes	—
16	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
17	C₂×C₈×D₇₉	—	4	12	no	—
18	#14579158	C₈	1	8	yes	complemented
19	C₄×D₃₁₆	—	4	6	no	—
20	C₄×D₃₁₆	—	4	6	no	—
21	C₂×C₄⋊Dic₇₉	C₂×C₄	1	6	yes	—
22	C₂×(C₂×D₇₉)⋊C₄	C₂×C₄	1	12	yes	—
23	(C₂×D₇₉)⋊C₈	—	4	12	no	—
24	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
25	C₂×C₄²×C₇₉	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
28	C₂³×D₇₉	C₂²×C₄	1	4	yes	—
29	C₂×D₃₁₆	—	4	6	no	—
30	C₂×D₃₁₆	—	4	6	no	—
31	C₂×D₃₁₆	C₂×C₈	1	24	yes	complemented
32	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
33	C₂×C₄×D₇₉	—	2	12	no	—
34	C₂×C₄×D₇₉	—	2	12	no	—
35	C₂×C₇₉⋊C₈	—	2	12	no	—
36	C₈×D₇₉	—	4	16	no	—
37	C₄⋊Dic₇₉	—	4	6	no	—
38	(C₂×D₇₉)⋊C₄	—	4	12	no	—
39	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
42	C₂×C₈×C₇₉	—	2	12	no	—
43	C₄²×C₇₉	—	2	6	no	—
44	C₄²×C₇₉	—	4	3	no	—
45	C₄²×C₇₉	C₂×C₈	1	4	yes	complemented
46	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
47	C₂²×D₇₉	—	2	12	no	—
48	C₂²×D₇₉	—	2	12	no	—
49	C₂²×D₇₉	C₂²×C₈	1	8	yes	—
50	C₂×C₄×C₇₉	—	2	3	no	—
51	C₂×C₄×C₇₉	—	2	3	no	—
52	C₂×C₄×C₇₉	—	2	3	no	—
53	C₂×C₄×C₇₉	—	2	6	no	—
54	C₂×C₄×C₇₉	C₂²×C₈	1	6	yes	—
55	C₂×C₄×C₇₉	H(5,𝔽₂)	1	6	yes	—
56	C₂×Dic₇₉	—	2	12	no	—
57	D₃₁₆	—	4	24	no	—
58	C₄×D₇₉	—	4	4	no	—
59	C₄×D₇₉	—	4	12	no	—
60	C₈×C₇₉	—	4	4	no	—
61	C₇₉⋊C₈	—	4	4	no	—
62	C₂²×C₇₉	#518	1	3	yes	—
63	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
64	C₂²×C₇₉	#568	1	3	yes	—
65	C₂×D₇₉	—	2	24	no	—
66	C₂×D₇₉	—	4	4	no	—
67	C₂×D₇₉	—	4	12	no	—
68	C₄×C₇₉	—	2	6	no	—
69	C₄×C₇₉	—	4	3	no	—
70	C₄×C₇₉	#560	1	4	yes	—
71	Dic₇₉	—	4	4	no	—
72	#23355	—	79	1	no	maximal, Sylow
73	C₂×C₇₉	#2380	1	1	yes	characteristic
74	C₂×C₇₉	#2972	1	3	yes	—
75	C₂×C₇₉	#2980	1	3	yes	—
76	D₇₉	—	4	8	no	—
77	C₄×#127	—	79	1	no	—
78	#1869	—	79	12	no	—
79	#1839	—	79	1	no	—
80	#1839	D₇₉	1	1	yes	characteristic, complemented
81	C₇₉	#23355	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	79	4	no	—
83	C₂×#127	—	79	1	no	—
84	C₂×C₄×D₄	—	79	6	no	—
85	C₂×C₂²⋊C₈	—	79	12	no	—
86	#390	—	79	6	no	—
87	#384	—	79	1	no	—
88	C₄³	C₂×D₇₉	1	1	yes	characteristic
89	#336	—	79	6	no	—
90	#336	C₂×D₇₉	1	6	yes	—
91	C₂²×D₄	—	79	6	no	—
92	C₂³×C₄	—	79	4	no	—
93	C₂²×C₈	—	316	12	no	—
94	C₂²×C₈	—	79	4	no	—
95	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
96	#127	—	79	8	no	—
97	C₄×D₄	—	316	6	no	—
98	C₄×D₄	—	316	6	no	—
99	C₂×C₄⋊C₄	—	79	6	no	—
100	C₂×C₂²⋊C₄	—	79	12	no	—
101	C₂×C₄²	C₄×D₇₉	1	3	yes	—
102	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₇₉	1	3	yes	—
104	C₂²⋊C₈	—	316	12	no	—
105	C₂⁴	—	79	4	no	—
106	C₂×D₄	—	316	6	no	—
107	C₂×D₄	—	316	6	no	—
108	C₂×D₄	—	79	24	no	—
109	C₂²×C₄	—	79	4	no	—
110	C₂²×C₄	—	158	12	no	—
111	C₂²×C₄	—	158	12	no	—
112	C₂²×C₄	C₂×C₄×D₇₉	1	3	yes	—
113	C₂²×C₄	C₂×C₄×D₇₉	1	3	yes	—
114	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
115	C₂×C₈	—	2	12	no	—
116	C₂×C₈	—	316	16	no	—
117	C₂×C₈	—	158	12	no	—
118	C₄⋊C₄	—	316	6	no	—
119	C₂²⋊C₄	—	316	12	no	—
120	C₄²	—	2	6	no	—
121	C₄²	—	4	3	no	—
122	C₄²	C₈×D₇₉	1	4	yes	complemented
123	C₂³	—	79	8	no	—
124	C₂³	—	158	12	no	—
125	C₂³	—	158	12	no	—
126	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
127	D₄	—	316	24	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	3	no	—
131	C₂×C₄	—	2	6	no	—
132	C₂×C₄	—	316	4	no	—
133	C₂×C₄	—	316	12	no	—
134	C₂×C₄	—	158	12	no	—
135	C₂×C₄	C₂×C₈×D₇₉	1	6	yes	—
136	C₂×C₄	#14579207	1	6	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	316	4	no	—
139	C₂²	—	316	4	no	—
140	C₂²	—	316	12	no	—
141	C₂²	—	158	24	no	—
142	C₂²	#19099037	1	3	yes	—
143	C₂²	C₂²×C₈×D₇₉	1	1	yes	characteristic
144	C₂²	#19099329	1	3	yes	—
145	C₄	—	2	6	no	—
146	C₄	—	4	3	no	—
147	C₄	—	316	4	no	—
148	C₄	#19099304	1	4	yes	—
149	C₂	—	316	8	no	—
150	C₂	#23766576	1	1	yes	characteristic
151	C₂	#23772423	1	3	yes	—
152	C₂	#23772470	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14579158

C₂

yes

maximal, characteristic

#23763691

C₂

yes

maximal, complemented

#23763543

C₂

yes

maximal, characteristic, complemented

C₇₉×#1839

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₇₉

C₂²

yes

—

C₂×#14579158

C₄

yes

characteristic

C₂×C₄×D₃₁₆

C₂²

yes

—

C₂×(C₂×D₇₉)⋊C₈

C₂²

yes

—

#19098465

C₄

yes

—

#19098433

C₄

yes

characteristic

#19098318

C₂²

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

C₂²

yes

—

C₂²×D₃₁₆

C₂×C₄

yes

—

C₂²×C₄×D₇₉

C₂³

yes

—

C₂²×C₇₉⋊C₈

C₂³

yes

—

C₂×C₈×D₇₉

—

#14579158

C₈

yes

complemented

C₄×D₃₁₆

—

C₄×D₃₁₆

—

C₂×C₄⋊Dic₇₉

C₂×C₄

yes

—

C₂×(C₂×D₇₉)⋊C₄

C₂×C₄

yes

—

(C₂×D₇₉)⋊C₈

—

C₂²×C₈×C₇₉

C₂³

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂³

yes

—

C₂³×D₇₉

C₂²×C₄

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂×C₈

yes

complemented

C₂²×Dic₇₉

C₂²×C₄

yes

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

—

C₂×C₇₉⋊C₈

—

C₈×D₇₉

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₈

yes

complemented

C₂³×C₇₉

C₂³×C₄

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

C₂²×C₈

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₈

yes

—

C₂×C₄×C₇₉

H(5,𝔽₂)

yes

—

C₂×Dic₇₉

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#518

yes

—

C₂²×C₇₉

C₂³×C₈

yes

characteristic

C₂²×C₇₉

#568

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#560

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2980

yes

—

D₇₉

—

—

—

—

D₇₉

yes

characteristic, complemented

C₇₉

#23355

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×#127

—

C₂×C₄×D₄

—

C₂×C₂²⋊C₈

—

#390

—

#384

—

C₄³

C₂×D₇₉

yes

characteristic

#336

—

#336

C₂×D₇₉

yes

—

C₂²×D₄

—

C₂³×C₄

—

C₂²×C₈

—

316

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₇₉

yes

—

#127

—

C₄×D₄

—

316

—

C₄×D₄

—

316

—

C₂×C₄⋊C₄

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₄²

C₄×D₇₉

yes

—

102

C₂×C₄²

C₄×D₇₉

yes

characteristic

103

C₂×C₄²

C₂²×D₇₉

yes

—

104

C₂²⋊C₈

—

316

—

105

C₂⁴

—

106

C₂×D₄

—

316

—

107

C₂×D₄

—

316

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

158

—

111

C₂²×C₄

—

158

—

112

C₂²×C₄

C₂×C₄×D₇₉

yes

—

113

C₂²×C₄

C₂×C₄×D₇₉

yes

—

114

C₂²×C₄

C₂³×D₇₉

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

316

—

117

C₂×C₈

—

158

—

118

C₄⋊C₄

—

316

—

119

C₂²⋊C₄

—

316

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₈×D₇₉

yes

complemented

123

C₂³

—

124

C₂³

—

158

—

125

C₂³

—

158

—

126

C₂³

C₂²×C₄×D₇₉

yes

characteristic

127

D₄

—

316

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

316

—

133

C₂×C₄

—

316

—

134

C₂×C₄

—

158

—

135

C₂×C₄

C₂×C₈×D₇₉

yes

—

136

C₂×C₄

#14579207

yes

—

137

C₈

—

138

C₈

—

316

—

139

C₂²

—

316

—

140

C₂²

—

316

—

141

C₂²

—

158

—

142

C₂²

#19099037

yes

—

143

C₂²

C₂²×C₈×D₇₉

yes

characteristic

144

C₂²

#19099329

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

316

—

148

C₄

#19099304

yes

—

149

C₂

—

316

—

150

C₂

#23766576

yes

characteristic

151

C₂

#23772423

yes

—

152

C₂

#23772470

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂³×C₈
Fitting	C₇₉×#1839	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂×#14579158	C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1839	D₇₉
hypocenter	C₇₉	#23355
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₇₉

commutator

C₂²×C₇₉

C₂³×C₈

Fitting

C₇₉×#1839

C₂

socle

C₂³×C₇₉

C₂³×C₄

prime radical

C₂×#14579158

C₄

cosocle

C₂²×C₄×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂²×C₄ ◁ #1839

N	H ≅ G/N	type
2. #23763691	C₂	semidirect
3. #23763543	C₂	semidirect
4. C₇₉×#1839	C₂	semidirect
18. #14579158	C₈	semidirect
31. C₂×D₃₁₆	C₂×C₈	semidirect
45. C₄²×C₇₉	C₂×C₈	semidirect
80. #1839	D₇₉	semidirect
81. C₇₉	#23355	semidirect
122. C₄²	C₈×D₇₉	semidirect

H ≅ G/N

type

2. #23763691

C₂

semidirect

3. #23763543

C₂

semidirect

4. C₇₉×#1839

C₂

semidirect

18. #14579158

C₈

semidirect

31. C₂×D₃₁₆

C₂×C₈

semidirect

45. C₄²×C₇₉

C₂×C₈

semidirect

80. #1839

D₇₉

semidirect

81. C₇₉

#23355

semidirect

122. C₄²

C₈×D₇₉

semidirect

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

subgroup	index
C₄×#14579158	2
#23763691	2
#23763543	2
C₇₉×#1839	2
#23355	79

subgroup

index

C₇₉×#1839

Subgroups — #34314814

Conjugacy classes of subgroups

Complements

Maximal subgroups