Group #34215531 — subgroups

Subgroups — #34215531

Order 20224.

1550 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 1550 conjugacy classes of subgroups — too many to draw. Only its 651 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×#19098433	C₂	1	3	yes	maximal
2	C₂×#19098429	C₂	1	3	yes	maximal
3	#23761665	C₂	1	8	yes	maximal
4	C₂×#376×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₇₉	C₂²	1	3	yes	—
7	C₂²×C₄⋊Dic₇₉	C₂²	1	12	yes	—
8	#19098433	—	2	12	no	—
9	#19098429	—	2	12	no	—
10	C₂×C₇₉⋊#95	C₂²	1	12	yes	—
11	C₂²×C₄²×C₇₉	C₂²	1	3	yes	—
12	#376×C₇₉	C₄	1	8	yes	complemented
13	C₂×#95×C₇₉	C₂²	1	4	yes	—
14	C₂³×Dic₇₉	C₂³	1	2	yes	—
15	C₂³×Dic₇₉	C₂³	1	6	yes	—
16	C₂×C₄⋊Dic₇₉	—	2	6	no	—
17	C₂×C₄⋊Dic₇₉	—	2	6	no	—
18	C₂×C₄⋊Dic₇₉	—	2	24	no	—
19	C₂×C₄⋊Dic₇₉	—	2	12	no	—
20	C₂×C₄⋊Dic₇₉	—	4	24	no	—
21	C₇₉⋊#95	—	2	24	no	—
22	C₂³×C₄×C₇₉	C₂³	1	6	yes	—
23	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	C₂×C₄	1	12	yes	—
25	C₂×C₄²×C₇₉	D₄	1	12	yes	—
26	C₂×C₄²×C₇₉	Q₈	1	12	yes	—
27	#95×C₇₉	C₂×C₄	1	16	yes	—
28	C₂²×Dic₇₉	—	2	6	no	—
29	C₂²×Dic₇₉	—	2	6	no	—
30	C₂²×Dic₇₉	—	2	12	no	—
31	C₂²×Dic₇₉	—	2	24	no	—
32	C₂²×Dic₇₉	—	2	8	no	—
33	C₄⋊Dic₇₉	—	4	48	no	—
34	C₄⋊Dic₇₉	—	4	4	no	—
35	C₄⋊Dic₇₉	—	4	12	no	—
36	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	—	2	12	no	—
38	C₂²×C₄×C₇₉	—	2	12	no	—
39	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
40	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₇₉	C₂×D₄	1	12	yes	—
42	C₂²×C₄×C₇₉	C₂×D₄	1	6	yes	—
43	C₂²×C₄×C₇₉	C₂×Q₈	1	12	yes	—
44	C₂²×C₄×C₇₉	C₂×Q₈	1	6	yes	—
45	C₄²×C₇₉	C₄⋊C₄	1	48	yes	complemented
46	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
47	C₂³×C₇₉	C₂²×D₄	1	3	yes	—
48	C₂³×C₇₉	C₂²×Q₈	1	3	yes	—
49	C₂³×C₇₉	H(5,𝔽₂)	1	4	yes	—
50	C₂³×C₇₉	2₋¹⁺⁴	1	4	yes	—
51	C₂×C₄×C₇₉	—	2	24	no	—
52	C₂×C₄×C₇₉	—	2	12	no	—
53	C₂×C₄×C₇₉	—	2	12	no	—
54	C₂×C₄×C₇₉	—	2	24	no	—
55	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	24	yes	—
56	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	12	yes	—
57	C₂×C₄×C₇₉	#127	1	4	yes	—
58	C₂×C₄×C₇₉	C₄⋊Q₈	1	12	yes	—
59	C₂×Dic₇₉	—	4	6	no	—
60	C₂×Dic₇₉	—	4	2	no	—
61	C₂×Dic₇₉	—	4	24	no	—
62	C₂×Dic₇₉	—	4	12	no	—
63	C₂×Dic₇₉	—	4	12	no	—
64	C₂²×C₇₉	C₂²×C₄⋊C₄	1	3	yes	—
65	C₂²×C₇₉	#520	1	4	yes	—
66	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
67	C₂²×C₇₉	C₂×C₄⋊Q₈	1	3	yes	—
68	C₂²×C₇₉	#535	1	6	yes	—
69	C₂²×C₇₉	#536	1	6	yes	—
70	C₂²×C₇₉	#544	1	12	yes	—
71	C₄×C₇₉	—	2	24	no	—
72	C₄×C₇₉	#384	1	8	yes	—
73	Dic₇₉	—	8	6	no	—
74	Dic₇₉	—	8	2	no	—
75	#14354	—	79	1	no	maximal, Sylow
76	C₂×C₇₉	C₂×#384	1	1	yes	characteristic
77	C₂×C₇₉	#2317	1	6	yes	—
78	C₂×C₇₉	#2657	1	1	yes	characteristic
79	C₂×C₇₉	#2658	1	1	yes	characteristic
80	C₂×C₇₉	#2660	1	6	yes	—
81	C₂×#384	—	79	3	no	—
82	C₂×#382	—	79	3	no	—
83	C₂×#376	D₇₉	1	1	yes	characteristic
84	#1444	—	79	8	no	—
85	C₇₉	#14354	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₄⋊C₄	—	79	12	no	—
87	C₂²×C₄⋊C₄	—	79	1	no	—
88	C₂²×C₄⋊C₄	—	79	3	no	—
89	C₂²×C₄²	C₂×D₇₉	1	3	yes	—
90	#384	—	158	12	no	—
91	#382	—	158	12	no	—
92	#376	Dic₇₉	1	8	yes	complemented
93	C₂×#95	—	79	12	no	—
94	C₂×#95	C₂×D₇₉	1	4	yes	—
95	C₂³×C₄	—	79	6	no	—
96	C₂³×C₄	—	79	2	no	—
97	C₂³×C₄	C₂²×D₇₉	1	6	yes	—
98	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
99	C₂×C₄⋊C₄	—	316	24	no	—
100	C₂×C₄⋊C₄	—	158	6	no	—
101	C₂×C₄⋊C₄	—	158	6	no	—
102	C₂×C₄⋊C₄	—	158	24	no	—
103	C₂×C₄⋊C₄	—	158	12	no	—
104	C₂×C₄²	Dic₁₅₈	1	12	yes	—
105	C₂×C₄²	D₃₁₆	1	12	yes	—
106	C₂×C₄²	C₂×Dic₇₉	1	12	yes	—
107	#95	—	158	24	no	—
108	#95	C₂×Dic₇₉	1	16	yes	—
109	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
110	C₂²×C₄	—	2	12	no	—
111	C₂²×C₄	—	2	12	no	—
112	C₂²×C₄	—	158	24	no	—
113	C₂²×C₄	—	158	8	no	—
114	C₂²×C₄	—	158	6	no	—
115	C₂²×C₄	—	158	6	no	—
116	C₂²×C₄	—	158	12	no	—
117	C₂²×C₄	C₂²×Dic₇₉	1	12	yes	—
118	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
119	C₂²×C₄	C₂×D₃₁₆	1	12	yes	—
120	C₂²×C₄	C₂×D₃₁₆	1	6	yes	—
121	C₂²×C₄	C₂×Dic₁₅₈	1	12	yes	—
122	C₂²×C₄	C₂×Dic₁₅₈	1	6	yes	—
123	C₄⋊C₄	—	316	48	no	—
124	C₄⋊C₄	—	316	4	no	—
125	C₄⋊C₄	—	316	12	no	—
126	C₄²	C₄⋊Dic₇₉	1	48	yes	complemented
127	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
128	C₂³	C₂²×D₃₁₆	1	3	yes	—
129	C₂³	C₂²×Dic₁₅₈	1	3	yes	—
130	C₂³	#14579207	1	4	yes	—
131	C₂³	#14579208	1	4	yes	—
132	C₂×C₄	—	2	12	no	—
133	C₂×C₄	—	2	24	no	—
134	C₂×C₄	—	2	24	no	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	—	316	12	no	—
137	C₂×C₄	—	316	6	no	—
138	C₂×C₄	—	316	2	no	—
139	C₂×C₄	—	316	24	no	—
140	C₂×C₄	—	316	12	no	—
141	C₂×C₄	C₂×C₄⋊Dic₇₉	1	24	yes	—
142	C₂×C₄	C₂×C₄⋊Dic₇₉	1	12	yes	—
143	C₂×C₄	#14579158	1	4	yes	—
144	C₂×C₄	#14579161	1	12	yes	—
145	C₂²	C₂²×C₄⋊Dic₇₉	1	3	yes	—
146	C₂²	#19099046	1	4	yes	—
147	C₂²	C₂×#14579158	1	1	yes	characteristic
148	C₂²	C₂×#14579161	1	3	yes	—
149	C₂²	#19099132	1	6	yes	—
150	C₂²	#19099139	1	6	yes	—
151	C₂²	#19099191	1	12	yes	—
152	C₄	—	2	24	no	—
153	C₄	—	632	6	no	—
154	C₄	—	632	2	no	—
155	C₄	#19098433	1	8	yes	—
156	C₂	C₂×#19098433	1	1	yes	characteristic
157	C₂	#23766023	1	6	yes	—
158	C₂	#23769506	1	1	yes	characteristic
159	C₂	#23769510	1	1	yes	characteristic
160	C₂	#23769519	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098433

C₂

yes

maximal

C₂×#19098429

C₂

yes

maximal

#23761665

C₂

yes

maximal

C₂×#376×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂²×C₄⋊Dic₇₉

C₂²

yes

—

C₂²×C₄⋊Dic₇₉

C₂²

yes

—

#19098433

—

#19098429

—

C₂×C₇₉⋊#95

C₂²

yes

—

C₂²×C₄²×C₇₉

C₂²

yes

—

#376×C₇₉

C₄

yes

complemented

C₂×#95×C₇₉

C₂²

yes

—

C₂³×Dic₇₉

C₂³

yes

—

C₂³×Dic₇₉

C₂³

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₇₉⋊#95

—

C₂³×C₄×C₇₉

C₂³

yes

—

C₂³×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

Q₈

yes

—

#95×C₇₉

C₂×C₄

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂⁴×C₇₉

C₂⁴

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×Q₈

yes

—

C₂²×C₄×C₇₉

C₂×Q₈

yes

—

C₄²×C₇₉

C₄⋊C₄

yes

complemented

C₂³×C₇₉

C₂³×C₄

yes

characteristic

C₂³×C₇₉

C₂²×D₄

yes

—

C₂³×C₇₉

C₂²×Q₈

yes

—

C₂³×C₇₉

H(5,𝔽₂)

yes

—

C₂³×C₇₉

2₋¹⁺⁴

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

#127

yes

—

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂²×C₇₉

C₂²×C₄⋊C₄

yes

—

C₂²×C₇₉

#520

yes

—

C₂²×C₇₉

C₂×#127

yes

characteristic

C₂²×C₇₉

C₂×C₄⋊Q₈

yes

—

C₂²×C₇₉

#535

yes

—

C₂²×C₇₉

#536

yes

—

C₂²×C₇₉

#544

yes

—

C₄×C₇₉

—

C₄×C₇₉

#384

yes

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×#384

—

C₂×#382

—

C₂×#376

D₇₉

yes

characteristic

#1444

—

C₇₉

#14354

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

C₂×D₇₉

yes

—

#384

—

158

—

#382

—

158

—

#376

Dic₇₉

yes

complemented

C₂×#95

—

C₂×#95

C₂×D₇₉

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₇₉

yes

—

C₂³×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₄⋊C₄

—

316

—

100

C₂×C₄⋊C₄

—

158

—

101

C₂×C₄⋊C₄

—

158

—

102

C₂×C₄⋊C₄

—

158

—

103

C₂×C₄⋊C₄

—

158

—

104

C₂×C₄²

Dic₁₅₈

yes

—

105

C₂×C₄²

D₃₁₆

yes

—

106

C₂×C₄²

C₂×Dic₇₉

yes

—

107

#95

—

158

—

108

#95

C₂×Dic₇₉

yes

—

109

C₂⁴

C₂³×D₇₉

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

158

—

113

C₂²×C₄

—

158

—

114

C₂²×C₄

—

158

—

115

C₂²×C₄

—

158

—

116

C₂²×C₄

—

158

—

117

C₂²×C₄

C₂²×Dic₇₉

yes

—

118

C₂²×C₄

C₂²×Dic₇₉

yes

—

119

C₂²×C₄

C₂×D₃₁₆

yes

—

120

C₂²×C₄

C₂×D₃₁₆

yes

—

121

C₂²×C₄

C₂×Dic₁₅₈

yes

—

122

C₂²×C₄

C₂×Dic₁₅₈

yes

—

123

C₄⋊C₄

—

316

—

124

C₄⋊C₄

—

316

—

125

C₄⋊C₄

—

316

—

126

C₄²

C₄⋊Dic₇₉

yes

complemented

127

C₂³

C₂³×Dic₇₉

yes

characteristic

128

C₂³

C₂²×D₃₁₆

yes

—

129

C₂³

C₂²×Dic₁₅₈

yes

—

130

C₂³

#14579207

yes

—

131

C₂³

#14579208

yes

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

316

—

137

C₂×C₄

—

316

—

138

C₂×C₄

—

316

—

139

C₂×C₄

—

316

—

140

C₂×C₄

—

316

—

141

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

—

142

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

—

143

C₂×C₄

#14579158

yes

—

144

C₂×C₄

#14579161

yes

—

145

C₂²

C₂²×C₄⋊Dic₇₉

yes

—

146

C₂²

#19099046

yes

—

147

C₂²

C₂×#14579158

yes

characteristic

148

C₂²

C₂×#14579161

yes

—

149

C₂²

#19099132

yes

—

150

C₂²

#19099139

yes

—

151

C₂²

#19099191

yes

—

152

C₄

—

153

C₄

—

632

—

154

C₄

—

632

—

155

C₄

#19098433

yes

—

156

C₂

C₂×#19098433

yes

characteristic

157

C₂

#23766023

yes

—

158

C₂

#23769506

yes

characteristic

159

C₂

#23769510

yes

characteristic

160

C₂

#23769519

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₇₉
commutator	C₂³×C₇₉	C₂³×C₄
Fitting	C₂×#376×C₇₉	C₂
socle, prime radical, cosocle	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂×#376	D₇₉
hypocenter	C₇₉	#14354
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₇₉

commutator

C₂³×C₇₉

C₂³×C₄

Fitting

C₂×#376×C₇₉

C₂

socle, prime radical, cosocle

C₂⁴×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂⁴ ◁ C₂×#376

N	H ≅ G/N	type
12. #376×C₇₉	C₄	semidirect
45. C₄²×C₇₉	C₄⋊C₄	semidirect
85. C₇₉	#14354	semidirect
92. #376	Dic₇₉	semidirect
126. C₄²	C₄⋊Dic₇₉	semidirect

H ≅ G/N

type

12. #376×C₇₉

C₄

semidirect

45. C₄²×C₇₉

C₄⋊C₄

semidirect

85. C₇₉

#14354

semidirect

92. #376

Dic₇₉

semidirect

126. C₄²

C₄⋊Dic₇₉

semidirect

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

subgroup	index
C₂×#19098433	2
C₂×#19098429	2
#23761665	2
C₂×#376×C₇₉	2
#14354	79

subgroup

index

C₂×#376×C₇₉

Subgroups — #34215531

Conjugacy classes of subgroups

Complements

Maximal subgroups