Group #34219620 — subgroups

Subgroups — #34219620

Order 20224.

1422 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1422 conjugacy classes of subgroups — too many to draw. Only its 411 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×#19098431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23761631	C₂	1	12	yes	maximal
4	C₂×#383×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₇₉⋊C₄	C₂²	1	4	yes	—
6	C₂²×C₄×Dic₇₉	C₂²	1	6	yes	—
7	#19098431	C₄	1	8	yes	complemented
8	C₂×C₇₉⋊#95	C₂²	1	12	yes	—
9	C₂×C₇₉⋊#95	C₂²	1	6	yes	—
10	C₄²×Dic₇₉	—	2	4	no	—
11	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
12	#383×C₇₉	—	2	4	no	—
13	C₂×#95×C₇₉	C₂²	1	6	yes	—
14	C₂³×Dic₇₉	C₂³	1	4	yes	—
15	C₂³×Dic₇₉	C₂³	1	4	yes	—
16	C₂×Dic₇₉⋊C₄	—	2	24	no	—
17	C₂×C₄×Dic₇₉	—	2	12	no	—
18	C₂×C₄×Dic₇₉	—	2	24	no	—
19	C₇₉⋊#95	—	2	24	no	—
20	C₇₉⋊#95	C₂×C₄	1	24	yes	—
21	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
22	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
23	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₇₉	D₄	1	4	yes	—
25	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
26	#95×C₇₉	—	2	12	no	—
27	C₂²×Dic₇₉	—	2	4	no	—
28	C₂²×Dic₇₉	—	2	12	no	—
29	C₂²×Dic₇₉	—	2	12	no	—
30	C₂²×Dic₇₉	—	2	12	no	—
31	C₂²×Dic₇₉	C₂²×C₄	1	8	yes	—
32	C₂²×Dic₇₉	#41	1	24	yes	—
33	Dic₇₉⋊C₄	—	4	16	no	—
34	C₄×Dic₇₉	—	2	48	no	—
35	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	—	2	12	no	—
37	C₂²×C₄×C₇₉	—	2	4	no	—
38	C₂²×C₄×C₇₉	—	2	12	no	—
39	C₂²×C₄×C₇₉	—	2	12	no	—
40	C₂²×C₄×C₇₉	C₂²×C₄	1	6	yes	—
41	C₂²×C₄×C₇₉	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₇₉	C₂×Q₈	1	6	yes	—
43	C₄²×C₇₉	C₄⋊C₄	1	16	yes	complemented
44	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
45	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
46	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
47	C₂³×C₇₉	C₂×#41	1	12	yes	—
48	C₂×C₄×C₇₉	—	2	12	no	—
49	C₂×C₄×C₇₉	—	2	12	no	—
50	C₂×C₄×C₇₉	—	2	12	no	—
51	C₂×C₄×C₇₉	—	4	12	no	—
52	C₂×C₄×C₇₉	—	4	4	no	—
53	C₂×C₄×C₇₉	—	4	12	no	—
54	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	12	yes	—
55	C₂×Dic₇₉	—	2	24	no	—
56	C₂×Dic₇₉	—	2	24	no	—
57	C₂×Dic₇₉	—	2	24	no	—
58	C₂×Dic₇₉	—	4	4	no	—
59	C₂×Dic₇₉	—	4	12	no	—
60	C₂×Dic₇₉	#126	1	16	yes	—
61	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₇₉	C₄×#41	1	6	yes	—
63	C₂²×C₇₉	#525	1	6	yes	—
64	C₂²×C₇₉	C₂×#126	1	4	yes	—
65	C₂²×C₇₉	#529	1	12	yes	—
66	C₂²×C₇₉	#533	1	6	yes	—
67	C₄×C₇₉	—	2	12	no	—
68	C₄×C₇₉	—	8	4	no	—
69	Dic₇₉	—	2	16	no	—
70	Dic₇₉	—	4	8	no	—
71	#14710	—	79	1	no	maximal, Sylow
72	C₂×C₇₉	#2296	1	3	yes	—
73	C₂×C₇₉	C₄×#126	1	4	yes	—
74	C₂×C₇₉	#2852	1	4	yes	—
75	C₂×C₇₉	#2863	1	4	yes	—
76	C₂×#383	—	79	1	no	—
77	C₂×#383	D₇₉	1	1	yes	characteristic
78	C₂×C₄³	—	79	1	no	—
79	#1436	—	79	12	no	—
80	C₇₉	#14710	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₄⋊C₄	—	79	4	no	—
82	C₂²×C₄²	—	79	6	no	—
83	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
84	#383	—	2	4	no	—
85	#383	—	79	8	no	—
86	C₂×#95	—	79	6	no	—
87	C₂×#95	—	79	12	no	—
88	C₂×#95	C₂×D₇₉	1	6	yes	—
89	C₄³	—	158	4	no	—
90	C₂³×C₄	—	79	4	no	—
91	C₂³×C₄	—	79	4	no	—
92	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
93	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
94	C₂×C₄⋊C₄	—	158	24	no	—
95	C₂×C₄²	—	158	24	no	—
96	C₂×C₄²	—	158	12	no	—
97	C₂×C₄²	Dic₁₅₈	1	4	yes	—
98	C₂×C₄²	C₄×D₇₉	1	4	yes	—
99	C₂×C₄²	C₂²⋊D₇₉	1	4	yes	—
100	#95	—	2	12	no	—
101	#95	—	79	24	no	—
102	#95	—	158	24	no	—
103	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
104	C₂²×C₄	—	2	12	no	—
105	C₂²×C₄	—	2	12	no	—
106	C₂²×C₄	—	2	12	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	—	79	24	no	—
109	C₂²×C₄	—	79	8	no	—
110	C₂²×C₄	—	158	12	no	—
111	C₂²×C₄	—	158	12	no	—
112	C₂²×C₄	—	158	12	no	—
113	C₂²×C₄	—	158	4	no	—
114	C₂²×C₄	C₂×C₄×D₇₉	1	6	yes	—
115	C₂²×C₄	C₂×C₂²⋊D₇₉	1	6	yes	—
116	C₂²×C₄	C₂×Dic₁₅₈	1	6	yes	—
117	C₄⋊C₄	—	316	16	no	—
118	C₄²	—	158	48	no	—
119	C₄²	Dic₇₉⋊C₄	1	16	yes	complemented
120	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
121	C₂³	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂³	C₂²×Dic₁₅₈	1	1	yes	characteristic
123	C₂³	#41×D₇₉	1	12	yes	—
124	C₂×C₄	—	2	12	no	—
125	C₂×C₄	—	2	12	no	—
126	C₂×C₄	—	2	12	no	—
127	C₂×C₄	—	4	12	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	4	12	no	—
130	C₂×C₄	—	316	4	no	—
131	C₂×C₄	—	316	12	no	—
132	C₂×C₄	—	79	16	no	—
133	C₂×C₄	—	158	24	no	—
134	C₂×C₄	—	158	24	no	—
135	C₂×C₄	—	158	24	no	—
136	C₂×C₄	C₂×Dic₇₉⋊C₄	1	12	yes	—
137	C₂²	C₂²×Dic₇₉⋊C₄	1	1	yes	characteristic
138	C₂²	#19099031	1	6	yes	—
139	C₂²	#19099070	1	6	yes	—
140	C₂²	#126×D₇₉	1	4	yes	—
141	C₂²	#19099096	1	12	yes	—
142	C₂²	#19099116	1	6	yes	—
143	C₄	—	2	12	no	—
144	C₄	—	8	4	no	—
145	C₄	—	316	8	no	—
146	C₄	—	158	16	no	—
147	C₂	#23765882	1	3	yes	—
148	C₂	#23765948	1	4	yes	—
149	C₂	#23771712	1	4	yes	—
150	C₂	#23771780	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098431

C₂

yes

maximal, characteristic

C₂×C₄²×Dic₇₉

C₂

yes

maximal, characteristic

#23761631

C₂

yes

maximal

C₂×#383×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₇₉⋊C₄

C₂²

yes

—

C₂²×C₄×Dic₇₉

C₂²

yes

—

#19098431

C₄

yes

complemented

C₂×C₇₉⋊#95

C₂²

yes

—

C₂×C₇₉⋊#95

C₂²

yes

—

C₄²×Dic₇₉

—

C₂²×C₄²×C₇₉

C₂²

yes

characteristic

#383×C₇₉

—

C₂×#95×C₇₉

C₂²

yes

—

C₂³×Dic₇₉

C₂³

yes

—

C₂³×Dic₇₉

C₂³

yes

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

—

C₂×C₄×Dic₇₉

—

C₇₉⋊#95

—

C₇₉⋊#95

C₂×C₄

yes

—

C₂³×C₄×C₇₉

C₂³

yes

—

C₂³×C₄×C₇₉

C₂³

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

Q₈

yes

—

#95×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

C₂²×C₄

yes

—

C₂²×Dic₇₉

#41

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂⁴×C₇₉

C₂⁴

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×Q₈

yes

—

C₄²×C₇₉

C₄⋊C₄

yes

complemented

C₂³×C₇₉

C₂³×C₄

yes

characteristic

C₂³×C₇₉

C₂²×D₄

yes

characteristic

C₂³×C₇₉

C₂²×Q₈

yes

characteristic

C₂³×C₇₉

C₂×#41

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

#126

yes

—

C₂²×C₇₉

C₂²×C₄⋊C₄

yes

characteristic

C₂²×C₇₉

C₄×#41

yes

—

C₂²×C₇₉

#525

yes

—

C₂²×C₇₉

C₂×#126

yes

—

C₂²×C₇₉

#529

yes

—

C₂²×C₇₉

#533

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

Dic₇₉

—

#14710

—

maximal, Sylow

C₂×C₇₉

#2296

yes

—

C₂×C₇₉

C₄×#126

yes

—

C₂×C₇₉

#2852

yes

—

C₂×C₇₉

#2863

yes

—

C₂×#383

—

C₂×#383

D₇₉

yes

characteristic

C₂×C₄³

—

#1436

—

C₇₉

#14710

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

C₂²×C₄²

C₂×D₇₉

yes

characteristic

#383

—

#383

—

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₇₉

yes

—

C₄³

—

158

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₇₉

yes

—

C₂³×C₄

C₂²×D₇₉

yes

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

—

158

—

C₂×C₄²

—

158

—

C₂×C₄²

Dic₁₅₈

yes

—

C₂×C₄²

C₄×D₇₉

yes

—

C₂×C₄²

C₂²⋊D₇₉

yes

—

100

#95

—

101

#95

—

102

#95

—

158

—

103

C₂⁴

C₂³×D₇₉

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

158

—

111

C₂²×C₄

—

158

—

112

C₂²×C₄

—

158

—

113

C₂²×C₄

—

158

—

114

C₂²×C₄

C₂×C₄×D₇₉

yes

—

115

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

116

C₂²×C₄

C₂×Dic₁₅₈

yes

—

117

C₄⋊C₄

—

316

—

118

C₄²

—

158

—

119

C₄²

Dic₇₉⋊C₄

yes

complemented

120

C₂³

C₂²×C₄×D₇₉

yes

characteristic

121

C₂³

C₂²×C₂²⋊D₇₉

yes

characteristic

122

C₂³

C₂²×Dic₁₅₈

yes

characteristic

123

C₂³

#41×D₇₉

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

316

—

131

C₂×C₄

—

316

—

132

C₂×C₄

—

133

C₂×C₄

—

158

—

134

C₂×C₄

—

158

—

135

C₂×C₄

—

158

—

136

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

—

137

C₂²

C₂²×Dic₇₉⋊C₄

yes

characteristic

138

C₂²

#19099031

yes

—

139

C₂²

#19099070

yes

—

140

C₂²

#126×D₇₉

yes

—

141

C₂²

#19099096

yes

—

142

C₂²

#19099116

yes

—

143

C₄

—

144

C₄

—

145

C₄

—

316

—

146

C₄

—

158

—

147

C₂

#23765882

yes

—

148

C₂

#23765948

yes

—

149

C₂

#23771712

yes

—

150

C₂

#23771780

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₇₉
commutator	C₂³×C₇₉	C₂³×C₄
Fitting	C₂×#383×C₇₉	C₂
socle, prime radical, cosocle	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂×#383	D₇₉
hypocenter	C₇₉	#14710
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#383

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₇₉

commutator

C₂³×C₇₉

C₂³×C₄

Fitting

C₂×#383×C₇₉

C₂

socle, prime radical, cosocle

C₂⁴×C₇₉

C₂⁴

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂⁴ ◁ C₂×#383

N	H ≅ G/N	type
7. #19098431	C₄	semidirect
43. C₄²×C₇₉	C₄⋊C₄	semidirect
80. C₇₉	#14710	semidirect
119. C₄²	Dic₇₉⋊C₄	semidirect

H ≅ G/N

type

7. #19098431

C₄

semidirect

43. C₄²×C₇₉

C₄⋊C₄

semidirect

80. C₇₉

#14710

semidirect

119. C₄²

Dic₇₉⋊C₄

semidirect

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

subgroup	index
C₂×#19098431	2
C₂×C₄²×Dic₇₉	2
#23761631	2
C₂×#383×C₇₉	2
#14710	79

subgroup

index

C₂×#19098431

C₂×C₄²×Dic₇₉

#23761631

C₂×#383×C₇₉

#14710

Subgroups — #34219620

Conjugacy classes of subgroups

Complements

Maximal subgroups