Group #34123733 — subgroups

Subgroups — C₇₉×#12044 #34123733

Order 20224.

222 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 114 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×Q₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₁₆×C₇₉	C₂	1	2	yes	maximal
3	C₇₉×#1372	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1332	C₂	1	2	yes	maximal
5	C₇₉×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₇₉	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₇₉	—	2	2	no	—
8	C₄×Q₁₆×C₇₉	C₂²	1	2	yes	—
9	#366×C₇₉	C₄	1	1	yes	characteristic
10	#358×C₇₉	C₄	1	2	yes	—
11	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	#335×C₇₉	C₂²	1	2	yes	—
14	Q₈⋊C₈×C₇₉	—	2	2	no	—
15	C₈²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₇₉	C₂×C₄	1	2	yes	—
17	C₄×Q₈×C₇₉	—	2	2	no	—
18	Q₃₂×C₇₉	C₈	1	4	yes	complemented
19	C₂×C₁₆×C₇₉	—	2	1	no	—
20	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
22	#107×C₇₉	C₂×C₄	1	2	yes	—
23	C₄⋊C₈×C₇₉	—	2	2	no	—
24	C₄⋊C₈×C₇₉	—	4	2	no	—
25	#103×C₇₉	—	2	2	no	—
26	C₄×C₈×C₇₉	—	4	2	no	—
27	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₇₉	—	2	2	no	—
31	Q₁₆×C₇₉	—	2	2	no	—
32	Q₁₆×C₇₉	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₇₉	—	2	1	no	—
34	C₂×C₈×C₇₉	—	4	2	no	—
35	C₂×C₈×C₇₉	D₈	1	2	yes	—
36	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
38	C₄⋊C₄×C₇₉	—	2	2	no	—
39	C₄⋊C₄×C₇₉	—	4	2	no	—
40	C₄²×C₇₉	—	4	2	no	—
41	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₉	—	2	2	no	—
43	C₁₆×C₇₉	—	2	1	no	—
44	C₁₆×C₇₉	C₂×C₈	1	2	yes	—
45	Q₈×C₇₉	—	2	4	no	—
46	Q₈×C₇₉	—	4	2	no	—
47	C₂×C₄×C₇₉	—	4	2	no	—
48	C₂×C₄×C₇₉	—	4	2	no	—
49	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
52	C₈×C₇₉	—	2	1	no	—
53	C₈×C₇₉	—	2	2	no	—
54	C₈×C₇₉	—	4	1	no	—
55	C₈×C₇₉	—	8	2	no	—
56	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₇₉	#131	1	1	yes	characteristic
58	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
59	C₄×C₇₉	—	2	1	no	—
60	C₄×C₇₉	—	4	4	no	—
61	C₄×C₇₉	—	8	2	no	—
62	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
63	C₄×C₇₉	#443	1	1	yes	characteristic
64	C₄×C₇₉	#510	1	2	yes	—
65	#12044	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₇₉	C₈×D₈	1	1	yes	characteristic
67	C₂×C₇₉	#2176	1	1	yes	characteristic
68	C₂×C₇₉	#2179	1	1	yes	characteristic
69	C₄×Q₃₂	C₂×C₇₉	1	1	yes	characteristic
70	C₈×Q₁₆	C₂×C₇₉	1	2	yes	—
71	#1372	C₂×C₇₉	1	1	yes	characteristic
72	#1332	C₂×C₇₉	1	2	yes	—
73	#1311	C₂×C₇₉	1	1	yes	characteristic
74	C₇₉	#12044	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×Q₃₂	C₄×C₇₉	1	1	yes	characteristic
76	C₈×Q₈	—	2	2	no	—
77	C₄×Q₁₆	C₂²×C₇₉	1	2	yes	—
78	#366	C₄×C₇₉	1	1	yes	characteristic
79	#358	C₄×C₇₉	1	2	yes	—
80	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
81	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
82	#335	C₂²×C₇₉	1	2	yes	—
83	Q₈⋊C₈	—	2	2	no	—
84	C₈²	C₂²×C₇₉	1	1	yes	characteristic
85	C₂×Q₁₆	C₂×C₄×C₇₉	1	2	yes	—
86	C₄×Q₈	—	2	2	no	—
87	Q₃₂	C₈×C₇₉	1	4	yes	complemented
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
90	C₂×C₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
91	#107	C₂×C₄×C₇₉	1	2	yes	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	—	4	2	no	—
94	#103	—	2	2	no	—
95	C₄×C₈	—	4	2	no	—
96	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
97	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
98	C₄×C₈	C₂³×C₇₉	1	1	yes	characteristic
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	C₂×C₈×C₇₉	1	4	yes	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	D₈×C₇₉	1	2	yes	—
105	C₂×C₈	C₂²×C₄×C₇₉	1	1	yes	characteristic
106	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	—	4	2	no	—
110	C₄²	C₂×D₄×C₇₉	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	C₂×C₈×C₇₉	1	2	yes	—
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	C₄×D₄×C₇₉	1	1	yes	characteristic
119	C₂×C₄	C₂×D₈×C₇₉	1	1	yes	characteristic
120	C₂×C₄	#135×C₇₉	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	2	no	—
123	C₈	—	4	1	no	—
124	C₈	—	8	2	no	—
125	C₈	C₂²×C₈×C₇₉	1	1	yes	characteristic
126	C₈	#131×C₇₉	1	1	yes	characteristic
127	C₂²	C₄×D₈×C₇₉	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	4	4	no	—
130	C₄	—	8	2	no	—
131	C₄	C₈×D₄×C₇₉	1	1	yes	characteristic
132	C₄	#443×C₇₉	1	1	yes	characteristic
133	C₄	#510×C₇₉	1	2	yes	—
134	C₂	C₈×D₈×C₇₉	1	1	yes	characteristic
135	C₂	C₇₉×#2176	1	1	yes	characteristic
136	C₂	C₇₉×#2179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₃₂×C₇₉

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₇₉

C₂

yes

maximal

C₇₉×#1372

C₂

yes

maximal, characteristic

C₇₉×#1332

C₂

yes

maximal

C₇₉×#1311

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₇₉

C₄

yes

characteristic

C₈×Q₈×C₇₉

—

C₄×Q₁₆×C₇₉

C₂²

yes

—

#366×C₇₉

C₄

yes

characteristic

#358×C₇₉

C₄

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

#335×C₇₉

C₂²

yes

—

Q₈⋊C₈×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Q₁₆×C₇₉

C₂×C₄

yes

—

C₄×Q₈×C₇₉

—

Q₃₂×C₇₉

C₈

yes

complemented

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

#107×C₇₉

C₂×C₄

yes

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

#103×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₈

yes

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂²×C₈

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₂²×C₇₉

C₄×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₈×D₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#12044

C₇₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₉

C₈×D₈

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₄×Q₃₂

C₂×C₇₉

yes

characteristic

C₈×Q₁₆

C₂×C₇₉

yes

—

#1372

C₂×C₇₉

yes

characteristic

#1332

C₂×C₇₉

yes

—

#1311

C₂×C₇₉

yes

characteristic

C₇₉

#12044

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₄×C₇₉

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

C₂²×C₇₉

yes

—

#366

C₄×C₇₉

yes

characteristic

#358

C₄×C₇₉

yes

—

C₄×C₁₆

C₂²×C₇₉

yes

characteristic

C₄×C₁₆

C₂²×C₇₉

yes

characteristic

#335

C₂²×C₇₉

yes

—

Q₈⋊C₈

—

C₈²

C₂²×C₇₉

yes

characteristic

C₂×Q₁₆

C₂×C₄×C₇₉

yes

—

C₄×Q₈

—

Q₃₂

C₈×C₇₉

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₇₉

yes

characteristic

C₂×C₁₆

C₂×C₄×C₇₉

yes

characteristic

#107

C₂×C₄×C₇₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

D₄×C₇₉

yes

characteristic

C₄×C₈

D₄×C₇₉

yes

characteristic

C₄×C₈

C₂³×C₇₉

yes

characteristic

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

C₂×C₈×C₇₉

yes

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₈×C₇₉

yes

—

105

C₂×C₈

C₂²×C₄×C₇₉

yes

characteristic

106

C₂×C₈

#41×C₇₉

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

C₂×D₄×C₇₉

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₂×C₈×C₇₉

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₄×D₄×C₇₉

yes

characteristic

119

C₂×C₄

C₂×D₈×C₇₉

yes

characteristic

120

C₂×C₄

#135×C₇₉

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₂²×C₈×C₇₉

yes

characteristic

126

C₈

#131×C₇₉

yes

characteristic

127

C₂²

C₄×D₈×C₇₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₈×D₄×C₇₉

yes

characteristic

132

C₄

#443×C₇₉

yes

characteristic

133

C₄

#510×C₇₉

yes

—

134

C₂

C₈×D₈×C₇₉

yes

characteristic

135

C₂

C₇₉×#2176

yes

characteristic

136

C₂

C₇₉×#2179

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₇₉	C₂×D₈
commutator	C₈	C₂²×C₈×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle, prime radical	C₂²×C₇₉	C₄×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₈²×C₇₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₇₉

C₂×D₈

commutator

C₈

C₂²×C₈×C₇₉

Frattini, cosocle

C₄×C₈

C₂³×C₇₉

socle, prime radical

C₂²×C₇₉

C₄×D₈

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₈²×C₇₉ ◁ G

N	H ≅ G/N	type
18. Q₃₂×C₇₉	C₈	semidirect
65. #12044	C₇₉	direct (N × H)
74. C₇₉	#12044	direct (N × H)
87. Q₃₂	C₈×C₇₉	semidirect

H ≅ G/N

type

18. Q₃₂×C₇₉

C₈

semidirect

65. #12044

C₇₉

direct (N × H)

74. C₇₉

#12044

direct (N × H)

87. Q₃₂

C₈×C₇₉

semidirect

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

subgroup	index
C₄×Q₃₂×C₇₉	2
C₈×Q₁₆×C₇₉	2
C₇₉×#1372	2
C₇₉×#1332	2
C₇₉×#1311	2
#12044	79

subgroup

index

C₄×Q₃₂×C₇₉

C₈×Q₁₆×C₇₉

C₇₉×#1372

C₇₉×#1332

C₇₉×#1311

#12044

Subgroups — C₇₉×#12044 #34123733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C79×#12044 #34123733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#12044 #34123733