Group #34182087 — subgroups

Subgroups — #34182087

Order 20224.

306 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 155 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	#23763538	C₂	1	2	yes	maximal
2	#23762159	C₂	1	2	yes	maximal
3	#23762156	C₂	1	2	yes	maximal
4	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19098587	C₂²	1	4	yes	—
6	#19098328	—	2	4	no	—
7	#19098316	—	2	4	no	—
8	#19098314	—	2	4	no	—
9	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#322×C₇₉	C₄	1	4	yes	—
13	C₂×#11929608	—	2	4	no	—
14	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
15	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
16	C₂²×C₈×C₇₉	D₄	1	2	yes	—
17	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
18	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
19	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₉	D₄	1	2	yes	—
24	C₄×C₈×C₇₉	D₄	1	2	yes	—
25	C₄×C₈×C₇₉	Q₈	1	2	yes	—
26	#11929608	—	4	8	no	—
27	C₂×C₇₉⋊C₈	—	4	4	no	—
28	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₇₉	—	2	4	no	—
32	C₂×C₈×C₇₉	—	4	2	no	—
33	C₂×C₈×C₇₉	—	4	2	no	—
34	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
37	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄×C₇₉	—	2	2	no	—
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#135	1	4	yes	—
51	C₈×C₇₉	—	2	4	no	—
52	C₈×C₇₉	—	2	4	no	—
53	C₈×C₇₉	—	2	4	no	—
54	C₇₉⋊C₈	—	8	4	no	—
55	C₂²×C₇₉	—	2	1	no	—
56	C₂²×C₇₉	—	2	1	no	—
57	C₂²×C₇₉	#384	1	1	yes	characteristic
58	C₂²×C₇₉	#499	1	2	yes	—
59	C₄×C₇₉	—	2	1	no	—
60	C₄×C₇₉	—	2	1	no	—
61	C₄×C₇₉	#427	1	2	yes	—
62	C₄×C₇₉	#430	1	2	yes	—
63	C₄×C₇₉	#430	1	2	yes	—
64	C₄×C₇₉	#495	1	2	yes	—
65	#8827	—	79	1	no	maximal, Sylow
66	C₂×C₇₉	—	2	2	no	—
67	C₂×C₇₉	#1846	1	1	yes	characteristic
68	C₂×C₇₉	#1849	1	1	yes	characteristic
69	C₂×C₇₉	#1849	1	1	yes	characteristic
70	#1837	—	79	2	no	—
71	#1567	—	79	2	no	—
72	#1566	—	79	2	no	—
73	C₂×#322	D₇₉	1	1	yes	characteristic
74	C₇₉	#8827	1	1	yes	characteristic, complemented, Sylow
75	#423	—	79	4	no	—
76	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
79	#341	—	158	4	no	—
80	#335	—	158	4	no	—
81	#334	—	158	4	no	—
82	#322	Dic₇₉	1	4	yes	—
83	C₂×M₁₆	—	158	4	no	—
84	C₂²×C₈	Dic₁₅₈	1	2	yes	—
85	C₂²×C₈	Dic₁₅₈	1	2	yes	—
86	C₂²×C₈	D₃₁₆	1	2	yes	—
87	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
88	C₄⋊C₈	—	158	4	no	—
89	C₄⋊C₈	—	158	4	no	—
90	C₄×C₈	Dic₁₅₈	1	2	yes	—
91	C₄×C₈	D₃₁₆	1	2	yes	—
92	C₄×C₈	D₃₁₆	1	2	yes	—
93	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
94	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
95	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
96	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
98	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
99	M₁₆	—	316	8	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	316	4	no	—
104	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
105	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
106	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
107	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
108	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
109	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
110	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
111	C₂³	#14579161	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
117	C₂×C₄	#14579158	1	1	yes	characteristic
118	C₂×C₄	#14579161	1	1	yes	characteristic
119	C₂×C₄	#14579161	1	1	yes	characteristic
120	C₂×C₄	#14579183	1	4	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	2	4	no	—
123	C₈	—	2	4	no	—
124	C₈	—	632	4	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#19098433	1	1	yes	characteristic
128	C₂²	#19098965	1	2	yes	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	#19098601	1	2	yes	—
132	C₄	#19098612	1	2	yes	—
133	C₄	#19098612	1	2	yes	—
134	C₄	#19098947	1	2	yes	—
135	C₂	—	2	2	no	—
136	C₂	#23763564	1	1	yes	characteristic
137	C₂	#23763575	1	1	yes	characteristic
138	C₂	#23763575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763538

C₂

yes

maximal

#23762159

C₂

yes

maximal

#23762156

C₂

yes

maximal

C₂×#322×C₇₉

C₂

yes

maximal, characteristic

#19098587

C₂²

yes

—

#19098328

—

#19098316

—

#19098314

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#322×C₇₉

C₄

yes

—

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

Q₈

yes

—

C₂²×C₈×C₇₉

Q₈

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#384

yes

characteristic

C₂²×C₇₉

#499

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#427

yes

—

C₄×C₇₉

#430

yes

—

C₄×C₇₉

#430

yes

—

C₄×C₇₉

#495

yes

—

#8827

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

—

D₇₉

yes

characteristic

C₇₉

#8827

yes

characteristic, complemented, Sylow

#423

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#341

—

158

—

#335

—

158

—

#334

—

158

—

#322

Dic₇₉

yes

—

C₂×M₁₆

—

158

—

C₂²×C₈

Dic₁₅₈

yes

—

C₂²×C₈

Dic₁₅₈

yes

—

C₂²×C₈

D₃₁₆

yes

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

158

—

C₄⋊C₈

—

158

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

D₃₁₆

yes

—

C₄×C₈

D₃₁₆

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

M₁₆

—

316

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

316

—

104

C₂×C₈

C₄⋊Dic₇₉

yes

—

105

C₂×C₈

C₄⋊Dic₇₉

yes

—

106

C₂×C₈

C₄⋊Dic₇₉

yes

—

107

C₄²

C₂²×Dic₇₉

yes

characteristic

108

C₄²

C₂×D₃₁₆

yes

characteristic

109

C₄²

C₂×D₃₁₆

yes

characteristic

110

C₄²

C₂×Dic₁₅₈

yes

characteristic

111

C₂³

#14579161

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

115

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

116

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

117

C₂×C₄

#14579158

yes

characteristic

118

C₂×C₄

#14579161

yes

characteristic

119

C₂×C₄

#14579161

yes

characteristic

120

C₂×C₄

#14579183

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

632

—

125

C₂²

—

126

C₂²

—

127

C₂²

#19098433

yes

characteristic

128

C₂²

#19098965

yes

—

129

C₄

—

130

C₄

—

131

C₄

#19098601

yes

—

132

C₄

#19098612

yes

—

133

C₄

#19098612

yes

—

134

C₄

#19098947

yes

—

135

C₂

—

136

C₂

#23763564

yes

characteristic

137

C₂

#23763575

yes

characteristic

138

C₂

#23763575

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle	C₂²×C₇₉	#384
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#8827
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14579158

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×#322×C₇₉

C₂

socle

C₂²×C₇₉

#384

prime radical

C₂³×C₇₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
74. C₇₉	#8827	semidirect

H ≅ G/N

type

74. C₇₉

#8827

semidirect

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

subgroup	index
#23763538	2
#23762159	2
#23762156	2
C₂×#322×C₇₉	2
#8827	79

subgroup

index

C₂×#322×C₇₉

Subgroups — #34182087

Conjugacy classes of subgroups

Complements

Maximal subgroups