Group #34175442 — subgroups

Subgroups — #34175442

Order 20224.

238 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 111 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×#19098384	C₂	1	2	yes	maximal
2	C₇₉×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579101	—	2	2	no	—
4	#19098384	—	2	2	no	—
5	#19098384	C₄	1	4	yes	complemented
6	C₂×#107×C₇₉	—	2	2	no	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
8	#384×C₇₉	C₄	1	2	yes	—
9	#14579101	—	2	4	no	—
10	#14579101	—	4	2	no	—
11	C₂×C₇₉⋊C₁₆	—	4	2	no	—
12	C₂²×C₈×C₇₉	D₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
14	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
15	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
16	#107×C₇₉	—	4	2	no	—
17	#107×C₇₉	—	4	2	no	—
18	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
20	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
21	C₇₉⋊C₁₆	—	4	4	no	—
22	C₂²×C₄×C₇₉	—	2	1	no	—
23	C₂²×C₄×C₇₉	—	4	2	no	—
24	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
26	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₉	D₈	1	2	yes	—
28	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
29	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
30	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
31	C₄⋊C₄×C₇₉	—	4	4	no	—
32	C₄⋊C₄×C₇₉	—	8	4	no	—
33	C₄²×C₇₉	—	2	1	no	—
34	C₄²×C₇₉	C₄²	1	1	yes	characteristic
35	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂³×C₇₉	#100	1	1	yes	characteristic
37	C₂×C₄×C₇₉	—	2	2	no	—
38	C₂×C₄×C₇₉	—	2	2	no	—
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	—	2	1	no	—
41	C₂×C₄×C₇₉	—	8	4	no	—
42	C₂×C₄×C₇₉	—	8	2	no	—
43	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
44	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
45	C₂×C₄×C₇₉	#102	1	2	yes	—
46	C₂×C₄×C₇₉	#103	1	2	yes	—
47	C₈×C₇₉	#102	1	2	yes	—
48	C₈×C₇₉	#103	1	2	yes	—
49	C₈×C₇₉	#106	1	2	yes	—
50	C₈×C₇₉	#107	1	2	yes	—
51	C₂²×C₇₉	—	2	1	no	—
52	C₂²×C₇₉	—	2	1	no	—
53	C₂²×C₇₉	#331	1	1	yes	characteristic
54	C₂²×C₇₉	#333	1	1	yes	characteristic
55	C₂²×C₇₉	#343	1	1	yes	characteristic
56	C₄×C₇₉	—	2	2	no	—
57	C₄×C₇₉	—	2	2	no	—
58	C₄×C₇₉	—	16	2	no	—
59	C₄×C₇₉	#332	1	2	yes	—
60	C₄×C₇₉	#340	1	2	yes	—
61	#7269	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	—	2	1	no	—
63	C₂×C₇₉	—	2	1	no	—
64	C₂×C₇₉	#1296	1	1	yes	characteristic
65	C₂×C₇₉	#1357	1	1	yes	characteristic
66	C₂×C₇₉	#1358	1	1	yes	characteristic
67	C₂×#364	—	79	2	no	—
68	#1845	D₇₉	1	1	yes	characteristic
69	C₇₉	#7269	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₃₂	—	158	2	no	—
71	C₂×#107	—	2	2	no	—
72	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
73	#384	Dic₇₉	1	2	yes	—
74	#364	—	79	4	no	—
75	#364	—	158	2	no	—
76	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	4	2	no	—
79	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
80	M₃₂	—	316	2	no	—
81	M₃₂	—	158	4	no	—
82	C₂×C₁₆	—	316	2	no	—
83	#107	—	4	2	no	—
84	#107	—	4	2	no	—
85	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
86	C₄×C₈	C₄×D₇₉	1	2	yes	—
87	C₄×C₈	D₃₁₆	1	1	yes	characteristic
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	2	no	—
90	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
91	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
92	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
93	C₂×C₈	#11929609	1	2	yes	—
94	C₂×C₈	#11929611	1	2	yes	—
95	C₂×C₈	#11929612	1	2	yes	—
96	C₂×C₈	#11929614	1	2	yes	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄²	—	2	1	no	—
100	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
101	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
102	C₁₆	—	316	4	no	—
103	C₂³	#14579078	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	8	4	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
111	C₂×C₄	#14579080	1	1	yes	characteristic
112	C₂×C₄	#14579082	1	2	yes	—
113	C₂×C₄	#14579085	1	2	yes	—
114	C₈	#14579081	1	2	yes	—
115	C₈	#14579084	1	2	yes	—
116	C₈	#14579092	1	2	yes	—
117	C₈	#14579094	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#19098307	1	1	yes	characteristic
121	C₂²	#19098312	1	1	yes	characteristic
122	C₂²	#19098332	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	16	2	no	—
126	C₄	#19098309	1	2	yes	—
127	C₄	#19098325	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#23761283	1	1	yes	characteristic
131	C₂	#23761437	1	1	yes	characteristic
132	C₂	#23761439	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098384

C₂

yes

maximal

C₇₉×#1845

C₂

yes

maximal, characteristic

C₂×#14579101

—

#19098384

—

#19098384

C₄

yes

complemented

C₂×#107×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#384×C₇₉

C₄

yes

—

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₄²

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₂³×C₇₉

#100

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#103

yes

—

C₈×C₇₉

#102

yes

—

C₈×C₇₉

#103

yes

—

C₈×C₇₉

#106

yes

—

C₈×C₇₉

#107

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#332

yes

—

C₄×C₇₉

#340

yes

—

#7269

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

C₇₉

#7269

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

158

—

C₂×#107

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#384

Dic₇₉

yes

—

#364

—

#364

—

158

—

C₂²×C₈

C₂²⋊D₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

M₃₂

—

316

—

M₃₂

—

158

—

C₂×C₁₆

—

316

—

#107

—

#107

—

C₄×C₈

Dic₁₅₈

yes

characteristic

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

D₃₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

C₂×C₈

Dic₇₉⋊C₄

yes

—

C₂×C₈

#11929609

yes

—

C₂×C₈

#11929611

yes

—

C₂×C₈

#11929612

yes

—

C₂×C₈

#11929614

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

C₄×Dic₇₉

yes

characteristic

101

C₄²

C₄⋊Dic₇₉

yes

characteristic

102

C₁₆

—

316

—

103

C₂³

#14579078

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₇₉⋊#95

yes

characteristic

111

C₂×C₄

#14579080

yes

characteristic

112

C₂×C₄

#14579082

yes

—

113

C₂×C₄

#14579085

yes

—

114

C₈

#14579081

yes

—

115

C₈

#14579084

yes

—

116

C₈

#14579092

yes

—

117

C₈

#14579094

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

#19098307

yes

characteristic

121

C₂²

#19098312

yes

characteristic

122

C₂²

#19098332

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#19098309

yes

—

127

C₄

#19098325

yes

—

128

C₂

—

129

C₂

—

130

C₂

#23761283

yes

characteristic

131

C₂

#23761437

yes

characteristic

132

C₂

#23761439

yes

characteristic

	subgroup	G/H
center	C₂²	#19098307
commutator	C₄²×C₇₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₇₉×#1845	C₂
socle	C₂²×C₇₉	#331
prime radical	C₂³×C₇₉	#100
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	#1845	D₇₉
hypocenter	C₇₉	#7269
LCS	G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂²

#19098307

commutator

C₄²×C₇₉

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₉

Fitting

C₇₉×#1845

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄×C₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
5. #19098384	C₄	semidirect
69. C₇₉	#7269	semidirect

H ≅ G/N

type

5. #19098384

C₄

semidirect

69. C₇₉

#7269

semidirect

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

subgroup	index
C₂×#19098384	2
C₇₉×#1845	2
#7269	79

subgroup

index

C₂×#19098384

C₇₉×#1845

#7269

Subgroups — #34175442

Conjugacy classes of subgroups

Complements

Maximal subgroups