Group #23762165 — subgroups

Subgroups — #23762165

Order 10112.

166 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19098620	C₂	1	1	yes	maximal, characteristic
2	C₄×#11929608	C₂	1	1	yes	maximal, characteristic
3	#19098315	C₂	1	2	yes	maximal, complemented
4	#19098313	C₂	1	2	yes	maximal, complemented
5	#423×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11929608	C₄	1	2	yes	—
7	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
10	#14579072	—	2	1	no	—
11	#14579072	C₄	1	1	yes	characteristic
12	#14579072	C₄	1	1	yes	characteristic
13	C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
14	C₂×M₁₆×C₇₉	—	2	1	no	—
15	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₉	—	2	1	no	—
17	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
18	#11929608	—	2	4	no	—
19	C₂×C₇₉⋊C₈	—	2	1	no	—
20	C₂×C₇₉⋊C₈	—	2	1	no	—
21	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	Q₈	1	1	yes	characteristic
26	M₁₆×C₇₉	—	4	2	no	—
27	C₂×C₈×C₇₉	—	2	2	no	—
28	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₇₉	D₄	1	1	yes	characteristic
30	C₄²×C₇₉	Q₈	1	1	yes	characteristic
31	C₄²×C₇₉	C₂³	1	1	yes	characteristic
32	C₂³×C₇₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₉	—	2	2	no	—
34	C₂×C₄×C₇₉	—	2	1	no	—
35	C₂×C₄×C₇₉	—	2	1	no	—
36	C₂×C₄×C₇₉	—	4	1	no	—
37	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₇₉	—	4	2	no	—
44	C₇₉⋊C₈	—	2	2	no	—
45	C₇₉⋊C₈	—	4	1	no	—
46	C₇₉⋊C₈	—	4	1	no	—
47	C₇₉⋊C₈	M₁₆	1	4	yes	complemented
48	C₂²×C₇₉	—	2	1	no	—
49	C₂²×C₇₉	—	2	1	no	—
50	C₂²×C₇₉	—	2	1	no	—
51	C₂²×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₇₉	—	2	1	no	—
53	C₄×C₇₉	—	2	1	no	—
54	C₄×C₇₉	—	2	1	no	—
55	C₄×C₇₉	—	2	1	no	—
56	C₄×C₇₉	C₂×M₁₆	1	2	yes	—
57	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₄×C₇₉	#137	1	1	yes	characteristic
59	C₂×C₇₉	—	4	1	no	—
60	C₂×C₇₉	#423	1	1	yes	characteristic
61	C₂×C₇₉	#428	1	1	yes	characteristic
62	C₂×C₇₉	#430	1	1	yes	characteristic
63	#1569	—	79	1	no	maximal, Sylow
64	C₇₉	#1569	1	1	yes	characteristic, complemented, Sylow
65	#432	—	79	1	no	—
66	#423	D₇₉	1	1	yes	characteristic
67	C₄×M₁₆	—	79	1	no	—
68	#335	—	79	2	no	—
69	#334	—	79	2	no	—
70	C₂×M₁₆	—	2	1	no	—
71	C₂×M₁₆	—	79	2	no	—
72	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
73	C₄⋊C₈	—	2	1	no	—
74	C₄⋊C₈	—	79	1	no	—
75	C₄⋊C₈	—	79	1	no	—
76	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
77	C₂²⋊C₈	—	158	1	no	—
78	#97	—	79	1	no	—
79	#97	—	79	1	no	—
80	#97	—	158	1	no	—
81	C₄×C₈	—	79	2	no	—
82	C₂²×C₄	Dic₁₅₈	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₇₉	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
85	M₁₆	—	4	2	no	—
86	M₁₆	—	158	4	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	79	2	no	—
89	C₂×C₈	—	79	2	no	—
90	C₂×C₈	—	158	1	no	—
91	C₂×C₈	—	158	1	no	—
92	C₄²	Dic₁₅₈	1	1	yes	characteristic
93	C₄²	C₄×D₇₉	1	1	yes	characteristic
94	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
95	C₄²	C₂²×D₇₉	1	1	yes	characteristic
96	C₂³	Dic₇₉⋊C₄	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
102	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
103	C₂×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	316	1	no	—
109	C₈	—	316	1	no	—
110	C₈	—	79	4	no	—
111	C₈	—	158	2	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	M₁₆×D₇₉	1	2	yes	—
121	C₄	#14579187	1	1	yes	characteristic
122	C₄	#14579192	1	1	yes	characteristic
123	C₂	—	4	1	no	—
124	C₂	#19098586	1	1	yes	characteristic
125	C₂	#19098603	1	1	yes	characteristic
126	C₂	#19098611	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098620

C₂

yes

maximal, characteristic

C₄×#11929608

C₂

yes

maximal, characteristic

#19098315

C₂

yes

maximal, complemented

#19098313

C₂

yes

maximal, complemented

#423×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₄

yes

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

#14579072

C₄

yes

characteristic

#14579072

C₄

yes

characteristic

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂×M₁₆×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

C₂²

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

Q₈

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

Q₈

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

M₁₆

yes

complemented

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×M₁₆

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#432

—

#423

D₇₉

yes

characteristic

C₄×M₁₆

—

#335

—

#334

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₉

yes

—

C₂²⋊C₈

—

158

—

#97

—

#97

—

#97

—

158

—

C₄×C₈

—

C₂²×C₄

Dic₁₅₈

yes

characteristic

C₂²×C₄

C₄×D₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

M₁₆

—

M₁₆

—

158

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄²

Dic₁₅₈

yes

characteristic

C₄²

C₄×D₇₉

yes

characteristic

C₄²

C₂²⋊D₇₉

yes

characteristic

C₄²

C₂²×D₇₉

yes

characteristic

C₂³

Dic₇₉⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

102

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

103

C₂×C₄

Dic₇₉⋊C₄

yes

characteristic

104

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

106

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

107

C₈

—

108

C₈

—

316

—

109

C₈

—

316

—

110

C₈

—

111

C₈

—

158

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×Dic₇₉⋊C₄

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

M₁₆×D₇₉

yes

—

121

C₄

#14579187

yes

characteristic

122

C₄

#14579192

yes

characteristic

123

C₂

—

124

C₂

#19098586

yes

characteristic

125

C₂

#19098603

yes

characteristic

126

C₂

#19098611

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₄²	C₂²×D₇₉
Fitting	#423×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄⋊C₄
prime radical	C₂³×C₇₉	C₄⋊C₄
cosocle	C₄²×C₇₉	C₂³
hypercenter	#423	D₇₉
hypocenter	C₇₉	#1569
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₄²

C₂²×D₇₉

Fitting

#423×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₄⋊C₄

prime radical

C₂³×C₇₉

C₄⋊C₄

cosocle

C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #19098315	C₂	semidirect
4. #19098313	C₂	semidirect
47. C₇₉⋊C₈	M₁₆	semidirect
64. C₇₉	#1569	semidirect

H ≅ G/N

type

3. #19098315

C₂

semidirect

4. #19098313

C₂

semidirect

47. C₇₉⋊C₈

M₁₆

semidirect

64. C₇₉

#1569

semidirect

Subgroups — #23762165

Conjugacy classes of subgroups

Complements