Group #23763527 — subgroups

Subgroups — #23763527

Order 10112.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#19098433	C₂	1	1	yes	maximal, characteristic
3	#19098325	C₂	1	4	yes	maximal
4	#384×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₇₉	—	2	1	no	—
7	C₂×C₄⋊Dic₇₉	—	2	1	no	—
8	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
9	C₄×C₇₉⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
11	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
12	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₇₉	—	2	2	no	—
15	C₂×C₇₉⋊C₈	—	2	2	no	—
16	C₂×C₇₉⋊C₈	—	2	2	no	—
17	C₂×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₇₉	—	2	4	no	—
19	C₄⋊Dic₇₉	—	4	2	no	—
20	C₄⋊Dic₇₉	—	4	2	no	—
21	C₂²×C₄×C₇₉	—	2	2	no	—
22	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₇₉	—	2	4	no	—
26	C₄⋊C₄×C₇₉	—	4	2	no	—
27	C₄⋊C₄×C₇₉	—	4	2	no	—
28	C₄²×C₇₉	C₂×C₄	1	2	yes	—
29	C₄²×C₇₉	Q₈	1	2	yes	—
30	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₇₉	—	4	2	no	—
32	C₂×C₄×C₇₉	—	4	2	no	—
33	C₂×C₄×C₇₉	—	4	2	no	—
34	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₇₉	D₈	1	2	yes	—
37	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
38	C₂×C₄×C₇₉	SD₁₆	1	2	yes	—
39	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
40	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₇₉	#41	1	2	yes	—
43	C₂×C₄×C₇₉	#41	1	2	yes	—
44	C₂×Dic₇₉	—	4	2	no	—
45	C₂×Dic₇₉	—	4	2	no	—
46	C₂×Dic₇₉	—	4	2	no	—
47	C₇₉⋊C₈	—	2	4	no	—
48	C₂²×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₇₉	#124	1	1	yes	characteristic
50	C₂²×C₇₉	#124	1	1	yes	characteristic
51	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₇₉	—	8	2	no	—
56	C₄×C₇₉	#106	1	4	yes	—
57	C₄×C₇₉	#107	1	4	yes	—
58	C₄×C₇₉	#117	1	2	yes	—
59	C₄×C₇₉	#125	1	2	yes	—
60	Dic₇₉	—	8	2	no	—
61	C₂×C₇₉	#382	1	1	yes	characteristic
62	C₂×C₇₉	C₂×#106	1	1	yes	characteristic
63	C₂×C₇₉	C₂×#107	1	1	yes	characteristic
64	C₂×C₇₉	#486	1	1	yes	characteristic
65	C₂×C₇₉	#486	1	1	yes	characteristic
66	C₂×C₇₉	#487	1	1	yes	characteristic
67	C₂×C₇₉	#487	1	1	yes	characteristic
68	#1834	—	79	1	no	maximal, Sylow
69	C₇₉	#1834	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄×C₈	—	79	1	no	—
71	#384	—	79	1	no	—
72	#384	D₇₉	1	1	yes	characteristic
73	#340	—	79	4	no	—
74	C₂²×C₈	—	79	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	79	2	no	—
78	C₂×C₄⋊C₄	—	158	1	no	—
79	C₂×C₄⋊C₄	—	158	1	no	—
80	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
81	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
82	C₄×C₈	—	79	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	158	2	no	—
85	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
88	C₂×C₈	—	79	4	no	—
89	C₂×C₈	—	158	2	no	—
90	C₂×C₈	—	158	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	316	2	no	—
95	C₄⋊C₄	—	316	2	no	—
96	C₄⋊C₄	—	158	4	no	—
97	C₄²	Dic₁₅₈	1	2	yes	—
98	C₄²	C₄×D₇₉	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	316	2	no	—
104	C₂×C₄	—	316	2	no	—
105	C₂×C₄	—	316	2	no	—
106	C₂×C₄	Dic₇₉⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₇₉⋊C₄	1	2	yes	—
108	C₂×C₄	#11929609	1	2	yes	—
109	C₂×C₄	#11929611	1	2	yes	—
110	C₂×C₄	#11929612	1	2	yes	—
111	C₂×C₄	#11929614	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
114	C₂×C₄	#11929623	1	2	yes	—
115	C₂×C₄	#11929625	1	2	yes	—
116	C₈	—	158	4	no	—
117	C₂²	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
118	C₂²	#14579148	1	1	yes	characteristic
119	C₂²	#14579150	1	1	yes	characteristic
120	C₂²	C₂×#11929609	1	1	yes	characteristic
121	C₂²	C₂×#11929611	1	1	yes	characteristic
122	C₂²	C₂×#11929612	1	1	yes	characteristic
123	C₂²	C₂×#11929614	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	632	2	no	—
126	C₄	#14579092	1	4	yes	—
127	C₄	#14579094	1	4	yes	—
128	C₄	#14579119	1	2	yes	—
129	C₄	#14579152	1	2	yes	—
130	C₂	#19098428	1	1	yes	characteristic
131	C₂	C₂×#14579092	1	1	yes	characteristic
132	C₂	C₂×#14579094	1	1	yes	characteristic
133	C₂	#19098909	1	1	yes	characteristic
134	C₂	#19098911	1	1	yes	characteristic
135	C₂	#19098913	1	1	yes	characteristic
136	C₂	#19098915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₇₉⋊C₈

C₂

yes

maximal, characteristic

#19098433

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal

#384×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₄×C₇₉⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

—

C₄²×C₇₉

Q₈

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₇₉

D₈

yes

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

Q₁₆

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×Q₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#106

yes

—

C₄×C₇₉

#107

yes

—

C₄×C₇₉

#117

yes

—

C₄×C₇₉

#125

yes

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

#384

—

#384

D₇₉

yes

characteristic

#340

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

158

—

C₄²

Dic₁₅₈

yes

—

C₄²

C₄×D₇₉

yes

—

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

316

—

104

C₂×C₄

—

316

—

105

C₂×C₄

—

316

—

106

C₂×C₄

Dic₇₉⋊C₄

yes

—

107

C₂×C₄

Dic₇₉⋊C₄

yes

—

108

C₂×C₄

#11929609

yes

—

109

C₂×C₄

#11929611

yes

—

110

C₂×C₄

#11929612

yes

—

111

C₂×C₄

#11929614

yes

—

112

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

113

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

114

C₂×C₄

#11929623

yes

—

115

C₂×C₄

#11929625

yes

—

116

C₈

—

158

—

117

C₂²

C₂×Dic₇₉⋊C₄

yes

characteristic

118

C₂²

#14579148

yes

characteristic

119

C₂²

#14579150

yes

characteristic

120

C₂²

C₂×#11929609

yes

characteristic

121

C₂²

C₂×#11929611

yes

characteristic

122

C₂²

C₂×#11929612

yes

characteristic

123

C₂²

C₂×#11929614

yes

characteristic

124

C₄

—

125

C₄

—

632

—

126

C₄

#14579092

yes

—

127

C₄

#14579094

yes

—

128

C₄

#14579119

yes

—

129

C₄

#14579152

yes

—

130

C₂

#19098428

yes

characteristic

131

C₂

C₂×#14579092

yes

characteristic

132

C₂

C₂×#14579094

yes

characteristic

133

C₂

#19098909

yes

characteristic

134

C₂

#19098911

yes

characteristic

135

C₂

#19098913

yes

characteristic

136

C₂

#19098915

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#384×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#384	D₇₉
hypocenter	C₇₉	#1834
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#384×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×C₇₉⋊C₈	C₄	semidirect
69. C₇₉	#1834	semidirect

H ≅ G/N

type

9. C₄×C₇₉⋊C₈

C₄

semidirect

69. C₇₉

#1834

semidirect

Subgroups — #23763527

Conjugacy classes of subgroups

Complements