Group #23764745 — subgroups

Subgroups — #23764745

Order 10112.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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	#19099194	C₂	1	1	yes	maximal, characteristic
2	#19098603	C₂	1	2	yes	maximal
3	#19098332	C₂	1	1	yes	maximal, characteristic
4	#19098323	C₂	1	2	yes	maximal
5	#384×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11929608	C₂²	1	2	yes	—
7	#14579160	—	2	2	no	—
8	#14579152	—	2	2	no	—
9	#14579140	—	2	2	no	—
10	#14579120	C₂²	1	2	yes	—
11	#14579094	—	2	2	no	—
12	#14579092	—	2	2	no	—
13	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
14	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₅₈	—	2	2	no	—
18	#11929608	—	2	2	no	—
19	#11929608	Q₈	1	4	yes	complemented
20	C₂×C₇₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₇₉	—	2	2	no	—
22	C₄⋊Dic₇₉	D₄	1	2	yes	—
23	C₄⋊Dic₇₉	D₄	1	2	yes	—
24	Dic₇₉⋊C₄	—	2	4	no	—
25	C₂²⋊Dic₇₉	—	2	2	no	—
26	C₄×Dic₇₉	—	2	2	no	—
27	C₂²×C₄×C₇₉	—	2	1	no	—
28	C₂²×C₄×C₇₉	—	2	1	no	—
29	C₂²×C₄×C₇₉	D₄	1	2	yes	—
30	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₇₉	—	2	2	no	—
32	C₄⋊C₄×C₇₉	—	2	2	no	—
33	C₄⋊C₄×C₇₉	—	4	4	no	—
34	C₄²×C₇₉	—	2	1	no	—
35	C₄²×C₇₉	Q₈	1	2	yes	complemented
36	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₉	—	2	2	no	—
38	C₂×C₄×C₇₉	—	2	2	no	—
39	C₂×C₄×C₇₉	—	2	2	no	—
40	C₂×C₄×C₇₉	—	2	2	no	—
41	C₂×C₄×C₇₉	—	4	2	no	—
42	C₂×C₄×C₇₉	—	4	2	no	—
43	C₂×C₄×C₇₉	—	4	1	no	—
44	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
48	C₂×Dic₇₉	—	2	2	no	—
49	C₂×Dic₇₉	—	2	2	no	—
50	Dic₁₅₈	—	8	2	no	—
51	C₇₉⋊C₈	—	2	4	no	—
52	C₂²×C₇₉	—	2	1	no	—
53	C₂²×C₇₉	—	2	1	no	—
54	C₂²×C₇₉	#120	1	1	yes	characteristic
55	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₇₉	—	2	4	no	—
58	C₄×C₇₉	—	4	2	no	—
59	C₄×C₇₉	—	8	1	no	—
60	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
62	Dic₇₉	—	4	2	no	—
63	Dic₇₉	—	4	2	no	—
64	C₂×C₇₉	—	2	1	no	—
65	C₂×C₇₉	—	2	1	no	—
66	C₂×C₇₉	#395	1	1	yes	characteristic
67	C₂×C₇₉	#453	1	1	yes	characteristic
68	C₂×C₇₉	#456	1	1	yes	characteristic
69	#2060	—	79	1	no	maximal, Sylow
70	C₇₉	#2060	1	1	yes	characteristic, complemented, Sylow
71	#544	—	79	1	no	—
72	#428	—	79	2	no	—
73	#384	D₇₉	1	1	yes	characteristic
74	#343	—	79	1	no	—
75	#339	—	79	2	no	—
76	C₂×M₁₆	—	79	2	no	—
77	C₄⋊Q₈	—	158	2	no	—
78	#125	—	158	2	no	—
79	C₂²⋊Q₈	—	158	2	no	—
80	#117	—	79	2	no	—
81	C₂×C₄⋊C₄	—	2	2	no	—
82	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
85	#107	—	158	2	no	—
86	#106	—	158	2	no	—
87	C₂×Q₈	—	158	2	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
91	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
92	M₁₆	—	79	4	no	—
93	M₁₆	—	158	2	no	—
94	C₂×C₈	—	158	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	79	2	no	—
99	C₄⋊C₄	—	79	2	no	—
100	C₄⋊C₄	—	158	4	no	—
101	C₄⋊C₄	—	158	2	no	—
102	C₂²⋊C₄	—	158	2	no	—
103	C₄²	—	2	1	no	—
104	C₄²	—	158	2	no	—
105	C₄²	Dic₁₅₈	1	2	yes	complemented
106	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
107	Q₈	—	632	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	158	2	no	—
116	C₂×C₄	—	158	2	no	—
117	C₂×C₄	D₄×D₇₉	1	2	yes	—
118	C₂×C₄	Q₈×D₇₉	1	2	yes	—
119	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
120	C₂×C₄	#11929624	1	1	yes	characteristic
121	C₈	—	158	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#14579129	1	1	yes	characteristic
125	C₂²	#14579140	1	1	yes	characteristic
126	C₂²	#14579159	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	4	2	no	—
129	C₄	—	8	1	no	—
130	C₄	—	316	2	no	—
131	C₄	—	316	2	no	—
132	C₄	#14579139	1	2	yes	—
133	C₄	#14579160	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#19098483	1	1	yes	characteristic
137	C₂	#19098730	1	1	yes	characteristic
138	C₂	#19098745	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099194

C₂

yes

maximal, characteristic

#19098603

C₂

yes

maximal

#19098332

C₂

yes

maximal, characteristic

#19098323

C₂

yes

maximal

#384×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

C₂²

yes

—

—

—

—

C₂²

yes

—

#14579094

—

#14579092

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

#11929608

—

#11929608

Q₈

yes

complemented

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

C₄⋊Dic₇₉

D₄

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

Q₈

yes

complemented

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

C₄⋊Q₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

C₄×C₇₉

C₄⋊Q₈

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#544

—

#428

—

#384

D₇₉

yes

characteristic

#343

—

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

158

—

#125

—

158

—

C₂²⋊Q₈

—

158

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

C₂×C₄²

C₂×D₇₉

yes

characteristic

#107

—

158

—

#106

—

158

—

C₂×Q₈

—

158

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

M₁₆

—

M₁₆

—

158

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

158

—

101

C₄⋊C₄

—

158

—

102

C₂²⋊C₄

—

158

—

103

C₄²

—

104

C₄²

—

158

—

105

C₄²

Dic₁₅₈

yes

complemented

106

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

107

Q₈

—

632

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

158

—

116

C₂×C₄

—

158

—

117

C₂×C₄

D₄×D₇₉

yes

—

118

C₂×C₄

Q₈×D₇₉

yes

—

119

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

120

C₂×C₄

#11929624

yes

characteristic

121

C₈

—

158

—

122

C₂²

—

123

C₂²

—

124

C₂²

#14579129

yes

characteristic

125

C₂²

#14579140

yes

characteristic

126

C₂²

#14579159

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

316

—

131

C₄

—

316

—

132

C₄

#14579139

yes

—

133

C₄

#14579160

yes

—

134

C₂

—

135

C₂

—

136

C₂

#19098483

yes

characteristic

137

C₂

#19098730

yes

characteristic

138

C₂

#19098745

yes

characteristic

	subgroup	G/H
center	C₂²	#14579129
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#384×C₇₉	C₂
socle	C₂²×C₇₉	#120
prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	#384	D₇₉
hypocenter	C₇₉	#2060
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #384

subgroup

G/H

center

C₂²

#14579129

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#384×C₇₉

C₂

socle

C₂²×C₇₉

#120

prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #384

N	H ≅ G/N	type
19. #11929608	Q₈	semidirect
35. C₄²×C₇₉	Q₈	semidirect
70. C₇₉	#2060	semidirect
105. C₄²	Dic₁₅₈	semidirect

H ≅ G/N

type

19. #11929608

Q₈

semidirect

35. C₄²×C₇₉

Q₈

semidirect

70. C₇₉

#2060

semidirect

105. C₄²

Dic₁₅₈

semidirect

Subgroups — #23764745

Conjugacy classes of subgroups

Complements