Group #23764548 — subgroups

Subgroups — #23764548

Order 10112.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14579084	C₂	1	2	yes	maximal
2	#19098469	C₂	1	1	yes	maximal, characteristic
3	#19098325	C₂	1	2	yes	maximal
4	#19098318	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
7	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
9	#14579084	—	2	4	no	—
10	C₇₉⋊#95	—	2	2	no	—
11	#125×C₇₉	—	2	4	no	—
12	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₅₈	—	2	2	no	—
16	C₂×Dic₁₅₈	—	2	2	no	—
17	C₂×Dic₁₅₈	—	2	1	no	—
18	C₂×Dic₁₅₈	Q₈	1	2	yes	—
19	C₂²×Dic₇₉	—	2	1	no	—
20	C₂²×Dic₇₉	—	2	1	no	—
21	C₂×C₇₉⋊C₈	—	2	2	no	—
22	C₂×C₇₉⋊C₈	—	2	2	no	—
23	C₂×C₇₉⋊C₈	—	2	2	no	—
24	C₄⋊Dic₇₉	—	2	2	no	—
25	C₂²×C₄×C₇₉	—	2	2	no	—
26	C₂²×C₄×C₇₉	D₄	1	2	yes	—
27	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₇₉	—	2	8	no	—
29	C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₄⋊C₄×C₇₉	D₄	1	4	yes	—
31	C₄²×C₇₉	—	2	2	no	—
32	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₇₉	—	2	2	no	—
34	C₂×C₄×C₇₉	—	2	4	no	—
35	C₂×C₄×C₇₉	—	2	2	no	—
36	C₂×C₄×C₇₉	—	2	2	no	—
37	C₂×C₄×C₇₉	—	4	2	no	—
38	C₂×C₄×C₇₉	—	4	2	no	—
39	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₉	#41	1	2	yes	—
42	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
43	C₂×Dic₇₉	—	2	2	no	—
44	C₂×Dic₇₉	—	4	1	no	—
45	C₂×Dic₇₉	—	4	1	no	—
46	C₂×Dic₇₉	—	4	1	no	—
47	C₂×Dic₇₉	—	4	2	no	—
48	Dic₁₅₈	—	2	4	no	—
49	Dic₁₅₈	—	4	2	no	—
50	C₇₉⋊C₈	—	4	2	no	—
51	C₂²×C₇₉	#120	1	1	yes	characteristic
52	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₇₉	#124	1	1	yes	characteristic
54	C₂²×C₇₉	#135	1	2	yes	—
55	C₂²×C₇₉	#137	1	2	yes	—
56	C₄×C₇₉	—	4	2	no	—
57	C₄×C₇₉	—	4	4	no	—
58	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₇₉	#124	1	2	yes	—
60	Dic₇₉	—	4	2	no	—
61	Dic₇₉	—	8	1	no	—
62	C₂×C₇₉	#393	1	1	yes	characteristic
63	C₂×C₇₉	#452	1	2	yes	—
64	C₂×C₇₉	#479	1	2	yes	—
65	C₂×C₇₉	#488	1	1	yes	characteristic
66	C₂×C₇₉	#491	1	1	yes	characteristic
67	#2026	—	79	1	no	maximal, Sylow
68	C₇₉	#2026	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	D₇₉	1	1	yes	characteristic
70	C₂×#103	—	79	2	no	—
71	#391	—	79	1	no	—
72	#340	—	79	2	no	—
73	#336	—	79	1	no	—
74	C₂²×Q₈	—	79	1	no	—
75	C₂²×C₈	—	79	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	79	1	no	—
79	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
80	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
81	#103	—	158	4	no	—
82	#95	—	158	2	no	—
83	C₂×Q₈	—	79	2	no	—
84	C₂×Q₈	—	158	2	no	—
85	C₂×Q₈	—	158	2	no	—
86	C₂×Q₈	—	158	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	158	1	no	—
89	C₂²×C₄	—	158	1	no	—
90	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
91	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
92	C₂×C₈	—	158	2	no	—
93	C₂×C₈	—	158	2	no	—
94	C₂×C₈	—	158	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	158	2	no	—
98	C₄⋊C₄	C₂²⋊D₇₉	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
101	Q₈	—	316	2	no	—
102	Q₈	—	158	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	316	1	no	—
110	C₂×C₄	—	316	1	no	—
111	C₂×C₄	—	316	1	no	—
112	C₂×C₄	—	316	2	no	—
113	C₂×C₄	—	158	2	no	—
114	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
115	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
116	C₂×C₄	#11929625	1	2	yes	—
117	C₂×C₄	#11929625	1	1	yes	characteristic
118	C₈	—	316	2	no	—
119	C₂²	#14579130	1	1	yes	characteristic
120	C₂²	#14579137	1	1	yes	characteristic
121	C₂²	#14579150	1	1	yes	characteristic
122	C₂²	#14579181	1	2	yes	—
123	C₂²	#14579191	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	316	2	no	—
127	C₄	—	632	1	no	—
128	C₄	#14579137	1	2	yes	—
129	C₄	#14579150	1	2	yes	—
130	C₂	#19098474	1	1	yes	characteristic
131	C₂	#19098725	1	2	yes	—
132	C₂	#19098867	1	2	yes	—
133	C₂	#19098920	1	1	yes	characteristic
134	C₂	#19098933	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579084

C₂

yes

maximal

#19098469

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal

#19098318

C₂

yes

maximal, characteristic

C₂×#125×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

#14579084

—

C₇₉⋊#95

—

#125×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

Q₈

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

—

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×D₄

yes

—

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#120

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#137

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

C₄×C₇₉

#124

yes

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#393

yes

characteristic

C₂×C₇₉

#452

yes

—

C₂×C₇₉

#479

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×#125

D₇₉

yes

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#103

—

158

—

#95

—

158

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₂×C₈

—

158

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄⋊C₄

C₂²⋊D₇₉

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

101

Q₈

—

316

—

102

Q₈

—

158

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

316

—

110

C₂×C₄

—

316

—

111

C₂×C₄

—

316

—

112

C₂×C₄

—

316

—

113

C₂×C₄

—

158

—

114

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

115

C₂×C₄

Q₈×D₇₉

yes

characteristic

116

C₂×C₄

#11929625

yes

—

117

C₂×C₄

#11929625

yes

characteristic

118

C₈

—

316

—

119

C₂²

#14579130

yes

characteristic

120

C₂²

#14579137

yes

characteristic

121

C₂²

#14579150

yes

characteristic

122

C₂²

#14579181

yes

—

123

C₂²

#14579191

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

316

—

127

C₄

—

632

—

128

C₄

#14579137

yes

—

129

C₄

#14579150

yes

—

130

C₂

#19098474

yes

characteristic

131

C₂

#19098725

yes

—

132

C₂

#19098867

yes

—

133

C₂

#19098920

yes

characteristic

134

C₂

#19098933

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×#125×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	C₂×#125	D₇₉
hypocenter	C₇₉	#2026
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×#125×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
68. C₇₉	#2026	semidirect

H ≅ G/N

type

68. C₇₉

#2026

semidirect

Subgroups — #23764548

Conjugacy classes of subgroups

Complements