Group #23765261 — subgroups

Subgroups — #23765261

Order 10112.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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×#14579105	C₂	1	1	yes	maximal, characteristic
2	#19098657	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098388	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098368	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098338	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
9	#14579126	—	2	1	no	—
10	D₄×Dic₇₉	—	2	1	no	—
11	#14579105	C₄	1	4	yes	complemented
12	C₂×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14579095	C₂²	1	1	yes	characteristic
14	#14579084	—	2	1	no	—
15	#14579082	—	2	1	no	—
16	C₈×Dic₇₉	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	#107×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂²×Dic₇₉	—	4	1	no	—
22	Dic₃₁₆	—	2	1	no	—
23	Dic₃₁₆	C₂×C₄	1	2	yes	complemented
24	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Dic₇₉	—	2	1	no	—
26	Dic₇₉⋊C₄	—	4	1	no	—
27	C₂²⋊Dic₇₉	—	4	1	no	—
28	C₄×Dic₇₉	—	4	1	no	—
29	C₄×Dic₇₉	D₄	1	1	yes	characteristic
30	C₇₉⋊C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×D₄×C₇₉	—	2	1	no	—
32	D₈×C₇₉	—	2	1	no	—
33	D₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₁₆×C₇₉	—	2	1	no	—
37	C₂³×C₇₉	—	4	1	no	—
38	D₄×C₇₉	—	2	2	no	—
39	D₄×C₇₉	—	4	1	no	—
40	C₂×C₄×C₇₉	—	4	1	no	—
41	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₇₉	—	4	1	no	—
43	C₂×Dic₇₉	—	4	1	no	—
44	C₂×Dic₇₉	—	8	2	no	—
45	C₂×Dic₇₉	D₈	1	1	yes	characteristic
46	C₂×Dic₇₉	D₈	1	1	yes	characteristic
47	Dic₁₅₈	—	2	2	no	—
48	Dic₁₅₈	—	4	1	no	—
49	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₇₉	#41	1	1	yes	characteristic
51	C₇₉⋊C₈	—	2	1	no	—
52	C₂²×C₇₉	—	4	2	no	—
53	C₂²×C₇₉	—	8	1	no	—
54	C₂²×C₇₉	—	8	1	no	—
55	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
56	C₄×C₇₉	—	8	1	no	—
57	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
58	C₄×C₇₉	#135	1	1	yes	characteristic
59	Dic₇₉	—	2	1	no	—
60	Dic₇₉	—	2	1	no	—
61	Dic₇₉	—	4	2	no	—
62	Dic₇₉	—	8	1	no	—
63	C₂×C₇₉	—	8	2	no	—
64	C₂×C₇₉	C₄×D₈	1	1	yes	characteristic
65	C₂×C₇₉	#509	1	1	yes	characteristic
66	C₂×C₇₉	#510	1	1	yes	characteristic
67	#2175	—	79	1	no	maximal, Sylow
68	C₇₉	#2175	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₃₂	—	79	1	no	—
70	C₄×Q₁₆	—	79	1	no	—
71	C₄×D₈	—	79	1	no	—
72	#366	—	79	1	no	—
73	#358	—	79	1	no	—
74	#357	D₇₉	1	1	yes	characteristic
75	#346	—	79	1	no	—
76	C₂×Q₁₆	—	79	1	no	—
77	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
78	C₄×Q₈	—	158	1	no	—
79	C₄×D₄	—	158	1	no	—
80	SD₃₂	—	79	4	no	—
81	C₂×C₁₆	—	79	1	no	—
82	C₂×C₁₆	C₂×D₇₉	1	1	yes	characteristic
83	#107	—	79	1	no	—
84	#107	C₂×D₇₉	1	1	yes	characteristic
85	#103	—	158	1	no	—
86	#102	—	158	1	no	—
87	C₄×C₈	—	79	1	no	—
88	C₂×Q₈	—	158	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂²×C₄	—	316	1	no	—
91	Q₁₆	—	79	2	no	—
92	Q₁₆	—	158	1	no	—
93	D₈	—	2	1	no	—
94	D₈	C₄×D₇₉	1	2	yes	—
95	C₂×C₈	—	79	1	no	—
96	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	316	1	no	—
99	C₄⋊C₄	—	158	1	no	—
100	C₂²⋊C₄	—	316	1	no	—
101	C₄²	—	316	1	no	—
102	C₄²	—	79	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	79	2	no	—
105	C₂³	—	4	1	no	—
106	Q₈	—	316	1	no	—
107	Q₈	—	158	2	no	—
108	D₄	—	2	2	no	—
109	D₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	316	1	no	—
112	C₂×C₄	—	316	1	no	—
113	C₂×C₄	—	79	1	no	—
114	C₂×C₄	—	79	1	no	—
115	C₂×C₄	—	632	2	no	—
116	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
117	C₈	—	158	1	no	—
118	C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
119	C₈	#11929623	1	1	yes	characteristic
120	C₂²	—	4	2	no	—
121	C₂²	—	8	1	no	—
122	C₂²	—	8	1	no	—
123	C₂²	D₈×D₇₉	1	1	yes	characteristic
124	C₄	—	8	1	no	—
125	C₄	—	316	2	no	—
126	C₄	—	632	1	no	—
127	C₄	—	158	1	no	—
128	C₄	—	158	1	no	—
129	C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
130	C₄	#14579182	1	1	yes	characteristic
131	C₂	—	8	2	no	—
132	C₂	#19098643	1	1	yes	characteristic
133	C₂	#19098997	1	1	yes	characteristic
134	C₂	#19099002	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579105

C₂

yes

maximal, characteristic

#19098657

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₉

C₂

yes

maximal, characteristic

#19098388

C₂

yes

maximal, characteristic, complemented

#19098368

C₂

yes

maximal, characteristic, complemented

#19098338

C₂

yes

maximal, characteristic, complemented

#357×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

characteristic

#14579126

—

D₄×Dic₇₉

—

#14579105

C₄

yes

complemented

C₂×C₇₉⋊C₁₆

C₂²

yes

characteristic

#14579095

C₂²

yes

characteristic

#14579084

—

#14579082

—

C₈×Dic₇₉

C₂²

yes

characteristic

C₂×D₈×C₇₉

C₂²

yes

characteristic

C₂×C₁₆×C₇₉

C₂²

yes

characteristic

#107×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

Dic₃₁₆

C₂×C₄

yes

complemented

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₄

yes

characteristic

C₇₉⋊C₁₆

C₂×C₄

yes

complemented

C₂×D₄×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

C₂×C₄

yes

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

D₈

yes

characteristic

C₂×Dic₇₉

D₈

yes

characteristic

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

characteristic

C₄×C₇₉

#135

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₄×D₈

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₄×Q₁₆

—

C₄×D₈

—

#366

—

#358

—

#357

D₇₉

yes

characteristic

#346

—

C₂×Q₁₆

—

C₂×D₈

C₂×D₇₉

yes

characteristic

C₄×Q₈

—

158

—

C₄×D₄

—

158

—

SD₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₉

yes

characteristic

#107

—

#107

C₂×D₇₉

yes

characteristic

#103

—

158

—

#102

—

158

—

C₄×C₈

—

C₂×Q₈

—

158

—

C₂×D₄

—

C₂²×C₄

—

316

—

Q₁₆

—

Q₁₆

—

158

—

D₈

—

D₈

C₄×D₇₉

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

316

—

C₄⋊C₄

—

158

—

100

C₂²⋊C₄

—

316

—

101

C₄²

—

316

—

102

C₄²

—

103

C₁₆

—

104

C₁₆

—

105

C₂³

—

106

Q₈

—

316

—

107

Q₈

—

158

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

316

—

112

C₂×C₄

—

316

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

632

—

116

C₂×C₄

D₄×D₇₉

yes

characteristic

117

C₈

—

158

—

118

C₈

C₂×C₄×D₇₉

yes

characteristic

119

C₈

#11929623

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

D₈×D₇₉

yes

characteristic

124

C₄

—

125

C₄

—

316

—

126

C₄

—

632

—

127

C₄

—

158

—

128

C₄

—

158

—

129

C₄

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

130

C₄

#14579182

yes

characteristic

131

C₂

—

132

C₂

#19098643

yes

characteristic

133

C₂

#19098997

yes

characteristic

134

C₂

#19099002

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₉
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#357×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
prime radical	C₂×D₈×C₇₉	C₂²
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	#357	D₇₉
hypocenter	C₇₉	#2175
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₉

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#357×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

prime radical

C₂×D₈×C₇₉

C₂²

cosocle

C₂×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
2. #19098657	C₂	semidirect
4. #19098388	C₂	semidirect
5. #19098368	C₂	semidirect
6. #19098338	C₂	semidirect
11. #14579105	C₄	semidirect
23. Dic₃₁₆	C₂×C₄	semidirect
30. C₇₉⋊C₁₆	C₂×C₄	semidirect
68. C₇₉	#2175	semidirect

H ≅ G/N

type

2. #19098657

C₂

semidirect

4. #19098388

C₂

semidirect

5. #19098368

C₂

semidirect

6. #19098338

C₂

semidirect

11. #14579105

C₄

semidirect

23. Dic₃₁₆

C₂×C₄

semidirect

30. C₇₉⋊C₁₆

C₂×C₄

semidirect

68. C₇₉

#2175

semidirect

Subgroups — #23765261

Conjugacy classes of subgroups

Complements