Group #23765333 — subgroups

Subgroups — #23765333

Order 10112.

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

normal subgroups only

This group has 216 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	C₂²×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579107	C₂	1	2	yes	maximal, complemented
3	#19098366	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	#467×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	—	2	1	no	—
7	C₂×Dic₃₁₆	—	2	1	no	—
8	C₂×Dic₃₁₆	—	2	4	no	—
9	C₂×Dic₃₁₆	C₂²	1	2	yes	—
10	#14579107	—	2	4	no	—
11	C₂×C₇₉⋊C₁₆	C₂²	1	2	yes	—
12	C₂×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈×C₇₉	—	2	1	no	—
15	#107×C₇₉	C₂²	1	1	yes	characteristic
16	#103×C₇₉	—	2	1	no	—
17	C₂×Dic₁₅₈	—	2	4	no	—
18	C₂×Dic₁₅₈	—	2	2	no	—
19	C₂×Dic₁₅₈	—	4	2	no	—
20	C₂×Dic₁₅₈	—	4	1	no	—
21	C₂²×Dic₇₉	—	4	1	no	—
22	Dic₃₁₆	—	2	4	no	—
23	Dic₃₁₆	—	2	2	no	—
24	Dic₃₁₆	D₄	1	4	yes	—
25	C₇₉⋊C₁₆	—	2	2	no	—
26	C₂×Q₈×C₇₉	—	2	1	no	—
27	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
28	Q₁₆×C₇₉	—	2	2	no	—
29	C₂×C₈×C₇₉	—	2	2	no	—
30	C₂×C₈×C₇₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₇₉	—	2	1	no	—
33	C₄⋊C₄×C₇₉	—	4	1	no	—
34	C₂²⋊C₄×C₇₉	—	4	1	no	—
35	C₂³×C₇₉	D₈	1	1	yes	characteristic
36	Q₈×C₇₉	—	4	1	no	—
37	Q₈×C₇₉	—	4	1	no	—
38	C₂×C₄×C₇₉	—	2	1	no	—
39	C₂×C₄×C₇₉	—	2	1	no	—
40	C₂×C₄×C₇₉	—	4	1	no	—
41	C₂×C₄×C₇₉	—	4	1	no	—
42	C₂×C₄×C₇₉	D₈	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₇₉	—	4	4	no	—
45	C₂×Dic₇₉	—	4	2	no	—
46	Dic₁₅₈	—	2	4	no	—
47	Dic₁₅₈	—	4	4	no	—
48	Dic₁₅₈	—	4	2	no	—
49	C₈×C₇₉	—	2	1	no	—
50	C₈×C₇₉	C₂×D₄	1	2	yes	—
51	C₂²×C₇₉	—	2	1	no	—
52	C₂²×C₇₉	—	2	1	no	—
53	C₂²×C₇₉	Q₃₂	1	2	yes	complemented
54	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
55	C₄×C₇₉	—	2	1	no	—
56	C₄×C₇₉	—	8	1	no	—
57	C₄×C₇₉	—	8	1	no	—
58	C₄×C₇₉	#120	1	1	yes	characteristic
59	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	Dic₇₉	—	4	4	no	—
61	C₂×C₇₉	—	2	2	no	—
62	C₂×C₇₉	#447	1	1	yes	characteristic
63	C₂×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
64	C₂×C₇₉	#510	1	1	yes	characteristic
65	#2188	—	79	1	no	maximal, Sylow
66	C₇₉	#2188	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₁₆	—	79	1	no	—
68	C₂×Q₃₂	—	79	2	no	—
69	#467	D₇₉	1	1	yes	characteristic
70	#358	—	79	2	no	—
71	C₂²⋊C₁₆	—	79	1	no	—
72	C₂²×Q₈	—	158	1	no	—
73	C₂×Q₁₆	—	79	2	no	—
74	C₂×Q₁₆	—	158	4	no	—
75	C₂×Q₁₆	—	158	1	no	—
76	C₂×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
78	C₂²⋊Q₈	—	2	1	no	—
79	Q₃₂	—	158	4	no	—
80	C₂×C₁₆	—	79	2	no	—
81	#107	C₂×D₇₉	1	1	yes	characteristic
82	#103	—	2	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	316	2	no	—
85	C₂×Q₈	—	316	1	no	—
86	C₂×Q₈	—	158	4	no	—
87	C₂×Q₈	—	158	2	no	—
88	C₂²×C₄	—	316	1	no	—
89	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
90	Q₁₆	—	2	2	no	—
91	Q₁₆	—	79	4	no	—
92	Q₁₆	—	158	4	no	—
93	Q₁₆	—	158	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₂²⋊C₄	—	4	1	no	—
100	C₁₆	—	158	2	no	—
101	C₂³	#11929609	1	1	yes	characteristic
102	Q₈	—	4	1	no	—
103	Q₈	—	4	1	no	—
104	Q₈	—	316	4	no	—
105	Q₈	—	316	2	no	—
106	Q₈	—	158	4	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	316	4	no	—
112	C₂×C₄	—	316	2	no	—
113	C₂×C₄	#11929609	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
115	C₈	—	2	1	no	—
116	C₈	D₄×D₇₉	1	2	yes	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#14579107	1	2	yes	complemented
120	C₂²	C₂×#11929609	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	8	1	no	—
123	C₄	—	8	1	no	—
124	C₄	—	316	4	no	—
125	C₄	#14579129	1	1	yes	characteristic
126	C₄	#14579186	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	#19098696	1	1	yes	characteristic
129	C₂	C₂×#14579107	1	1	yes	characteristic
130	C₂	#19099003	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₁₆

C₂

yes

maximal, characteristic

C₂×#14579107

C₂

yes

maximal, complemented

#19098366

C₂

yes

maximal, complemented

C₂²⋊(C₇₉⋊C₁₆)

C₂

yes

maximal, characteristic

#467×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

—

C₂×Dic₃₁₆

C₂²

yes

—

#14579107

—

C₂×C₇₉⋊C₁₆

C₂²

yes

—

C₂×Q₁₆×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₉

—

#107×C₇₉

C₂²

yes

characteristic

#103×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

Dic₃₁₆

—

Dic₃₁₆

D₄

yes

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂³×C₇₉

D₈

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×D₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

Q₃₂

yes

complemented

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#447

yes

characteristic

C₂×C₇₉

C₂×Q₃₂

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×Q₃₂

—

#467

D₇₉

yes

characteristic

#358

—

C₂²⋊C₁₆

—

C₂²×Q₈

—

158

—

C₂×Q₁₆

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

—

158

—

C₂×Q₁₆

C₂×D₇₉

yes

characteristic

C₂²×C₈

C₂×D₇₉

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

158

—

C₂×C₁₆

—

#107

C₂×D₇₉

yes

characteristic

#103

—

C₂×Q₈

—

C₂×Q₈

—

316

—

C₂×Q₈

—

316

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂²×C₄

—

316

—

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

158

—

Q₁₆

—

158

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₁₆

—

158

—

101

C₂³

#11929609

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

316

—

105

Q₈

—

316

—

106

Q₈

—

158

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

316

—

112

C₂×C₄

—

316

—

113

C₂×C₄

#11929609

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

115

C₈

—

116

C₈

D₄×D₇₉

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

#14579107

yes

complemented

120

C₂²

C₂×#11929609

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

316

—

125

C₄

#14579129

yes

characteristic

126

C₄

#14579186

yes

characteristic

127

C₂

—

128

C₂

#19098696

yes

characteristic

129

C₂

C₂×#14579107

yes

characteristic

130

C₂

#19099003

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₇₉
Fitting	#467×C₇₉	C₂
socle	C₂²×C₇₉	C₂×D₈
prime radical	C₂³×C₇₉	D₈
hypercenter	#467	D₇₉
hypocenter	C₇₉	#2188
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

subgroup

G/H

center

C₂²

C₂×#11929609

commutator, cosocle

C₂×C₈×C₇₉

C₂³

Frattini

C₂×C₈

C₂²×D₇₉

Fitting

#467×C₇₉

C₂

socle

C₂²×C₇₉

C₂×D₈

prime radical

C₂³×C₇₉

D₈

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

N	H ≅ G/N	type
2. C₂×#14579107	C₂	semidirect
3. #19098366	C₂	semidirect
53. C₂²×C₇₉	Q₃₂	semidirect
66. C₇₉	#2188	semidirect
119. C₂²	#14579107	semidirect

H ≅ G/N

type

2. C₂×#14579107

C₂

semidirect

3. #19098366

C₂

semidirect

53. C₂²×C₇₉

Q₃₂

semidirect

66. C₇₉

#2188

semidirect

119. C₂²

#14579107

semidirect

Subgroups — #23765333

Conjugacy classes of subgroups

Complements