Group #22625484 — subgroups

Subgroups — #22625484

Order 9344.

180 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 84 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	#17989603	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
3	#17989451	C₂	1	2	yes	maximal, complemented
4	#17989449	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596833	—	2	1	no	—
7	C₄⋊#10596833	—	2	1	no	—
8	C₄⋊#10596833	C₂²	1	1	yes	characteristic
9	C₄⋊#10596833	C₂²	1	2	yes	—
10	C₄⋊#10596833	C₂²	1	1	yes	characteristic
11	C₂²⋊#10596833	—	2	1	no	—
12	C₄×#10596833	—	2	1	no	—
13	C₂×#11924012	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
15	#14556501	C₄	1	2	yes	—
16	C₈×Dic₇₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#10596833	—	2	2	no	—
19	C₂×#10596833	—	2	1	no	—
20	C₂×#10596833	—	2	1	no	—
21	C₂×#10596833	—	4	2	no	—
22	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
23	C₂²×Dic₇₃	Q₈	1	1	yes	characteristic
24	#11924012	C₈	1	4	yes	complemented
25	C₂×#10596831	C₂×C₄	1	2	yes	—
26	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₇₃	D₄	1	1	yes	characteristic
28	C₄×Dic₇₃	Q₈	1	1	yes	characteristic
29	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
31	M₁₆×C₇₃	—	2	2	no	—
32	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
33	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₃	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₇₃	M₁₆	1	2	yes	—
37	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₇₃	—	2	1	no	—
39	C₂×Dic₇₃	—	2	1	no	—
40	C₂×Dic₇₃	—	4	1	no	—
41	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
45	#10596833	—	4	1	no	—
46	#10596833	—	4	2	no	—
47	#10596833	—	4	1	no	—
48	C₈×C₇₃	—	2	2	no	—
49	#10596831	C₂×C₈	1	4	yes	complemented
50	C₂²×C₇₃	—	2	1	no	—
51	C₂²×C₇₃	—	2	1	no	—
52	C₂²×C₇₃	C₄⋊C₈	1	2	yes	—
53	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₇₃	C₄⋊C₈	1	2	yes	—
55	C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
56	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
57	Dic₇₃	—	2	1	no	—
58	Dic₇₃	—	2	1	no	—
59	Dic₇₃	—	2	1	no	—
60	Dic₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	Dic₇₃	#137	1	1	yes	characteristic
62	C₂×C₇₃	—	2	2	no	—
63	C₂×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₇₃	#428	1	1	yes	characteristic
65	C₂×C₇₃	#430	1	1	yes	characteristic
66	#1565	—	73	1	no	maximal, Sylow
67	C₇₃	#1565	1	1	yes	characteristic, complemented, Sylow
68	#431	—	73	1	no	—
69	C₂×C₄⋊C₈	—	73	1	no	—
70	C₄×M₁₆	—	73	1	no	—
71	#335	—	73	2	no	—
72	#334	—	73	2	no	—
73	C₂×M₁₆	—	73	1	no	—
74	C₂×M₁₆	#64408	1	1	yes	characteristic
75	C₂²×C₈	—	146	1	no	—
76	C₂×C₄²	—	73	1	no	—
77	C₄⋊C₈	—	73	1	no	—
78	C₄⋊C₈	—	73	2	no	—
79	C₄⋊C₈	—	73	1	no	—
80	C₄⋊C₈	—	146	1	no	—
81	C₂²⋊C₈	—	146	1	no	—
82	#97	—	73	2	no	—
83	C₄×C₈	—	73	2	no	—
84	C₄×C₈	—	146	1	no	—
85	C₂²×C₄	—	73	1	no	—
86	C₂²×C₄	—	73	1	no	—
87	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	73	4	no	—
90	C₂×C₈	—	292	2	no	—
91	C₂×C₈	—	73	2	no	—
92	C₂×C₈	—	146	1	no	—
93	C₂×C₈	—	146	1	no	—
94	C₂×C₈	—	146	2	no	—
95	C₂×C₈	C₂×#64408	1	2	yes	—
96	C₄²	—	73	1	no	—
97	C₄²	—	73	1	no	—
98	C₄²	—	73	1	no	—
99	C₄²	—	73	1	no	—
100	C₂³	C₄⋊#64408	1	1	yes	characteristic
101	C₂×C₄	—	292	1	no	—
102	C₂×C₄	—	73	1	no	—
103	C₂×C₄	—	73	1	no	—
104	C₂×C₄	—	73	1	no	—
105	C₂×C₄	—	73	1	no	—
106	C₂×C₄	—	146	1	no	—
107	C₂×C₄	—	146	1	no	—
108	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
109	C₂×C₄	#11924036	1	2	yes	—
110	C₂×C₄	#11924037	1	2	yes	—
111	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	292	2	no	—
114	C₈	—	292	1	no	—
115	C₈	—	292	1	no	—
116	C₈	—	73	4	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#14556658	1	2	yes	—
120	C₂²	C₂×C₄⋊#64408	1	1	yes	characteristic
121	C₄	—	73	1	no	—
122	C₄	—	73	1	no	—
123	C₄	—	146	1	no	—
124	C₄	—	146	1	no	—
125	C₄	—	146	1	no	—
126	C₄	#14556658	1	2	yes	—
127	C₄	C₂×#11924036	1	1	yes	characteristic
128	C₄	C₂×#11924037	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×#14556658	1	1	yes	characteristic
131	C₂	#17989598	1	1	yes	characteristic
132	C₂	#17989602	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989603

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596833

C₂

yes

maximal, characteristic

#17989451

C₂

yes

maximal, complemented

#17989449

C₂

yes

maximal, complemented

M₁₆×Dic₇₃

C₂

yes

maximal, characteristic

C₂²×#10596833

—

C₄⋊#10596833

—

C₄⋊#10596833

C₂²

yes

characteristic

C₄⋊#10596833

C₂²

yes

—

C₄⋊#10596833

C₂²

yes

—

—

C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

#14556501

C₄

yes

—

C₈×Dic₇₃

C₂²

yes

—

C₂×M₁₆×C₇₃

C₄

yes

—

—

—

—

C₂²×Dic₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

Q₈

yes

characteristic

#11924012

C₈

yes

complemented

C₂×#10596831

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

Q₈

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×Q₈

yes

—

—

—

C₈×C₇₃

—

#10596831

C₂×C₈

yes

complemented

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄⋊C₈

yes

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₄⋊C₈

yes

—

C₄×C₇₃

C₂²×C₈

yes

characteristic

C₄×C₇₃

C₂×M₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄⋊C₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

#335

—

#334

—

C₂×M₁₆

—

C₂×M₁₆

#64408

yes

characteristic

C₂²×C₈

—

146

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

#97

—

C₄×C₈

—

C₄×C₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₂×#64408

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

100

C₂³

C₄⋊#64408

yes

characteristic

101

C₂×C₄

—

292

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

146

—

107

C₂×C₄

—

146

—

108

C₂×C₄

C₄⋊#64408

yes

characteristic

109

C₂×C₄

#11924036

yes

—

110

C₂×C₄

#11924037

yes

—

111

C₂×C₄

C₂²×#64408

yes

characteristic

112

C₈

—

113

C₈

—

292

—

114

C₈

—

292

—

115

C₈

—

292

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

#14556658

yes

—

120

C₂²

C₂×C₄⋊#64408

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

146

—

124

C₄

—

146

—

125

C₄

—

146

—

126

C₄

#14556658

yes

—

127

C₄

C₂×#11924036

yes

characteristic

128

C₄

C₂×#11924037

yes

characteristic

129

C₂

—

130

C₂

C₂×#14556658

yes

characteristic

131

C₂

#17989598

yes

characteristic

132

C₂

#17989602

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#64408
commutator	C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂×M₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₄
hypercenter	C₂×M₁₆	#64408
hypocenter	C₇₃	#1565
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂²

C₂×C₄⋊#64408

commutator

C₄×C₇₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂×M₁₆×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄⋊C₄

C₂×M₁₆

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
3. #17989451	C₂	semidirect
4. #17989449	C₂	semidirect
24. #11924012	C₈	semidirect
49. #10596831	C₂×C₈	semidirect
67. C₇₃	#1565	semidirect

H ≅ G/N

type

3. #17989451

C₂

semidirect

4. #17989449

C₂

semidirect

24. #11924012

C₈

semidirect

49. #10596831

C₂×C₈

semidirect

67. C₇₃

#1565

semidirect

Subgroups — #22625484

Conjugacy classes of subgroups

Complements