Group #22626338 — subgroups

Subgroups — #22626338

Order 9344.

142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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	#17989633	C₂	1	1	yes	maximal, characteristic
2	#17989627	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989511	C₂	1	1	yes	maximal, characteristic, complemented
4	#17989506	C₂	1	1	yes	maximal, characteristic, complemented
5	#17989504	C₂	1	1	yes	maximal, characteristic
6	#17989477	C₂	1	1	yes	maximal, characteristic, complemented
7	#17989029	C₂	1	1	yes	maximal, characteristic
8	#14556679	—	2	1	no	—
9	Q₈×#64408	—	2	1	no	—
10	#14556666	C₂²	1	1	yes	characteristic
11	#14556661	C₂²	1	1	yes	characteristic
12	#14556659	C₂²	1	1	yes	characteristic
13	#14556656	—	2	1	no	—
14	#14556654	—	2	1	no	—
15	#14556645	C₂²	1	1	yes	characteristic
16	#14556614	C₂²	1	1	yes	characteristic
17	Q₁₆×D₇₃	C₂²	1	1	yes	characteristic
18	#14556537	C₄	1	1	yes	characteristic, complemented
19	#14556536	C₄	1	1	yes	characteristic, complemented
20	#14556534	C₄	1	1	yes	characteristic, complemented
21	#14556530	C₂²	1	1	yes	characteristic
22	Q₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
23	#11924038	—	2	1	no	—
24	#11924038	—	4	1	no	—
25	#11924036	—	4	1	no	—
26	#11924036	D₄	1	1	yes	characteristic
27	C₄⋊#64408	—	2	1	no	—
28	C₄⋊#64408	—	4	1	no	—
29	C₄×#64408	—	4	1	no	—
30	C₄×#64408	D₄	1	1	yes	characteristic
31	#11924031	—	2	1	no	—
32	#11924029	—	2	1	no	—
33	Q₈×D₇₃	—	2	1	no	—
34	Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
35	#11924018	—	2	1	no	—
36	#11924015	—	2	1	no	—
37	D₅₈₄	C₂×C₄	1	1	yes	characteristic
38	C₈×D₇₃	C₂³	1	1	yes	characteristic
39	#11924003	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
41	Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#64408	—	2	1	no	—
44	C₂×#64408	—	4	1	no	—
45	Q₈×C₇₃	—	2	1	no	—
46	Q₈×C₇₃	—	2	1	no	—
47	D₂₉₂	—	2	1	no	—
48	D₂₉₂	—	4	1	no	—
49	C₄×D₇₃	—	4	1	no	—
50	C₄×D₇₃	—	4	1	no	—
51	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
52	Dic₁₄₆	—	2	1	no	—
53	Dic₁₄₆	—	4	1	no	—
54	#10596833	—	2	1	no	—
55	#10596833	—	4	1	no	—
56	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	#10596831	#41	1	1	yes	characteristic
58	C₂×D₇₃	—	4	1	no	—
59	C₂×D₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#64408	—	4	1	no	—
61	#64408	—	8	1	no	—
62	C₄×C₇₃	—	4	1	no	—
63	C₄×C₇₃	—	4	1	no	—
64	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
65	Dic₇₃	—	4	1	no	—
66	Dic₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₇₃	#442	1	1	yes	characteristic
68	D₇₃	—	2	1	no	—
69	D₇₃	—	8	1	no	—
70	#2180	—	73	1	no	maximal, Sylow
71	C₇₃	#2180	1	1	yes	characteristic, complemented, Sylow
72	#510	—	73	1	no	—
73	#444	—	73	1	no	—
74	#441	—	73	1	no	—
75	#365	—	73	1	no	—
76	#362	—	73	1	no	—
77	#360	—	73	1	no	—
78	#347	—	73	1	no	—
79	#135	—	73	1	no	—
80	C₂×Q₁₆	—	73	1	no	—
81	#131	—	146	1	no	—
82	C₄×Q₈	—	146	1	no	—
83	Q₃₂	—	73	1	no	—
84	Q₃₂	#64408	1	1	yes	characteristic, complemented
85	SD₃₂	—	73	1	no	—
86	SD₃₂	—	73	1	no	—
87	M₃₂	—	73	1	no	—
88	M₃₂	—	73	1	no	—
89	#108	—	73	1	no	—
90	#106	—	73	1	no	—
91	#104	—	146	1	no	—
92	#103	—	146	1	no	—
93	#97	—	73	1	no	—
94	#41	—	146	1	no	—
95	C₂×Q₈	—	146	1	no	—
96	Q₁₆	—	73	1	no	—
97	Q₁₆	—	146	1	no	—
98	Q₁₆	C₂×#64408	1	1	yes	characteristic
99	Q₁₆	C₂×#64408	1	1	yes	characteristic
100	SD₁₆	—	146	1	no	—
101	D₈	—	73	1	no	—
102	M₁₆	—	292	1	no	—
103	M₁₆	—	146	1	no	—
104	C₂×C₈	—	292	1	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	73	1	no	—
107	C₄⋊C₄	—	292	1	no	—
108	C₄⋊C₄	—	146	1	no	—
109	C₄²	—	292	1	no	—
110	C₄²	—	73	1	no	—
111	C₁₆	—	73	1	no	—
112	C₁₆	—	146	1	no	—
113	C₁₆	C₂×#64408	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	292	1	no	—
117	Q₈	—	146	1	no	—
118	D₄	—	292	1	no	—
119	D₄	—	146	1	no	—
120	C₂×C₄	—	292	1	no	—
121	C₂×C₄	—	292	1	no	—
122	C₂×C₄	—	292	1	no	—
123	C₂×C₄	—	73	1	no	—
124	C₂×C₄	—	146	1	no	—
125	C₈	—	292	1	no	—
126	C₈	—	73	1	no	—
127	C₈	—	146	1	no	—
128	C₈	C₂²×#64408	1	1	yes	characteristic
129	C₂²	—	292	1	no	—
130	C₂²	—	73	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	292	1	no	—
134	C₄	—	292	1	no	—
135	C₄	—	584	1	no	—
136	C₄	—	73	1	no	—
137	C₄	D₄×#64408	1	1	yes	characteristic
138	C₂	—	584	1	no	—
139	C₂	—	146	1	no	—
140	C₂	#17989628	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989633

C₂

yes

maximal, characteristic

#17989627

C₂

yes

maximal, characteristic, complemented

#17989511

C₂

yes

maximal, characteristic, complemented

#17989506

C₂

yes

maximal, characteristic, complemented

#17989504

C₂

yes

maximal, characteristic

#17989477

C₂

yes

maximal, characteristic, complemented

#17989029

C₂

yes

maximal, characteristic

#14556679

—

Q₈×#64408

—

#14556666

C₂²

yes

characteristic

#14556661

C₂²

yes

characteristic

#14556659

C₂²

yes

—

—

C₂²

yes

characteristic

#14556614

C₂²

yes

characteristic

Q₁₆×D₇₃

C₂²

yes

characteristic

#14556537

C₄

yes

characteristic, complemented

#14556536

C₄

yes

characteristic, complemented

#14556534

C₄

yes

characteristic, complemented

#14556530

C₂²

yes

characteristic

Q₃₂×C₇₃

C₄

yes

characteristic, complemented

—

—

—

D₄

yes

—

—

—

D₄

yes

characteristic

#11924031

—

#11924029

—

Q₈×D₇₃

—

Dic₂₉₂

C₂×C₄

yes

characteristic

#11924018

—

#11924015

—

D₅₈₄

C₂×C₄

yes

characteristic

C₈×D₇₃

C₂³

yes

characteristic

#11924003

C₂×C₄

yes

characteristic

Q₁₆×C₇₃

C₂×C₄

yes

characteristic

Q₁₆×C₇₃

C₂×C₄

yes

characteristic

C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×#64408

—

C₂×#64408

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₄

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₈×C₇₃

C₂²×C₄

yes

yes

C₂×D₇₃

—

C₂×D₇₃

yes

—

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

characteristic

Dic₇₃

—

Dic₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#2180

—

maximal, Sylow

C₇₃

#2180

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

—

C₂×Q₁₆

—

#131

—

146

—

C₄×Q₈

—

146

—

Q₃₂

—

Q₃₂

#64408

yes

characteristic, complemented

SD₃₂

—

SD₃₂

—

M₃₂

—

M₃₂

—

#108

—

#106

—

#104

—

146

—

#103

—

146

—

#97

—

#41

—

146

—

C₂×Q₈

—

146

—

Q₁₆

—

Q₁₆

—

146

—

Q₁₆

yes

Q₁₆

yes

100

SD₁₆

—

146

—

101

D₈

—

102

M₁₆

—

292

—

103

M₁₆

—

146

—

104

C₂×C₈

—

292

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₄⋊C₄

—

292

—

108

C₄⋊C₄

—

146

—

109

C₄²

—

292

—

110

C₄²

—

111

C₁₆

—

112

C₁₆

—

146

—

113

C₁₆

C₂×#64408

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

292

—

117

Q₈

—

146

—

118

D₄

—

292

—

119

D₄

—

146

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

292

—

122

C₂×C₄

—

292

—

123

C₂×C₄

—

124

C₂×C₄

—

146

—

125

C₈

—

292

—

126

C₈

—

127

C₈

—

146

—

128

C₈

C₂²×#64408

yes

characteristic

129

C₂²

—

292

—

130

C₂²

—

131

C₄

—

132

C₄

—

133

C₄

—

292

—

134

C₄

—

292

—

135

C₄

—

584

—

136

C₄

—

137

C₄

D₄×#64408

yes

characteristic

138

C₂

—

584

—

139

C₂

—

146

—

140

C₂

#17989628

yes

characteristic

	subgroup	G/H
center	C₂	#17989628
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₈	C₂²×#64408
Fitting	Q₃₂×C₇₃	C₄
socle	C₂×C₇₃	#442
hypercenter	Q₃₂	#64408
hypocenter	C₇₃	#2180
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

#17989628

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₈

C₂²×#64408

Fitting

Q₃₂×C₇₃

C₄

C₂×C₇₃

Q₃₂

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
2. #17989627	C₂	semidirect
3. #17989511	C₂	semidirect
4. #17989506	C₂	semidirect
6. #17989477	C₂	semidirect
18. #14556537	C₄	semidirect
19. #14556536	C₄	semidirect
20. #14556534	C₄	semidirect
22. Q₃₂×C₇₃	C₄	semidirect
71. C₇₃	#2180	semidirect
84. Q₃₂	#64408	semidirect

H ≅ G/N

type

2. #17989627

C₂

semidirect

3. #17989511

C₂

semidirect

4. #17989506

C₂

semidirect

6. #17989477

C₂

semidirect

18. #14556537

C₄

semidirect

19. #14556536

C₄

semidirect

20. #14556534

C₄

semidirect

22. Q₃₂×C₇₃

C₄

semidirect

71. C₇₃

#2180

semidirect

84. Q₃₂

#64408

semidirect

Subgroups — #22626338

Conjugacy classes of subgroups

Complements