Finite Group Databasebeta

← #36364280

Subgroups — #36364280

Order 22144.

402 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012046C211yesmaximal, characteristic, complemented
2#26012039C211yesmaximal, characteristic
3#26011894C211yesmaximal, characteristic, complemented
4#26011863C211yesmaximal, characteristic, complemented
5#26011842C211yesmaximal, characteristic, complemented
6#26011736C214yesmaximal, complemented
7#26011588C214yesmaximal, complemented
8C4×#14583809C211yesmaximal, characteristic, complemented
9#486×C173C211yesmaximal, characteristic, complemented
10C2×#14583825—21no—
11C2×#14583814C2214yes—
12C2×#14583809C2212yes—
13#19149679C2211yescharacteristic
14#19149679C2211yescharacteristic
15#19149670—21no—
16#19149666—21no—
17#19149659C2212yes—
18#19149655—21no—
19#19149650—21no—
20#19149650C2212yes—
21Q8×Dic173—21no—
22#19149644C2212yes—
23D4×Dic173—21no—
24D4×Dic173C2212yes—
25#19149637—21no—
26C42×D173C2211yescharacteristic
27#19149612C2214yes—
28#19149604C2214yes—
29#19149589C2212yes—
30C4×#13063436C2211yescharacteristic
31C4⋊Q8×C173C2211yescharacteristic
32#127×C173C2211yescharacteristic
33#102×C173C2214yes—
34C4×C8×C173C2211yescharacteristic
35#14583825—44no—
36C2×Dic346—21no—
37C2×Dic346C2312yes—
38C2×C22⋊D173—22no—
39C22×Dic173—22no—
40C2×C4×D173D411yescharacteristic
41C2×C4×D173D411yescharacteristic
42C2×C4×D173C2311yescharacteristic
43#14583814—28no—
44#14583809—24no—
45C2×#13063436C2312yes—
46C4⋊Dic173—22no—
47C4⋊Dic173—21no—
48C4⋊Dic173—21no—
49C4⋊Dic173—21no—
50C4⋊Dic173C2312yes—
51#14583805—22no—
52C22⋊Dic173—22no—
53C22⋊Dic173—22no—
54#14583803—22no—
55C4×Dic173—22no—
56C4×Dic173D411yescharacteristic
57C4×Dic173D411yescharacteristic
58C4×Dic173C2311yescharacteristic
59C2×Q8×C173—21no—
60C2×D4×C173—21no—
61C2×D4×C173—21no—
62C2×D4×C173C2312yes—
63C2×C8×C173C2312yes—
64C4⋊C4×C173—21no—
65C4⋊C4×C173C2312yes—
66C42×C173C2311yescharacteristic
67C22×D173C2×D411yescharacteristic
68Dic346—24no—
69Dic346—42no—
70C22⋊D173—44no—
71C2×Dic173—22no—
72C2×Dic173—22no—
73C2×Dic173—44no—
74C2×Dic173C2×D411yescharacteristic
75C2×Dic173C2×D411yescharacteristic
76C2×Dic173C2×D411yescharacteristic
77C4×D173—22no—
78C4×D173—22no—
79C4×D173—22no—
80#13063436#4114yescomplemented
81C23×C173—22no—
82Q8×C173—42no—
83D4×C173—24no—
84D4×C173—42no—
85D4×C173—42no—
86C2×C4×C173—22no—
87C2×C4×C173C2×D411yescharacteristic
88C2×C4×C173C2×D411yescharacteristic
89C2×C4×C173C2411yescharacteristic
90C8×C173—22no—
91C22×C173—42no—
92C22×C173—42no—
93C22×C173—42no—
94C22×C173C22×D411yescharacteristic
95C2×D173—21no—
96C2×D173—21no—
97C2×D173—21no—
98C4×C173—42no—
99C4×C173AGL(1,ℤ/8)12yes—
100C4×C173#13712yes—
101C4×C173C2×#4112yes—
102Dic173—21no—
103Dic173—21no—
104Dic173—21no—
105Dic173—42no—
106Dic173—42no—
107C2×C173—82no—
108C2×C173#53211yescharacteristic
109C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C173C2×#13711yescharacteristic
111D173—41no—
112C173#318211yescharacteristic, complemented, Sylow
113#3182—1731nomaximal, Sylow
114#533—1731no—
115#532—1731no—
116#500—1731no—
117#492—1731no—
118#487—1731no—
119#486D17311yescharacteristic, complemented
120#468—1734no—
121#440—1734no—
122C4×M16—1731no—
123C2×#41—3461no—
124C2×SD16—1734no—
125C2×M16—1732no—
126C4⋊Q8—1731no—
127C4⋊Q8—1731no—
128C4⋊Q8C2×D17311yescharacteristic
129#127C2×D17311yescharacteristic
130#125—3461no—
131#124—3461no—
132C22⋊Q8—3461no—
133C22⋊Q8—1732no—
134#121—3461no—
135#121—1732no—
136C4×Q8—3461no—
137C4×Q8—1732no—
138C4×D4—3461no—
139C4×D4—1732no—
140#117—3461no—
141C2×C42—1731no—
142#107—1734no—
143#103—1734no—
144#102C2×D17314yes—
145#97—1732no—
146C4×C8—1731no—
147C4×C8C2×D17311yescharacteristic
148#41—6924no—
149C2×Q8—21no—
150C2×Q8—3461no—
151C2×Q8—1732no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—3462no—
155C2×D4C22×D17312yes—
156C22×C4—3462no—
157C22×C4—1731no—
158C22×C4—1731no—
159C22×C4—1731no—
160SD16—3468no—
161M16—3464no—
162C2×C8—1732no—
163C2×C8C22×D17312yes—
164C4⋊C4—21no—
165C4⋊C4—3462no—
166C4⋊C4—3462no—
167C4⋊C4—3461no—
168C4⋊C4—3461no—
169C4⋊C4—3461no—
170C4⋊C4—1732no—
171C4⋊C4C22×D17312yes—
172C22⋊C4—3462no—
173C22⋊C4—3462no—
174C22⋊C4—3462no—
175C42—3462no—
176C42—1731no—
177C42—1731no—
178C42—1731no—
179C42C22×D17311yescharacteristic
180C23—22no—
181C23—1731no—
182Q8—42no—
183Q8—3464no—
184Q8—6922no—
185D4—24no—
186D4—42no—
187D4—42no—
188D4—6924no—
189C2×C4—22no—
190C2×C4—3462no—
191C2×C4—3462no—
192C2×C4—3462no—
193C2×C4—3462no—
194C2×C4—3462no—
195C2×C4—1731no—
196C2×C4—1731no—
197C2×C4—1731no—
198C2×C4—6924no—
199C2×C4D4×D17311yescharacteristic
200C2×C4D4×D17311yescharacteristic
201C2×C4C23×D17311yescharacteristic
202C8—22no—
203C8—1734no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—3461no—
208C22—3461no—
209C22—3461no—
210C22C2×D4×D17311yescharacteristic
211C4—42no—
212C4—3461no—
213C4—3461no—
214C4—3461no—
215C4—6922no—
216C4—6922no—
217C4#1914970312yes—
218C4#1914970812yes—
219C4#41×D17312yes—
220C2—82no—
221C2—6921no—
222C2#2601204011yescharacteristic
223C2C2×#1914970311yescharacteristic
224C2C2×#1914970811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#486×C173C2
socleC22×C173C22×D4
prime radical#26012039C2
hypercenter#486D173
hypocenterC173#3182
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26012046C2semidirect
3. #26011894C2semidirect
4. #26011863C2semidirect
5. #26011842C2semidirect
6. #26011736C2semidirect
7. #26011588C2semidirect
8. C4×#14583809C2semidirect
9. #486×C173C2semidirect
80. #13063436#41semidirect
112. C173#3182semidirect
119. #486D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260120462
#260120392
#260118942
#260118632
#260118422
#260117362
#260115882
C4×#145838092
#486×C1732
#3182173