Finite Group Databasebeta

← #36362916

Subgroups — #36362916

Order 22144.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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#26012327C211yesmaximal, characteristic, complemented
2#26012291C212yesmaximal, complemented
3#26012037C211yesmaximal, characteristic, complemented
4#26011676C212yesmaximal, complemented
5#26011671C212yesmaximal, complemented
6#26011666C212yesmaximal, complemented
7#26011661C212yesmaximal, complemented
8C2×#19149605C212yesmaximal, complemented
9#413×C173C211yesmaximal, characteristic, complemented
10#19149727C2212yes—
11#19149726—22no—
12#19149725—22no—
13#19149724—22no—
14#19149724C2212yes—
15C2×#14583826—22no—
16C2×#14583826C2212yes—
17C2×#14583826C2211yescharacteristic, complemented
18#41×D173—24no—
19#19149712—22no—
20#19149712C2214yescomplemented
21#19149707—22no—
22#19149707C2214yescomplemented
23#19149701—24no—
24#19149677C2211yescharacteristic, complemented
25#19149675C2211yescharacteristic, complemented
26#19149668C2211yescharacteristic, complemented
27#19149666C2211yescharacteristic, complemented
28#19149651—22no—
29C4×C22⋊D173—22no—
30C2×C4×Dic173C2211yescharacteristic, complemented
31#19149605C2214yescomplemented
32#19149605C2214yescomplemented
33C2×#41×C173C2211yescharacteristic
34C2×M16×C173C2212yescomplemented
35#101×C173C2212yescomplemented
36#100×C173C2212yescomplemented
37#14583827—24no—
38#14583826—24no—
39#14583826—24no—
40#14583826—22no—
41#14583826—24no—
42#14583826—22no—
43#14583826—22no—
44#14583826—22no—
45#14583826D414yescomplemented
46#14583826C2314yes—
47#14583825—28no—
48#14583825—24no—
49C2×Dic346—22no—
50C2×Dic346D412yescomplemented
51C2×Dic346C2311yescharacteristic
52Q8×D173—24no—
53C2×D692—22no—
54C2×D692D412yescomplemented
55C2×D692C2311yescharacteristic
56C2×C22⋊D173—24no—
57C2×C22⋊D173—22no—
58C2×C22⋊D173—22no—
59C2×C22⋊D173—21no—
60D4×D173—28no—
61D4×D173—24no—
62C22×Dic173—21no—
63C2×C4×D173—24no—
64C2×C4×D173—22no—
65C2×C4×D173—21no—
66Dic692—24no—
67#14583815—24no—
68#14583815—24no—
69D1384—24no—
70#14583805—22no—
71C22⋊Dic173—22no—
72#14583803—22no—
73C4×Dic173—21no—
74C4×Dic173C2311yescharacteristic
75C4×Dic173C2311yescharacteristic
76#41×C173—22no—
77#41×C173—22no—
78C2×Q8×C173C2311yescharacteristic
79C2×D4×C173D412yes—
80C2×D4×C173C2311yescharacteristic
81C22×C4×C173D412yes—
82C22×C4×C173C2311yescharacteristic
83M16×C173—22no—
84M16×C173C2314yes—
85C2×C8×C173—22no—
86C22×D173—24no—
87C22×D173—22no—
88C22×D173—21no—
89Dic346—24no—
90Dic346—22no—
91Dic346—22no—
92Dic346C2×D414yes—
93D692—24no—
94D692—22no—
95D692—22no—
96D692C2×D414yes—
97C22⋊D173—28no—
98C22⋊D173—24no—
99C22⋊D173—24no—
100C22⋊D173—44no—
101C22⋊D173—42no—
102C2×Dic173—24no—
103C2×Dic173—22no—
104C2×Dic173—21no—
105C2×Dic173—21no—
106C2×Dic173—21no—
107C2×Dic173—42no—
108C4×D173—28no—
109C4×D173—24no—
110C4×D173—24no—
111C4×D173—41no—
112C4×D173—41no—
113C23×C173C2×D412yes—
114C23×C173C2×D411yescharacteristic
115Q8×C173—22no—
116D4×C173—24no—
117D4×C173—22no—
118C2×C4×C173—22no—
119C2×C4×C173—22no—
120C2×C4×C173C2×D412yes—
121C2×C4×C173C2×D412yes—
122C2×C4×C173C2×D412yes—
123C2×C4×C173C2×D411yescharacteristic
124C2×C4×C173C2411yescharacteristic
125C8×C173—24no—
126C22×C173—22no—
127C22×C173—41no—
128C22×C173—42no—
129C22×C173#12012yes—
130C22×C173C22×D411yescharacteristic
131C2×D173—24no—
132C2×D173—42no—
133C2×D173—41no—
134C2×D173—82no—
135C2×D173—84no—
136C4×C173—22no—
137C4×C173#12011yescharacteristic
138C4×C173#12011yescharacteristic
139C4×C173C22×D412yes—
140Dic173—22no—
141Dic173—24no—
142Dic173—41no—
143Dic173—41no—
144C2×C173—21no—
145C2×C173—22no—
146C2×C173—42no—
147C2×C173C2×#12011yescharacteristic
148D173—44no—
149D173—81no—
150C173#301811yescharacteristic, complemented, Sylow
151#3018—1731nomaximal, Sylow
152#585—1731no—
153#575—1732no—
154#532—1731no—
155#456—1732no—
156#455—1732no—
157#454—1732no—
158#453—1732no—
159C2×#104—1732no—
160#413D17311yescharacteristic, complemented
1612−1+4—3462no—
1622−1+4—1732no—
163H(5,𝔽2)—3462no—
164H(5,𝔽2)—3462no—
165H(5,𝔽2)—1732no—
166C2×#41—3464no—
167C2×#41—3462no—
168C2×#41—1731no—
169C2×#41—1732no—
170C2×#41C2×D17311yescharacteristic
171#137—3462no—
172#137—1734no—
173AGL(1,ℤ/8)—3462no—
174AGL(1,ℤ/8)—1734no—
175#135—3464no—
176C2×M16C2×D17312yescomplemented
177C4⋊Q8—1731no—
178#127—1731no—
179#124—1731no—
180#124—1731no—
181#121—3462no—
182C4×D4—3462no—
183C2×C42—1731no—
184#104—1734no—
185#104—1734no—
186#101C2×D17312yescomplemented
187#100C2×D17312yescomplemented
188#41—22no—
189#41—22no—
190#41—3464no—
191#41—3464no—
192#41—3468no—
193#41—3464no—
194#41—3464no—
195#41—3462no—
196#41—3462no—
197#41—3464no—
198#41—3462no—
199#41—3462no—
200#41—1734no—
201#41—1734no—
202C2×Q8—3462no—
203C2×Q8—3464no—
204C2×Q8—1732no—
205C2×Q8—1731no—
206C2×Q8C22×D17311yescharacteristic
207C2×D4—3462no—
208C2×D4—3468no—
209C2×D4—3464no—
210C2×D4—3464no—
211C2×D4—3462no—
212C2×D4—3462no—
213C2×D4—3461no—
214C2×D4—1732no—
215C2×D4—1731no—
216C2×D4D69212yes—
217C2×D4C22×D17311yescharacteristic
218C22×C4—3464no—
219C22×C4—3462no—
220C22×C4—3461no—
221C22×C4—3461no—
222C22×C4D69212yes—
223C22×C4C22×D17311yescharacteristic
224Q16—3464no—
225SD16—3464no—
226SD16—3464no—
227D8—3464no—
228M16—22no—
229M16C22×D17314yes—
230C2×C8—22no—
231C4⋊C4—3462no—
232C22⋊C4—3462no—
233C22⋊C4—3462no—
234C42—3461no—
235C42—1731no—
236C42—1731no—
237C23—3464no—
238C23—3462no—
239C23—3461no—
240C23C2×D69212yes—
241C23C2×D69211yescharacteristic
242Q8—22no—
243Q8—3464no—
244Q8—3462no—
245Q8—3462no—
246Q8—1734no—
247D4—24no—
248D4—22no—
249D4—3464no—
250D4—3468no—
251D4—3464no—
252D4—3464no—
253D4—3462no—
254D4—3462no—
255D4—1734no—
256D4—6924no—
257D4—6922no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—3468no—
261C2×C4—3464no—
262C2×C4—3462no—
263C2×C4—3461no—
264C2×C4—3464no—
265C2×C4—3464no—
266C2×C4—3461no—
267C2×C4—3461no—
268C2×C4—6922no—
269C2×C4—6921no—
270C2×C4—6921no—
271C2×C4D4×D17312yes—
272C2×C4D4×D17312yes—
273C2×C4C2×D69212yes—
274C2×C4C2×D69211yescharacteristic
275C2×C4C23×D17311yescharacteristic
276C8—24no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—3464no—
281C22—13842no—
282C22—13844no—
283C22—6922no—
284C22—6921no—
285C22#1914964912yes—
286C22C22×D69211yescharacteristic
287C4—22no—
288C4—3462no—
289C4—3464no—
290C4—6921no—
291C4—6921no—
292C4#1914964911yescharacteristic
293C4#1914964911yescharacteristic
294C4C2×D4×D17312yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—13841no—
299C2—6924no—
300C2C2×#1914964911yescharacteristic
subgroupG/H
centerC4#19149649
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#413×C173C2
socleC2×C173C2×#120
hypercenter#413D173
hypocenterC173#3018
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #26012327C2semidirect
2. #26012291C2semidirect
3. #26012037C2semidirect
4. #26011676C2semidirect
5. #26011671C2semidirect
6. #26011666C2semidirect
7. #26011661C2semidirect
8. C2×#19149605C2semidirect
9. #413×C173C2semidirect
17. C2×#14583826C22semidirect
20. #19149712C22semidirect
22. #19149707C22semidirect
24. #19149677C22semidirect
25. #19149675C22semidirect
26. #19149668C22semidirect
27. #19149666C22semidirect
30. C2×C4×Dic173C22semidirect
31. #19149605C22semidirect
32. #19149605C22semidirect
34. C2×M16×C173C22semidirect
35. #101×C173C22semidirect
36. #100×C173C22semidirect
45. #14583826D4semidirect
50. C2×Dic346D4semidirect
54. C2×D692D4semidirect
150. C173#3018semidirect
160. #413D173semidirect
176. C2×M16C2×D173semidirect
186. #101C2×D173semidirect
187. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260122912
#260120372
#260116762
#260116712
#260116662
#260116612
C2×#191496052
#413×C1732
#3018173