Finite Group Databasebeta

← #36363141

Subgroups — #36363141

Order 22144.

520 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19149699C211yesmaximal, characteristic, complemented
2C23×#13063436C211yesmaximal, characteristic, complemented
3#26012002C211yesmaximal, characteristic, complemented
4#26011730C212yesmaximal, complemented
5#26011725C212yesmaximal, complemented
6#26011722C212yesmaximal, complemented
7#26011718C212yesmaximal, complemented
8#26011527C211yesmaximal, characteristic, complemented
9#26011487C211yesmaximal, characteristic, complemented
10#26011485C211yesmaximal, characteristic, complemented
11#525×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13#19149699—24no—
14C2×#14583816C2211yescharacteristic, complemented
15C2×#14583814C2211yescharacteristic, complemented
16C2×#14583813C2211yescharacteristic, complemented
17C2×#14583811C2211yescharacteristic, complemented
18C22×#13063436—24no—
19C22×#13063436C2212yes—
20C22×#13063436C2212yes—
21C22×#13063436C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149662—21no—
24#19149658C2212yes—
25#19149653C2212yes—
26#19149648—21no—
27C4×C22⋊D173—22no—
28#19149638C2211yescharacteristic
29#19149612C2212yescomplemented
30#19149610C2212yescomplemented
31#19149603C2212yescomplemented
32#19149602C2212yescomplemented
33#19149600C2212yescomplemented
34#19149599C2212yescomplemented
35C2×#41×C173C2211yescharacteristic
36C23×C4×C173C2211yescharacteristic
37#123×C173—21no—
38C22⋊Q8×C173C2212yes—
39#121×C173C2212yes—
40#120×C173—21no—
41C4×D4×C173—22no—
42#117×C173C2211yescharacteristic
43#14583826—22no—
44#14583826—22no—
45C2×Dic346C2311yescharacteristic
46C2×D692C2311yescharacteristic
47C2×C22⋊D173—22no—
48C2×C22⋊D173—21no—
49C2×C4×D173—21no—
50#14583816—22no—
51#14583814—22no—
52#14583813—22no—
53#14583811—22no—
54C2×#13063436—24no—
55C2×#13063436—24no—
56C2×#13063436—22no—
57C2×#13063436D414yescomplemented
58C2×#13063436C2312yes—
59C2×#13063436C2311yescharacteristic
60C2×#13063436C2311yescharacteristic
61C4⋊Dic173C2312yes—
62#14583805—22no—
63C22⋊Dic173—21no—
64C22⋊Dic173—22no—
65#14583803—22no—
66C4×Dic173—21no—
67C24×C173D411yescharacteristic
68#41×C173—22no—
69#41×C173—22no—
70C2×Q8×C173C2311yescharacteristic
71C2×D4×C173—21no—
72C2×D4×C173—22no—
73C2×D4×C173C2311yescharacteristic
74C22×C4×C173—22no—
75C22×C4×C173—22no—
76C22×C4×C173—21no—
77C22×C4×C173D412yes—
78C22×C4×C173D411yescharacteristic
79C22×C4×C173C2312yes—
80C22×C4×C173C2311yescharacteristic
81C4⋊C4×C173—22no—
82C4⋊C4×C173C2312yes—
83C22⋊C4×C173—21no—
84C22⋊C4×C173—22no—
85C22⋊C4×C173—22no—
86C42×C173—21no—
87C22×D173—21no—
88Dic346—21no—
89Dic346—21no—
90D692—21no—
91D692—21no—
92C22⋊D173—42no—
93C22⋊D173—44no—
94C2×Dic173—21no—
95C2×Dic173—22no—
96C4×D173—42no—
97#13063436—22no—
98#13063436C2×D412yes—
99#13063436C2×D412yes—
100C23×C173—22no—
101C23×C173—24no—
102C23×C173—21no—
103C23×C173C2×D411yescharacteristic
104C23×C173C2×D412yes—
105Q8×C173—21no—
106Q8×C173—21no—
107D4×C173—21no—
108D4×C173—21no—
109D4×C173—42no—
110D4×C173—44no—
111C2×C4×C173—22no—
112C2×C4×C173—24no—
113C2×C4×C173—22no—
114C2×C4×C173—22no—
115C2×C4×C173—22no—
116C2×C4×C173—21no—
117C2×C4×C173—42no—
118C2×C4×C173C2×D412yes—
119C2×C4×C173C2×D412yes—
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173#4112yes—
122C2×C4×C173C2411yescharacteristic
123C22×C173—28no—
124C22×C173—22no—
125C22×C173—22no—
126C22×C173—22no—
127C22×C173—41no—
128C22×C173—42no—
129C22×C173#12112yes—
130C22×C173#13514yescomplemented
131C22×C173C22×D411yescharacteristic
132C2×D173—41no—
133C2×D173—42no—
134C4×C173—22no—
135C4×C173—41no—
136C4×C173—42no—
137C4×C173#12112yes—
138C4×C173C22×D411yescharacteristic
139C4×C173C2×#4111yescharacteristic
140Dic173—41no—
141Dic173—42no—
142C2×C173—22no—
143C2×C173—24no—
144C2×C173—81no—
145C2×C173C2×#12111yescharacteristic
146C2×C173C2×#13512yes—
147D173—81no—
148C173#305011yescharacteristic, complemented, Sylow
149#3050—1731nomaximal, Sylow
150C2×#135—1731no—
151C23×C8—1731no—
152#525—1731no—
153#525D17311yescharacteristic, complemented
154#467—1732no—
155#466—1732no—
156#465—1732no—
157#465—1732no—
158#427—1731no—
159#416—1731no—
160#416—1731no—
161C2×#41—1731no—
162C2×#41C2×D17311yescharacteristic
163C23×C4C2×D17311yescharacteristic
164#135—3464no—
165C2×Q16—1731no—
166C2×SD16—1731no—
167C2×SD16—1731no—
168C2×D8—1731no—
169C22×C8—3464no—
170C22×C8—1731no—
171C22×C8—1732no—
172C22×C8—1732no—
173C22×C8—1731no—
174#123—21no—
175#123—3461no—
176C22⋊Q8—1732no—
177C22⋊Q8C2×D17312yes—
178#121—1732no—
179#121C2×D17312yes—
180#120—21no—
181#120—3461no—
182C4×D4—22no—
183C4×D4—3462no—
184#117—1731no—
185#117C2×D17311yescharacteristic
186#107—1732no—
187#106—1732no—
188#103—1732no—
189#103—1732no—
190#102—1732no—
191#102—1732no—
192C24C22⋊D17311yescharacteristic
193#41—22no—
194#41—22no—
195#41—3462no—
196#41—3462no—
197C2×Q8—1731no—
198C2×Q8C22×D17311yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—3462no—
202C2×D4—3461no—
203C2×D4—1731no—
204C2×D4C22×D17311yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—3461no—
209C22×C4C22⋊D17311yescharacteristic
210C22×C4C22⋊D17312yes—
211C22×C4C22×D17312yes—
212C22×C4C22×D17311yescharacteristic
213Q16—3462no—
214SD16—3462no—
215SD16—3462no—
216D8—3462no—
217C2×C8—3462no—
218C2×C8—3464no—
219C2×C8—3464no—
220C2×C8—1732no—
221C2×C8—1734no—
222C2×C8—1731no—
223C2×C8—1731no—
224C4⋊C4—22no—
225C4⋊C4—3462no—
226C4⋊C4—1732no—
227C4⋊C4C22×D17312yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—3461no—
232C22⋊C4—3462no—
233C22⋊C4—3462no—
234C42—21no—
235C42—3461no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—3461no—
240C23C2×C22⋊D17311yescharacteristic
241C23C2×C22⋊D17312yes—
242Q8—21no—
243Q8—21no—
244Q8—3461no—
245Q8—3461no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—3461no—
251D4—3461no—
252D4—6922no—
253D4—6924no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—3462no—
262C2×C4—3461no—
263C2×C4—6922no—
264C2×C4D4×D17312yes—
265C2×C4C2×C22⋊D17312yes—
266C2×C4C2×C22⋊D17311yescharacteristic
267C2×C4#1458382512yes—
268C2×C4C23×D17311yescharacteristic
269C8—3462no—
270C8—1732no—
271C8—1732no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—42no—
278C22—6921no—
279C22—6922no—
280C22#1914965112yes—
281C22#1914969914yescomplemented
282C22C22×C22⋊D17311yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—6921no—
287C4—6922no—
288C4#1914965112yes—
289C4C2×D4×D17311yescharacteristic
290C4C2×#1458382511yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—13841no—
295C2C2×#1914965111yescharacteristic
296C2C2×#1914969912yes—
subgroupG/H
centerC2×C4C2×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#525×C173C2
socleC22×C173C22×D4
hypercenter#525D173
hypocenterC173#3050
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#19149699C2semidirect
2. C23×#13063436C2semidirect
3. #26012002C2semidirect
4. #26011730C2semidirect
5. #26011725C2semidirect
6. #26011722C2semidirect
7. #26011718C2semidirect
8. #26011527C2semidirect
9. #26011487C2semidirect
10. #26011485C2semidirect
11. #525×C173C2semidirect
14. C2×#14583816C22semidirect
15. C2×#14583814C22semidirect
16. C2×#14583813C22semidirect
17. C2×#14583811C22semidirect
29. #19149612C22semidirect
30. #19149610C22semidirect
31. #19149603C22semidirect
32. #19149602C22semidirect
33. #19149600C22semidirect
34. #19149599C22semidirect
57. C2×#13063436D4semidirect
130. C22×C173#135semidirect
148. C173#3050semidirect
153. #525D173semidirect
281. C22#19149699semidirect

Maximal subgroups

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

subgroupindex
C2×#191496992
C23×#130634362
#260120022
#260117302
#260117252
#260117222
#260117182
#260115272
#260114872
#260114852
#525×C1732
#3050173