Finite Group Databasebeta

← #36362911

Subgroups — #36362911

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#26012289C212yesmaximal, complemented
3#26011673C212yesmaximal, complemented
4#26011668C212yesmaximal, complemented
5#26011663C212yesmaximal, complemented
6#26011658C212yesmaximal, complemented
7C2×#19149607C212yesmaximal, complemented
8#26011475C211yesmaximal, characteristic, complemented
9#532×C173C211yesmaximal, characteristic, complemented
10#19149727C2212yes—
11#19149726—22no—
12#19149725—22no—
13#19149724—22no—
14#19149724C2212yes—
15C2×#14583826—22no—
16C2×#14583826C2212yes—
17#41×D173—24no—
18#19149710C2214yescomplemented
19#19149709—22no—
20#19149705C2214yescomplemented
21#19149704—22no—
22#19149699—24no—
23C2×#14583810C2212yescomplemented
24#19149607C2214yescomplemented
25#19149607C2214yescomplemented
26#19149598C2212yescomplemented
27#19149596C2212yescomplemented
28C2×#41×C173C2211yescharacteristic
29C2×#41×C173C2211yescharacteristic, complemented
30C4⋊Q8×C173C2211yescharacteristic, complemented
31#127×C173C2211yescharacteristic, complemented
32#124×C173C2211yescharacteristic, complemented
33#124×C173C2211yescharacteristic, complemented
34#121×C173—22no—
35C4×D4×C173—22no—
36C2×C42×C173C2211yescharacteristic, complemented
37#14583827—24no—
38#14583826—24no—
39#14583826—24no—
40#14583826—22no—
41#14583826—24no—
42#14583826—22no—
43#14583826D414yescomplemented
44#14583826C2314yes—
45#14583825—28no—
46#14583825—24no—
47C2×Dic346—22no—
48C2×Dic346D412yescomplemented
49Q8×D173—24no—
50C2×D692—22no—
51C2×D692D412yescomplemented
52C2×C22⋊D173—24no—
53C2×C22⋊D173—22no—
54D4×D173—28no—
55D4×D173—24no—
56C2×C4×D173—24no—
57C2×C4×D173—22no—
58#14583816—24no—
59#14583814—24no—
60#14583813—24no—
61#14583811—24no—
62#14583810—22no—
63#14583810C2314yes—
64C2×#13063436—22no—
65#41×C173—22no—
66#41×C173—22no—
67#41×C173—22no—
68#41×C173—22no—
69C2×Q8×C173C2311yescharacteristic
70C2×Q8×C173C2311yescharacteristic
71C2×D4×C173—21no—
72C2×D4×C173—22no—
73C2×D4×C173D412yes—
74C2×D4×C173C2311yescharacteristic
75C2×D4×C173C2311yescharacteristic
76C22×C4×C173—21no—
77C22×C4×C173—21no—
78C22×C4×C173D412yes—
79C22×C4×C173C2311yescharacteristic
80C4⋊C4×C173—22no—
81C22⋊C4×C173—22no—
82C22⋊C4×C173—22no—
83C42×C173—21no—
84C42×C173C2311yescharacteristic
85C42×C173C2311yescharacteristic
86C22×D173—24no—
87C22×D173—22no—
88Dic346—24no—
89Dic346—22no—
90Dic346C2×D414yes—
91D692—24no—
92D692—22no—
93D692C2×D414yes—
94C22⋊D173—28no—
95C22⋊D173—24no—
96C22⋊D173—24no—
97C2×Dic173—24no—
98C2×Dic173—22no—
99C4×D173—28no—
100C4×D173—24no—
101C4×D173—24no—
102#13063436—24no—
103C23×C173—21no—
104C23×C173C2×D412yes—
105C23×C173C2×D411yescharacteristic
106Q8×C173—22no—
107Q8×C173—22no—
108D4×C173—24no—
109D4×C173—22no—
110D4×C173—22no—
111D4×C173—44no—
112D4×C173—42no—
113C2×C4×C173—21no—
114C2×C4×C173—22no—
115C2×C4×C173—22no—
116C2×C4×C173—21no—
117C2×C4×C173—21no—
118C2×C4×C173—42no—
119C2×C4×C173—41no—
120C2×C4×C173—41no—
121C2×C4×C173C2×D412yes—
122C2×C4×C173C2×D412yes—
123C2×C4×C173C2×D412yes—
124C2×C4×C173C2×D411yescharacteristic
125C2×C4×C173C2411yescharacteristic
126C22×C173—22no—
127C22×C173—41no—
128C22×C173—42no—
129C22×C173—42no—
130C22×C173—41no—
131C22×C173#12012yes—
132C22×C173C22×D411yescharacteristic
133C2×D173—24no—
134C2×D173—82no—
135C2×D173—84no—
136C4×C173—22no—
137C4×C173—22no—
138C4×C173—41no—
139C4×C173—41no—
140C4×C173#12011yescharacteristic
141C4×C173#12011yescharacteristic
142C4×C173C22×D412yes—
143Dic173—24no—
144C2×C173—21no—
145C2×C173—22no—
146C2×C173—42no—
147C2×C173—81no—
148C2×C173C2×#12011yescharacteristic
149D173—44no—
150C173#301811yescharacteristic, complemented, Sylow
151#3018—1731nomaximal, Sylow
152#585—1731no—
153#575—1732no—
154#532D17311yescharacteristic, complemented
155#456—1732no—
156#455—1732no—
157#454—1732no—
158#453—1732no—
159C2×#104—1732no—
160#413—1731no—
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—1732no—
169C2×#41C2×D17311yescharacteristic
170C2×#41C2×D17311yescharacteristic, complemented
171#137—3462no—
172#137—1734no—
173AGL(1,ℤ/8)—3462no—
174AGL(1,ℤ/8)—1734no—
175#135—3464no—
176C2×M16—1732no—
177C4⋊Q8C2×D17311yescharacteristic, complemented
178#127C2×D17311yescharacteristic, complemented
179#124C2×D17311yescharacteristic, complemented
180#124C2×D17311yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D17311yescharacteristic, complemented
184#104—1734no—
185#104—1734no—
186#101—1732no—
187#100—1732no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—3464no—
193#41—3464no—
194#41—3468no—
195#41—3464no—
196#41—3464no—
197#41—3462no—
198#41—3462no—
199#41—3464no—
200#41—1734no—
201#41—1734no—
202C2×Q8—3462no—
203C2×Q8—3464no—
204C2×Q8—1732no—
205C2×Q8C22×D17311yescharacteristic
206C2×Q8C22×D17311yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—3462no—
210C2×D4—3468no—
211C2×D4—3464no—
212C2×D4—3464no—
213C2×D4—3462no—
214C2×D4—1732no—
215C2×D4C22⋊D17312yes—
216C2×D4C22×D17311yescharacteristic
217C2×D4C22×D17311yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—3464no—
221C22×C4—3462no—
222C22×C4C22⋊D17312yes—
223C22×C4C22×D17311yescharacteristic
224Q16—3464no—
225SD16—3464no—
226SD16—3464no—
227D8—3464no—
228M16—3462no—
229M16—1734no—
230C2×C8—3462no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D17311yescharacteristic
236C42C22×D17311yescharacteristic
237C23—21no—
238C23—3464no—
239C23—3462no—
240C23C2×C22⋊D17312yes—
241C23C2×C22⋊D17311yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—3464no—
245Q8—3462no—
246Q8—1734no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—3464no—
253D4—3468no—
254D4—3464no—
255D4—3464no—
256D4—3462no—
257D4—1734no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—3468no—
267C2×C4—3464no—
268C2×C4—3462no—
269C2×C4—3464no—
270C2×C4—3464no—
271C2×C4D4×D17312yes—
272C2×C4D4×D17312yes—
273C2×C4C2×C22⋊D17312yes—
274C2×C4C2×C22⋊D17311yescharacteristic
275C2×C4C23×D17311yescharacteristic
276C8—3464no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—3464no—
283C22—13842no—
284C22—13844no—
285C22#1914964712yes—
286C22C22×C22⋊D17311yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—3464no—
292C4#1914964711yescharacteristic
293C4#1914964711yescharacteristic
294C4C2×D4×D17312yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—6924no—
300C2C2×#1914964711yescharacteristic
subgroupG/H
centerC4#19149647
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#532×C173C2
socleC2×C173C2×#120
hypercenter#532D173
hypocenterC173#3018
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #26012327C2semidirect
2. #26012289C2semidirect
3. #26011673C2semidirect
4. #26011668C2semidirect
5. #26011663C2semidirect
6. #26011658C2semidirect
7. C2×#19149607C2semidirect
8. #26011475C2semidirect
9. #532×C173C2semidirect
18. #19149710C22semidirect
20. #19149705C22semidirect
23. C2×#14583810C22semidirect
24. #19149607C22semidirect
25. #19149607C22semidirect
26. #19149598C22semidirect
27. #19149596C22semidirect
29. C2×#41×C173C22semidirect
30. C4⋊Q8×C173C22semidirect
31. #127×C173C22semidirect
32. #124×C173C22semidirect
33. #124×C173C22semidirect
36. C2×C42×C173C22semidirect
43. #14583826D4semidirect
48. C2×Dic346D4semidirect
51. C2×D692D4semidirect
150. C173#3018semidirect
154. #532D173semidirect
170. C2×#41C2×D173semidirect
177. C4⋊Q8C2×D173semidirect
178. #127C2×D173semidirect
179. #124C2×D173semidirect
180. #124C2×D173semidirect
183. C2×C42C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260122892
#260116732
#260116682
#260116632
#260116582
C2×#191496072
#260114752
#532×C1732
#3018173