Finite Group Databasebeta

← #36362912

Subgroups — #36362912

Order 22144.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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#26012289C212yesmaximal, complemented
2#26012037C211yesmaximal, characteristic, complemented
3#26011674C212yesmaximal, complemented
4#26011669C212yesmaximal, complemented
5#26011664C212yesmaximal, complemented
6#26011659C212yesmaximal, complemented
7C2×#19149606C212yesmaximal, complemented
8#26011475C211yesmaximal, characteristic, complemented
9#585×C173C211yesmaximal, characteristic, complemented
10C2×#14583826C2211yescharacteristic, complemented
11#19149710—22no—
12#19149709C2214yescomplemented
13#19149705—22no—
14#19149704C2214yescomplemented
15#19149699—24no—
16C2×#14583810C2212yescomplemented
17#19149677C2211yescharacteristic, complemented
18#19149675C2211yescharacteristic, complemented
19#19149668C2211yescharacteristic, complemented
20#19149666C2211yescharacteristic, complemented
21#19149651—22no—
22C4×C22⋊D173—22no—
23C2×C4×Dic173C2211yescharacteristic, complemented
24#19149606C2214yescomplemented
25#19149606C2214yescomplemented
26#19149598C2212yescomplemented
27#19149596C2212yescomplemented
282−1+4×C173—22no—
292−1+4×C173C2212yes—
30H(5,𝔽2)×C173—22no—
31H(5,𝔽2)×C173—22no—
32H(5,𝔽2)×C173C2212yes—
33C2×#41×C173—24no—
34C2×#41×C173—22no—
35C2×#41×C173C2212yes—
36C2×#41×C173C2211yescharacteristic
37#14583826—22no—
38#14583826—22no—
39C2×Dic346C2311yescharacteristic
40C2×D692C2311yescharacteristic
41C2×C22⋊D173—22no—
42C2×C22⋊D173—21no—
43C22×Dic173—21no—
44C2×C4×D173—21no—
45#14583816—24no—
46#14583814—24no—
47#14583813—24no—
48#14583811—24no—
49#14583810—22no—
50#14583810C2314yes—
51C2×#13063436—22no—
52#14583805—22no—
53C22⋊Dic173—22no—
54#14583803—22no—
55C4×Dic173—21no—
56C4×Dic173C2311yescharacteristic
57C4×Dic173C2311yescharacteristic
58#41×C173—24no—
59#41×C173—28no—
60#41×C173—24no—
61#41×C173—24no—
62#41×C173—24no—
63#41×C173—22no—
64#41×C173—22no—
65#41×C173—24no—
66#41×C173—22no—
67#41×C173—22no—
68#41×C173D414yescomplemented
69#41×C173C2314yes—
70C2×Q8×C173—22no—
71C2×Q8×C173—24no—
72C2×Q8×C173D412yescomplemented
73C2×Q8×C173C2311yescharacteristic
74C2×D4×C173—28no—
75C2×D4×C173—22no—
76C2×D4×C173—24no—
77C2×D4×C173—22no—
78C2×D4×C173—24no—
79C2×D4×C173D412yes—
80C2×D4×C173D412yescomplemented
81C2×D4×C173C2311yescharacteristic
82C22×C4×C173—24no—
83C22×C4×C173—22no—
84C22×C4×C173D412yes—
85C22×C4×C173C2311yescharacteristic
86C22×D173—21no—
87Dic346—22no—
88D692—22no—
89C22⋊D173—44no—
90C22⋊D173—42no—
91C2×Dic173—21no—
92C2×Dic173—21no—
93C2×Dic173—21no—
94C2×Dic173—42no—
95C4×D173—41no—
96C4×D173—41no—
97#13063436—24no—
98C23×C173—24no—
99C23×C173—22no—
100C23×C173C2×D412yes—
101C23×C173C2×D411yescharacteristic
102Q8×C173—24no—
103Q8×C173—22no—
104Q8×C173—22no—
105Q8×C173C2×D414yes—
106D4×C173—28no—
107D4×C173—24no—
108D4×C173—24no—
109D4×C173—24no—
110D4×C173—24no—
111D4×C173—22no—
112D4×C173—22no—
113D4×C173C2×D414yes—
114C2×C4×C173—24no—
115C2×C4×C173—28no—
116C2×C4×C173—22no—
117C2×C4×C173—24no—
118C2×C4×C173—22no—
119C2×C4×C173—22no—
120C2×C4×C173—24no—
121C2×C4×C173C2×D412yes—
122C2×C4×C173C2×D411yescharacteristic
123C2×C4×C173C2×D412yes—
124C2×C4×C173C2×D412yes—
125C2×C4×C173C2411yescharacteristic
126C22×C173—22no—
127C22×C173—24no—
128C22×C173—41no—
129C22×C173—42no—
130C22×C173—82no—
131C22×C173—84no—
132C22×C173#12012yes—
133C22×C173C22×D411yescharacteristic
134C2×D173—42no—
135C2×D173—41no—
136C4×C173—24no—
137C4×C173—22no—
138C4×C173#12011yescharacteristic
139C4×C173#12011yescharacteristic
140C4×C173C22×D412yes—
141Dic173—22no—
142Dic173—41no—
143Dic173—41no—
144C2×C173—21no—
145C2×C173—22no—
146C2×C173—42no—
147C2×C173—44no—
148C2×C173C2×#12011yescharacteristic
149D173—81no—
150C173#301811yescharacteristic, complemented, Sylow
151#3018—1731nomaximal, Sylow
152#585D17311yescharacteristic, complemented
153#575—1732no—
154#532—1731no—
155#456—1732no—
156#455—1732no—
157#454—1732no—
158#453—1732no—
159C2×#104—1732no—
160#413—1731no—
1612−1+4—22no—
1622−1+4C2×D17312yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D17312yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—1731no—
169C2×#41C2×D17312yes—
170C2×#41C2×D17311yescharacteristic
171#137—3462no—
172#137—1734no—
173AGL(1,ℤ/8)—3462no—
174AGL(1,ℤ/8)—1734no—
175#135—3464no—
176C2×M16—1732no—
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#101—1732no—
187#100—1732no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—3462no—
199#41—3462no—
200#41C22⋊D17314yescomplemented
201#41C22×D17314yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—1731no—
205C2×Q8C22⋊D17312yescomplemented
206C2×Q8C22×D17311yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—3462no—
213C2×D4—3461no—
214C2×D4—1731no—
215C2×D4C22⋊D17312yes—
216C2×D4C22⋊D17312yescomplemented
217C2×D4C22×D17311yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—3461no—
221C22×C4—3461no—
222C22×C4C22⋊D17312yes—
223C22×C4C22×D17311yescharacteristic
224Q16—3464no—
225SD16—3464no—
226SD16—3464no—
227D8—3464no—
228M16—3462no—
229M16—1734no—
230C2×C8—3462no—
231C4⋊C4—3462no—
232C22⋊C4—3462no—
233C22⋊C4—3462no—
234C42—3461no—
235C42—1731no—
236C42—1731no—
237C23—24no—
238C23—22no—
239C23—3461no—
240C23C2×C22⋊D17312yes—
241C23C2×C22⋊D17311yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—3462no—
246Q8C2×C22⋊D17314yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—3462no—
255D4—6924no—
256D4—6922no—
257D4C2×C22⋊D17314yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—3461no—
266C2×C4—3461no—
267C2×C4—3461no—
268C2×C4—6922no—
269C2×C4—6921no—
270C2×C4—6921no—
271C2×C4C2×C22⋊D17312yes—
272C2×C4C2×C22⋊D17311yescharacteristic
273C2×C4C2×C22⋊D17312yes—
274C2×C4C2×C22⋊D17312yes—
275C2×C4C23×D17311yescharacteristic
276C8—3464no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—6922no—
284C22—6921no—
285C22#1914964812yes—
286C22C22×C22⋊D17311yescharacteristic
287C4—24no—
288C4—22no—
289C4—3462no—
290C4—6921no—
291C4—6921no—
292C4#1914964811yescharacteristic
293C4#1914964811yescharacteristic
294C4C22×C22⋊D17312yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—13841no—
300C2C2×#1914964811yescharacteristic
subgroupG/H
centerC4#19149648
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#585×C173C2
socleC2×C173C2×#120
hypercenter#585D173
hypocenterC173#3018
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #26012289C2semidirect
2. #26012037C2semidirect
3. #26011674C2semidirect
4. #26011669C2semidirect
5. #26011664C2semidirect
6. #26011659C2semidirect
7. C2×#19149606C2semidirect
8. #26011475C2semidirect
9. #585×C173C2semidirect
10. C2×#14583826C22semidirect
12. #19149709C22semidirect
14. #19149704C22semidirect
16. C2×#14583810C22semidirect
17. #19149677C22semidirect
18. #19149675C22semidirect
19. #19149668C22semidirect
20. #19149666C22semidirect
23. C2×C4×Dic173C22semidirect
24. #19149606C22semidirect
25. #19149606C22semidirect
26. #19149598C22semidirect
27. #19149596C22semidirect
68. #41×C173D4semidirect
72. C2×Q8×C173D4semidirect
80. C2×D4×C173D4semidirect
150. C173#3018semidirect
152. #585D173semidirect
200. #41C22⋊D173semidirect
205. C2×Q8C22⋊D173semidirect
216. C2×D4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260122892
#260120372
#260116742
#260116692
#260116642
#260116592
C2×#191496062
#260114752
#585×C1732
#3018173