Finite Group Databasebeta

← #36362021

Subgroups — #36362021

Order 22144.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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×#19149724C211yesmaximal, characteristic
2#26011964C212yesmaximal, complemented
3#26011463C214yesmaximal, complemented
4C2×#19149594C211yesmaximal, characteristic, complemented
5C2×#19149593C212yesmaximal, complemented
6#99×D173C214yesmaximal, complemented
7C2×#99×C173C211yesmaximal, characteristic, complemented
8#19149724—24no—
9#19149724—24no—
10C2×#14583826C412yescomplemented
11C2×#14583825C412yescomplemented
12C2×#14583825C2212yes—
13C22×C22⋊D173C412yescomplemented
14C22×C22⋊D173C2211yescharacteristic
15C22×C22⋊D173C2211yescharacteristic
16C2×D4×D173C412yescomplemented
17C2×D4×D173C2212yes—
18C4×C22⋊D173—24no—
19#19149639—24no—
20#19149635C2214yescomplemented
21C2×C22⋊Dic173C2212yescomplemented
22C22⋊C4×D173C2214yescomplemented
23#19149594C2214yescomplemented
24#19149593C2218yescomplemented
25C22×D4×C173C2211yescharacteristic
26C2×C22⋊C4×C173C2212yescomplemented
27#99×C173C2214yescomplemented
28C23×D173C2×C411yescharacteristic, complemented
29C23×D173C2×C412yescomplemented
30C23×D173C2311yescharacteristic
31#14583826—24no—
32#14583826—24no—
33#14583825—28no—
34#14583825—28no—
35C2×Dic346C2×C412yescomplemented
36C2×D692C2×C412yescomplemented
37C2×C22⋊D173—24no—
38C2×C22⋊D173—24no—
39C2×C22⋊D173—22no—
40C2×C22⋊D173—22no—
41C2×C22⋊D173C2×C412yes—
42C2×C22⋊D173C2×C412yes—
43C2×C22⋊D173C2×C412yescomplemented
44C2×C22⋊D173C2×C412yescomplemented
45C2×C22⋊D173C2×C412yescomplemented
46C2×C22⋊D173D414yes—
47C2×C22⋊D173D414yes—
48C2×C22⋊D173C2312yes—
49D4×D173—28no—
50D4×D173—28no—
51C22×Dic173—22no—
52C22×Dic173C2×C412yescomplemented
53C22×Dic173C2×C411yescharacteristic, complemented
54C22×Dic173C2311yescharacteristic
55C2×C4×D173—24no—
56C2×C4×D173C2×C412yescomplemented
57C2×C4×D173C2×C412yescomplemented
58#14583805—24no—
59C22⋊Dic173—22no—
60C22⋊Dic173C2314yes—
61#14583803—24no—
62C4×Dic173—24no—
63C24×C173C2×C411yescharacteristic, complemented
64C24×C173C2311yescharacteristic
65C2×D4×C173—22no—
66C2×D4×C173—22no—
67C2×D4×C173C2×C412yescomplemented
68C2×D4×C173C2312yes—
69C22×C4×C173—22no—
70C22×C4×C173C2×C411yescharacteristic, complemented
71C22⋊C4×C173—22no—
72C22⋊C4×C173C2314yes—
73C22×D173—22no—
74C22×D173—24no—
75C22×D173—81no—
76C22×D173—82no—
77C22×D173—81no—
78C22×D173C22×C412yes—
79C22×D173C22×C412yes—
80C22×D173C2×D414yes—
81Dic346—24no—
82D692—24no—
83C22⋊D173—22no—
84C22⋊D173—22no—
85C22⋊D173—24no—
86C22⋊D173—24no—
87C22⋊D173—24no—
88C22⋊D173C22⋊C418yes—
89C22⋊D173C22⋊C414yes—
90C22⋊D173C22⋊C414yes—
91C2×Dic173—22no—
92C2×Dic173—24no—
93C2×Dic173—24no—
94C2×Dic173—82no—
95C2×Dic173C22×C412yes—
96C2×Dic173C22×C412yes—
97C2×Dic173C2×D414yes—
98C4×D173—24no—
99C4×D173—24no—
100C4×D173—84no—
101C23×C173—21no—
102C23×C173—21no—
103C23×C173—81no—
104C23×C173—81no—
105C23×C173C22×C412yes—
106C23×C173C22×C411yescharacteristic
107C23×C173C22×C411yescharacteristic
108C23×C173C2×D412yes—
109C23×C173C2×D412yes—
110C23×C173C2411yescharacteristic
111D4×C173—24no—
112D4×C173—24no—
113C2×C4×C173—21no—
114C2×C4×C173—21no—
115C2×C4×C173—24no—
116C2×C4×C173—82no—
117C2×C4×C173C22×C412yes—
118C22×C173—22no—
119C22×C173—42no—
120C22×C173—41no—
121C22×C173—41no—
122C22×C173—42no—
123C22×C173—41no—
124C22×C173—42no—
125C22×C173—81no—
126C22×C173—81no—
127C22×C173C2×C22⋊C411yescharacteristic
128C22×C173C2×C22⋊C411yescharacteristic
129C22×C173C2×C22⋊C412yes—
130C22×C173C23×C411yescharacteristic
131C22×C173C22×D412yes—
132C2×D173—22no—
133C2×D173—44no—
134C2×D173—42no—
135C2×D173—42no—
136C2×D173—84no—
137C2×D173—82no—
138C2×D173C2×C22⋊C414yes—
139C4×C173—22no—
140C4×C173—44no—
141Dic173—22no—
142Dic173—44no—
143Dic173C2×C22⋊C414yes—
144C2×C173—22no—
145C2×C173—21no—
146C2×C173—22no—
147C2×C173—21no—
148C2×C173—21no—
149C2×C173—42no—
150C2×C173C22×C22⋊C411yescharacteristic
151D173—24no—
152D173—42no—
153C173#288311yescharacteristic, complemented, Sylow
154#2883—1731nomaximal, Sylow
155C2×H(5,𝔽2)—1731no—
156#518—1732no—
157#410—1734no—
158C2×#99—1731no—
159C2×#99—1732no—
160C2×#99—1734no—
161C2×#99D17311yescharacteristic, complemented
162H(5,𝔽2)—3464no—
163H(5,𝔽2)—3464no—
164C2×#41—1732no—
165C2×#41—1732no—
166C2×#41—1732no—
167C22×D4—1732no—
168C22×D4—1731no—
169C22×D4—1732no—
170C22×D4—1732no—
171C22×D4—1731no—
172C22×D4C2×D17311yescharacteristic
173C4×D4—3464no—
174C4×D4—3464no—
175#117—1734no—
176C2×C22⋊C4—1732no—
177C2×C22⋊C4—1734no—
178C2×C22⋊C4C2×D17312yescomplemented
179#99—1734no—
180#99—1738no—
181#99C2×D17314yescomplemented
182C24—1732no—
183C24—1731no—
184C24—1731no—
185C24C4×D17311yescharacteristic, complemented
186C24C22×D17311yescharacteristic
187#41—3464no—
188#41—3464no—
189#41—3468no—
190#41—3468no—
191C2×Q8—1732no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—3468no—
195C2×D4—3468no—
196C2×D4—3462no—
197C2×D4—3462no—
198C2×D4—3464no—
199C2×D4—3464no—
200C2×D4—1732no—
201C2×D4—1732no—
202C2×D4—1732no—
203C2×D4—1732no—
204C2×D4—1732no—
205C2×D4—1732no—
206C2×D4—1734no—
207C2×D4—1734no—
208C2×D4—1732no—
209C2×D4C4×D17312yescomplemented
210C2×D4C22×D17312yes—
211C22×C4—22no—
212C22×C4—3464no—
213C22×C4—3462no—
214C22×C4—1731no—
215C22×C4—1732no—
216C22×C4—1732no—
217C22×C4—1732no—
218C22×C4—1731no—
219C22×C4C4×D17311yescharacteristic, complemented
220C4⋊C4—3464no—
221C22⋊C4—22no—
222C22⋊C4—3464no—
223C22⋊C4—3462no—
224C22⋊C4—1734no—
225C22⋊C4C22×D17314yes—
226C42—3464no—
227C23—21no—
228C23—21no—
229C23—81no—
230C23—81no—
231C23—3464no—
232C23—3462no—
233C23—13841no—
234C23—13842no—
235C23—13841no—
236C23—1732no—
237C23—1732no—
238C23—1734no—
239C23C2×C4×D17312yes—
240C23C2×C4×D17311yescharacteristic
241C23C2×C4×D17311yescharacteristic
242C23D4×D17312yes—
243C23D4×D17312yes—
244C23C23×D17311yescharacteristic
245Q8—3464no—
246D4—24no—
247D4—24no—
248D4—3464no—
249D4—3464no—
250D4—3462no—
251D4—3462no—
252D4—3464no—
253D4—3464no—
254D4—1734no—
255D4—1734no—
256D4—1738no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—82no—
261C2×C4—3464no—
262C2×C4—3464no—
263C2×C4—3462no—
264C2×C4—3464no—
265C2×C4—3464no—
266C2×C4—13844no—
267C2×C4—13842no—
268C2×C4—1732no—
269C2×C4—1732no—
270C2×C4—1734no—
271C2×C4C2×C4×D17312yes—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—3462no—
282C22—13842no—
283C22—13844no—
284C22—1734no—
285C22—6924no—
286C22—6922no—
287C22—6922no—
288C22C22⋊C4×D17311yescharacteristic
289C22C22⋊C4×D17311yescharacteristic
290C22C22⋊C4×D17312yes—
291C22C22×C4×D17311yescharacteristic
292C22C2×D4×D17312yes—
293C4—22no—
294C4—44no—
295C4—3462no—
296C4—1734no—
297C4—6924no—
298C2—22no—
299C2—21no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—42no—
304C2—3464no—
305C2—6922no—
306C2C2×C22⋊C4×D17311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D173
commutatorC22×C173C23×C4
FrattiniC23C23×D173
FittingC2×#99×C173C2
socleC2×C173C22×C22⋊C4
prime radicalC2×#19149724C2
cosocleC23×C173C24
hypercenterC2×#99D173
hypocenterC173#2883
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #26011964C2semidirect
3. #26011463C2semidirect
4. C2×#19149594C2semidirect
5. C2×#19149593C2semidirect
6. #99×D173C2semidirect
7. C2×#99×C173C2semidirect
10. C2×#14583826C4semidirect
11. C2×#14583825C4semidirect
13. C22×C22⋊D173C4semidirect
16. C2×D4×D173C4semidirect
20. #19149635C22semidirect
21. C2×C22⋊Dic173C22semidirect
22. C22⋊C4×D173C22semidirect
23. #19149594C22semidirect
24. #19149593C22semidirect
26. C2×C22⋊C4×C173C22semidirect
27. #99×C173C22semidirect
28. C23×D173C2×C4semidirect
29. C23×D173C2×C4semidirect
35. C2×Dic346C2×C4semidirect
36. C2×D692C2×C4semidirect
43. C2×C22⋊D173C2×C4semidirect
44. C2×C22⋊D173C2×C4semidirect
45. C2×C22⋊D173C2×C4semidirect
52. C22×Dic173C2×C4semidirect
53. C22×Dic173C2×C4semidirect
56. C2×C4×D173C2×C4semidirect
57. C2×C4×D173C2×C4semidirect
63. C24×C173C2×C4semidirect
67. C2×D4×C173C2×C4semidirect
70. C22×C4×C173C2×C4semidirect
153. C173#2883semidirect
161. C2×#99D173semidirect
178. C2×C22⋊C4C2×D173semidirect
181. #99C2×D173semidirect
185. C24C4×D173semidirect
209. C2×D4C4×D173semidirect
219. C22×C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497242
#260119642
#260114632
C2×#191495942
C2×#191495932
#99×D1732
C2×#99×C1732
#2883173