Finite Group Databasebeta

← #36364347

Subgroups — #36364347

Order 22144.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#26012004C211yesmaximal, characteristic
2C2×#19149655C211yesmaximal, characteristic, complemented
3#26011831C212yesmaximal, complemented
4#26011819C211yesmaximal, characteristic, complemented
5#26011805C211yesmaximal, characteristic, complemented
6#26011729C212yesmaximal, complemented
7#26011723C212yesmaximal, complemented
8#26011652C212yesmaximal, complemented
9#26011630C211yesmaximal, characteristic, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#451×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic
13C22×C4×D173C2211yescharacteristic
14C2×#14583813C2212yescomplemented
15C2×C8×D173C2212yes—
16#19149664—21no—
17#19149655—21no—
18#19149655—21no—
19#19149655C2212yes—
20#19149655C2212yes—
21#19149655C2211yescharacteristic
22#19149654C2211yescharacteristic
23#19149653C2211yescharacteristic
24#19149649—21no—
25#19149641—21no—
26#19149639—21no—
27#19149637C2211yescharacteristic
28C2×C4⋊Dic173C2211yescharacteristic
29C2×#14583805—21no—
30C2×#14583803—21no—
31#19149613C2212yescomplemented
32#19149610C2212yescomplemented
33#19149603C2212yescomplemented
34#19149603C2212yescomplemented
35#19149601C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yes—
38C22×Q8×C173C2211yescharacteristic
39C2×Q16×C173C2212yescomplemented
40C22⋊Q8×C173C2211yescharacteristic
41#103×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44#14583827—42no—
45C2×D692—21no—
46C2×D692—21no—
47C2×D692C2311yescharacteristic
48C2×C22⋊D173—21no—
49C22×Dic173—21no—
50C22×Dic173D411yescharacteristic
51C2×C4×D173—21no—
52C2×C4×D173—42no—
53C2×C4×D173D411yescharacteristic
54C2×C4×D173D411yescharacteristic
55C2×C4×D173C2311yescharacteristic
56C2×C4×D173C2311yescharacteristic
57#14583813—42no—
58C2×#13063436C2312yes—
59C8×D173—42no—
60C4⋊Dic173—21no—
61C4⋊Dic173C2312yes—
62C4⋊Dic173C2311yescharacteristic
63#14583805—22no—
64#14583805—22no—
65#14583805—21no—
66#14583803—22no—
67#14583803—22no—
68#14583803—21no—
69#14583803—21no—
70#14583803—21no—
71#14583803—21no—
72C4×Dic173—21no—
73C2×Q8×C173—21no—
74C2×Q8×C173—42no—
75C2×Q8×C173C2311yescharacteristic
76C2×Q8×C173C2312yes—
77C22×C4×C173—21no—
78C22×C4×C173C2311yescharacteristic
79Q16×C173—42no—
80C2×C8×C173C2312yes—
81C4⋊C4×C173—21no—
82C4⋊C4×C173C2311yescharacteristic
83C22⋊C4×C173—21no—
84C22×D173—22no—
85C22×D173—22no—
86C22×D173—21no—
87C22×D173—41no—
88C22×D173C2×D411yescharacteristic
89C22×D173C2×D411yescharacteristic
90D692—22no—
91D692—42no—
92D692—42no—
93C22⋊D173—42no—
94C2×Dic173—22no—
95C2×Dic173—21no—
96C2×Dic173—41no—
97C2×Dic173—42no—
98C2×Dic173C2×D411yescharacteristic
99C2×Dic173C2×D411yescharacteristic
100C4×D173—22no—
101C4×D173—22no—
102C4×D173—41no—
103C4×D173—41no—
104C4×D173—42no—
105#13063436—22no—
106C23×C173C2×D411yescharacteristic
107Q8×C173—22no—
108Q8×C173—22no—
109Q8×C173—22no—
110Q8×C173—42no—
111C2×C4×C173—22no—
112C2×C4×C173—21no—
113C2×C4×C173—21no—
114C2×C4×C173—22no—
115C2×C4×C173—42no—
116C2×C4×C173C2×D411yescharacteristic
117C2×C4×C173C2411yescharacteristic
118C8×C173—22no—
119C22×C173—21no—
120C22×C173—21no—
121C22×C173#13512yes—
122C22×C173C22×D411yescharacteristic
123C2×D173—21no—
124C2×D173—21no—
125C2×D173—21no—
126C2×D173—21no—
127C2×D173—21no—
128C2×D173—44no—
129C2×D173—41no—
130C2×D173—41no—
131C2×D173—41no—
132C2×D173#13712yes—
133C4×C173—21no—
134C4×C173—42no—
135C4×C173—41no—
136C4×C173—41no—
137C4×C173H(5,𝔽2)12yes—
138Dic173—21no—
139Dic173—21no—
140Dic173—42no—
141Dic173—41no—
142C2×C173—22no—
143C2×C173#53411yescharacteristic
144C2×C173C2×#13511yescharacteristic
145C2×C173C2×#13711yescharacteristic
146D173—22no—
147D173—41no—
148D173—81no—
149C173#319011yescharacteristic, complemented, Sylow
150#3190—1731nomaximal, Sylow
151#525—1731no—
152C2×C22⋊Q8—1731no—
153#485—1732no—
154#483—1731no—
155#480—1731no—
156#467—1732no—
157#465—1732no—
158#452—1732no—
159#451D17311yescharacteristic, complemented
160#448—1731no—
161C2×C22⋊C8—1731no—
162C2×#41—1731no—
163C22×Q8C2×D17311yescharacteristic
164C23×C4—1731no—
165C2×Q16C2×D17312yescomplemented
166C2×SD16—1732no—
167C22×C8—1732no—
168#123—3461no—
169C22⋊Q8—3461no—
170C22⋊Q8—3461no—
171C22⋊Q8—1732no—
172C22⋊Q8—1732no—
173C22⋊Q8—1731no—
174C22⋊Q8C2×D17311yescharacteristic
175#121—1731no—
176#121—1731no—
177#120—3461no—
178C4×D4—3461no—
179C4×D4—3461no—
180#117—1731no—
181C2×C4⋊C4—3461no—
182C2×C4⋊C4—1731no—
183C2×C22⋊C4—3461no—
184#107—1732no—
185#106—1732no—
186#103—1732no—
187#103—1732no—
188#103C2×D17312yescomplemented
189#102—1732no—
190C22⋊C8—1731no—
191C22⋊C8—1732no—
192C22⋊C8C2×D17311yescharacteristic
193C24—1731no—
194#41—6922no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8C22×D17311yescharacteristic
198C2×Q8C22×D17312yes—
199C2×D4—3461no—
200C2×D4—3461no—
201C2×D4—3461no—
202C2×D4—1731no—
203C22×C4—21no—
204C22×C4—3461no—
205C22×C4—3461no—
206C22×C4—1731no—
207C22×C4—1731no—
208C22×C4—1731no—
209C22×C4—1731no—
210C22×C4—1731no—
211C22×C4—6922no—
212C22×C4C22×D17311yescharacteristic
213Q16—42no—
214SD16—6922no—
215C2×C8—1732no—
216C2×C8—6922no—
217C2×C8C22×D17312yes—
218C4⋊C4—21no—
219C4⋊C4—3462no—
220C4⋊C4—3462no—
221C4⋊C4—3461no—
222C4⋊C4—3461no—
223C4⋊C4—1732no—
224C4⋊C4—1731no—
225C4⋊C4C22×D17311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3462no—
228C22⋊C4—3462no—
229C22⋊C4—3461no—
230C22⋊C4—3461no—
231C22⋊C4—3461no—
232C22⋊C4—3461no—
233C42—3461no—
234C23—3462no—
235C23—3462no—
236C23—3461no—
237C23—1731no—
238C23—1731no—
239C23—6921no—
240C23D4×D17311yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—22no—
244Q8—42no—
245D4—3462no—
246D4—6922no—
247D4—6922no—
248D4—6922no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—3462no—
255C2×C4—3462no—
256C2×C4—3461no—
257C2×C4—3462no—
258C2×C4—1731no—
259C2×C4—1731no—
260C2×C4—6921no—
261C2×C4—6921no—
262C2×C4—6922no—
263C2×C4—6921no—
264C2×C4—6922no—
265C2×C4D4×D17311yescharacteristic
266C2×C4C23×D17311yescharacteristic
267C8—22no—
268C8—3462no—
269C22—21no—
270C22—21no—
271C22—3461no—
272C22—3461no—
273C22—3461no—
274C22—3461no—
275C22—3461no—
276C22—1732no—
277C22—6924no—
278C22—6921no—
279C22—6921no—
280C22—6921no—
281C22#1914970212yes—
282C22C2×D4×D17311yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—3461no—
288C4—3461no—
289C4—6922no—
290C4—6921no—
291C4#1914972412yes—
292C2—22no—
293C2—3462no—
294C2—13841no—
295C2—6921no—
296C2#2601205111yescharacteristic
297C2C2×#1914970211yescharacteristic
298C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#451×C173C2
socleC22×C173C22×D4
prime radical#26012004C2
hypercenter#451D173
hypocenterC173#3190
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#19149655C2semidirect
3. #26011831C2semidirect
4. #26011819C2semidirect
5. #26011805C2semidirect
6. #26011729C2semidirect
7. #26011723C2semidirect
8. #26011652C2semidirect
9. #26011630C2semidirect
10. C22⋊C8×D173C2semidirect
11. #451×C173C2semidirect
14. C2×#14583813C22semidirect
31. #19149613C22semidirect
32. #19149610C22semidirect
33. #19149603C22semidirect
34. #19149603C22semidirect
35. #19149601C22semidirect
39. C2×Q16×C173C22semidirect
41. #103×C173C22semidirect
149. C173#3190semidirect
159. #451D173semidirect
165. C2×Q16C2×D173semidirect
188. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120042
C2×#191496552
#260118312
#260118192
#260118052
#260117292
#260117232
#260116522
#260116302
C22⋊C8×D1732
#451×C1732
#3190173