Finite Group Databasebeta

← #36363484

Subgroups — #36363484

Order 22144.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#26011998C211yesmaximal, characteristic
2C2×C4⋊C4×D173C211yesmaximal, characteristic, complemented
3#26011825C212yesmaximal, complemented
4#26011820C212yesmaximal, complemented
5#26011808C212yesmaximal, complemented
6#26011803C212yesmaximal, complemented
7#26011533C211yesmaximal, characteristic, complemented
8#26011531C211yesmaximal, characteristic, complemented
9#26011492C211yesmaximal, characteristic, complemented
10#26011454C211yesmaximal, characteristic, complemented
11#417×C173C211yesmaximal, characteristic, complemented
12C22×C4×D173—21no—
13C22×C4×D173C2211yescharacteristic
14C2×#14583810C2211yescharacteristic
15#19149660—21no—
16#19149655C2211yescharacteristic
17#19149655C2211yescharacteristic
18#19149650C2211yescharacteristic
19#19149650C2211yescharacteristic
20#19149647—21no—
21C4×C22⋊D173—21no—
22C4×C22⋊D173—21no—
23#19149638C2211yescharacteristic
24C2×C4⋊Dic173C2211yescharacteristic
25C2×#14583805—21no—
26C4⋊C4×D173—21no—
27C4⋊C4×D173—21no—
28C4⋊C4×D173C2212yes—
29C4⋊C4×D173C2212yes—
30#19149613C2212yescomplemented
31#19149612C2212yescomplemented
32#19149611C2212yescomplemented
33#19149610C2212yescomplemented
34#19149603C2212yescomplemented
35#19149600C2212yescomplemented
36#19149591C2212yes—
37#19149591C2212yes—
38C2×#41×C173C2211yescharacteristic
39C2×M16×C173C2211yescharacteristic
40C2×C4⋊C4×C173C2211yescharacteristic
41#103×C173C2212yescomplemented
42#102×C173C2212yescomplemented
43C23×D173D411yescharacteristic
44C2×C22⋊D173—21no—
45C2×C22⋊D173—21no—
46C22×Dic173—21no—
47C22×Dic173D411yescharacteristic
48C2×C4×D173—22no—
49C2×C4×D173—22no—
50C2×C4×D173—22no—
51C2×C4×D173—22no—
52C2×C4×D173—44no—
53C2×C4×D173D411yescharacteristic
54C2×C4×D173D411yescharacteristic
55C2×C4×D173C2311yescharacteristic
56C2×C4×D173C2311yescharacteristic
57#14583810—22no—
58C2×#13063436C2312yes—
59C4⋊Dic173—21no—
60C4⋊Dic173C2312yes—
61C4⋊Dic173C2311yescharacteristic
62C4⋊Dic173C2311yescharacteristic
63#14583805—22no—
64#14583805—22no—
65#14583805—21no—
66#14583805—21no—
67C22⋊Dic173—21no—
68C22⋊Dic173—21no—
69C22⋊Dic173—21no—
70#14583803—21no—
71#14583803—21no—
72C4×Dic173—21no—
73#41×C173—22no—
74#41×C173—22no—
75C2×Q8×C173C2311yescharacteristic
76C2×D4×C173—21no—
77C2×D4×C173C2311yescharacteristic
78C22×C4×C173—21no—
79C22×C4×C173—21no—
80C22×C4×C173C2311yescharacteristic
81M16×C173—22no—
82C2×C8×C173C2312yes—
83C4⋊C4×C173—21no—
84C4⋊C4×C173C2312yes—
85C22×D173—22no—
86C22×D173—22no—
87C22×D173—22no—
88C22×D173C2×D411yescharacteristic
89C22×D173C2×D411yescharacteristic
90C22⋊D173—42no—
91C22⋊D173—42no—
92C2×Dic173—22no—
93C2×Dic173—22no—
94C2×Dic173—21no—
95C2×Dic173—21no—
96C2×Dic173—42no—
97C2×Dic173C2×D411yescharacteristic
98C2×Dic173C2×D411yescharacteristic
99C4×D173—22no—
100C4×D173—22no—
101C4×D173—21no—
102C4×D173—21no—
103C4×D173—21no—
104C4×D173—21no—
105C4×D173—44no—
106C4×D173—44no—
107#13063436—22no—
108C23×C173—21no—
109C23×C173C2×D411yescharacteristic
110Q8×C173—21no—
111Q8×C173—21no—
112D4×C173—21no—
113D4×C173—21no—
114D4×C173—42no—
115C2×C4×C173—22no—
116C2×C4×C173—21no—
117C2×C4×C173—42no—
118C2×C4×C173—42no—
119C2×C4×C173C2×D411yescharacteristic
120C2×C4×C173#4112yes—
121C2×C4×C173#4112yes—
122C2×C4×C173C2411yescharacteristic
123C8×C173—22no—
124C22×C173—21no—
125C22×C173—21no—
126C22×C173—41no—
127C22×C173—41no—
128C22×C173—41no—
129C22×C173#12312yes—
130C22×C173C22×D411yescharacteristic
131C2×D173—28no—
132C2×D173—21no—
133C2×D173—21no—
134C2×D173—21no—
135C2×D173—21no—
136C2×D173AGL(1,ℤ/8)12yes—
137C2×D173#13712yes—
138C4×C173—41no—
139C4×C173—42no—
140C4×C173#12312yes—
141C4×C173C2×#4111yescharacteristic
142C4×C173C2×#4111yescharacteristic
143Dic173—21no—
144Dic173—21no—
145Dic173—42no—
146Dic173—41no—
147Dic173—41no—
148C2×C173—22no—
149C2×C173—81no—
150C2×C173C2×#12311yescharacteristic
151C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C173C2×#13711yescharacteristic
153D173—22no—
154D173—22no—
155C173#309211yescharacteristic, complemented, Sylow
156#3092—1731nomaximal, Sylow
157#525—1731no—
158C22×C4⋊C4—1731no—
159#484—1732no—
160#483—1732no—
161#481—1732no—
162#480—1732no—
163#428—1731no—
164#428—1731no—
165#417—1731no—
166#417D17311yescharacteristic, complemented
167#407—1731no—
168C2×#41C2×D17311yescharacteristic
169C23×C4—3461no—
170C23×C4—1731no—
171C2×M16—1731no—
172C2×M16C2×D17311yescharacteristic
173#123—3461no—
174C22⋊Q8—1731no—
175C22⋊Q8—1731no—
176#121—1731no—
177#121—1731no—
178#120—3461no—
179C4×D4—3461no—
180C4×D4—3461no—
181#117—1731no—
182C2×C4⋊C4—3461no—
183C2×C4⋊C4—3461no—
184C2×C4⋊C4—3461no—
185C2×C4⋊C4—1731no—
186C2×C4⋊C4—1732no—
187C2×C4⋊C4—1732no—
188C2×C4⋊C4C2×D17311yescharacteristic
189#107—1732no—
190#107—1732no—
191#106—1732no—
192#106—1732no—
193#103—1732no—
194#103C2×D17312yescomplemented
195#102—1732no—
196#102C2×D17312yescomplemented
197C22⋊C8—1732no—
198C22⋊C8—1732no—
199C24—1731no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D17311yescharacteristic
203C2×D4—21no—
204C2×D4—3461no—
205C2×D4—3461no—
206C2×D4C22×D17311yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—3462no—
210C22×C4—3462no—
211C22×C4—3462no—
212C22×C4—3462no—
213C22×C4—3461no—
214C22×C4—1731no—
215C22×C4—1731no—
216C22×C4—1731no—
217C22×C4—1731no—
218C22×C4—1731no—
219C22×C4—6924no—
220C22×C4C22×D17311yescharacteristic
221M16—22no—
222M16—3462no—
223C2×C8—1732no—
224C2×C8C22×D17312yes—
225C4⋊C4—21no—
226C4⋊C4—3462no—
227C4⋊C4—3462no—
228C4⋊C4—3461no—
229C4⋊C4—3461no—
230C4⋊C4—3461no—
231C4⋊C4—1732no—
232C4⋊C4—1731no—
233C4⋊C4—1731no—
234C4⋊C4C22×D17312yes—
235C22⋊C4—3461no—
236C22⋊C4—3461no—
237C22⋊C4—3461no—
238C22⋊C4—3461no—
239C22⋊C4—3461no—
240C42—3461no—
241C23—21no—
242C23—3462no—
243C23—3462no—
244C23—3462no—
245C23—1731no—
246C23—1731no—
247C23D4×D17311yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—6922no—
254D4—6922no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—3462no—
260C2×C4—3462no—
261C2×C4—3462no—
262C2×C4—3461no—
263C2×C4—3461no—
264C2×C4—3461no—
265C2×C4—3461no—
266C2×C4—3461no—
267C2×C4—3461no—
268C2×C4—3462no—
269C2×C4—1731no—
270C2×C4—1731no—
271C2×C4—6924no—
272C2×C4—6924no—
273C2×C4—6922no—
274C2×C4D4×D17311yescharacteristic
275C2×C4#1458382512yes—
276C2×C4#1458382612yes—
277C2×C4C23×D17311yescharacteristic
278C8—22no—
279C8—3462no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—3468no—
286C22—3461no—
287C22—3461no—
288C22—3461no—
289C22—3461no—
290C22—1732no—
291C22—1732no—
292C22#1914966112yes—
293C22C2×D4×D17311yescharacteristic
294C4—41no—
295C4—42no—
296C4—3461no—
297C4—3461no—
298C4—6922no—
299C4—6921no—
300C4—6921no—
301C4#1914966112yes—
302C4C2×#1458382511yescharacteristic
303C4C2×#1458382611yescharacteristic
304C2—22no—
305C2—81no—
306C2—3462no—
307C2—3462no—
308C2C2×#1914966111yescharacteristic
309C2AGL(1,ℤ/8)×D17311yescharacteristic
310C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#417×C173C2
socleC22×C173C22×D4
prime radical#26011998C2
hypercenter#417D173
hypocenterC173#3092
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D173C2semidirect
3. #26011825C2semidirect
4. #26011820C2semidirect
5. #26011808C2semidirect
6. #26011803C2semidirect
7. #26011533C2semidirect
8. #26011531C2semidirect
9. #26011492C2semidirect
10. #26011454C2semidirect
11. #417×C173C2semidirect
30. #19149613C22semidirect
31. #19149612C22semidirect
32. #19149611C22semidirect
33. #19149610C22semidirect
34. #19149603C22semidirect
35. #19149600C22semidirect
41. #103×C173C22semidirect
42. #102×C173C22semidirect
155. C173#3092semidirect
166. #417D173semidirect
194. #103C2×D173semidirect
196. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260119982
C2×C4⋊C4×D1732
#260118252
#260118202
#260118082
#260118032
#260115332
#260115312
#260114922
#260114542
#417×C1732
#3092173