Finite Group Databasebeta

← #36363474

Subgroups — #36363474

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26011998C211yesmaximal, characteristic
2#117×D173C211yesmaximal, characteristic, complemented
3#26011831C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011825C211yesmaximal, characteristic, complemented
6#26011820C211yesmaximal, characteristic, complemented
7#26011814C211yesmaximal, characteristic, complemented
8#26011813C211yesmaximal, characteristic, complemented
9#26011808C211yesmaximal, characteristic, complemented
10#26011803C211yesmaximal, characteristic, complemented
11#26011529C211yesmaximal, characteristic, complemented
12#26011527C211yesmaximal, characteristic, complemented
13#26011487C211yesmaximal, characteristic, complemented
14C2×#19149591C211yesmaximal, characteristic, complemented
15#416×C173C211yesmaximal, characteristic, complemented
16C22×C4×D173C2211yescharacteristic
17C22×#13063436C2211yescharacteristic
18#19149660—21no—
19#19149655C2211yescharacteristic
20#19149655C2211yescharacteristic
21#19149650C2211yescharacteristic
22#19149650C2211yescharacteristic
23#19149647—21no—
24C4×C22⋊D173—21no—
25C4×C22⋊D173—21no—
26#19149638C2211yescharacteristic
27#19149638C2211yescharacteristic
28#19149637C2211yescharacteristic
29#19149637C2211yescharacteristic
30#19149636—21no—
31#19149635—21no—
32C4⋊C4×D173C2211yescharacteristic
33C4⋊C4×D173C2211yescharacteristic
34C22⋊C4×D173—21no—
35C42×D173—21no—
36#19149613C2211yescharacteristic, complemented
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149611C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149610C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic, complemented
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54#117×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173D411yescharacteristic
60C2×C22⋊D173—21no—
61C2×C22⋊D173—21no—
62C22×Dic173D411yescharacteristic
63C2×C4×D173—21no—
64C2×C4×D173—21no—
65C2×C4×D173—22no—
66C2×C4×D173—22no—
67C2×C4×D173D411yescharacteristic
68C2×C4×D173D411yescharacteristic
69C2×C4×D173C2311yescharacteristic
70C2×C4×D173C2311yescharacteristic
71C2×#13063436—22no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94#41×C173—22no—
95#41×C173—22no—
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173—21no—
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173—21no—
100C22×C4×C173C2311yescharacteristic
101C2×C8×C173—22no—
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173—21no—
108C22×D173—22no—
109C22×D173—22no—
110C22×D173—41no—
111C22×D173C2×D411yescharacteristic
112C22×D173C2×D411yescharacteristic
113C22⋊D173—42no—
114C22⋊D173—42no—
115C2×Dic173—22no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C4×D173—22no—
123C4×D173—22no—
124C4×D173—21no—
125C4×D173—21no—
126C4×D173—21no—
127C4×D173—21no—
128C4×D173—42no—
129C4×D173—42no—
130#13063436—21no—
131#13063436—21no—
132C23×C173—21no—
133C23×C173C2×D411yescharacteristic
134Q8×C173—21no—
135Q8×C173—21no—
136D4×C173—21no—
137D4×C173—21no—
138D4×C173—42no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—42no—
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173#4112yes—
145C2×C4×C173#4112yes—
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173#12312yes—
155C22×C173C22×D411yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—44no—
162C2×D173#13512yes—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173#12312yes—
167C4×C173C2×#4111yescharacteristic
168C4×C173C2×#4111yescharacteristic
169Dic173—21no—
170Dic173—21no—
171Dic173—41no—
172Dic173—41no—
173Dic173—41no—
174Dic173—41no—
175C2×C173—22no—
176C2×C173—81no—
177C2×C173C2×#12311yescharacteristic
178C2×C173C2×#13511yescharacteristic
179C2×C173#57511yescharacteristic
180D173—22no—
181D173—41no—
182C173#309111yescharacteristic, complemented, Sylow
183#3091—1731nomaximal, Sylow
184#525—1731no—
185C2×#117—1731no—
186#485—1731no—
187#485—1731no—
188#484—1731no—
189#483—1731no—
190#482—1731no—
191#482—1731no—
192#481—1731no—
193#480—1731no—
194#427—1731no—
195#427—1731no—
196#416—1731no—
197#416D17311yescharacteristic, complemented
198C2×C22⋊C8—1731no—
199C2×#41C2×D17311yescharacteristic
200C23×C4—1731no—
201C22×C8—1731no—
202C22×C8C2×D17311yescharacteristic
203#123—3461no—
204C22⋊Q8—1731no—
205C22⋊Q8—1731no—
206#121—1731no—
207#121—1731no—
208#120—3461no—
209C4×D4—3461no—
210C4×D4—3461no—
211#117—3461no—
212#117—3461no—
213#117—1731no—
214#117—1731no—
215#117—1731no—
216#117—1731no—
217#117C2×D17311yescharacteristic
218C2×C4⋊C4—1731no—
219C2×C4⋊C4—1731no—
220C2×C22⋊C4—3461no—
221C2×C42—3461no—
222#107—1731no—
223#107—1731no—
224#107—1731no—
225#107—1731no—
226#106—1731no—
227#106—1731no—
228#106—1731no—
229#106—1731no—
230#103—1731no—
231#103—1731no—
232#103C2×D17311yescharacteristic, complemented
233#103C2×D17311yescharacteristic, complemented
234#102—1731no—
235#102—1731no—
236#102C2×D17311yescharacteristic, complemented
237#102C2×D17311yescharacteristic, complemented
238C22⋊C8—1731no—
239C22⋊C8—1731no—
240C22⋊C8—1731no—
241C22⋊C8—1731no—
242C24—1731no—
243#41—22no—
244#41—22no—
245C2×Q8C22×D17311yescharacteristic
246C2×D4—21no—
247C2×D4—3461no—
248C2×D4—3461no—
249C2×D4C22×D17311yescharacteristic
250C22×C4—21no—
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4—3462no—
254C22×C4—3462no—
255C22×C4—1731no—
256C22×C4—1731no—
257C22×C4—1731no—
258C22×C4—1731no—
259C22×C4—1731no—
260C22×C4C22×D17311yescharacteristic
261C2×C8—22no—
262C2×C8—3462no—
263C2×C8—1731no—
264C2×C8—1731no—
265C2×C8C22×D17311yescharacteristic
266C2×C8C22×D17311yescharacteristic
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4C22×D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C42—21no—
287C42—3461no—
288C42—3461no—
289C42—3461no—
290C42—3461no—
291C23—21no—
292C23—3462no—
293C23—3462no—
294C23—1731no—
295C23—1731no—
296C23—6921no—
297C23D4×D17311yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—6922no—
304D4—6922no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—3462no—
310C2×C4—3462no—
311C2×C4—3462no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4—6922no—
323C2×C4—6922no—
324C2×C4D4×D17311yescharacteristic
325C2×C4#1458382512yes—
326C2×C4#1458382612yes—
327C2×C4C23×D17311yescharacteristic
328C8—21no—
329C8—21no—
330C8—3461no—
331C8—3461no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—3461no—
338C22—3461no—
339C22—3461no—
340C22—3461no—
341C22—3461no—
342C22—1732no—
343C22—6924no—
344C22#1914966112yes—
345C22C2×D4×D17311yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3461no—
350C4—3461no—
351C4—6921no—
352C4—6921no—
353C4—6921no—
354C4—6921no—
355C4#1914966112yes—
356C4C2×#1458382511yescharacteristic
357C4C2×#1458382611yescharacteristic
358C2—22no—
359C2—81no—
360C2—3462no—
361C2—6921no—
362C2C2×#1914966111yescharacteristic
363C2#135×D17311yescharacteristic
364C2#2601228811yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
prime radical#26011998C2
hypercenter#416D173
hypocenterC173#3091
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #117×D173C2semidirect
3. #26011831C2semidirect
4. #26011830C2semidirect
5. #26011825C2semidirect
6. #26011820C2semidirect
7. #26011814C2semidirect
8. #26011813C2semidirect
9. #26011808C2semidirect
10. #26011803C2semidirect
11. #26011529C2semidirect
12. #26011527C2semidirect
13. #26011487C2semidirect
14. C2×#19149591C2semidirect
15. #416×C173C2semidirect
36. #19149613C22semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149612C22semidirect
40. #19149611C22semidirect
41. #19149611C22semidirect
42. #19149610C22semidirect
43. #19149610C22semidirect
44. #19149603C22semidirect
45. #19149603C22semidirect
46. #19149600C22semidirect
47. #19149600C22semidirect
55. #103×C173C22semidirect
56. #103×C173C22semidirect
57. #102×C173C22semidirect
58. #102×C173C22semidirect
182. C173#3091semidirect
197. #416D173semidirect
232. #103C2×D173semidirect
233. #103C2×D173semidirect
236. #102C2×D173semidirect
237. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260119982
#117×D1732
#260118312
#260118302
#260118252
#260118202
#260118142
#260118132
#260118082
#260118032
#260115292
#260115272
#260114872
C2×#191495912
#416×C1732
#3091173