Finite Group Databasebeta

← #36363514

Subgroups — #36363514

Order 22144.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012057C211yesmaximal, characteristic
2#26011972C211yesmaximal, characteristic, complemented
3#26011818C211yesmaximal, characteristic, complemented
4#26011816C211yesmaximal, characteristic, complemented
5#26011813C211yesmaximal, characteristic, complemented
6#26011812C211yesmaximal, characteristic, complemented
7#26011809C211yesmaximal, characteristic, complemented
8#26011808C211yesmaximal, characteristic, complemented
9#26011806C211yesmaximal, characteristic, complemented
10#26011532C211yesmaximal, characteristic, complemented
11#26011528C211yesmaximal, characteristic, complemented
12#26011494C211yesmaximal, characteristic, complemented
13#102×D173C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#480×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149675—21no—
21#19149665—21no—
22#19149654C2211yescharacteristic
23#19149653C2211yescharacteristic
24#19149650C2211yescharacteristic
25#19149649—21no—
26#19149647—21no—
27#19149644—21no—
28#19149641—21no—
29C4×C22⋊D173—21no—
30C4×C22⋊D173—21no—
31#19149638C2211yescharacteristic, complemented
32#19149637C2211yescharacteristic
33#19149637C2211yescharacteristic
34#19149637C2211yescharacteristic, complemented
35C4⋊C4×D173C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic, complemented
37#19149613C2211yescharacteristic, complemented
38#19149612C2211yescharacteristic, complemented
39#19149611C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52#121×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic, complemented
54#107×C173C2211yescharacteristic, complemented
55#107×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—22no—
61#14583826—22no—
62C2×Dic346D411yescharacteristic
63C2×D692—21no—
64C2×D692D411yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173D411yescharacteristic
70C2×C22⋊D173D411yescharacteristic
71D4×D173—22no—
72D4×D173—22no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76C2×C4×D173C2311yescharacteristic
77#14583809—22no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—22no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C2×D4×C173—21no—
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173—21no—
100C22×C4×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173—21no—
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22×D173—21no—
109C22×D173—42no—
110C22×D173—81no—
111C22×D173—81no—
112C22×D173C2×D411yescharacteristic
113C22×D173C2×D411yescharacteristic
114Dic346—22no—
115D692—22no—
116D692—21no—
117D692—21no—
118D692—81no—
119C22⋊D173—22no—
120C22⋊D173—22no—
121C22⋊D173—42no—
122C22⋊D173—81no—
123C22⋊D173—81no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—81no—
132C4×D173—81no—
133C4×D173#4112yes—
134C4×D173#4112yes—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138C23×C173C2×D411yescharacteristic
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—81no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—81no—
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—84no—
168C2×D173#12312yes—
169C4×C173—21no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173—41no—
173C4×C173C2×#4111yescharacteristic
174C4×C173C2×#4111yescharacteristic
175Dic173—21no—
176Dic173—41no—
177Dic173—41no—
178Dic173—41no—
179Dic173#12312yes—
180C2×C173—41no—
181C2×C173—81no—
182C2×C173C2×#12311yescharacteristic
183C2×C173#57611yescharacteristic
184C2×C173#57711yescharacteristic
185D173—22no—
186D173—41no—
187D173—81no—
188C173#309611yescharacteristic, complemented, Sylow
189#3096—1731nomaximal, Sylow
190#535—1731no—
191#520—1731no—
192#482—1731no—
193#482—1731no—
194#482—1731no—
195#482—1731no—
196#481—1731no—
197#481—1731no—
198#480—1731no—
199#480D17311yescharacteristic, complemented
200#428—1731no—
201#427—1731no—
202#418—1731no—
203C2×#102—1731no—
204#408—1731no—
205C2×#41—1731no—
206C22×D4—1731no—
207C2×M16—1731no—
208C22×C8—1731no—
209#127—3461no—
210#124—3461no—
211#121—1731no—
212#121—1731no—
213#121—1731no—
214#121C2×D17311yescharacteristic
215#120—3461no—
216#120—3461no—
217C4×Q8—3461no—
218C4×D4—3461no—
219C4×D4—3461no—
220C4×D4—3461no—
221#117—1731no—
222#117—1731no—
223#117—1731no—
224#117—1731no—
225C2×C4⋊C4—1731no—
226C2×C4⋊C4—1731no—
227C2×C4⋊C4C2×D17311yescharacteristic, complemented
228#107—1731no—
229#107—1731no—
230#107C2×D17311yescharacteristic, complemented
231#107C2×D17311yescharacteristic, complemented
232#106—1731no—
233#106—1731no—
234#106—1731no—
235#106—1731no—
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242#102C2×D17311yescharacteristic
243#102C2×D17311yescharacteristic
244C22⋊C8—1731no—
245C22⋊C8—1731no—
246C22⋊C8—1731no—
247C22⋊C8C2×D17311yescharacteristic, complemented
248C24—3461no—
249#41—3462no—
250#41—3462no—
251C2×Q8—1731no—
252C2×D4—21no—
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—3461no—
256C2×D4—3462no—
257C2×D4—3462no—
258C2×D4—3461no—
259C2×D4—1731no—
260C2×D4—1731no—
261C2×D4—1731no—
262C2×D4—1731no—
263C2×D4C22×D17311yescharacteristic
264C22×C4—21no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—1731no—
268C22×C4—1731no—
269C22×C4C22×D17311yescharacteristic
270M16—3462no—
271C2×C8—3462no—
272C2×C8—1731no—
273C2×C8—1731no—
274C2×C8C22×D17311yescharacteristic
275C2×C8C22×D17311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—1731no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4C22×D17311yescharacteristic
283C4⋊C4C22×D17311yescharacteristic
284C4⋊C4C22×D17311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C42—3461no—
294C42—3461no—
295C42—3461no—
296C42—3461no—
297C23—21no—
298C23—3461no—
299C23—13841no—
300C23—13841no—
301C23—1731no—
302C23—1731no—
303C23—6922no—
304C23D4×D17311yescharacteristic
305Q8—3462no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—3462no—
310D4—3462no—
311D4—3462no—
312D4—3461no—
313D4—3461no—
314D4—13841no—
315D4—13841no—
316D4—13841no—
317D4—6922no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—13841no—
330C2×C4—13841no—
331C2×C4—1731no—
332C2×C4—1731no—
333C2×C4—1732no—
334C2×C4—1732no—
335C2×C4D4×D17311yescharacteristic
336C2×C4C23×D17311yescharacteristic
337C8—21no—
338C8—21no—
339C8—3461no—
340C8—3461no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—3461no—
348C22—3461no—
349C22—3461no—
350C22—3461no—
351C22—3461no—
352C22—13844no—
353C22—1732no—
354C22—6921no—
355C22—6921no—
356C22—6921no—
357C22C2×D4×D17311yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—3461no—
363C4—1732no—
364C4—6921no—
365C4—6921no—
366C4—6921no—
367C4#41×D17311yescharacteristic
368C4#41×D17311yescharacteristic
369C2—41no—
370C2—81no—
371C2—3462no—
372C2—13841no—
373C2—6921no—
374C2#123×D17311yescharacteristic
375C2#2601229211yescharacteristic
376C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#480×C173C2
socleC22×C173C22×D4
prime radical#26012057C2
hypercenter#480D173
hypocenterC173#3096
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #26011972C2semidirect
3. #26011818C2semidirect
4. #26011816C2semidirect
5. #26011813C2semidirect
6. #26011812C2semidirect
7. #26011809C2semidirect
8. #26011808C2semidirect
9. #26011806C2semidirect
10. #26011532C2semidirect
11. #26011528C2semidirect
12. #26011494C2semidirect
13. #102×D173C2semidirect
14. #26011456C2semidirect
15. #480×C173C2semidirect
31. #19149638C22semidirect
34. #19149637C22semidirect
36. C4⋊C4×D173C22semidirect
37. #19149613C22semidirect
38. #19149612C22semidirect
39. #19149611C22semidirect
40. #19149610C22semidirect
41. #19149610C22semidirect
42. #19149610C22semidirect
49. C22⋊#13063436C22semidirect
50. #19149591C22semidirect
51. #19149591C22semidirect
53. C2×C4⋊C4×C173C22semidirect
54. #107×C173C22semidirect
55. #107×C173C22semidirect
58. C22⋊C8×C173C22semidirect
188. C173#3096semidirect
199. #480D173semidirect
227. C2×C4⋊C4C2×D173semidirect
230. #107C2×D173semidirect
231. #107C2×D173semidirect
247. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120572
#260119722
#260118182
#260118162
#260118132
#260118122
#260118092
#260118082
#260118062
#260115322
#260115282
#260114942
#102×D1732
#260114562
#480×C1732
#3096173