Finite Group Databasebeta

← #36363709

Subgroups — #36363709

Order 22144.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012036C211yesmaximal, characteristic, complemented
2#26011959C211yesmaximal, characteristic, complemented
3#26011815C211yesmaximal, characteristic, complemented
4#26011775C211yesmaximal, characteristic, complemented
5#26011770C211yesmaximal, characteristic, complemented
6#26011690C211yesmaximal, characteristic, complemented
7#26011685C211yesmaximal, characteristic, complemented
8#26011638C211yesmaximal, characteristic, complemented
9#26011634C211yesmaximal, characteristic, complemented
10#26011602C211yesmaximal, characteristic, complemented
11#26011600C211yesmaximal, characteristic, complemented
12#26011590C211yesmaximal, characteristic, complemented
13#26011588C211yesmaximal, characteristic, complemented
14#26011545C211yesmaximal, characteristic, complemented
15#482×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825—21no—
18C2×#14583815C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×D1384C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22#19149678C2211yescharacteristic
23#19149675C2211yescharacteristic
24#19149665—21no—
25#19149653C2211yescharacteristic
26#19149651—21no—
27#19149644C2211yescharacteristic
28#19149644C2211yescharacteristic
29D4×Dic173C2211yescharacteristic
30#19149641C2211yescharacteristic
31#19149641C2211yescharacteristic
32#19149641C2211yescharacteristic
33#19149639—21no—
34#19149639—21no—
35#19149639—21no—
36#19149636—21no—
37C2×C4×Dic173C2211yescharacteristic
38C42×D173—21no—
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic
51#19149589C2211yescharacteristic
52#121×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—22no—
61#14583825—44no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C22×Dic173—21no—
71C22×Dic173D411yescharacteristic
72C22×Dic173D411yescharacteristic
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77#14583815—22no—
78#14583814—22no—
79D1384—22no—
80#14583811—22no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173D411yescharacteristic
95C4×Dic173D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97C4×Dic173C2311yescharacteristic
98C2×D4×C173—21no—
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C42×C173—21no—
109C22×D173—21no—
110C22×D173—21no—
111Dic346—21no—
112Dic346—42no—
113Dic346#4112yescomplemented
114D692—21no—
115D692—21no—
116D692—21no—
117D692#4112yescomplemented
118C22⋊D173—24no—
119C22⋊D173—42no—
120C22⋊D173—42no—
121C22⋊D173—42no—
122C2×Dic173—21no—
123C2×Dic173—22no—
124C2×Dic173—21no—
125C2×Dic173—41no—
126C2×Dic173—42no—
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C4×D173—24no—
132C4×D173—42no—
133C4×D173—42no—
134C4×D173—42no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138C23×C173C2×D411yescharacteristic
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—42no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C8×C173—21no—
150C8×C173—21no—
151C22×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173C22×D411yescharacteristic
158C2×D173—22no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C4×C173—21no—
165C4×C173—41no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173C2×#4111yescharacteristic
169C4×C173C2×#4111yescharacteristic
170Dic173—22no—
171Dic173—21no—
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173AGL(1,ℤ/8)12yes—
176C2×C173—41no—
177C2×C173—81no—
178C2×C173#52511yescharacteristic
179C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C173#57511yescharacteristic
181D173—42no—
182D173—81no—
183C173#311511yescharacteristic, complemented, Sylow
184#3115—1731nomaximal, Sylow
185#532—1731no—
186C4×#41—1731no—
187#482—1731no—
188#482D17311yescharacteristic, complemented
189#475—1731no—
190#474—1731no—
191#460—1731no—
192#459—1731no—
193#449—1731no—
194#449—1731no—
195#442—1731no—
196#442—1731no—
197#440—1731no—
198#440—1731no—
199#432—1731no—
200C2×#41—3461no—
201C2×#41—1731no—
202C2×SD16—1731no—
203C2×SD16—1731no—
204C2×D8—1731no—
205C2×D8—1731no—
206C4⋊Q8—1731no—
207#127—1731no—
208#124—3461no—
209#121—3461no—
210#121—1731no—
211#121C2×D17311yescharacteristic
212C4×Q8—1731no—
213C4×Q8—1731no—
214C4×D4—3461no—
215C4×D4—3461no—
216C4×D4—3461no—
217C4×D4—1731no—
218C4×D4—1731no—
219C4×D4—1731no—
220C4×D4—1731no—
221#117—3461no—
222#117C2×D17311yescharacteristic
223C2×C42—3461no—
224C2×C42—1731no—
225#107—1731no—
226#107C2×D17311yescharacteristic
227#106—1731no—
228#106C2×D17311yescharacteristic
229C4⋊C8—1731no—
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102—1731no—
237#102C2×D17311yescharacteristic
238#102C2×D17311yescharacteristic
239C22⋊C8—1731no—
240C22⋊C8C2×D17311yescharacteristic
241#97—1731no—
242#97—1731no—
243#41—3462no—
244#41—3462no—
245#41—6924no—
246C2×Q8—3461no—
247C2×Q8—1731no—
248C2×D4—21no—
249C2×D4—3461no—
250C2×D4—3461no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—1731no—
254C2×D4—1731no—
255C2×D4C22×D17311yescharacteristic
256C22×C4—3461no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—1731no—
262C22×C4—1731no—
263C22×C4C22×D17311yescharacteristic
264SD16—3462no—
265SD16—3462no—
266D8—3462no—
267D8—3462no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8C22×D17311yescharacteristic
271C2×C8C22×D17311yescharacteristic
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—1731no—
276C4⋊C4C22×D17311yescharacteristic
277C4⋊C4C22×D17311yescharacteristic
278C4⋊C4C22×D17311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C42—21no—
287C42—3461no—
288C42—3461no—
289C42—1731no—
290C42—1731no—
291C42—1731no—
292C42—1731no—
293C23—21no—
294C23—3461no—
295C23—3461no—
296C23D4×D17311yescharacteristic
297Q8—3461no—
298Q8—1732no—
299Q8—6922no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—3464no—
304D4—3461no—
305D4—3461no—
306D4—3461no—
307D4—1732no—
308D4—6922no—
309D4—6922no—
310D4—6922no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—3464no—
317C2×C4—3461no—
318C2×C4—3462no—
319C2×C4—3461no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4—1731no—
324C2×C4—6922no—
325C2×C4—6922no—
326C2×C4—6922no—
327C2×C4—6921no—
328C2×C4—6922no—
329C2×C4D4×D17311yescharacteristic
330C2×C4C23×D17311yescharacteristic
331C8—21no—
332C8—21no—
333C8—3461no—
334C8—3461no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—3462no—
342C22—6921no—
343C22—6921no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—3462no—
353C4—3461no—
354C4—3461no—
355C4—3461no—
356C4—1732no—
357C4—6921no—
358C4#41×D17311yescharacteristic
359C4#41×D17311yescharacteristic
360C2—41no—
361C2—81no—
362C2—13841no—
363C2—6922no—
364C2#2601199911yescharacteristic
365C2AGL(1,ℤ/8)×D17311yescharacteristic
366C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#482×C173C2
socleC22×C173C22×D4
hypercenter#482D173
hypocenterC173#3115
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #26012036C2semidirect
2. #26011959C2semidirect
3. #26011815C2semidirect
4. #26011775C2semidirect
5. #26011770C2semidirect
6. #26011690C2semidirect
7. #26011685C2semidirect
8. #26011638C2semidirect
9. #26011634C2semidirect
10. #26011602C2semidirect
11. #26011600C2semidirect
12. #26011590C2semidirect
13. #26011588C2semidirect
14. #26011545C2semidirect
15. #482×C173C2semidirect
113. Dic346#41semidirect
117. D692#41semidirect
183. C173#3115semidirect
188. #482D173semidirect

Maximal subgroups

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

subgroupindex
#260120362
#260119592
#260118152
#260117752
#260117702
#260116902
#260116852
#260116382
#260116342
#260116022
#260116002
#260115902
#260115882
#260115452
#482×C1732
#3115173