Finite Group Databasebeta

← #36365274

Subgroups — #36365274

Order 22144.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149698C211yesmaximal, characteristic, complemented
2C22×#14583814C211yesmaximal, characteristic, complemented
3#26012147C211yesmaximal, characteristic, complemented
4#26012135C211yesmaximal, characteristic, complemented
5#26011874C211yesmaximal, characteristic, complemented
6#26011719C211yesmaximal, characteristic, complemented
7#26011718C211yesmaximal, characteristic, complemented
8#26011712C211yesmaximal, characteristic, complemented
9#26011704C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011649C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic, complemented
13#26011582C211yesmaximal, characteristic, complemented
14#26011554C211yesmaximal, characteristic, complemented
15#447×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic, complemented
18C22×Dic346C2211yescharacteristic
19#19149698—24no—
20C2×Dic692C2211yescharacteristic, complemented
21C2×Dic692C2211yescharacteristic, complemented
22C2×#14583814—24no—
23C2×#14583814C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583814C2211yescharacteristic
27C2×#14583814C2211yescharacteristic, complemented
28C22×#13063436C2211yescharacteristic
29C2×C8×D173C2211yescharacteristic, complemented
30#19149666C2211yescharacteristic
31#19149666C2211yescharacteristic, complemented
32#19149665—21no—
33#19149660—21no—
34#19149659C2211yescharacteristic, complemented
35#19149658C2211yescharacteristic
36#19149656—21no—
37#19149651—21no—
38#19149651—21no—
39#19149650C2211yescharacteristic
40#19149648—21no—
41#19149644C2211yescharacteristic, complemented
42D4×Dic173—21no—
43D4×Dic173C2211yescharacteristic
44C4×C22⋊D173C2211yescharacteristic
45C2×C22⋊Dic173—21no—
46#19149610C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149604C2211yescharacteristic, complemented
50#19149602C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53C8×Dic173C2211yescharacteristic, complemented
54C22×D4×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic
56C2×D8×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic
60C22⋊C8×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—22no—
63#14583825—41no—
64#14583825—41no—
65C2×Dic346—22no—
66C2×Dic346—21no—
67C2×Dic346—41no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70C2×Dic346C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173D411yescharacteristic
75C22×Dic173—21no—
76C22×Dic173—21no—
77C22×Dic173—21no—
78C22×Dic173—21no—
79C2×C4×D173C2311yescharacteristic
80Dic692—22no—
81Dic692—41no—
82#14583814—22no—
83#14583814—22no—
84#14583814—22no—
85#14583814—22no—
86#14583814D414yescomplemented
87C2×#13063436—22no—
88C2×#13063436C2311yescharacteristic
89C2×#13063436C2311yescharacteristic
90C8×D173—22no—
91C4⋊Dic173—21no—
92C4⋊Dic173C2311yescharacteristic
93#14583805—21no—
94#14583805—21no—
95#14583805D411yescharacteristic
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173—21no—
99C22⋊Dic173—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803D411yescharacteristic
106C4×Dic173—21no—
107C4×Dic173C2311yescharacteristic
108C24×C173—21no—
109C2×D4×C173—22no—
110C2×D4×C173—21no—
111C2×D4×C173—21no—
112C2×D4×C173—41no—
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C2×D4×C173C2311yescharacteristic
116C22×C4×C173C2311yescharacteristic
117D8×C173—22no—
118D8×C173—41no—
119C2×C8×C173C2311yescharacteristic
120C2×C8×C173C2311yescharacteristic
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—42no—
130Dic346C2×D412yes—
131C22⋊D173—22no—
132C22⋊D173—42no—
133C22⋊D173—42no—
134C22⋊D173—81no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—42no—
140C2×Dic173—42no—
141C2×Dic173—42no—
142C2×Dic173—81no—
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C4×D173—21no—
147C4×D173—21no—
148#13063436—21no—
149#13063436C2×D412yes—
150C23×C173—21no—
151C23×C173—21no—
152C23×C173—21no—
153C23×C173—42no—
154C23×C173—42no—
155C23×C173—81no—
156C23×C173C2×D411yescharacteristic
157D4×C173—21no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—42no—
163D4×C173—42no—
164D4×C173C2×D412yes—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173—21no—
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—22no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—84no—
184C22×C173#13512yescomplemented
185C22×C173C22×D411yescharacteristic
186C2×D173—21no—
187C2×D173—21no—
188C2×D173—21no—
189C4×C173—21no—
190C4×C173—41no—
191C4×C173C22×D411yescharacteristic
192C4×C173H(5,𝔽2)11yescharacteristic
193Dic173—21no—
194Dic173—22no—
195Dic173—21no—
196Dic173—21no—
197Dic173—41no—
198Dic173—41no—
199Dic173—41no—
200C2×C173—22no—
201C2×C173—42no—
202C2×C173—81no—
203C2×C173—81no—
204C2×C173D4211yescharacteristic
205C2×C173C2×#13511yescharacteristic
206C2×C173#57711yescharacteristic
207D173—41no—
208C173#328211yescharacteristic, complemented, Sylow
209#3282—1731nomaximal, Sylow
210C2×#135—1731no—
211C22×SD16—1731no—
212#548—1731no—
213#546—1731no—
214#495—1731no—
215#465—1731no—
216#465—1731no—
217#464—1731no—
218#463—1731no—
219#452—1731no—
220#451—1731no—
221#449—1731no—
222#447D17311yescharacteristic, complemented
223C4×SD16—1731no—
224C8×D4—1731no—
225C2×#41—1731no—
226C2×#41—1731no—
227C22×Q8—1731no—
228C22×D4C2×D17311yescharacteristic
229#135—3464no—
230C2×Q16—1731no—
231C2×Q16—1731no—
232C2×SD16—3464no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16—1731no—
236C2×SD16—1731no—
237C2×SD16—1731no—
238C2×D8C2×D17311yescharacteristic
239C2×D8C2×D17311yescharacteristic
240C22×C8—1731no—
241C22×C8—1731no—
242#124—3461no—
243#124—1731no—
244#124—1731no—
245#123—3461no—
246C22⋊Q8—3461no—
247C22⋊Q8—1731no—
248C22⋊Q8—1731no—
249#121—3461no—
250#121—3461no—
251#121—1731no—
252#121C2×D17311yescharacteristic
253#120—3461no—
254C4×Q8—1731no—
255C4×D4—3461no—
256C4×D4—1731no—
257C4×D4—1731no—
258C2×C22⋊C4—3461no—
259#106—1731no—
260C4⋊C8—1731no—
261#103—1731no—
262#103—1731no—
263#103—1731no—
264#102—1731no—
265#102C2×D17311yescharacteristic
266#102C2×D17311yescharacteristic
267C22⋊C8—1731no—
268C22⋊C8C2×D17311yescharacteristic
269C4×C8—1731no—
270C24—21no—
271#41—3462no—
272#41—3462no—
273#41—6921no—
274#41—6921no—
275C2×Q8—3462no—
276C2×Q8—3461no—
277C2×Q8—1731no—
278C2×Q8—1731no—
279C2×Q8—1731no—
280C2×Q8—6921no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—3461no—
288C2×D4—1731no—
289C2×D4C22×D17311yescharacteristic
290C2×D4C22×D17311yescharacteristic
291C2×D4C22×D17311yescharacteristic
292C22×C4—3461no—
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—1731no—
297C22×C4C22×D17311yescharacteristic
298Q16—3462no—
299Q16—6921no—
300SD16—3462no—
301SD16—3462no—
302SD16—3462no—
303SD16—3462no—
304SD16—1734no—
305D8—22no—
306D8—41no—
307C2×C8—3462no—
308C2×C8—3462no—
309C2×C8—1731no—
310C2×C8—1731no—
311C2×C8C22×D17311yescharacteristic
312C2×C8C22×D17311yescharacteristic
313C4⋊C4—3461no—
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—1731no—
317C4⋊C4—1731no—
318C4⋊C4C22×D17311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—1731no—
329C22⋊C4—1731no—
330C42—3461no—
331C42—1731no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1731no—
339C23D4×D17311yescharacteristic
340Q8—3461no—
341Q8—3461no—
342Q8—3461no—
343Q8—3461no—
344Q8—3461no—
345Q8—1732no—
346Q8—6922no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—42no—
354D4—3462no—
355D4—13841no—
356D4—6922no—
357D4—6922no—
358D4D4×D17312yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—13841no—
369C2×C4—1731no—
370C2×C4—1731no—
371C2×C4—1731no—
372C2×C4—6922no—
373C2×C4—6922no—
374C2×C4—6922no—
375C2×C4D4×D17311yescharacteristic
376C2×C4C23×D17311yescharacteristic
377C8—21no—
378C8—21no—
379C8—3461no—
380C8—1732no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—84no—
393C22—3461no—
394C22—3461no—
395C22—3461no—
396C22#1914969812yescomplemented
397C22C2×D4×D17311yescharacteristic
398C4—21no—
399C4—41no—
400C4—3461no—
401C4—3461no—
402C4—3461no—
403C4—3462no—
404C4—6921no—
405C4—6921no—
406C4—6921no—
407C4C2×D4×D17311yescharacteristic
408C4#1914972411yescharacteristic
409C2—22no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—6921no—
414C2#2601212311yescharacteristic
415C2C2×#1914969811yescharacteristic
416C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#447×C173C2
socleC22×C173C22×D4
hypercenter#447D173
hypocenterC173#3282
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#19149698C2semidirect
2. C22×#14583814C2semidirect
3. #26012147C2semidirect
4. #26012135C2semidirect
5. #26011874C2semidirect
6. #26011719C2semidirect
7. #26011718C2semidirect
8. #26011712C2semidirect
9. #26011704C2semidirect
10. #26011654C2semidirect
11. #26011649C2semidirect
12. #26011633C2semidirect
13. #26011582C2semidirect
14. #26011554C2semidirect
15. #447×C173C2semidirect
17. C2×#14583825C22semidirect
20. C2×Dic692C22semidirect
21. C2×Dic692C22semidirect
27. C2×#14583814C22semidirect
29. C2×C8×D173C22semidirect
31. #19149666C22semidirect
34. #19149659C22semidirect
41. #19149644C22semidirect
46. #19149610C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149604C22semidirect
50. #19149602C22semidirect
51. #19149600C22semidirect
52. #19149591C22semidirect
53. C8×Dic173C22semidirect
86. #14583814D4semidirect
184. C22×C173#135semidirect
208. C173#3282semidirect
222. #447D173semidirect
396. C22#19149698semidirect

Maximal subgroups

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

subgroupindex
C2×#191496982
C22×#145838142
#260121472
#260121352
#260118742
#260117192
#260117182
#260117122
#260117042
#260116542
#260116492
#260116332
#260115822
#260115542
#447×C1732
#3282173