Finite Group Databasebeta

← #36364633

Subgroups — #36364633

Order 22144.

396 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012101C211yesmaximal, characteristic, complemented
2#26012055C211yesmaximal, characteristic
3#26011884C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011854C211yesmaximal, characteristic, complemented
6#26011847C211yesmaximal, characteristic, complemented
7#26011716C211yesmaximal, characteristic, complemented
8#26011715C211yesmaximal, characteristic, complemented
9#26011690C211yesmaximal, characteristic, complemented
10#26011689C211yesmaximal, characteristic, complemented
11#26011653C211yesmaximal, characteristic, complemented
12#26011644C211yesmaximal, characteristic, complemented
13#26011641C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C2×#14583816C2211yescharacteristic, complemented
18C2×#14583815C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583814C2211yescharacteristic, complemented
21C2×#14583813C2211yescharacteristic
22C2×#14583813C2211yescharacteristic
23#19149677C2211yescharacteristic, complemented
24#19149675C2211yescharacteristic
25#19149675C2211yescharacteristic
26#19149672—21no—
27#19149672—21no—
28#19149670—21no—
29#19149670C2211yescharacteristic, complemented
30#19149669—21no—
31#19149668C2211yescharacteristic
32#19149668C2211yescharacteristic
33#19149658C2211yescharacteristic
34#19149653—21no—
35#19149653—21no—
36#19149647—21no—
37#19149647—21no—
38Q8×Dic173C2211yescharacteristic, complemented
39#19149644C2211yescharacteristic, complemented
40#19149638C2211yescharacteristic
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C2×#41×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic, complemented
57#102×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic
59C23×D173—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692—21no—
62C2×D692—41no—
63C2×D692—41no—
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70#14583816—41no—
71#14583815—41no—
72#14583815—41no—
73#14583814—41no—
74#14583813—41no—
75#14583813—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173—21no—
79C4⋊Dic173—21no—
80C4⋊Dic173D411yescharacteristic
81C4⋊Dic173D411yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173D411yescharacteristic
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173—21no—
106C22×C4×C173C2311yescharacteristic
107Q16×C173—41no—
108SD16×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—42no—
117C22×D173—81no—
118C22×D173—81no—
119Dic346—21no—
120Dic346—21no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—21no—
125D692—42no—
126C22⋊D173—81no—
127C22⋊D173—81no—
128C22⋊D173—81no—
129C22⋊D173—81no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C2×Dic173C2×D411yescharacteristic
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173C2×D411yescharacteristic
140Q8×C173—21no—
141Q8×C173—21no—
142Q8×C173—21no—
143Q8×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—42no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—42no—
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C8×C173—21no—
156C8×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173C22×D411yescharacteristic
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—84no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173H(5,𝔽2)11yescharacteristic
176C4×C173H(5,𝔽2)11yescharacteristic
177Dic173—21no—
178Dic173—21no—
179Dic173—21no—
180Dic173—21no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—41no—
184C2×C173—81no—
185C2×C173#53511yescharacteristic
186C2×C173#57711yescharacteristic
187C2×C173#57711yescharacteristic
188D173—81no—
189D173—81no—
190C173#322211yescharacteristic, complemented, Sylow
191#3222—1731nomaximal, Sylow
192#541—1731no—
193#535—1731no—
194#497—1731no—
195#492—1731no—
196#490—1731no—
197#488—1731no—
198#464—1731no—
199#464—1731no—
200#460—1731no—
201#460—1731no—
202#452—1731no—
203#452D17311yescharacteristic, complemented
204#450—1731no—
205#450—1731no—
206#433—1731no—
207C2×#41C2×D17311yescharacteristic
208C22×D4—1731no—
209C2×Q16—1731no—
210C2×Q16C2×D17311yescharacteristic, complemented
211C2×SD16—1731no—
212C2×SD16—1731no—
213C2×SD16—1731no—
214C2×SD16—1731no—
215C2×SD16—1731no—
216C2×SD16C2×D17311yescharacteristic, complemented
217C4⋊Q8—1731no—
218#127—1731no—
219#127—1731no—
220#126—3461no—
221#126—3461no—
222#125—3461no—
223#125—3461no—
224#125—1731no—
225#124—1731no—
226#124—1731no—
227C22⋊Q8—1731no—
228C22⋊Q8C2×D17311yescharacteristic
229#121—3461no—
230#121—3461no—
231#120—3461no—
232#120—3461no—
233C4×Q8—1731no—
234C4×Q8—1731no—
235#117—1731no—
236C4⋊C8—1731no—
237C4⋊C8—1731no—
238#103—1731no—
239#103C2×D17311yescharacteristic, complemented
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102C2×D17311yescharacteristic, complemented
246C22⋊C8—1731no—
247C22⋊C8C2×D17311yescharacteristic
248#97—1731no—
249C4×C8—1731no—
250C24—3461no—
251#41—41no—
252#41—41no—
253C2×Q8—1731no—
254C2×Q8C22×D17311yescharacteristic
255C2×Q8C22×D17311yescharacteristic
256C2×D4—21no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—1731no—
263C2×D4—1731no—
264C2×D4—6921no—
265C2×D4—6921no—
266C2×D4C22×D17311yescharacteristic
267C22×C4—21no—
268C22×C4C22×D17311yescharacteristic
269Q16—41no—
270Q16—6921no—
271SD16—41no—
272SD16—6921no—
273SD16—6921no—
274SD16—6921no—
275SD16—6921no—
276SD16—6921no—
277C2×C8—1731no—
278C2×C8—1731no—
279C2×C8C22×D17311yescharacteristic
280C2×C8C22×D17311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—1731no—
290C4⋊C4—1731no—
291C4⋊C4—1731no—
292C4⋊C4C22×D17311yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—1731no—
300C22⋊C4—1731no—
301C42—3461no—
302C42—3461no—
303C42—1731no—
304C42—1731no—
305C23—21no—
306C23—3461no—
307C23—3461no—
308C23—13841no—
309C23—13841no—
310C23—6922no—
311C23D4×D17311yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—21no—
316Q8—3461no—
317Q8—3461no—
318D4—21no—
319D4—21no—
320D4—42no—
321D4—3461no—
322D4—3461no—
323D4—3461no—
324D4—3461no—
325D4—13841no—
326D4—13841no—
327D4—13841no—
328D4—13841no—
329D4—6922no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—1731no—
339C2×C4—1731no—
340C2×C4—1731no—
341C2×C4—1731no—
342C2×C4D4×D17311yescharacteristic
343C2×C4C23×D17311yescharacteristic
344C8—21no—
345C8—21no—
346C8—3461no—
347C8—3461no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—13844no—
355C22—6921no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C2×D4×D17311yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—3461no—
367C4—3461no—
368C4—3461no—
369C4—3461no—
370C4—6921no—
371C4—6921no—
372C4#1914972411yescharacteristic
373C4#1914972411yescharacteristic
374C2—41no—
375C2—81no—
376C2—13841no—
377C2—13841no—
378C2#2601205611yescharacteristic
379C2#2601229811yescharacteristic
380C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
prime radical#26012055C2
hypercenter#452D173
hypocenterC173#3222
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #26012101C2semidirect
3. #26011884C2semidirect
4. #26011862C2semidirect
5. #26011854C2semidirect
6. #26011847C2semidirect
7. #26011716C2semidirect
8. #26011715C2semidirect
9. #26011690C2semidirect
10. #26011689C2semidirect
11. #26011653C2semidirect
12. #26011644C2semidirect
13. #26011641C2semidirect
14. #26011549C2semidirect
15. #452×C173C2semidirect
17. C2×#14583816C22semidirect
20. C2×#14583814C22semidirect
23. #19149677C22semidirect
29. #19149670C22semidirect
38. Q8×Dic173C22semidirect
39. #19149644C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
43. #19149603C22semidirect
46. #19149600C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
53. C2×Q16×C173C22semidirect
54. C2×SD16×C173C22semidirect
56. #103×C173C22semidirect
57. #102×C173C22semidirect
190. C173#3222semidirect
203. #452D173semidirect
210. C2×Q16C2×D173semidirect
216. C2×SD16C2×D173semidirect
239. #103C2×D173semidirect
245. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121012
#260120552
#260118842
#260118622
#260118542
#260118472
#260117162
#260117152
#260116902
#260116892
#260116532
#260116442
#260116412
#260115492
#452×C1732
#3222173