Finite Group Databasebeta

← #36364631

Subgroups — #36364631

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#26012058C211yesmaximal, characteristic
3#26011882C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011854C211yesmaximal, characteristic, complemented
6#26011844C211yesmaximal, characteristic, complemented
7#26011712C211yesmaximal, characteristic, complemented
8#26011711C211yesmaximal, characteristic, complemented
9#26011690C211yesmaximal, characteristic, complemented
10#26011689C211yesmaximal, characteristic, complemented
11#26011655C211yesmaximal, characteristic, complemented
12#26011651C211yesmaximal, characteristic, complemented
13#26011642C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Dic692C2211yescharacteristic, complemented
18C2×#14583816C2211yescharacteristic, complemented
19C2×#14583815C2211yescharacteristic, complemented
20C2×#14583814C2211yescharacteristic
21C2×#14583814C2211yescharacteristic
22C2×#14583813C2211yescharacteristic, complemented
23#19149677C2211yescharacteristic, complemented
24#19149675C2211yescharacteristic
25#19149675C2211yescharacteristic
26#19149672—21no—
27#19149672—21no—
28#19149670—21no—
29#19149670C2211yescharacteristic, complemented
30#19149669—21no—
31#19149666C2211yescharacteristic
32#19149666C2211yescharacteristic
33#19149658C2211yescharacteristic
34#19149650—21no—
35#19149650—21no—
36#19149648—21no—
37#19149648—21no—
38Q8×Dic173C2211yescharacteristic, complemented
39#19149644C2211yescharacteristic, complemented
40#19149638C2211yescharacteristic
41#19149608C2211yescharacteristic, complemented
42#19149608C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346C2311yescharacteristic
62C2×Dic346C2311yescharacteristic
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C4×D173—21no—
70Dic692—41no—
71#14583816—41no—
72#14583815—41no—
73#14583814—41no—
74#14583814—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⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173D411yescharacteristic
94C22⋊Dic173D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99C24×C173—21no—
100C2×Q8×C173C2311yescharacteristic
101C2×D4×C173—21no—
102C2×D4×C173—41no—
103C2×D4×C173—41no—
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C22×C4×C173C2311yescharacteristic
107SD16×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—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119D692—21no—
120D692—21no—
121C22⋊D173—42no—
122C22⋊D173—81no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C22⋊D173—81no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—42no—
134#13063436—21no—
135#13063436—21no—
136C23×C173—21no—
137C23×C173—21no—
138C23×C173—42no—
139C23×C173—81no—
140C23×C173—81no—
141C23×C173C2×D411yescharacteristic
142Q8×C173—21no—
143Q8×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148D4×C173—42no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
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×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—84no—
167C22×C173C22×D411yescharacteristic
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173H(5,𝔽2)11yescharacteristic
175C4×C173H(5,𝔽2)11yescharacteristic
176Dic173—21no—
177Dic173—21no—
178Dic173—21no—
179Dic173—21no—
180Dic173—41no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—41no—
184C2×C173—81no—
185C2×C173—81no—
186C2×C173#53511yescharacteristic
187C2×C173#57711yescharacteristic
188C2×C173#57711yescharacteristic
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#452—1731no—
204#450—1731no—
205#450D17311yescharacteristic, complemented
206#433—1731no—
207C2×#41—1731no—
208C22×D4C2×D17311yescharacteristic
209C2×Q16—1731no—
210C2×Q16—1731no—
211C2×SD16—1731no—
212C2×SD16—1731no—
213C2×SD16—1731no—
214C2×SD16—1731no—
215C2×SD16C2×D17311yescharacteristic
216C2×SD16C2×D17311yescharacteristic
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#103—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102C2×D17311yescharacteristic
245#102C2×D17311yescharacteristic
246C22⋊C8—1731no—
247C22⋊C8C2×D17311yescharacteristic
248#97—1731no—
249C4×C8—1731no—
250C24—21no—
251#41—6921no—
252#41—6921no—
253C2×Q8—1731no—
254C2×Q8—1731no—
255C2×Q8C22×D17311yescharacteristic
256C2×D4—21no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—1731no—
265C2×D4C22×D17311yescharacteristic
266C2×D4C22×D17311yescharacteristic
267C22×C4—3461no—
268C22×C4C22×D17311yescharacteristic
269Q16—6921no—
270Q16—6921no—
271SD16—41no—
272SD16—41no—
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—21no—
307C23—42no—
308C23—81no—
309C23—81no—
310C23—3461no—
311C23D4×D17311yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—3461no—
315Q8—3461no—
316Q8—3461no—
317Q8—3461no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
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—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—1731no—
338C2×C4—1731no—
339C2×C4—1731no—
340C2×C4—1731no—
341C2×C4—6922no—
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—41no—
355C22—41no—
356C22—41no—
357C22—84no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C2×D4×D17311yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—3461no—
366C4—3461no—
367C4—3461no—
368C4—3461no—
369C4—6921no—
370C4—6921no—
371C4—6921no—
372C4#1914972411yescharacteristic
373C4#1914972411yescharacteristic
374C2—41no—
375C2—81no—
376C2—81no—
377C2—13841no—
378C2#2601205611yescharacteristic
379C2#2601229711yescharacteristic
380C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radical#26012058C2
hypercenter#450D173
hypocenterC173#3222
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012101C2semidirect
3. #26011882C2semidirect
4. #26011862C2semidirect
5. #26011854C2semidirect
6. #26011844C2semidirect
7. #26011712C2semidirect
8. #26011711C2semidirect
9. #26011690C2semidirect
10. #26011689C2semidirect
11. #26011655C2semidirect
12. #26011651C2semidirect
13. #26011642C2semidirect
14. #26011549C2semidirect
15. #450×C173C2semidirect
17. C2×Dic692C22semidirect
18. C2×#14583816C22semidirect
19. C2×#14583815C22semidirect
22. C2×#14583813C22semidirect
23. #19149677C22semidirect
29. #19149670C22semidirect
38. Q8×Dic173C22semidirect
39. #19149644C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
45. #19149601C22semidirect
48. #19149599C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
190. C173#3222semidirect
205. #450D173semidirect

Maximal subgroups

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

subgroupindex
#260121012
#260120582
#260118822
#260118622
#260118542
#260118442
#260117122
#260117112
#260116902
#260116892
#260116552
#260116512
#260116422
#260115492
#450×C1732
#3222173