Finite Group Databasebeta

← #36364653

Subgroups — #36364653

Order 22144.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012108C211yesmaximal, characteristic, complemented
2#26012058C211yesmaximal, characteristic, complemented
3#26011879C211yesmaximal, characteristic, complemented
4#26011874C211yesmaximal, characteristic, complemented
5#26011849C211yesmaximal, characteristic, complemented
6#26011836C211yesmaximal, characteristic, complemented
7#26011712C211yesmaximal, characteristic, complemented
8#26011704C211yesmaximal, characteristic, complemented
9#26011690C211yesmaximal, characteristic, complemented
10#26011684C211yesmaximal, characteristic, complemented
11#26011655C211yesmaximal, characteristic, complemented
12#26011642C211yesmaximal, characteristic, complemented
13#26011634C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#447×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Dic692C2211yescharacteristic, complemented
18C2×#14583815C2211yescharacteristic, complemented
19C2×#14583814C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583814C2211yescharacteristic, complemented
22C2×#14583811C2211yescharacteristic, complemented
23#19149678C2211yescharacteristic, complemented
24#19149675C2211yescharacteristic
25#19149675C2211yescharacteristic
26#19149672—21no—
27#19149666—21no—
28#19149666C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149666C2211yescharacteristic, complemented
31#19149660—21no—
32#19149658C2211yescharacteristic
33#19149656—21no—
34#19149650—21no—
35#19149650—21no—
36#19149648—21no—
37#19149648—21no—
38#19149644C2211yescharacteristic, complemented
39D4×Dic173C2211yescharacteristic, 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×D8×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C22×Dic173—21no—
71C2×C4×D173—21no—
72Dic692—41no—
73#14583815—41no—
74#14583814—41no—
75#14583814—41no—
76#14583814—41no—
77#14583811—41no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C4⋊Dic173D411yescharacteristic
81C4⋊Dic173D411yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173D411yescharacteristic
95C22⋊Dic173D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99C24×C173—21no—
100C2×D4×C173—21no—
101C2×D4×C173—21no—
102C2×D4×C173—41no—
103C2×D4×C173—41no—
104C2×D4×C173C2311yescharacteristic
105C2×D4×C173C2311yescharacteristic
106C2×D4×C173C2311yescharacteristic
107C22×C4×C173C2311yescharacteristic
108D8×C173—41no—
109D8×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—81no—
120D692—21no—
121D692—21no—
122C22⋊D173—42no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C22⋊D173—81no—
126C22⋊D173—81no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—81no—
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C4×D173—42no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173—21no—
140C23×C173—21no—
141C23×C173—42no—
142C23×C173—81no—
143C23×C173—81no—
144C23×C173C2×D411yescharacteristic
145D4×C173—21no—
146D4×C173—21no—
147D4×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—21no—
151D4×C173—42no—
152D4×C173—81no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2411yescharacteristic
158C8×C173—21no—
159C8×C173—21no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—84no—
173C22×C173C22×D411yescharacteristic
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C4×C173—21no—
178C4×C173—41no—
179C4×C173H(5,𝔽2)11yescharacteristic
180C4×C173H(5,𝔽2)11yescharacteristic
181Dic173—21no—
182Dic173—21no—
183Dic173—21no—
184Dic173—21no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188C2×C173—41no—
189C2×C173—81no—
190C2×C173—81no—
191C2×C173—81no—
192C2×C173#53511yescharacteristic
193C2×C173#57611yescharacteristic
194C2×C173#57711yescharacteristic
195D173—81no—
196C173#322411yescharacteristic, complemented, Sylow
197#3224—1731nomaximal, Sylow
198#542—1731no—
199#535—1731no—
200#496—1731no—
201#495—1731no—
202#489—1731no—
203#486—1731no—
204#464—1731no—
205#463—1731no—
206#460—1731no—
207#459—1731no—
208#452—1731no—
209#450—1731no—
210#449—1731no—
211#447D17311yescharacteristic, complemented
212#433—1731no—
213C2×#41—1731no—
214C22×D4C2×D17311yescharacteristic
215C2×Q16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×D8—1731no—
221C2×D8C2×D17311yescharacteristic
222C2×D8C2×D17311yescharacteristic
223C4⋊Q8—1731no—
224#127—1731no—
225#127—1731no—
226#126—3461no—
227#124—3461no—
228#124—1731no—
229#124—1731no—
230#124—1731no—
231#123—3461no—
232C22⋊Q8—3461no—
233C22⋊Q8—1731no—
234#121—3461no—
235#121—3461no—
236#121C2×D17311yescharacteristic
237#120—3461no—
238#120—3461no—
239C4×Q8—1731no—
240C4×D4—1731no—
241#117—1731no—
242C4⋊C8—1731no—
243C4⋊C8—1731no—
244#103—1731no—
245#103—1731no—
246#102—1731no—
247#102—1731no—
248#102—1731no—
249#102—1731no—
250#102C2×D17311yescharacteristic
251#102C2×D17311yescharacteristic
252C22⋊C8—1731no—
253C22⋊C8C2×D17311yescharacteristic
254#97—1731no—
255C4×C8—1731no—
256C24—21no—
257#41—6921no—
258#41—6921no—
259C2×Q8—3461no—
260C2×Q8—1731no—
261C2×Q8—1731no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4C22×D17311yescharacteristic
273C2×D4C22×D17311yescharacteristic
274C2×D4C22×D17311yescharacteristic
275C22×C4—3461no—
276C22×C4—3461no—
277C22×C4C22×D17311yescharacteristic
278Q16—6921no—
279SD16—6921no—
280SD16—6921no—
281SD16—6921no—
282SD16—6921no—
283D8—41no—
284D8—41no—
285D8—6921no—
286C2×C8—1731no—
287C2×C8—1731no—
288C2×C8C22×D17311yescharacteristic
289C2×C8C22×D17311yescharacteristic
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—3461no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4—1731no—
296C4⋊C4C22×D17311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—1731no—
307C22⋊C4—1731no—
308C42—3461no—
309C42—1731no—
310C42—1731no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—3461no—
318C23D4×D17311yescharacteristic
319Q8—3461no—
320Q8—3461no—
321Q8—3461no—
322Q8—3461no—
323Q8—13841no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—81no—
332D4—3461no—
333D4—3461no—
334D4—13841no—
335D4—13841no—
336D4—13841no—
337D4—13841no—
338D4—6922no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—13841no—
346C2×C4—1731no—
347C2×C4—1731no—
348C2×C4—1731no—
349C2×C4—1731no—
350C2×C4—6922no—
351C2×C4D4×D17311yescharacteristic
352C2×C4C23×D17311yescharacteristic
353C8—21no—
354C8—21no—
355C8—3461no—
356C8—3461no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—84no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C2×D4×D17311yescharacteristic
374C4—21no—
375C4—41no—
376C4—3461no—
377C4—3461no—
378C4—3461no—
379C4—3461no—
380C4—6921no—
381C4—6921no—
382C4—6921no—
383C4#1914972411yescharacteristic
384C4#1914972411yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—81no—
389C2—13841no—
390C2#2601205611yescharacteristic
391C2#2601229211yescharacteristic
392C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#447×C173C2
socleC22×C173C22×D4
hypercenter#447D173
hypocenterC173#3224
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. #26012108C2semidirect
2. #26012058C2semidirect
3. #26011879C2semidirect
4. #26011874C2semidirect
5. #26011849C2semidirect
6. #26011836C2semidirect
7. #26011712C2semidirect
8. #26011704C2semidirect
9. #26011690C2semidirect
10. #26011684C2semidirect
11. #26011655C2semidirect
12. #26011642C2semidirect
13. #26011634C2semidirect
14. #26011549C2semidirect
15. #447×C173C2semidirect
17. C2×Dic692C22semidirect
18. C2×#14583815C22semidirect
21. C2×#14583814C22semidirect
22. C2×#14583811C22semidirect
23. #19149678C22semidirect
30. #19149666C22semidirect
38. #19149644C22semidirect
39. D4×Dic173C22semidirect
41. #19149608C22semidirect
42. #19149608C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
45. #19149601C22semidirect
48. #19149599C22semidirect
50. #19149589C22semidirect
51. C8×Dic173C22semidirect
196. C173#3224semidirect
211. #447D173semidirect

Maximal subgroups

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

subgroupindex
#260121082
#260120582
#260118792
#260118742
#260118492
#260118362
#260117122
#260117042
#260116902
#260116842
#260116552
#260116422
#260116342
#260115492
#447×C1732
#3224173