Finite Group Databasebeta

← #36364657

Subgroups — #36364657

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#26012112C211yesmaximal, characteristic, complemented
2#26011878C211yesmaximal, characteristic, complemented
3#26011873C211yesmaximal, characteristic, complemented
4#26011850C211yesmaximal, characteristic, complemented
5#26011837C211yesmaximal, characteristic, complemented
6#26011710C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011688C211yesmaximal, characteristic, complemented
9#26011683C211yesmaximal, characteristic, complemented
10#26011653C211yesmaximal, characteristic, complemented
11#26011642C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011625C211yesmaximal, characteristic, complemented
14#26011552C211yesmaximal, characteristic, complemented
15#535×C173C211yesmaximal, characteristic, complemented
16C2×#14583816C2211yescharacteristic, complemented
17C2×#14583814C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583814C2211yescharacteristic, complemented
20C2×#14583813C2211yescharacteristic, complemented
21C2×#14583811C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23C2×#14583811C2211yescharacteristic, complemented
24#19149679C2211yescharacteristic, complemented
25#19149674—21no—
26#19149667C2211yescharacteristic, complemented
27#19149665—21no—
28#19149661—21no—
29#19149658C2211yescharacteristic
30#19149657—21no—
31#19149653C2211yescharacteristic
32C4×Dic346C2211yescharacteristic, complemented
33C4×D692C2211yescharacteristic, complemented
34C4⋊#13063436C2211yescharacteristic, complemented
35C4⋊#13063436C2211yescharacteristic, complemented
36#19149603C2211yescharacteristic, complemented
37#19149603C2211yescharacteristic, complemented
38#19149600C2211yescharacteristic
39#19149600C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149600C2211yescharacteristic, complemented
43#19149600C2211yescharacteristic, complemented
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46#19149590C2211yescharacteristic, complemented
47C4×#13063436C2211yescharacteristic, complemented
48C2×#41×C173C2211yescharacteristic
49C22×D4×C173C2211yescharacteristic
50#127×C173C2211yescharacteristic
51#127×C173C2211yescharacteristic
52#124×C173C2211yescharacteristic
53#124×C173C2211yescharacteristic
54#121×C173—21no—
55#121×C173—21no—
56#120×C173—21no—
57#120×C173—21no—
58#117×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C4×D173—21no—
64#14583816—41no—
65#14583814—41no—
66#14583814—41no—
67#14583814—41no—
68#14583813—41no—
69#14583811—41no—
70#14583811—41no—
71#14583811—41no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81C22⋊Dic173—21no—
82#14583803—21no—
83#14583803—21no—
84#14583803—21no—
85#14583803—21no—
86C4×Dic173—21no—
87C24×C173—21no—
88#41×C173—41no—
89#41×C173—41no—
90C2×Q8×C173C2311yescharacteristic
91C2×D4×C173—21no—
92C2×D4×C173—21no—
93C2×D4×C173—21no—
94C2×D4×C173—21no—
95C2×D4×C173—21no—
96C2×D4×C173—21no—
97C2×D4×C173—41no—
98C2×D4×C173—41no—
99C2×D4×C173C2311yescharacteristic
100C2×D4×C173C2311yescharacteristic
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104C4⋊C4×C173D411yescharacteristic
105C4⋊C4×C173D411yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C22⋊C4×C173—21no—
110C22⋊C4×C173D411yescharacteristic
111C22⋊C4×C173D411yescharacteristic
112C42×C173C2311yescharacteristic
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—81no—
118D692—21no—
119D692—21no—
120C22⋊D173—81no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C4×D173—81no—
125#13063436—21no—
126#13063436—21no—
127#13063436—21no—
128#13063436—21no—
129C23×C173—21no—
130C23×C173—21no—
131C23×C173—21no—
132C23×C173—42no—
133C23×C173—81no—
134C23×C173—81no—
135C23×C173C2×D411yescharacteristic
136Q8×C173—21no—
137Q8×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173—42no—
145D4×C173—42no—
146D4×C173—81no—
147D4×C173—81no—
148D4×C173—81no—
149D4×C173—81no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—42no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2411yescharacteristic
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—21no—
179C4×C173—21no—
180C4×C173—21no—
181C4×C173—21no—
182C4×C173—41no—
183C4×C173H(5,𝔽2)11yescharacteristic
184C4×C173H(5,𝔽2)11yescharacteristic
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#535D17311yescharacteristic, complemented
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#447—1731no—
212#433—1731no—
213C2×#41C2×D17311yescharacteristic
214C22×D4C2×D17311yescharacteristic
215C2×Q16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×D8—1731no—
221C2×D8—1731no—
222C2×D8—1731no—
223C4⋊Q8—1731no—
224#127C2×D17311yescharacteristic
225#127C2×D17311yescharacteristic
226#126—3461no—
227#124—3461no—
228#124—1731no—
229#124C2×D17311yescharacteristic
230#124C2×D17311yescharacteristic
231#123—3461no—
232C22⋊Q8—3461no—
233C22⋊Q8—1731no—
234#121—21no—
235#121—21no—
236#121—1731no—
237#120—21no—
238#120—21no—
239C4×Q8—1731no—
240C4×D4—1731no—
241#117C2×D17311yescharacteristic
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#102—1731no—
251#102—1731no—
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254#97—1731no—
255C4×C8—1731no—
256C24—21no—
257#41—41no—
258#41—41no—
259C2×Q8—3461no—
260C2×Q8—1731no—
261C2×Q8C22×D17311yescharacteristic
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4C22×D17311yescharacteristic
273C2×D4C22×D17311yescharacteristic
274C2×D4C22×D17311yescharacteristic
275C22×C4—21no—
276C22×C4—3461no—
277C22×C4C22×D17311yescharacteristic
278Q16—6921no—
279SD16—6921no—
280SD16—6921no—
281SD16—6921no—
282SD16—6921no—
283D8—6921no—
284D8—6921no—
285D8—6921no—
286C2×C8—1731no—
287C2×C8—1731no—
288C2×C8—1731no—
289C2×C8—1731no—
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—3461no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C4⋊C4C22⋊D17311yescharacteristic
296C4⋊C4C22⋊D17311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4C22⋊D17311yescharacteristic
307C22⋊C4C22⋊D17311yescharacteristic
308C42—3461no—
309C42C22×D17311yescharacteristic
310C42C22×D17311yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—3461no—
318C23C2×C22⋊D17311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—3461no—
322Q8—3461no—
323Q8—13841no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—3461no—
337D4—3461no—
338D4—13841no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—13841no—
347C2×C4C2×C22⋊D17311yescharacteristic
348C2×C4C2×C22⋊D17311yescharacteristic
349C2×C4C2×C22⋊D17311yescharacteristic
350C2×C4C2×C22⋊D17311yescharacteristic
351C2×C4C2×C22⋊D17311yescharacteristic
352C2×C4C23×D17311yescharacteristic
353C8—3461no—
354C8—3461no—
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—
373C22C22×C22⋊D17311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—6921no—
381C4—6921no—
382C4—6921no—
383C4#1914972411yescharacteristic
384C4#1914972411yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—81no—
389C2—13841no—
390C2#2601205811yescharacteristic
391C2#2601229411yescharacteristic
392C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#535×C173C2
socleC22×C173C22×D4
hypercenter#535D173
hypocenterC173#3224
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #26012112C2semidirect
2. #26011878C2semidirect
3. #26011873C2semidirect
4. #26011850C2semidirect
5. #26011837C2semidirect
6. #26011710C2semidirect
7. #26011703C2semidirect
8. #26011688C2semidirect
9. #26011683C2semidirect
10. #26011653C2semidirect
11. #26011642C2semidirect
12. #26011636C2semidirect
13. #26011625C2semidirect
14. #26011552C2semidirect
15. #535×C173C2semidirect
16. C2×#14583816C22semidirect
19. C2×#14583814C22semidirect
20. C2×#14583813C22semidirect
23. C2×#14583811C22semidirect
24. #19149679C22semidirect
26. #19149667C22semidirect
32. C4×Dic346C22semidirect
33. C4×D692C22semidirect
34. C4⋊#13063436C22semidirect
35. C4⋊#13063436C22semidirect
36. #19149603C22semidirect
37. #19149603C22semidirect
42. #19149600C22semidirect
43. #19149600C22semidirect
46. #19149590C22semidirect
47. C4×#13063436C22semidirect
196. C173#3224semidirect
199. #535D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260118782
#260118732
#260118502
#260118372
#260117102
#260117032
#260116882
#260116832
#260116532
#260116422
#260116362
#260116252
#260115522
#535×C1732
#3224173