Finite Group Databasebeta

← #36364358

Subgroups — #36364358

Order 22144.

424 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012004C211yesmaximal, characteristic, complemented
2#26012003C211yesmaximal, characteristic, complemented
3#26011834C211yesmaximal, characteristic, complemented
4#26011817C211yesmaximal, characteristic, complemented
5#26011815C211yesmaximal, characteristic, complemented
6#26011731C211yesmaximal, characteristic, complemented
7#26011728C211yesmaximal, characteristic, complemented
8#26011723C211yesmaximal, characteristic, complemented
9#26011721C211yesmaximal, characteristic, complemented
10#26011656C211yesmaximal, characteristic, complemented
11#26011651C211yesmaximal, characteristic, complemented
12#26011639C211yesmaximal, characteristic, complemented
13#26011638C211yesmaximal, characteristic, complemented
14C22⋊C8×D173C211yesmaximal, characteristic, complemented
15#485×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C2×#14583816C2211yescharacteristic, complemented
20C2×#14583815C2211yescharacteristic, complemented
21C2×#14583813C2211yescharacteristic, complemented
22C2×D1384C2211yescharacteristic, complemented
23C2×C8×D173C2211yescharacteristic
24C2×C8×D173C2211yescharacteristic
25#19149664—21no—
26#19149663—21no—
27#19149659C2211yescharacteristic
28#19149659C2211yescharacteristic
29#19149655C2211yescharacteristic
30#19149654C2211yescharacteristic
31#19149654C2211yescharacteristic
32#19149654C2211yescharacteristic
33#19149653C2211yescharacteristic
34#19149649—21no—
35#19149649—21no—
36C4×D692—21no—
37C4×D692—21no—
38#19149641—21no—
39#19149639—21no—
40#19149637C2211yescharacteristic
41#19149612C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic, complemented
46#19149599C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic, complemented
48#19149599C2211yescharacteristic, complemented
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic, complemented
55#106×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic
59C23×D173D411yescharacteristic
60#14583827—41no—
61#14583827—41no—
62#14583826—41no—
63#14583826—41no—
64C2×Dic346C2311yescharacteristic
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—21no—
69C2×D692C2311yescharacteristic
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C22×Dic173D411yescharacteristic
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173—41no—
77C2×C4×D173—41no—
78C2×C4×D173D411yescharacteristic
79C2×C4×D173D411yescharacteristic
80C2×C4×D173C2311yescharacteristic
81C2×C4×D173C2311yescharacteristic
82#14583816—41no—
83#14583815—41no—
84#14583813—41no—
85D1384—41no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C8×D173—41no—
89C8×D173—41no—
90C4⋊Dic173—21no—
91C4⋊Dic173—21no—
92C4⋊Dic173C2311yescharacteristic
93#14583805—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102C4×Dic173—21no—
103C2×Q8×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173—21no—
114C22×D173—22no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—41no—
118C22×D173—41no—
119C22×D173C2×D411yescharacteristic
120C22×D173C2×D411yescharacteristic
121Dic346—21no—
122Dic346—21no—
123D692—21no—
124D692—21no—
125D692—21no—
126D692—21no—
127D692—42no—
128D692—42no—
129D692—42no—
130D692—42no—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—41no—
136C2×Dic173C2×D411yescharacteristic
137C2×Dic173C2×D411yescharacteristic
138C4×D173—21no—
139C4×D173—21no—
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—41no—
143C4×D173—41no—
144C4×D173—42no—
145C4×D173—42no—
146#13063436—21no—
147#13063436—21no—
148C23×C173C2×D411yescharacteristic
149Q8×C173—21no—
150Q8×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2411yescharacteristic
159C8×C173—21no—
160C8×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—21no—
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—44no—
171C2×D173—41no—
172C2×D173—41no—
173C2×D173—41no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173#13512yes—
178C4×C173—21no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173H(5,𝔽2)11yescharacteristic
184C4×C173H(5,𝔽2)11yescharacteristic
185Dic173—21no—
186Dic173—21no—
187Dic173—41no—
188Dic173—41no—
189C2×C173—41no—
190C2×C173#53411yescharacteristic
191C2×C173C2×#13511yescharacteristic
192C2×C173#57511yescharacteristic
193D173—22no—
194D173—41no—
195D173—81no—
196D173—81no—
197C173#319111yescharacteristic, complemented, Sylow
198#3191—1731nomaximal, Sylow
199#525—1731no—
200#525—1731no—
201#485—1731no—
202#485D17311yescharacteristic, complemented
203#482—1731no—
204#482—1731no—
205#467—1731no—
206#466—1731no—
207#465—1731no—
208#465—1731no—
209#452—1731no—
210#452—1731no—
211#449—1731no—
212#449—1731no—
213C2×C22⋊C8—1731no—
214C2×#41—1731no—
215C2×#41—1731no—
216C23×C4—1731no—
217C2×Q16—1731no—
218C2×SD16—1731no—
219C2×SD16—1731no—
220C2×D8—1731no—
221C22×C8—1731no—
222C22×C8—1731no—
223#123—3461no—
224#123—3461no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227C22⋊Q8—1731no—
228C22⋊Q8C2×D17311yescharacteristic
229#121—1731no—
230#121—1731no—
231#121—1731no—
232#121—1731no—
233#120—3461no—
234#120—3461no—
235C4×D4—3461no—
236C4×D4—3461no—
237C4×D4—3461no—
238C4×D4—3461no—
239#117—1731no—
240#117C2×D17311yescharacteristic
241#107—1731no—
242#107C2×D17311yescharacteristic, complemented
243#106—1731no—
244#106C2×D17311yescharacteristic, complemented
245#103—1731no—
246#103—1731no—
247#103C2×D17311yescharacteristic, complemented
248#103C2×D17311yescharacteristic, complemented
249#102—1731no—
250#102—1731no—
251#102—1731no—
252#102—1731no—
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256C22⋊C8C2×D17311yescharacteristic
257C24—1731no—
258#41—6921no—
259#41—6921no—
260#41—6921no—
261#41—6921no—
262C2×Q8—1731no—
263C2×Q8C22×D17311yescharacteristic
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—1731no—
271C2×D4—1731no—
272C22×C4—3461no—
273C22×C4—3461no—
274C22×C4—1731no—
275C22×C4—1731no—
276C22×C4—1731no—
277C22×C4—1731no—
278C22×C4—1731no—
279C22×C4—6921no—
280C22×C4—6921no—
281C22×C4C22×D17311yescharacteristic
282Q16—6921no—
283SD16—6921no—
284SD16—6921no—
285D8—6921no—
286C2×C8—1731no—
287C2×C8—1731no—
288C2×C8—6921no—
289C2×C8—6921no—
290C2×C8C22×D17311yescharacteristic
291C2×C8C22×D17311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—3461no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—1731no—
297C4⋊C4C22×D17311yescharacteristic
298C4⋊C4C22×D17311yescharacteristic
299C4⋊C4C22×D17311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C42—21no—
311C42—3461no—
312C23—3462no—
313C23—3461no—
314C23—3461no—
315C23—1731no—
316C23—1731no—
317C23—6921no—
318C23—6921no—
319C23D4×D17311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—3461no—
323Q8—3461no—
324D4—3461no—
325D4—3461no—
326D4—3461no—
327D4—3461no—
328D4—6922no—
329D4—6922no—
330D4—6922no—
331D4—6922no—
332D4—6922no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—1731no—
347C2×C4—1731no—
348C2×C4—6921no—
349C2×C4—6921no—
350C2×C4—6922no—
351C2×C4—6922no—
352C2×C4—6921no—
353C2×C4D4×D17311yescharacteristic
354C2×C4C23×D17311yescharacteristic
355C8—21no—
356C8—21no—
357C8—3461no—
358C8—3461no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—3461no—
363C22—3461no—
364C22—3461no—
365C22—3461no—
366C22—3461no—
367C22—1732no—
368C22—6924no—
369C22—6921no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22—6921no—
374C22—6921no—
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3461no—
382C4—3461no—
383C4—6921no—
384C4—6921no—
385C4#1914972511yescharacteristic
386C4#1914972511yescharacteristic
387C2—41no—
388C2—3462no—
389C2—13841no—
390C2—13841no—
391C2—6921no—
392C2#2601205211yescharacteristic
393C2#135×D17311yescharacteristic
394C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#485×C173C2
socleC22×C173C22×D4
hypercenter#485D173
hypocenterC173#3191
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #26012004C2semidirect
2. #26012003C2semidirect
3. #26011834C2semidirect
4. #26011817C2semidirect
5. #26011815C2semidirect
6. #26011731C2semidirect
7. #26011728C2semidirect
8. #26011723C2semidirect
9. #26011721C2semidirect
10. #26011656C2semidirect
11. #26011651C2semidirect
12. #26011639C2semidirect
13. #26011638C2semidirect
14. C22⋊C8×D173C2semidirect
15. #485×C173C2semidirect
19. C2×#14583816C22semidirect
20. C2×#14583815C22semidirect
21. C2×#14583813C22semidirect
22. C2×D1384C22semidirect
41. #19149612C22semidirect
42. #19149610C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
45. #19149601C22semidirect
46. #19149599C22semidirect
47. #19149599C22semidirect
48. #19149599C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
56. #103×C173C22semidirect
57. #103×C173C22semidirect
197. C173#3191semidirect
202. #485D173semidirect
242. #107C2×D173semidirect
244. #106C2×D173semidirect
247. #103C2×D173semidirect
248. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120042
#260120032
#260118342
#260118172
#260118152
#260117312
#260117282
#260117232
#260117212
#260116562
#260116512
#260116392
#260116382
C22⋊C8×D1732
#485×C1732
#3191173