Finite Group Databasebeta

← #36364348

Subgroups — #36364348

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012003C211yesmaximal, characteristic
2C22⋊Q8×D173C211yesmaximal, characteristic, complemented
3#26011832C211yesmaximal, characteristic, complemented
4#26011823C211yesmaximal, characteristic, complemented
5#26011806C211yesmaximal, characteristic, complemented
6#26011733C211yesmaximal, characteristic, complemented
7#26011731C211yesmaximal, characteristic, complemented
8#26011723C211yesmaximal, characteristic, complemented
9#26011721C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011651C211yesmaximal, characteristic, complemented
12#26011648C211yesmaximal, characteristic, complemented
13#26011631C211yesmaximal, characteristic, complemented
14C22⋊C8×D173C211yesmaximal, characteristic, complemented
15#485×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C2×Dic692C2211yescharacteristic, complemented
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583815C2211yescharacteristic, complemented
22C2×#14583813C2211yescharacteristic, complemented
23C2×C8×D173C2211yescharacteristic
24C2×C8×D173C2211yescharacteristic
25#19149663—21no—
26#19149659C2211yescharacteristic
27#19149659C2211yescharacteristic
28#19149659C2211yescharacteristic
29#19149658C2211yescharacteristic
30#19149657—21no—
31#19149656—21no—
32#19149655C2211yescharacteristic
33#19149654C2211yescharacteristic
34#19149654C2211yescharacteristic
35#19149649—21no—
36C4×D692—21no—
37C4×D692—21no—
38C4⋊C4×D173—21no—
39C4⋊C4×D173C2211yescharacteristic
40C22⋊C4×D173—21no—
41#19149612C2211yescharacteristic, complemented
42#19149610C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149602C2211yescharacteristic, complemented
47#19149601C2211yescharacteristic, 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#14583826—41no—
61#14583826—41no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65Q8×D173—41no—
66Q8×D173—41no—
67C2×D692—21no—
68C2×D692—21no—
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C22×Dic173D411yescharacteristic
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—41no—
76C2×C4×D173—41no—
77C2×C4×D173D411yescharacteristic
78C2×C4×D173D411yescharacteristic
79C2×C4×D173C2311yescharacteristic
80C2×C4×D173C2311yescharacteristic
81Dic692—41no—
82#14583816—41no—
83#14583815—41no—
84#14583813—41no—
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87C8×D173—41no—
88C8×D173—41no—
89C4⋊Dic173—21no—
90C4⋊Dic173—21no—
91C4⋊Dic173—21no—
92C4⋊Dic173C2311yescharacteristic
93#14583805—21no—
94#14583805—21no—
95#14583805—21no—
96C22⋊Dic173—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803—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—22no—
116C22×D173—21no—
117C22×D173—41no—
118C22×D173C2×D411yescharacteristic
119C22×D173C2×D411yescharacteristic
120Dic346—21no—
121Dic346—21no—
122Dic346—21no—
123Dic346—21no—
124Dic346—42no—
125D692—21no—
126D692—21no—
127D692—42no—
128D692—42no—
129C22⋊D173—42no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—41no—
134C2×Dic173C2×D411yescharacteristic
135C2×Dic173C2×D411yescharacteristic
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—21no—
139C4×D173—21no—
140C4×D173—41no—
141C4×D173—41no—
142C4×D173—42no—
143C4×D173—42no—
144C4×D173—42no—
145#13063436—21no—
146#13063436—21no—
147C23×C173C2×D411yescharacteristic
148Q8×C173—21no—
149Q8×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
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×C173C22×D411yescharacteristic
164C2×D173—21no—
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—21no—
168C2×D173—44no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C2×D173#13512yes—
173C2×D173#13712yes—
174C4×C173—21no—
175C4×C173—41no—
176C4×C173—41no—
177C4×C173—41no—
178C4×C173—41no—
179C4×C173H(5,𝔽2)11yescharacteristic
180C4×C173H(5,𝔽2)11yescharacteristic
181Dic173—21no—
182Dic173—21no—
183Dic173—41no—
184Dic173—41no—
185Dic173—41no—
186C2×C173—41no—
187C2×C173#53411yescharacteristic
188C2×C173C2×#13511yescharacteristic
189C2×C173C2×#13711yescharacteristic
190D173—22no—
191D173—22no—
192D173—81no—
193C173#319011yescharacteristic, complemented, Sylow
194#3190—1731nomaximal, Sylow
195#525—1731no—
196C2×C22⋊Q8—1731no—
197#485—1731no—
198#485D17311yescharacteristic, complemented
199#483—1731no—
200#480—1731no—
201#467—1731no—
202#467—1731no—
203#465—1731no—
204#465—1731no—
205#452—1731no—
206#452—1731no—
207#451—1731no—
208#448—1731no—
209C2×C22⋊C8—1731no—
210C2×#41—1731no—
211C22×Q8—1731no—
212C23×C4—1731no—
213C2×Q16—1731no—
214C2×Q16—1731no—
215C2×SD16—1731no—
216C2×SD16—1731no—
217C22×C8—1731no—
218C22×C8—1731no—
219#123—3461no—
220C22⋊Q8—3461no—
221C22⋊Q8—3461no—
222C22⋊Q8—1731no—
223C22⋊Q8—1731no—
224C22⋊Q8—1731no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227C22⋊Q8C2×D17311yescharacteristic
228#121—1731no—
229#121—1731no—
230#120—3461no—
231C4×D4—3461no—
232C4×D4—3461no—
233#117C2×D17311yescharacteristic
234C2×C4⋊C4—3461no—
235C2×C4⋊C4—1731no—
236C2×C22⋊C4—3461no—
237#107—1731no—
238#107C2×D17311yescharacteristic, complemented
239#106—1731no—
240#106C2×D17311yescharacteristic, complemented
241#103—1731no—
242#103—1731no—
243#103—1731no—
244#103—1731no—
245#103C2×D17311yescharacteristic, complemented
246#103C2×D17311yescharacteristic, complemented
247#102—1731no—
248#102—1731no—
249C22⋊C8—1731no—
250C22⋊C8—1731no—
251C22⋊C8—1731no—
252C22⋊C8C2×D17311yescharacteristic
253C24—1731no—
254#41—6921no—
255#41—6921no—
256C2×Q8—3461no—
257C2×Q8—1731no—
258C2×Q8—1731no—
259C2×Q8—6921no—
260C2×Q8—6921no—
261C2×Q8C22×D17311yescharacteristic
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4—1731no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—1731no—
270C22×C4—1731no—
271C22×C4—1731no—
272C22×C4—1731no—
273C22×C4—1731no—
274C22×C4—6921no—
275C22×C4—6921no—
276C22×C4C22×D17311yescharacteristic
277Q16—6921no—
278Q16—6921no—
279SD16—6921no—
280SD16—6921no—
281C2×C8—1731no—
282C2×C8—1731no—
283C2×C8—6921no—
284C2×C8—6921no—
285C2×C8C22×D17311yescharacteristic
286C2×C8C22×D17311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—3461no—
293C4⋊C4—3461no—
294C4⋊C4—1731no—
295C4⋊C4C22×D17311yescharacteristic
296C4⋊C4C22×D17311yescharacteristic
297C4⋊C4C22×D17311yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C42—21no—
308C23—3462no—
309C23—3462no—
310C23—3461no—
311C23—1731no—
312C23—1731no—
313C23—6921no—
314C23D4×D17311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—3461no—
318Q8—3461no—
319Q8—3461no—
320Q8—3461no—
321Q8—6922no—
322D4—3461no—
323D4—3461no—
324D4—6922no—
325D4—6922no—
326D4—6922no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—3461no—
340C2×C4—1731no—
341C2×C4—1731no—
342C2×C4—6921no—
343C2×C4—6921no—
344C2×C4—6922no—
345C2×C4—6922no—
346C2×C4—6922no—
347C2×C4—6921no—
348C2×C4D4×D17311yescharacteristic
349C2×C4C23×D17311yescharacteristic
350C8—21no—
351C8—21no—
352C8—3461no—
353C8—3461no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—3461no—
358C22—3461no—
359C22—3461no—
360C22—3461no—
361C22—1732no—
362C22—1732no—
363C22—6924no—
364C22—6921no—
365C22—6921no—
366C22—6921no—
367C22C2×D4×D17311yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—3461no—
374C4—3461no—
375C4—6921no—
376C4—6921no—
377C4—6921no—
378C4#1914972511yescharacteristic
379C4#1914972511yescharacteristic
380C2—41no—
381C2—3462no—
382C2—3462no—
383C2—13841no—
384C2#2601205211yescharacteristic
385C2#135×D17311yescharacteristic
386C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#485×C173C2
socleC22×C173C22×D4
prime radical#26012003C2
hypercenter#485D173
hypocenterC173#3190
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
2. C22⋊Q8×D173C2semidirect
3. #26011832C2semidirect
4. #26011823C2semidirect
5. #26011806C2semidirect
6. #26011733C2semidirect
7. #26011731C2semidirect
8. #26011723C2semidirect
9. #26011721C2semidirect
10. #26011655C2semidirect
11. #26011651C2semidirect
12. #26011648C2semidirect
13. #26011631C2semidirect
14. C22⋊C8×D173C2semidirect
15. #485×C173C2semidirect
19. C2×Dic692C22semidirect
20. C2×#14583816C22semidirect
21. C2×#14583815C22semidirect
22. C2×#14583813C22semidirect
41. #19149612C22semidirect
42. #19149610C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
45. #19149602C22semidirect
46. #19149602C22semidirect
47. #19149601C22semidirect
48. #19149599C22semidirect
54. #107×C173C22semidirect
55. #106×C173C22semidirect
56. #103×C173C22semidirect
57. #103×C173C22semidirect
193. C173#3190semidirect
198. #485D173semidirect
238. #107C2×D173semidirect
240. #106C2×D173semidirect
245. #103C2×D173semidirect
246. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120032
C22⋊Q8×D1732
#260118322
#260118232
#260118062
#260117332
#260117312
#260117232
#260117212
#260116552
#260116512
#260116482
#260116312
C22⋊C8×D1732
#485×C1732
#3190173