Finite Group Databasebeta

← #36365413

Subgroups — #36365413

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012187C211yesmaximal, characteristic, complemented
2#26011845C211yesmaximal, characteristic, complemented
3#26011824C211yesmaximal, characteristic, complemented
4#26011812C211yesmaximal, characteristic, complemented
5#26011791C211yesmaximal, characteristic, complemented
6#26011725C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011703C211yesmaximal, characteristic, complemented
9#26011636C211yesmaximal, characteristic, complemented
10#26011625C211yesmaximal, characteristic, complemented
11C4×#14583811C211yesmaximal, characteristic, complemented
12D4×#13063436C211yesmaximal, characteristic, complemented
13#26011487C211yesmaximal, characteristic, complemented
14C2×#19149600C211yesmaximal, characteristic, complemented
15#546×C173C211yesmaximal, characteristic, complemented
16C2×#14583813C2211yescharacteristic, complemented
17C2×#14583811C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C22×#13063436C2211yescharacteristic
20C22×#13063436C2211yescharacteristic, complemented
21#19149674—21no—
22#19149671C2211yescharacteristic, complemented
23#19149661—21no—
24#19149653C2211yescharacteristic
25#19149653C2211yescharacteristic, complemented
26#19149651—21no—
27C4×D692C2211yescharacteristic, complemented
28#19149639—21no—
29#19149638C2211yescharacteristic, complemented
30C2×C4⋊Dic173C2211yescharacteristic
31#19149612C2211yescharacteristic, complemented
32#19149612C2211yescharacteristic, complemented
33#19149610C2211yescharacteristic, complemented
34C4⋊#13063436C2211yescharacteristic, complemented
35#19149603C2211yescharacteristic, complemented
36#19149603C2211yescharacteristic, complemented
37#19149603C2211yescharacteristic, complemented
38#19149600C2211yescharacteristic
39#19149600C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149599C2211yescharacteristic, complemented
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic, complemented
47C4×#13063436C2211yescharacteristic, complemented
48C2×#41×C173C2211yescharacteristic
49C22×D4×C173C2211yescharacteristic
50#124×C173C2211yescharacteristic
51#123×C173—21no—
52C22⋊Q8×C173C2211yescharacteristic
53#121×C173—21no—
54#121×C173C2211yescharacteristic
55#120×C173—21no—
56C4×D4×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic
58C2×C22⋊C4×C173—21no—
59C2×D692C2311yescharacteristic
60C2×C22⋊D173—21no—
61C2×C22⋊D173—21no—
62C22×Dic173—21no—
63C2×C4×D173—21no—
64#14583813—41no—
65#14583811—22no—
66#14583811—41no—
67C2×#13063436—41no—
68C2×#13063436—41no—
69C2×#13063436C2311yescharacteristic
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C4⋊Dic173—21no—
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79C22⋊Dic173—21no—
80#14583803—21no—
81#14583803—21no—
82#14583803—21no—
83C4×Dic173—21no—
84C24×C173—21no—
85#41×C173—22no—
86#41×C173—41no—
87C2×Q8×C173C2311yescharacteristic
88C2×D4×C173—22no—
89C2×D4×C173—21no—
90C2×D4×C173—21no—
91C2×D4×C173—21no—
92C2×D4×C173—41no—
93C2×D4×C173D411yescharacteristic
94C2×D4×C173C2311yescharacteristic
95C2×D4×C173C2311yescharacteristic
96C22×C4×C173—21no—
97C22×C4×C173—21no—
98C22×C4×C173C2311yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173D411yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112D692—21no—
113D692—21no—
114C22⋊D173—42no—
115C22⋊D173—81no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—81no—
120C4×D173—81no—
121#13063436—21no—
122#13063436—21no—
123#13063436—21no—
124#13063436—21no—
125C23×C173—21no—
126C23×C173—21no—
127C23×C173—42no—
128C23×C173—42no—
129C23×C173—81no—
130C23×C173C2×D411yescharacteristic
131C23×C173C2×D411yescharacteristic
132Q8×C173—21no—
133Q8×C173—21no—
134D4×C173—22no—
135D4×C173—21no—
136D4×C173—21no—
137D4×C173—21no—
138D4×C173—42no—
139D4×C173—42no—
140D4×C173—81no—
141D4×C173#4112yes—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—42no—
149C2×C4×C173—81no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—22no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—84no—
167C22×C173#13512yes—
168C22×C173C22×D411yescharacteristic
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—41no—
172C4×C173—21no—
173C4×C173—21no—
174C4×C173—21no—
175C4×C173—21no—
176C4×C173—41no—
177C4×C173—41no—
178C4×C173C2×#4111yescharacteristic
179C4×C173H(5,𝔽2)11yescharacteristic
180Dic173—41no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—22no—
184C2×C173—41no—
185C2×C173—42no—
186C2×C173—81no—
187C2×C173#54611yescharacteristic
188C2×C173C2×#13511yescharacteristic
189C2×C173#57611yescharacteristic
190D173—81no—
191C173#329711yescharacteristic, complemented, Sylow
192#3297—1731nomaximal, Sylow
193#553—1731no—
194#546D17311yescharacteristic, complemented
195#488—1731no—
196#484—1731no—
197#482—1731no—
198#478—1731no—
199#466—1731no—
200#465—1731no—
201#463—1731no—
202#449—1731no—
203#447—1731no—
204C4×D8—1731no—
205C8×D4—1731no—
206#416—1731no—
207C2×#102—1731no—
208C2×#41C2×D17311yescharacteristic
209C22×D4C2×D17311yescharacteristic
210C2×SD16—1731no—
211C2×D8—1731no—
212C2×D8—1731no—
213C22×C8—1731no—
214C22×C8—1731no—
215#126—3461no—
216#125—1731no—
217#124C2×D17311yescharacteristic
218#123—21no—
219#123—3461no—
220C22⋊Q8C2×D17311yescharacteristic
221#121—21no—
222#121—3461no—
223#121—1731no—
224#121—1731no—
225#121C2×D17311yescharacteristic
226#120—21no—
227C4×D4—3461no—
228C4×D4—1731no—
229C4×D4C2×D17311yescharacteristic
230C4×D4C2×D17311yescharacteristic
231#117—1731no—
232C2×C4⋊C4—1731no—
233C2×C22⋊C4—21no—
234#107—1731no—
235#107—1731no—
236#106—1731no—
237C4⋊C8—1731no—
238#103—1731no—
239#103—1731no—
240#103—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102—1731no—
246#102—1731no—
247#102—1731no—
248C22⋊C8—1731no—
249C22⋊C8—1731no—
250C4×C8—1731no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D17311yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—1731no—
263C2×D4C22⋊D17311yescharacteristic
264C2×D4C22×D17311yescharacteristic
265C2×D4C22×D17311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4C22×D17311yescharacteristic
271C22×C4C22×D17311yescharacteristic
272SD16—6921no—
273D8—3462no—
274D8—6921no—
275C2×C8—1731no—
276C2×C8—1731no—
277C2×C8—1731no—
278C2×C8—1731no—
279C2×C8—6921no—
280C2×C8—6921no—
281C4⋊C4—21no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4—1731no—
288C4⋊C4C22⋊D17311yescharacteristic
289C4⋊C4C22×D17311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4C22⋊D17311yescharacteristic
300C22⋊C4C22⋊D17311yescharacteristic
301C42—3461no—
302C42C22×D17311yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—3461no—
309C23C2×C22⋊D17311yescharacteristic
310C23C2×C22⋊D17311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—3461no—
321D4—3461no—
322D4—13841no—
323D4—6922no—
324D4#1458382512yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—13841no—
337C2×C4—13841no—
338C2×C4C2×C22⋊D17311yescharacteristic
339C2×C4C2×C22⋊D17311yescharacteristic
340C2×C4C2×C22⋊D17311yescharacteristic
341C2×C4C2×C22⋊D17311yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—3461no—
344C8—3461no—
345C8—3461no—
346C8—3461no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—6921no—
360C22—6921no—
361C22—6921no—
362C22#1914969912yes—
363C22C22×C22⋊D17311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—6921no—
371C4—6921no—
372C4—6921no—
373C4C2×#1458382511yescharacteristic
374C4#1914972411yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—13841no—
380C2#2601213511yescharacteristic
381C2C2×#1914969911yescharacteristic
382C2#2601229411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#546×C173C2
socleC22×C173C22×D4
hypercenter#546D173
hypocenterC173#3297
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #26012187C2semidirect
2. #26011845C2semidirect
3. #26011824C2semidirect
4. #26011812C2semidirect
5. #26011791C2semidirect
6. #26011725C2semidirect
7. #26011722C2semidirect
8. #26011703C2semidirect
9. #26011636C2semidirect
10. #26011625C2semidirect
11. C4×#14583811C2semidirect
12. D4×#13063436C2semidirect
13. #26011487C2semidirect
14. C2×#19149600C2semidirect
15. #546×C173C2semidirect
16. C2×#14583813C22semidirect
20. C22×#13063436C22semidirect
22. #19149671C22semidirect
25. #19149653C22semidirect
27. C4×D692C22semidirect
29. #19149638C22semidirect
31. #19149612C22semidirect
32. #19149612C22semidirect
33. #19149610C22semidirect
34. C4⋊#13063436C22semidirect
35. #19149603C22semidirect
36. #19149603C22semidirect
37. #19149603C22semidirect
44. #19149599C22semidirect
46. C22⋊#13063436C22semidirect
47. C4×#13063436C22semidirect
191. C173#3297semidirect
194. #546D173semidirect

Maximal subgroups

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

subgroupindex
#260121872
#260118452
#260118242
#260118122
#260117912
#260117252
#260117222
#260117032
#260116362
#260116252
C4×#145838112
D4×#130634362
#260114872
C2×#191496002
#546×C1732
#3297173