Finite Group Databasebeta

← #36365411

Subgroups — #36365411

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#26012189C211yesmaximal, characteristic, complemented
2#26012132C211yesmaximal, characteristic, complemented
3#26011847C211yesmaximal, characteristic, complemented
4#26011818C211yesmaximal, characteristic, complemented
5#26011794C211yesmaximal, characteristic, complemented
6#26011728C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011709C211yesmaximal, characteristic, complemented
9#26011639C211yesmaximal, characteristic, complemented
10#26011627C211yesmaximal, characteristic, complemented
11#26011573C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14C2×#19149599C211yesmaximal, characteristic, complemented
15#484×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583813C2211yescharacteristic, complemented
19C2×D1384C2211yescharacteristic
20C2×D1384C2211yescharacteristic
21C22×#13063436C2211yescharacteristic
22C2×C8×D173C2211yescharacteristic, complemented
23#19149674—21no—
24#19149670C2211yescharacteristic, complemented
25#19149668C2211yescharacteristic
26#19149664—21no—
27#19149664—21no—
28#19149654—21no—
29#19149654C2211yescharacteristic
30#19149654C2211yescharacteristic, complemented
31#19149653—21no—
32#19149653C2211yescharacteristic
33#19149649—21no—
34#19149641C2211yescharacteristic
35#19149641C2211yescharacteristic, complemented
36C4×C22⋊D173—21no—
37C4×C22⋊D173C2211yescharacteristic
38#19149637C2211yescharacteristic, complemented
39C22⋊C4×D173—21no—
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149599C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C22⋊Q8×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic, complemented
55#107×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic
59C23×D173—21no—
60#14583827—22no—
61#14583827—41no—
62C2×D692—22no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—41no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×D692C2311yescharacteristic
71C2×C22⋊D173—21no—
72C2×C22⋊D173D411yescharacteristic
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77#14583813—41no—
78D1384—22no—
79D1384—41no—
80C2×#13063436—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C8×D173—41no—
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805D411yescharacteristic
88C22⋊Dic173D411yescharacteristic
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803D411yescharacteristic
98C4×Dic173—21no—
99C4×Dic173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101C22×C4×C173—21no—
102C22×C4×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—42no—
115C22×D173—42no—
116C22×D173—81no—
117C22×D173C2×D411yescharacteristic
118D692—21no—
119D692—21no—
120D692—21no—
121D692—21no—
122D692—21no—
123D692—42no—
124D692—42no—
125D692—81no—
126D692—81no—
127D692#4112yes—
128C22⋊D173—22no—
129C22⋊D173—42no—
130C2×Dic173—21no—
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—21no—
135C4×D173—21no—
136C4×D173—42no—
137C4×D173—81no—
138C4×D173—81no—
139#13063436—21no—
140#13063436—21no—
141C23×C173C2×D411yescharacteristic
142Q8×C173—21no—
143Q8×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—81no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173#13512yes—
158C22×C173C22×D411yescharacteristic
159C2×D173—21no—
160C2×D173—22no—
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C2×D173—84no—
172C4×C173—21no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173—41no—
176C4×C173—41no—
177C4×C173C2×#4111yescharacteristic
178C4×C173H(5,𝔽2)11yescharacteristic
179Dic173—21no—
180Dic173—21no—
181Dic173—21no—
182Dic173—41no—
183C2×C173—22no—
184C2×C173#54611yescharacteristic
185C2×C173C2×#13511yescharacteristic
186C2×C173#57611yescharacteristic
187D173—41no—
188D173—42no—
189D173—81no—
190D173—81no—
191C173#329711yescharacteristic, complemented, Sylow
192#3297—1731nomaximal, Sylow
193#553—1731no—
194#546—1731no—
195#488—1731no—
196#484D17311yescharacteristic, complemented
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×#41—1731no—
209C22×D4—1731no—
210C2×SD16—1731no—
211C2×D8—1731no—
212C2×D8—1731no—
213C22×C8—1731no—
214C22×C8—1731no—
215#126—3461no—
216#125—1731no—
217#124—1731no—
218#123—3461no—
219#123—3461no—
220C22⋊Q8C2×D17311yescharacteristic
221#121—3461no—
222#121—3461no—
223#121—1731no—
224#121—1731no—
225#121—1731no—
226#120—3461no—
227C4×D4—3461no—
228C4×D4—1731no—
229C4×D4—1731no—
230C4×D4—1731no—
231#117—1731no—
232C2×C4⋊C4C2×D17311yescharacteristic
233C2×C22⋊C4—3461no—
234#107C2×D17311yescharacteristic, complemented
235#107C2×D17311yescharacteristic, complemented
236#106—1731no—
237C4⋊C8—1731no—
238#103—1731no—
239#103C2×D17311yescharacteristic, complemented
240#103C2×D17311yescharacteristic, complemented
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⋊C8C2×D17311yescharacteristic
250C4×C8—1731no—
251C24—3461no—
252#41—3462no—
253#41—6921no—
254C2×Q8C22×D17311yescharacteristic
255C2×D4—3462no—
256C2×D4—3461no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—1731no—
262C2×D4—1731no—
263C2×D4—1731no—
264C2×D4—1731no—
265C2×D4—6921no—
266C22×C4—21no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—1731no—
271C22×C4C22×D17311yescharacteristic
272SD16—6921no—
273D8—3462no—
274D8—6921no—
275C2×C8—1731no—
276C2×C8—1731no—
277C2×C8—6921no—
278C2×C8—6921no—
279C2×C8C22×D17311yescharacteristic
280C2×C8C22×D17311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4C22×D17311yescharacteristic
288C4⋊C4C22×D17311yescharacteristic
289C4⋊C4C22×D17311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—1731no—
300C22⋊C4—1731no—
301C42—3461no—
302C42—1731no—
303C23—3461no—
304C23—3461no—
305C23—3461no—
306C23—13841no—
307C23—1731no—
308C23—6922no—
309C23—6922no—
310C23D4×D17311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—3462no—
314D4—3461no—
315D4—3461no—
316D4—3461no—
317D4—3461no—
318D4—3461no—
319D4—13841no—
320D4—13841no—
321D4—1732no—
322D4—6922no—
323D4—6922no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—13841no—
336C2×C4—13841no—
337C2×C4—1731no—
338C2×C4—1731no—
339C2×C4—1731no—
340C2×C4—6922no—
341C2×C4D4×D17311yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3461no—
346C8—3461no—
347C22—21no—
348C22—21no—
349C22—3461no—
350C22—3461no—
351C22—3461no—
352C22—3462no—
353C22—13844no—
354C22—6921no—
355C22—6921no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22—6921no—
362C22#1914970212yes—
363C22C2×D4×D17311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—3461no—
370C4—3461no—
371C4—3461no—
372C4—6921no—
373C4#41×D17311yescharacteristic
374C4#1914972511yescharacteristic
375C2—22no—
376C2—13841no—
377C2—13841no—
378C2—6921no—
379C2—6922no—
380C2#2601213311yescharacteristic
381C2C2×#1914970211yescharacteristic
382C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#484×C173C2
socleC22×C173C22×D4
hypercenter#484D173
hypocenterC173#3297
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #26012189C2semidirect
2. #26012132C2semidirect
3. #26011847C2semidirect
4. #26011818C2semidirect
5. #26011794C2semidirect
6. #26011728C2semidirect
7. #26011722C2semidirect
8. #26011709C2semidirect
9. #26011639C2semidirect
10. #26011627C2semidirect
11. #26011573C2semidirect
12. #26011554C2semidirect
13. #26011486C2semidirect
14. C2×#19149599C2semidirect
15. #484×C173C2semidirect
18. C2×#14583813C22semidirect
22. C2×C8×D173C22semidirect
24. #19149670C22semidirect
30. #19149654C22semidirect
35. #19149641C22semidirect
38. #19149637C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149603C22semidirect
49. #19149599C22semidirect
50. #19149591C22semidirect
51. C8×Dic173C22semidirect
54. #107×C173C22semidirect
55. #107×C173C22semidirect
56. #103×C173C22semidirect
57. #103×C173C22semidirect
191. C173#3297semidirect
196. #484D173semidirect
234. #107C2×D173semidirect
235. #107C2×D173semidirect
239. #103C2×D173semidirect
240. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121892
#260121322
#260118472
#260118182
#260117942
#260117282
#260117222
#260117092
#260116392
#260116272
#260115732
#260115542
#260114862
C2×#191495992
#484×C1732
#3297173