Finite Group Databasebeta

← #36365417

Subgroups — #36365417

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#26012193C211yesmaximal, characteristic, complemented
2#26012135C211yesmaximal, characteristic, complemented
3#26011844C211yesmaximal, characteristic, complemented
4#26011827C211yesmaximal, characteristic, complemented
5#26011813C211yesmaximal, characteristic, complemented
6#26011789C211yesmaximal, characteristic, complemented
7#26011724C211yesmaximal, characteristic, complemented
8#26011718C211yesmaximal, characteristic, complemented
9#26011704C211yesmaximal, characteristic, complemented
10#26011633C211yesmaximal, characteristic, complemented
11D8×Dic173C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic, complemented
13#26011484C211yesmaximal, characteristic, complemented
14C2×#19149600C211yesmaximal, characteristic, complemented
15#447×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583814C2211yescharacteristic, complemented
18C22×#13063436C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic, complemented
20#19149672—21no—
21#19149670C2211yescharacteristic, complemented
22#19149666C2211yescharacteristic
23#19149663—21no—
24#19149660—21no—
25#19149658C2211yescharacteristic
26#19149651—21no—
27#19149651—21no—
28#19149650C2211yescharacteristic
29#19149650C2211yescharacteristic, complemented
30#19149648—21no—
31D4×Dic173—21no—
32D4×Dic173C2211yescharacteristic
33D4×Dic173C2211yescharacteristic, complemented
34C4×C22⋊D173C2211yescharacteristic
35#19149637C2211yescharacteristic, complemented
36C2×C4⋊Dic173C2211yescharacteristic
37C2×C22⋊Dic173—21no—
38#19149613C2211yescharacteristic, complemented
39#19149613C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic, complemented
50#19149591C2211yescharacteristic, complemented
51C8×Dic173C2211yescharacteristic, complemented
52C22×D4×C173C2211yescharacteristic
53C2×D8×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173D411yescharacteristic
66C22×Dic173—21no—
67C22×Dic173—21no—
68C22×Dic173—21no—
69C22×Dic173—21no—
70C2×C4×D173C2311yescharacteristic
71#14583814—41no—
72C2×#13063436—41no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C8×D173—41no—
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C24×C173—21no—
97C2×D4×C173—22no—
98C2×D4×C173—21no—
99C2×D4×C173—21no—
100C2×D4×C173—41no—
101C2×D4×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105D8×C173—22no—
106D8×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114C22⋊D173—22no—
115C22⋊D173—42no—
116C22⋊D173—81no—
117C22⋊D173—81no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—42no—
123C2×Dic173—81no—
124C2×Dic173—81no—
125C2×Dic173—81no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131#13063436—21no—
132#13063436—21no—
133C23×C173—21no—
134C23×C173—21no—
135C23×C173—21no—
136C23×C173—42no—
137C23×C173—42no—
138C23×C173—81no—
139C23×C173C2×D411yescharacteristic
140D4×C173—21no—
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—42no—
146D4×C173—42no—
147D4×C173#4112yes—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2411yescharacteristic
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—22no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
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—21no—
170C2×D173—21no—
171C2×D173—21no—
172C4×C173—21no—
173C4×C173—41no—
174C4×C173C2×#4111yescharacteristic
175C4×C173H(5,𝔽2)11yescharacteristic
176Dic173—21no—
177Dic173—21no—
178Dic173—21no—
179Dic173—41no—
180Dic173—41no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—22no—
184C2×C173—42no—
185C2×C173—81no—
186C2×C173—81no—
187C2×C173#54611yescharacteristic
188C2×C173C2×#13511yescharacteristic
189C2×C173#57611yescharacteristic
190D173—41no—
191C173#329711yescharacteristic, complemented, Sylow
192#3297—1731nomaximal, Sylow
193#553—1731no—
194#546—1731no—
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#447D17311yescharacteristic, complemented
204C4×D8—1731no—
205C8×D4—1731no—
206#416—1731no—
207C2×#102—1731no—
208C2×#41—1731no—
209C22×D4C2×D17311yescharacteristic
210C2×SD16—1731no—
211C2×D8C2×D17311yescharacteristic
212C2×D8C2×D17311yescharacteristic
213C22×C8—1731no—
214C22×C8—1731no—
215#126—3461no—
216#125—1731no—
217#124—1731no—
218#123—3461no—
219#123—3461no—
220C22⋊Q8—1731no—
221#121—3461no—
222#121—3461no—
223#121—1731no—
224#121—1731no—
225#121C2×D17311yescharacteristic
226#120—3461no—
227C4×D4—3461no—
228C4×D4—1731no—
229C4×D4—1731no—
230C4×D4—1731no—
231#117—1731no—
232C2×C4⋊C4—1731no—
233C2×C22⋊C4—3461no—
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#102C2×D17311yescharacteristic
247#102C2×D17311yescharacteristic
248C22⋊C8—1731no—
249C22⋊C8C2×D17311yescharacteristic
250C4×C8—1731no—
251C24—21no—
252#41—3462no—
253#41—6921no—
254C2×Q8—1731no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—1731no—
263C2×D4C22×D17311yescharacteristic
264C2×D4C22×D17311yescharacteristic
265C2×D4C22×D17311yescharacteristic
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—1731no—
271C22×C4C22×D17311yescharacteristic
272SD16—6921no—
273D8—22no—
274D8—41no—
275C2×C8—1731no—
276C2×C8—1731no—
277C2×C8—6921no—
278C2×C8—6921no—
279C2×C8C22×D17311yescharacteristic
280C2×C8C22×D17311yescharacteristic
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—3461no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4—1731no—
288C4⋊C4—1731no—
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—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23—1731no—
310C23D4×D17311yescharacteristic
311Q8—3461no—
312Q8—3461no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—42no—
319D4—42no—
320D4—3462no—
321D4—13841no—
322D4—13841no—
323D4—6922no—
324D4#1458382512yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—13841no—
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—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—3461no—
360C22—3461no—
361C22—3461no—
362C22#1914969812yes—
363C22C2×D4×D17311yescharacteristic
364C4—21no—
365C4—41no—
366C4—3461no—
367C4—3461no—
368C4—3461no—
369C4—6921no—
370C4—6921no—
371C4—6921no—
372C4—6921no—
373C4C2×#1458382511yescharacteristic
374C4#1914972411yescharacteristic
375C2—22no—
376C2—42no—
377C2—81no—
378C2—81no—
379C2—6921no—
380C2#2601212811yescharacteristic
381C2C2×#1914969811yescharacteristic
382C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#447×C173C2
socleC22×C173C22×D4
hypercenter#447D173
hypocenterC173#3297
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #26012193C2semidirect
2. #26012135C2semidirect
3. #26011844C2semidirect
4. #26011827C2semidirect
5. #26011813C2semidirect
6. #26011789C2semidirect
7. #26011724C2semidirect
8. #26011718C2semidirect
9. #26011704C2semidirect
10. #26011633C2semidirect
11. D8×Dic173C2semidirect
12. #26011554C2semidirect
13. #26011484C2semidirect
14. C2×#19149600C2semidirect
15. #447×C173C2semidirect
17. C2×#14583814C22semidirect
19. C2×C8×D173C22semidirect
21. #19149670C22semidirect
29. #19149650C22semidirect
33. D4×Dic173C22semidirect
35. #19149637C22semidirect
38. #19149613C22semidirect
39. #19149613C22semidirect
40. #19149610C22semidirect
41. #19149608C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149602C22semidirect
49. #19149600C22semidirect
50. #19149591C22semidirect
51. C8×Dic173C22semidirect
191. C173#3297semidirect
203. #447D173semidirect

Maximal subgroups

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

subgroupindex
#260121932
#260121352
#260118442
#260118272
#260118132
#260117892
#260117242
#260117182
#260117042
#260116332
D8×Dic1732
#260115542
#260114842
C2×#191496002
#447×C1732
#3297173