Finite Group Databasebeta

← #36365508

Subgroups — #36365508

Order 22144.

408 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012205C211yesmaximal, characteristic, complemented
2#26012159C211yesmaximal, characteristic, complemented
3#26011836C211yesmaximal, characteristic, complemented
4#26011818C211yesmaximal, characteristic, complemented
5#26011812C211yesmaximal, characteristic, complemented
6#26011785C211yesmaximal, characteristic, complemented
7#26011726C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011634C211yesmaximal, characteristic, complemented
10#26011633C211yesmaximal, characteristic, complemented
11#26011582C211yesmaximal, characteristic, complemented
12C8×C22⋊D173C211yesmaximal, characteristic, complemented
13#26011488C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#466×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149675—21no—
23#19149675C2211yescharacteristic
24#19149666—21no—
25#19149665—21no—
26#19149654C2211yescharacteristic
27#19149652—21no—
28#19149651—21no—
29#19149651—21no—
30#19149650C2211yescharacteristic
31#19149644C2211yescharacteristic
32D4×Dic173C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149639—21no—
35#19149638C2211yescharacteristic
36#19149637C2211yescharacteristic
37#19149635—21no—
38#19149612C2211yescharacteristic
39#19149610C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×D8×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54#121×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—41no—
61#14583825—22no—
62#14583825—41no—
63C2×Dic346—21no—
64C2×Dic346C2311yescharacteristic
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173D411yescharacteristic
73C22×Dic173—21no—
74C22×Dic173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77#14583814—22no—
78#14583811—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C8×D173—41no—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C2×D4×C173—21no—
99C2×D4×C173—21no—
100C2×D4×C173C2311yescharacteristic
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103D8×C173—41no—
104C2×C8×C173—41no—
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—81no—
115Dic346#4112yes—
116D692—21no—
117D692—21no—
118D692—81no—
119C22⋊D173—22no—
120C22⋊D173—42no—
121C22⋊D173—42no—
122C22⋊D173—42no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—42no—
129C2×Dic173—42no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—42no—
136#13063436—21no—
137#13063436—21no—
138C23×C173—21no—
139C23×C173—21no—
140C23×C173C2×D411yescharacteristic
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—81no—
146D4×C173—81no—
147C2×C4×C173—21no—
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—21no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173C22×D411yescharacteristic
165C2×D173—21no—
166C2×D173—21no—
167C2×D173—21no—
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173C2×#4111yescharacteristic
175C4×C173H(5,𝔽2)11yescharacteristic
176Dic173—21no—
177Dic173—22no—
178Dic173—21no—
179Dic173—41no—
180Dic173—41no—
181Dic173#13512yes—
182C2×C173—41no—
183C2×C173—81no—
184C2×C173—81no—
185C2×C173#54611yescharacteristic
186C2×C173C2×#13511yescharacteristic
187C2×C173#57611yescharacteristic
188D173—41no—
189D173—81no—
190C173#330511yescharacteristic, complemented, Sylow
191#3305—1731nomaximal, Sylow
192#555—1731no—
193#550—1731no—
194#486—1731no—
195#482—1731no—
196#482—1731no—
197#477—1731no—
198#466—1731no—
199#466D17311yescharacteristic, complemented
200#459—1731no—
201#449—1731no—
202#449—1731no—
203C4×SD16—1731no—
204C8×D4—1731no—
205#416—1731no—
206#416—1731no—
207C2×#41—1731no—
208C2×#41—1731no—
209C2×SD16—1731no—
210C2×D8—1731no—
211C2×D8C2×D17311yescharacteristic
212C22×C8—1731no—
213C22×C8C2×D17311yescharacteristic
214C4⋊Q8—1731no—
215#127—3461no—
216#127—1731no—
217#124—3461no—
218#124—3461no—
219#121—3461no—
220#121—3461no—
221#121—3461no—
222#121—1731no—
223#121—1731no—
224#121C2×D17311yescharacteristic
225#121C2×D17311yescharacteristic
226C4×Q8—1731no—
227C4×D4—3461no—
228C4×D4—1731no—
229C4×D4—1731no—
230#117—3461no—
231#117—1731no—
232#117—1731no—
233#107—1731no—
234#107C2×D17311yescharacteristic
235#106—1731no—
236C4⋊C8—1731no—
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244#102—1731no—
245#102C2×D17311yescharacteristic
246#102C2×D17311yescharacteristic
247C22⋊C8—1731no—
248C22⋊C8—1731no—
249C4×C8—1731no—
250#41—3462no—
251#41—3462no—
252#41—6921no—
253#41—6921no—
254C2×Q8—3461no—
255C2×Q8—1731no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—1731no—
265C2×D4—1731no—
266C2×D4C22×D17311yescharacteristic
267C2×D4C22×D17311yescharacteristic
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4—1731no—
272C22×C4C22×D17311yescharacteristic
273SD16—3462no—
274D8—41no—
275D8—6921no—
276C2×C8—41no—
277C2×C8—1731no—
278C2×C8—1731no—
279C2×C8—6921no—
280C2×C8C22×D17311yescharacteristic
281C2×C8C22×D17311yescharacteristic
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4C22×D17311yescharacteristic
288C4⋊C4C22×D17311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—1731no—
297C22⋊C4—1731no—
298C42—3461no—
299C42—3461no—
300C42—3461no—
301C42—1731no—
302C23—21no—
303C23—21no—
304C23—3461no—
305C23—1731no—
306C23D4×D17311yescharacteristic
307Q8—3461no—
308Q8—13841no—
309Q8—1732no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—81no—
315D4—81no—
316D4—3462no—
317D4—3461no—
318D4—3461no—
319D4—13841no—
320D4—13841no—
321D4—13841no—
322D4—6922no—
323D4—6922no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—1731no—
335C2×C4—1731no—
336C2×C4—1731no—
337C2×C4—6922no—
338C2×C4—6922no—
339C2×C4—6922no—
340C2×C4D4×D17311yescharacteristic
341C2×C4C23×D17311yescharacteristic
342C8—21no—
343C8—21no—
344C8—3461no—
345C8—3461no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—3461no—
356C22—3461no—
357C22—3461no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C2×D4×D17311yescharacteristic
362C4—21no—
363C4—41no—
364C4—41no—
365C4—3461no—
366C4—3461no—
367C4—3462no—
368C4—1732no—
369C4—6921no—
370C4—6921no—
371C4#41×D17311yescharacteristic
372C4#1914972411yescharacteristic
373C2—41no—
374C2—81no—
375C2—81no—
376C2—13841no—
377C2—6921no—
378C2#2601213011yescharacteristic
379C2#135×D17311yescharacteristic
380C2#2601229211yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#466×C173C2
socleC22×C173C22×D4
hypercenter#466D173
hypocenterC173#3305
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #26012205C2semidirect
2. #26012159C2semidirect
3. #26011836C2semidirect
4. #26011818C2semidirect
5. #26011812C2semidirect
6. #26011785C2semidirect
7. #26011726C2semidirect
8. #26011684C2semidirect
9. #26011634C2semidirect
10. #26011633C2semidirect
11. #26011582C2semidirect
12. C8×C22⋊D173C2semidirect
13. #26011488C2semidirect
14. #26011484C2semidirect
15. #466×C173C2semidirect
190. C173#3305semidirect
199. #466D173semidirect

Maximal subgroups

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

subgroupindex
#260122052
#260121592
#260118362
#260118182
#260118122
#260117852
#260117262
#260116842
#260116342
#260116332
#260115822
C8×C22⋊D1732
#260114882
#260114842
#466×C1732
#3305173