Finite Group Databasebeta

← #36364445

Subgroups — #36364445

Order 22144.

390 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#26012069C211yesmaximal, characteristic
2#26011832C211yesmaximal, characteristic, complemented
3#26011824C211yesmaximal, characteristic, complemented
4#26011819C211yesmaximal, characteristic, complemented
5#26011815C211yesmaximal, characteristic, complemented
6#26011747C211yesmaximal, characteristic, complemented
7#26011739C211yesmaximal, characteristic, complemented
8#26011730C211yesmaximal, characteristic, complemented
9#26011722C211yesmaximal, characteristic, complemented
10#26011651C211yesmaximal, characteristic, complemented
11#26011651C211yesmaximal, characteristic, complemented
12#26011647C211yesmaximal, characteristic, complemented
13#26011630C211yesmaximal, characteristic, complemented
14#26011458C211yesmaximal, characteristic, complemented
15#536×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583810C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149658C2211yescharacteristic
24#19149658C2211yescharacteristic
25#19149655—21no—
26#19149653C2211yescharacteristic
27#19149653C2211yescharacteristic
28#19149651—21no—
29#19149651—21no—
30#19149650—21no—
31C2×C4⋊Dic173C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149612C2211yescharacteristic
34#19149610C2211yescharacteristic
35#19149610C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149603C2211yescharacteristic
39#19149603C2211yescharacteristic
40#19149602C2211yescharacteristic
41#19149602C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47C22⋊#13063436C2211yescharacteristic
48C2×#41×C173C2211yescharacteristic
49C22×Q8×C173C2211yescharacteristic
50C4⋊Q8×C173—21no—
51#124×C173—21no—
52#123×C173—21no—
53#123×C173—21no—
54C22⋊Q8×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56C22⋊Q8×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58#117×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C22×Dic173—21no—
67C2×C4×D173—21no—
68#14583816—41no—
69#14583816—41no—
70#14583813—41no—
71#14583813—41no—
72#14583810—41no—
73C2×#13063436—41no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85#14583803—21no—
86#14583803—21no—
87#41×C173—41no—
88#41×C173—41no—
89C2×Q8×C173—21no—
90C2×Q8×C173—41no—
91C2×Q8×C173—41no—
92C2×Q8×C173D411yescharacteristic
93C2×Q8×C173C2311yescharacteristic
94C2×Q8×C173C2311yescharacteristic
95C2×D4×C173D411yescharacteristic
96C2×D4×C173D411yescharacteristic
97C2×D4×C173D411yescharacteristic
98C22×C4×C173—21no—
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173—21no—
104C4⋊C4×C173—21no—
105C4⋊C4×C173—21no—
106C4⋊C4×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C22⋊C4×C173—21no—
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173—21no—
114C22×D173—21no—
115Dic346—21no—
116Dic346—21no—
117D692—21no—
118D692—21no—
119C22⋊D173—42no—
120C22⋊D173—81no—
121C22⋊D173—81no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—81no—
126C4×D173—42no—
127#13063436—21no—
128#13063436—21no—
129#13063436—21no—
130#13063436—21no—
131C23×C173C2×D411yescharacteristic
132C23×C173C2×D411yescharacteristic
133C23×C173C2×D411yescharacteristic
134Q8×C173—21no—
135Q8×C173—21no—
136Q8×C173—21no—
137Q8×C173—21no—
138Q8×C173—41no—
139Q8×C173—42no—
140D4×C173—41no—
141D4×C173—41no—
142D4×C173—41no—
143C2×C4×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—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—42no—
154C2×C4×C173C2×D411yescharacteristic
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2411yescharacteristic
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173C22×D411yescharacteristic
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C4×C173—21no—
172C4×C173—21no—
173C4×C173—21no—
174C4×C173—41no—
175C4×C173—41no—
176C4×C173—41no—
177C4×C173—41no—
178C4×C173H(5,𝔽2)11yescharacteristic
179C4×C173H(5,𝔽2)11yescharacteristic
180Dic173—41no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—41no—
184C2×C173—41no—
185C2×C173—41no—
186C2×C173#53411yescharacteristic
187C2×C173#57711yescharacteristic
188C2×C173#57811yescharacteristic
189D173—81no—
190C173#320111yescharacteristic, complemented, Sylow
191#3201—1731nomaximal, Sylow
192#537—1731no—
193#536D17311yescharacteristic, complemented
194#485—1731no—
195#484—1731no—
196#483—1731no—
197#482—1731no—
198#470—1731no—
199#468—1731no—
200#467—1731no—
201#465—1731no—
202#452—1731no—
203#452—1731no—
204#451—1731no—
205#448—1731no—
206#408—1731no—
207C2×#41—1731no—
208C2×#41C2×D17311yescharacteristic
209C22×Q8C2×D17311yescharacteristic
210C2×Q16—1731no—
211C2×Q16—1731no—
212C2×SD16—1731no—
213C2×SD16—1731no—
214C2×M16—1731no—
215C22×C8—1731no—
216C4⋊Q8—21no—
217#124—21no—
218#123—21no—
219#123—21no—
220C22⋊Q8—3461no—
221C22⋊Q8—1731no—
222C22⋊Q8—1731no—
223C22⋊Q8C2×D17311yescharacteristic
224C22⋊Q8C2×D17311yescharacteristic
225C22⋊Q8C2×D17311yescharacteristic
226C22⋊Q8C2×D17311yescharacteristic
227#121—3461no—
228#121—3461no—
229#121—3461no—
230#121—1731no—
231#121—1731no—
232#117C2×D17311yescharacteristic
233C2×C4⋊C4—1731no—
234#107—1731no—
235#107—1731no—
236#106—1731no—
237#106—1731no—
238#103—1731no—
239#103—1731no—
240#103—1731no—
241#103—1731no—
242#103—1731no—
243#103—1731no—
244#102—1731no—
245#102—1731no—
246C22⋊C8—1731no—
247C22⋊C8—1731no—
248C22⋊C8—1731no—
249C22⋊C8—1731no—
250#41—41no—
251#41—41no—
252#41—6921no—
253#41—6921no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—1731no—
258C2×Q8C22⋊D17311yescharacteristic
259C2×Q8C22×D17311yescharacteristic
260C2×Q8C22×D17311yescharacteristic
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—1731no—
265C2×D4C22⋊D17311yescharacteristic
266C2×D4C22⋊D17311yescharacteristic
267C2×D4C22⋊D17311yescharacteristic
268C22×C4—21no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4C22×D17311yescharacteristic
272C22×C4C22×D17311yescharacteristic
273C22×C4C22×D17311yescharacteristic
274Q16—6921no—
275Q16—6921no—
276SD16—6921no—
277SD16—6921no—
278M16—6921no—
279C2×C8—1731no—
280C2×C8—1731no—
281C2×C8—1731no—
282C2×C8—1731no—
283C2×C8—6921no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—1731no—
292C4⋊C4—1731no—
293C4⋊C4C22×D17311yescharacteristic
294C4⋊C4C22×D17311yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C42—21no—
305C23—3461no—
306C23C2×C22⋊D17311yescharacteristic
307C23C2×C22⋊D17311yescharacteristic
308C23C2×C22⋊D17311yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—41no—
314Q8—42no—
315Q8—3461no—
316Q8—3461no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—3461no—
321D4—3461no—
322D4—13841no—
323D4—13841no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—13841no—
340C2×C4—6922no—
341C2×C4C2×C22⋊D17311yescharacteristic
342C2×C4C2×C22⋊D17311yescharacteristic
343C2×C4C2×C22⋊D17311yescharacteristic
344C2×C4C23×D17311yescharacteristic
345C8—3461no—
346C8—3461no—
347C8—3461no—
348C8—3461no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C22×C22⋊D17311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—6921no—
370C4—6921no—
371C4—6921no—
372C4#1914972411yescharacteristic
373C4#1914972411yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—13841no—
378C2#2601205011yescharacteristic
379C2#2601229911yescharacteristic
380C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#536×C173C2
socleC22×C173C22×D4
prime radical#26012069C2
hypercenter#536D173
hypocenterC173#3201
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #26011832C2semidirect
3. #26011824C2semidirect
4. #26011819C2semidirect
5. #26011815C2semidirect
6. #26011747C2semidirect
7. #26011739C2semidirect
8. #26011730C2semidirect
9. #26011722C2semidirect
10. #26011651C2semidirect
11. #26011651C2semidirect
12. #26011647C2semidirect
13. #26011630C2semidirect
14. #26011458C2semidirect
15. #536×C173C2semidirect
190. C173#3201semidirect
193. #536D173semidirect

Maximal subgroups

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

subgroupindex
#260120692
#260118322
#260118242
#260118192
#260118152
#260117472
#260117392
#260117302
#260117222
#260116512
#260116472
#260116302
#260114582
#536×C1732
#3201173