Finite Group Databasebeta

← #36364460

Subgroups — #36364460

Order 22144.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012069C211yesmaximal, characteristic, complemented
2#26011826C211yesmaximal, characteristic, complemented
3#26011824C211yesmaximal, characteristic, complemented
4#26011804C211yesmaximal, characteristic, complemented
5#26011802C211yesmaximal, characteristic, complemented
6#26011739C211yesmaximal, characteristic, complemented
7#26011735C211yesmaximal, characteristic, complemented
8#26011730C211yesmaximal, characteristic, complemented
9#26011725C211yesmaximal, characteristic, complemented
10#26011653C211yesmaximal, characteristic, complemented
11#26011651C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011634C211yesmaximal, characteristic, complemented
14#26011458C211yesmaximal, characteristic, complemented
15#537×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
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#19149612C2211yescharacteristic
35#19149612C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149602C2211yescharacteristic
39#19149602C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47C22⋊#13063436C2211yescharacteristic
48C2×#41×C173C2211yescharacteristic
49C2×#41×C173C2211yescharacteristic
50C22⋊Q8×C173—21no—
51C22⋊Q8×C173C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53#121×C173—21no—
54#121×C173—21no—
55#121×C173—21no—
56#121×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58C2×C4⋊C4×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#14583814—41no—
70#14583813—41no—
71#14583811—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—
89#41×C173—41no—
90#41×C173—41no—
91C2×Q8×C173D411yescharacteristic
92C2×Q8×C173C2311yescharacteristic
93C2×D4×C173—21no—
94C2×D4×C173—21no—
95C2×D4×C173—21no—
96C2×D4×C173D411yescharacteristic
97C2×D4×C173D411yescharacteristic
98C2×D4×C173D411yescharacteristic
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173—21no—
101C22×C4×C173—21no—
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173—21no—
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C22⋊C4×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×C173—21no—
132C23×C173C2×D411yescharacteristic
133C23×C173C2×D411yescharacteristic
134C23×C173C2×D411yescharacteristic
135Q8×C173—21no—
136Q8×C173—21no—
137Q8×C173—41no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—41no—
141D4×C173—41no—
142D4×C173—41no—
143D4×C173—42no—
144D4×C173—81no—
145D4×C173—81no—
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—21no—
154C2×C4×C173—21no—
155C2×C4×C173—42no—
156C2×C4×C173—81no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2411yescharacteristic
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—41no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173C22×D411yescharacteristic
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C4×C173—21no—
178C4×C173—21no—
179C4×C173—21no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173—41no—
183C4×C173H(5,𝔽2)11yescharacteristic
184C4×C173H(5,𝔽2)11yescharacteristic
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188C2×C173—41no—
189C2×C173—41no—
190C2×C173—41no—
191C2×C173—81no—
192C2×C173#53411yescharacteristic
193C2×C173#57611yescharacteristic
194C2×C173#57811yescharacteristic
195D173—81no—
196C173#320311yescharacteristic, complemented, Sylow
197#3203—1731nomaximal, Sylow
198#537—1731no—
199#537D17311yescharacteristic, complemented
200#484—1731no—
201#484—1731no—
202#480—1731no—
203#480—1731no—
204#468—1731no—
205#468—1731no—
206#467—1731no—
207#466—1731no—
208#452—1731no—
209#452—1731no—
210#449—1731no—
211#449—1731no—
212#408—1731no—
213C2×#41—1731no—
214C2×#41C2×D17311yescharacteristic
215C2×#41C2×D17311yescharacteristic
216C2×Q16—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×D8—1731no—
220C2×M16—1731no—
221C22×C8—1731no—
222C22⋊Q8—21no—
223C22⋊Q8—3461no—
224C22⋊Q8—1731no—
225C22⋊Q8—1731no—
226C22⋊Q8C2×D17311yescharacteristic
227C22⋊Q8C2×D17311yescharacteristic
228#121—21no—
229#121—21no—
230#121—21no—
231#121—3461no—
232#121—3461no—
233#121—3461no—
234#121—1731no—
235#121—1731no—
236#121C2×D17311yescharacteristic
237#121C2×D17311yescharacteristic
238C2×C4⋊C4—1731no—
239C2×C4⋊C4C2×D17311yescharacteristic
240#107—1731no—
241#107—1731no—
242#107—1731no—
243#107—1731no—
244#103—1731no—
245#103—1731no—
246#103—1731no—
247#103—1731no—
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102—1731no—
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8—1731no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—6921no—
261#41—6921no—
262C2×Q8—1731no—
263C2×Q8C22⋊D17311yescharacteristic
264C2×Q8C22×D17311yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4C22⋊D17311yescharacteristic
273C2×D4C22⋊D17311yescharacteristic
274C2×D4C22⋊D17311yescharacteristic
275C2×D4C22×D17311yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—3461no—
279C22×C4—3461no—
280C22×C4C22×D17311yescharacteristic
281C22×C4C22×D17311yescharacteristic
282C22×C4C22×D17311yescharacteristic
283Q16—6921no—
284SD16—6921no—
285SD16—6921no—
286D8—6921no—
287M16—6921no—
288C2×C8—1731no—
289C2×C8—1731no—
290C2×C8—1731no—
291C2×C8—1731no—
292C2×C8—6921no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—3461no—
297C4⋊C4—3461no—
298C4⋊C4—3461no—
299C4⋊C4—1731no—
300C4⋊C4—1731no—
301C4⋊C4C22×D17311yescharacteristic
302C4⋊C4C22×D17311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C23—21no—
312C23—3461no—
313C23C2×C22⋊D17311yescharacteristic
314C23C2×C22⋊D17311yescharacteristic
315C23C2×C22⋊D17311yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—41no—
319Q8—3461no—
320Q8—3461no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—3461no—
330D4—3461no—
331D4—13841no—
332D4—13841no—
333D4—6922no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—13841no—
349C2×C4—6922no—
350C2×C4C2×C22⋊D17311yescharacteristic
351C2×C4C2×C22⋊D17311yescharacteristic
352C2×C4C2×C22⋊D17311yescharacteristic
353C2×C4C23×D17311yescharacteristic
354C8—3461no—
355C8—3461no—
356C8—3461no—
357C8—3461no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C22×C22⋊D17311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—6921no—
381C4—6921no—
382C4—6921no—
383C4#1914972411yescharacteristic
384C4#1914972411yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—81no—
389C2—13841no—
390C2#2601205011yescharacteristic
391C2#2601229411yescharacteristic
392C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#537×C173C2
socleC22×C173C22×D4
hypercenter#537D173
hypocenterC173#3203
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #26012069C2semidirect
2. #26011826C2semidirect
3. #26011824C2semidirect
4. #26011804C2semidirect
5. #26011802C2semidirect
6. #26011739C2semidirect
7. #26011735C2semidirect
8. #26011730C2semidirect
9. #26011725C2semidirect
10. #26011653C2semidirect
11. #26011651C2semidirect
12. #26011636C2semidirect
13. #26011634C2semidirect
14. #26011458C2semidirect
15. #537×C173C2semidirect
196. C173#3203semidirect
199. #537D173semidirect

Maximal subgroups

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

subgroupindex
#260120692
#260118262
#260118242
#260118042
#260118022
#260117392
#260117352
#260117302
#260117252
#260116532
#260116512
#260116362
#260116342
#260114582
#537×C1732
#3203173