Finite Group Databasebeta

← #36364667

Subgroups — #36364667

Order 22144.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012110C211yesmaximal, characteristic, complemented
2#26012066C211yesmaximal, characteristic, complemented
3#26011873C211yesmaximal, characteristic, complemented
4#26011851C211yesmaximal, characteristic, complemented
5#26011841C211yesmaximal, characteristic, complemented
6#26011715C211yesmaximal, characteristic, complemented
7#26011708C211yesmaximal, characteristic, complemented
8#26011699C211yesmaximal, characteristic, complemented
9#26011694C211yesmaximal, characteristic, complemented
10#26011652C211yesmaximal, characteristic, complemented
11#26011645C211yesmaximal, characteristic, complemented
12#26011635C211yesmaximal, characteristic
13#26011628C211yesmaximal, characteristic, complemented
14#26011553C211yesmaximal, characteristic, complemented
15#497×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583811C2211yescharacteristic
22#19149679C2211yescharacteristic
23#19149677C2211yescharacteristic
24#19149672—21no—
25#19149668C2211yescharacteristic
26#19149667C2211yescharacteristic
27#19149665—21no—
28#19149664—21no—
29#19149664—21no—
30#19149663—21no—
31#19149657—21no—
32#19149657—21no—
33#19149656—21no—
34#19149655C2211yescharacteristic
35#19149650C2211yescharacteristic
36Q8×Dic173C2211yescharacteristic
37D4×Dic173C2211yescharacteristic
38#19149636C2211yescharacteristic
39#19149608C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51C4×#13063436C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C4⋊Q8×C173C2211yescharacteristic
57#124×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583827—41no—
60#14583827—41no—
61C2×Dic346—21no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64Q8×D173—41no—
65Q8×D173—41no—
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73#14583816—41no—
74#14583814—41no—
75#14583813—41no—
76#14583811—41no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87#14583805D411yescharacteristic
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96#14583803D411yescharacteristic
97C4×Dic173—21no—
98C4×Dic173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173C2311yescharacteristic
103Q16×C173—41no—
104Q16×C173—41no—
105SD16×C173—41no—
106SD16×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173—21no—
110C4⋊C4×C173—21no—
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173C2×D411yescharacteristic
116Dic346—21no—
117Dic346—21no—
118Dic346—42no—
119Dic346—81no—
120D692—21no—
121D692—21no—
122D692—42no—
123C22⋊D173—81no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—81no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—42no—
134C4×D173—42no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138Q8×C173—21no—
139Q8×C173—21no—
140Q8×C173—21no—
141Q8×C173—21no—
142Q8×C173—21no—
143Q8×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173C22×D411yescharacteristic
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C4×C173—21no—
165C4×C173—21no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173H(5,𝔽2)11yescharacteristic
170C4×C173H(5,𝔽2)11yescharacteristic
171Dic173—21no—
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—81no—
178C2×C173#53511yescharacteristic
179C2×C173#57711yescharacteristic
180C2×C173#57811yescharacteristic
181D173—41no—
182D173—81no—
183C173#322511yescharacteristic, complemented, Sylow
184#3225—1731nomaximal, Sylow
185#542—1731no—
186#536—1731no—
187#497D17311yescharacteristic, complemented
188#495—1731no—
189#489—1731no—
190#487—1731no—
191#464—1731no—
192#463—1731no—
193#462—1731no—
194#461—1731no—
195#452—1731no—
196#451—1731no—
197#449—1731no—
198#448—1731no—
199#433—1731no—
200C2×#41—1731no—
201C22×Q8—1731no—
202C2×Q16—1731no—
203C2×Q16C2×D17311yescharacteristic
204C2×Q16C2×D17311yescharacteristic
205C2×SD16—1731no—
206C2×SD16—1731no—
207C2×SD16C2×D17311yescharacteristic
208C2×SD16C2×D17311yescharacteristic
209C2×D8—1731no—
210C4⋊Q8—1731no—
211C4⋊Q8—1731no—
212C4⋊Q8C2×D17311yescharacteristic
213#126—3461no—
214#124—3461no—
215#124—1731no—
216#124—1731no—
217#124C2×D17311yescharacteristic
218#123—3461no—
219#123—3461no—
220#123—3461no—
221C22⋊Q8—3461no—
222C22⋊Q8—3461no—
223C22⋊Q8—3461no—
224C22⋊Q8—1731no—
225#121—1731no—
226C4×Q8—1731no—
227C4×D4—1731no—
228#117—1731no—
229C4⋊C8—1731no—
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#102—1731no—
238#102—1731no—
239C22⋊C8—1731no—
240C22⋊C8—1731no—
241#97C2×D17311yescharacteristic
242C4×C8—1731no—
243#41—6921no—
244#41—6921no—
245C2×Q8—3461no—
246C2×Q8—3461no—
247C2×Q8—1731no—
248C2×Q8—6921no—
249C2×Q8—6921no—
250C2×Q8C22×D17311yescharacteristic
251C2×Q8C22×D17311yescharacteristic
252C2×Q8C22×D17311yescharacteristic
253C2×D4—3461no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4C22×D17311yescharacteristic
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—1731no—
261Q16—41no—
262Q16—41no—
263Q16—6921no—
264SD16—41no—
265SD16—41no—
266SD16—6921no—
267SD16—6921no—
268D8—6921no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4—1731no—
283C4⋊C4—1731no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—1731no—
294C22⋊C4—1731no—
295C42—3461no—
296C42—1731no—
297C42C22×D17311yescharacteristic
298C23—21no—
299C23—3461no—
300C23—1731no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—3461no—
308Q8—3461no—
309Q8—13841no—
310Q8—6922no—
311D4—21no—
312D4—21no—
313D4—3461no—
314D4—3461no—
315D4—13841no—
316D4—6922no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—13841no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—6922no—
330C2×C4—6922no—
331C2×C4D4×D17311yescharacteristic
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3461no—
337C8—3461no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—3461no—
342C22—3461no—
343C22—3461no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—3461no—
354C4—3461no—
355C4—3461no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4#1914972411yescharacteristic
360C4#1914972411yescharacteristic
361C2—81no—
362C2—13841no—
363C2—6921no—
364C2#2601205411yescharacteristic
365C2#2601230011yescharacteristic
366C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#497×C173C2
socleC22×C173C22×D4
prime radical#26011635C2
hypercenter#497D173
hypocenterC173#3225
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #26012110C2semidirect
2. #26012066C2semidirect
3. #26011873C2semidirect
4. #26011851C2semidirect
5. #26011841C2semidirect
6. #26011715C2semidirect
7. #26011708C2semidirect
8. #26011699C2semidirect
9. #26011694C2semidirect
10. #26011652C2semidirect
11. #26011645C2semidirect
13. #26011628C2semidirect
14. #26011553C2semidirect
15. #497×C173C2semidirect
183. C173#3225semidirect
187. #497D173semidirect

Maximal subgroups

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

subgroupindex
#260121102
#260120662
#260118732
#260118512
#260118412
#260117152
#260117082
#260116992
#260116942
#260116522
#260116452
#260116352
#260116282
#260115532
#497×C1732
#3225173