Finite Group Databasebeta

← #36363834

Subgroups — #36363834

Order 22144.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26011971C211yesmaximal, characteristic, complemented
2#26011819C211yesmaximal, characteristic, complemented
3#26011807C211yesmaximal, characteristic, complemented
4#26011778C211yesmaximal, characteristic, complemented
5#26011773C211yesmaximal, characteristic, complemented
6#26011710C211yesmaximal, characteristic, complemented
7#26011703C211yesmaximal, characteristic, complemented
8#26011651C211yesmaximal, characteristic, complemented
9#26011634C211yesmaximal, characteristic
10#26011601C211yesmaximal, characteristic, complemented
11#26011596C211yesmaximal, characteristic, complemented
12C4×#14583813C211yesmaximal, characteristic, complemented
13C4×#14583814C211yesmaximal, characteristic, complemented
14#26011552C211yesmaximal, characteristic, complemented
15#542×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C4×Dic346C2211yescharacteristic
22C4×Dic346C2211yescharacteristic
23C4×D692C2211yescharacteristic
24C4×D692C2211yescharacteristic
25#19149639—21no—
26#19149639—21no—
27#19149637—21no—
28C2×C4⋊Dic173C2211yescharacteristic
29C4⋊C4×D173—21no—
30#19149610C2211yescharacteristic
31#19149610C2211yescharacteristic
32#19149610C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35C4⋊#13063436C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149602C2211yescharacteristic
39#19149602C2211yescharacteristic
40#19149600C2211yescharacteristic
41#19149600C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44C22⋊#13063436C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46#19149590C2211yescharacteristic
47C4×#13063436C2211yescharacteristic
48C4⋊Q8×C173C2211yescharacteristic
49#126×C173—21no—
50#124×C173—21no—
51#124×C173C2211yescharacteristic
52#123×C173—21no—
53C22⋊Q8×C173—21no—
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56C4×Q8×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic
58#117×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—22no—
61C2×Dic346C2311yescharacteristic
62C2×D692C2311yescharacteristic
63C2×C22⋊D173—21no—
64C22×Dic173—21no—
65C2×C4×D173—21no—
66C2×C4×D173—21no—
67C2×C4×D173—21no—
68#14583816—22no—
69#14583814—22no—
70#14583813—22no—
71#14583811—22no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583803—21no—
82#14583803—21no—
83C4×Dic173—21no—
84C4×Dic173—21no—
85C2×Q8×C173—21no—
86C2×Q8×C173C2311yescharacteristic
87C2×D4×C173—21no—
88C2×D4×C173C2311yescharacteristic
89C22×C4×C173—21no—
90C22×C4×C173C2311yescharacteristic
91C4⋊C4×C173—21no—
92C4⋊C4×C173—21no—
93C4⋊C4×C173—21no—
94C4⋊C4×C173D411yescharacteristic
95C4⋊C4×C173D411yescharacteristic
96C4⋊C4×C173C2311yescharacteristic
97C4⋊C4×C173C2311yescharacteristic
98C22⋊C4×C173—21no—
99C22⋊C4×C173—21no—
100C22⋊C4×C173—21no—
101C22⋊C4×C173—21no—
102C22⋊C4×C173—21no—
103C22⋊C4×C173D411yescharacteristic
104C22⋊C4×C173D411yescharacteristic
105C42×C173—21no—
106C42×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109Dic346—21no—
110Dic346#4112yes—
111D692—21no—
112D692#4112yes—
113C22⋊D173—24no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—81no—
118C4×D173—24no—
119C4×D173—81no—
120C4×D173—81no—
121#13063436—21no—
122#13063436—21no—
123#13063436—21no—
124#13063436—21no—
125C23×C173—21no—
126C23×C173C2×D411yescharacteristic
127Q8×C173—21no—
128Q8×C173—21no—
129Q8×C173—81no—
130D4×C173—21no—
131D4×C173—21no—
132D4×C173—81no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—81no—
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2411yescharacteristic
145C22×C173—21no—
146C22×C173—21no—
147C22×C173—21no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173C22×D411yescharacteristic
152C2×D173—22no—
153C2×D173—41no—
154C2×D173—41no—
155C4×C173—21no—
156C4×C173—21no—
157C4×C173—21no—
158C4×C173—21no—
159C4×C173—21no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173C2×#4111yescharacteristic
164C4×C173C2×#4111yescharacteristic
165Dic173—22no—
166Dic173—41no—
167Dic173—41no—
168C2×C173—41no—
169C2×C173—81no—
170C2×C173#52511yescharacteristic
171C2×C173#57711yescharacteristic
172C2×C173#57711yescharacteristic
173D173—42no—
174C173#312511yescharacteristic, complemented, Sylow
175#3125—1731nomaximal, Sylow
176#542D17311yescharacteristic, complemented
177#520—1731no—
178#483—1731no—
179#481—1731no—
180#476—1731no—
181#475—1731no—
182#464—1731no—
183#463—1731no—
184#452—1731no—
185#449—1731no—
186#442—1731no—
187#441—1731no—
188C4×SD16—1731no—
189C4×SD16—1731no—
190#433—1731no—
191C2×#41—1731no—
192C2×Q16—1731no—
193C2×SD16—1731no—
194C2×SD16—1731no—
195C2×D8—1731no—
196C4⋊Q8C2×D17311yescharacteristic
197#126—21no—
198#124—21no—
199#124C2×D17311yescharacteristic
200#123—21no—
201C22⋊Q8—21no—
202C22⋊Q8C2×D17311yescharacteristic
203#121C2×D17311yescharacteristic
204C4×Q8—1731no—
205C4×Q8—1731no—
206C4×Q8C2×D17311yescharacteristic
207C4×D4—3461no—
208C4×D4—3461no—
209C4×D4—1731no—
210C4×D4—1731no—
211C4×D4C2×D17311yescharacteristic
212#117—3461no—
213#117C2×D17311yescharacteristic
214C2×C4⋊C4—3461no—
215C2×C4⋊C4—1731no—
216#106—1731no—
217#106—1731no—
218#106—1731no—
219#106—1731no—
220C4⋊C8—1731no—
221C4⋊C8—1731no—
222#103—1731no—
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#102—1731no—
227#102—1731no—
228#102—1731no—
229#102—1731no—
230C22⋊C8—1731no—
231C22⋊C8—1731no—
232#97—1731no—
233C4×C8—1731no—
234#41—3462no—
235#41—3462no—
236C2×Q8—21no—
237C2×Q8—1731no—
238C2×Q8C22×D17311yescharacteristic
239C2×D4—21no—
240C2×D4—3461no—
241C2×D4—1731no—
242C2×D4C22×D17311yescharacteristic
243C22×C4—21no—
244C22×C4—3461no—
245C22×C4—3461no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4C22×D17311yescharacteristic
249Q16—3462no—
250SD16—3462no—
251SD16—3462no—
252D8—3462no—
253C2×C8—1731no—
254C2×C8—1731no—
255C2×C8—1731no—
256C2×C8—1731no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—1731no—
264C4⋊C4—1731no—
265C4⋊C4C22⋊D17311yescharacteristic
266C4⋊C4C22⋊D17311yescharacteristic
267C4⋊C4C22×D17311yescharacteristic
268C4⋊C4C22×D17311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4C22⋊D17311yescharacteristic
277C22⋊C4C22⋊D17311yescharacteristic
278C42—21no—
279C42—3461no—
280C42—3461no—
281C42C22×D17311yescharacteristic
282C42C22×D17311yescharacteristic
283C23—21no—
284C23—3461no—
285C23C2×C22⋊D17311yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—81no—
289Q8—3461no—
290Q8—1732no—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—3464no—
295D4—3461no—
296D4—1732no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—3464no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—13841no—
308C2×C4—13841no—
309C2×C4—13841no—
310C2×C4C2×C22⋊D17311yescharacteristic
311C2×C4C2×C22⋊D17311yescharacteristic
312C2×C4C2×C22⋊D17311yescharacteristic
313C2×C4C2×C22⋊D17311yescharacteristic
314C2×C4C2×C22⋊D17311yescharacteristic
315C2×C4C23×D17311yescharacteristic
316C8—3461no—
317C8—3461no—
318C8—3461no—
319C8—3461no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—3462no—
327C22—6921no—
328C22—6921no—
329C22C22×C22⋊D17311yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—3462no—
339C4—6921no—
340C4—6921no—
341C4#41×D17311yescharacteristic
342C4#41×D17311yescharacteristic
343C2—41no—
344C2—81no—
345C2—6922no—
346C2#2601199811yescharacteristic
347C2#2601229911yescharacteristic
348C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#542×C173C2
socleC22×C173C22×D4
prime radical#26011634C2
hypercenter#542D173
hypocenterC173#3125
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #26011971C2semidirect
2. #26011819C2semidirect
3. #26011807C2semidirect
4. #26011778C2semidirect
5. #26011773C2semidirect
6. #26011710C2semidirect
7. #26011703C2semidirect
8. #26011651C2semidirect
10. #26011601C2semidirect
11. #26011596C2semidirect
12. C4×#14583813C2semidirect
13. C4×#14583814C2semidirect
14. #26011552C2semidirect
15. #542×C173C2semidirect
174. C173#3125semidirect
176. #542D173semidirect

Maximal subgroups

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

subgroupindex
#260119712
#260118192
#260118072
#260117782
#260117732
#260117102
#260117032
#260116512
#260116342
#260116012
#260115962
C4×#145838132
C4×#145838142
#260115522
#542×C1732
#3125173