Finite Group Databasebeta

← #36363821

Subgroups — #36363821

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#26012106C211yesmaximal, characteristic, complemented
2#26011970C211yesmaximal, characteristic, complemented
3#26011828C211yesmaximal, characteristic, complemented
4#26011803C211yesmaximal, characteristic, complemented
5#26011785C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011712C211yesmaximal, characteristic, complemented
8#26011704C211yesmaximal, characteristic, complemented
9#26011654C211yesmaximal, characteristic, complemented
10#26011633C211yesmaximal, characteristic
11#26011591C211yesmaximal, characteristic, complemented
12#26011591C211yesmaximal, characteristic, complemented
13C4×Dic692C211yesmaximal, characteristic, complemented
14#26011550C211yesmaximal, characteristic, complemented
15C4×D8×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149672—21no—
22#19149666C2211yescharacteristic
23#19149665—21no—
24#19149662—21no—
25#19149659C2211yescharacteristic
26#19149658—21no—
27#19149650C2211yescharacteristic
28C4×Dic346C2211yescharacteristic
29C4×Dic346C2211yescharacteristic
30#19149644C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32C4×C22⋊D173—21no—
33#19149639—21no—
34#19149636C2211yescharacteristic
35#19149635—21no—
36C2×#14583805—21no—
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic
52C2×D8×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—22no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C22×Dic173—21no—
67C22×Dic173—21no—
68C2×C4×D173—21no—
69C2×C4×D173C2311yescharacteristic
70Dic692—22no—
71#14583814—22no—
72#14583814—22no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C4⋊Dic173—21no—
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89#14583803—21no—
90#14583803D411yescharacteristic
91#14583803D411yescharacteristic
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C2×D4×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C22×C4×C173—21no—
100D8×C173—22no—
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173C2×D411yescharacteristic
109Dic346—21no—
110Dic346—21no—
111Dic346—21no—
112Dic346—81no—
113Dic346#4112yes—
114C22⋊D173—24no—
115C22⋊D173—81no—
116C2×Dic173—24no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—81no—
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—21no—
126C4×D173—21no—
127C4×D173—81no—
128#13063436—21no—
129#13063436—21no—
130C23×C173—21no—
131C23×C173—21no—
132D4×C173—21no—
133D4×C173—21no—
134D4×C173—21no—
135D4×C173#4112yes—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—81no—
139C2×C4×C173—81no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173—22no—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173C22×D411yescharacteristic
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—21no—
155C4×C173—21no—
156C4×C173—21no—
157C4×C173—41no—
158C4×C173—41no—
159C4×C173C2×#4111yescharacteristic
160C4×C173C2×#4111yescharacteristic
161Dic173—21no—
162Dic173—22no—
163Dic173—21no—
164Dic173—21no—
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168C2×C173—42no—
169C2×C173—81no—
170C2×C173#52511yescharacteristic
171C2×C173#57611yescharacteristic
172C2×C173#57811yescharacteristic
173D173—41no—
174C173#312411yescharacteristic, complemented, Sylow
175#3124—1731nomaximal, Sylow
176#542—1731no—
177#520—1731no—
178#484—1731no—
179#480—1731no—
180#477—1731no—
181#474—1731no—
182#464—1731no—
183#463—1731no—
184#452—1731no—
185#449—1731no—
186#440—1731no—
187#440—1731no—
188C4×Q16—1731no—
189C4×D8D17311yescharacteristic, complemented
190#433—1731no—
191C2×#41—1731no—
192C2×Q16—1731no—
193C2×SD16—1731no—
194C2×SD16—1731no—
195C2×D8C2×D17311yescharacteristic
196C4⋊Q8—1731no—
197#126—3461no—
198#124—3461no—
199#124—1731no—
200#123—3461no—
201C22⋊Q8—3461no—
202C22⋊Q8—1731no—
203#121—1731no—
204C4×Q8—1731no—
205C4×Q8—1731no—
206C4×Q8—1731no—
207C4×D4—3461no—
208C4×D4—3461no—
209C4×D4—1731no—
210C4×D4C2×D17311yescharacteristic
211C4×D4C2×D17311yescharacteristic
212#117—3461no—
213#117—1731no—
214C2×C4⋊C4—3461no—
215C2×C4⋊C4—1731no—
216#107—1731no—
217#107—1731no—
218#107—1731no—
219#107C2×D17311yescharacteristic
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#102C2×D17311yescharacteristic
229#102C2×D17311yescharacteristic
230C22⋊C8—1731no—
231C22⋊C8—1731no—
232#97—1731no—
233C4×C8C2×D17311yescharacteristic
234#41—3462no—
235#41—3462no—
236C2×Q8—3461no—
237C2×Q8—1731no—
238C2×Q8—1731no—
239C2×D4—3461no—
240C2×D4—3461no—
241C2×D4C22×D17311yescharacteristic
242C2×D4C22×D17311yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—3461no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4—1731no—
249Q16—3462no—
250SD16—3462no—
251SD16—3462no—
252D8—22no—
253C2×C8—1731no—
254C2×C8—1731no—
255C2×C8C22×D17311yescharacteristic
256C2×C8C22×D17311yescharacteristic
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—1731no—
264C4⋊C4—1731no—
265C4⋊C4—1731no—
266C4⋊C4—1731no—
267C4⋊C4C22×D17311yescharacteristic
268C4⋊C4C22×D17311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—1731no—
277C22⋊C4—1731no—
278C42—3461no—
279C42—3461no—
280C42—3461no—
281C42—1731no—
282C42C22×D17311yescharacteristic
283C23—21no—
284C23—21no—
285C23—1731no—
286Q8—3461no—
287Q8—3461no—
288Q8—3461no—
289Q8—13841no—
290Q8—1732no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—3464no—
295D4—13841no—
296D4#1458382612yes—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—3464no—
302C2×C4—3461no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—13841no—
309C2×C4—13841no—
310C2×C4—1731no—
311C2×C4—1731no—
312C2×C4—1731no—
313C2×C4D4×D17311yescharacteristic
314C2×C4D4×D17311yescharacteristic
315C2×C4C23×D17311yescharacteristic
316C8—21no—
317C8—21no—
318C8—3461no—
319C8—3461no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—3461no—
327C22—3461no—
328C22—3461no—
329C22C2×D4×D17311yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—3461no—
335C4—3462no—
336C4—3461no—
337C4—3461no—
338C4—6921no—
339C4—6921no—
340C4—6921no—
341C4#41×D17311yescharacteristic
342C4C2×#1458382611yescharacteristic
343C2—42no—
344C2—81no—
345C2—6921no—
346C2#2601200011yescharacteristic
347C2#2601229211yescharacteristic
348C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×D8×C173C2
socleC22×C173C22×D4
prime radical#26011633C2
hypercenterC4×D8D173
hypocenterC173#3124
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #26012106C2semidirect
2. #26011970C2semidirect
3. #26011828C2semidirect
4. #26011803C2semidirect
5. #26011785C2semidirect
6. #26011769C2semidirect
7. #26011712C2semidirect
8. #26011704C2semidirect
9. #26011654C2semidirect
11. #26011591C2semidirect
12. #26011591C2semidirect
13. C4×Dic692C2semidirect
14. #26011550C2semidirect
15. C4×D8×C173C2semidirect
174. C173#3124semidirect
189. C4×D8D173semidirect

Maximal subgroups

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

subgroupindex
#260121062
#260119702
#260118282
#260118032
#260117852
#260117692
#260117122
#260117042
#260116542
#260116332
#260115912
C4×Dic6922
#260115502
C4×D8×C1732
#3124173