Finite Group Databasebeta

← #36363848

Subgroups — #36363848

Order 22144.

382 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#26012094C211yesmaximal, characteristic
2#26011969C211yesmaximal, characteristic, complemented
3#26011816C211yesmaximal, characteristic, complemented
4#26011812C211yesmaximal, characteristic, complemented
5#26011800C211yesmaximal, characteristic, complemented
6#26011796C211yesmaximal, characteristic, complemented
7#26011689C211yesmaximal, characteristic, complemented
8#26011689C211yesmaximal, characteristic, complemented
9#26011630C211yesmaximal, characteristic, complemented
10#26011594C211yesmaximal, characteristic, complemented
11#26011592C211yesmaximal, characteristic, complemented
12SD16×Dic173C211yesmaximal, characteristic, complemented
13#26011576C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#448×C173C211yesmaximal, characteristic, complemented
16C2×#14583813C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18#19149675C2211yescharacteristic
19#19149670C2211yescharacteristic
20#19149663—21no—
21#19149661—21no—
22#19149653C2211yescharacteristic
23#19149651—21no—
24#19149650—21no—
25Q8×Dic173—21no—
26Q8×Dic173—21no—
27Q8×Dic173C2211yescharacteristic
28Q8×Dic173C2211yescharacteristic
29Q8×Dic173C2211yescharacteristic
30Q8×Dic173C2211yescharacteristic
31D4×Dic173C2211yescharacteristic
32#19149641C2211yescharacteristic
33#19149638—21no—
34#19149638—21no—
35#19149638C2211yescharacteristic
36#19149638C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×D692C2311yescharacteristic
60C2×C22⋊D173—21no—
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63C22×Dic173—21no—
64C2×C4×D173—21no—
65#14583813—22no—
66#14583813—22no—
67C2×#13063436C2311yescharacteristic
68C2×#13063436C2311yescharacteristic
69C4⋊Dic173—21no—
70C4⋊Dic173—21no—
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173D411yescharacteristic
75C4⋊Dic173D411yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80C22⋊Dic173—21no—
81C22⋊Dic173—21no—
82C22⋊Dic173D411yescharacteristic
83C22⋊Dic173D411yescharacteristic
84#14583803—21no—
85#14583803—21no—
86C4×Dic173—21no—
87C4×Dic173—21no—
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173C2311yescharacteristic
92C4×Dic173C2311yescharacteristic
93C2×Q8×C173—22no—
94C2×Q8×C173—22no—
95C2×Q8×C173—21no—
96C2×Q8×C173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98C2×D4×C173—21no—
99C2×D4×C173C2311yescharacteristic
100C22×C4×C173—21no—
101C22×C4×C173C2311yescharacteristic
102SD16×C173—22no—
103SD16×C173—22no—
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22×D173—21no—
109D692—21no—
110D692—21no—
111C22⋊D173—81no—
112C22⋊D173—81no—
113C22⋊D173—81no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—81no—
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C4×D173—81no—
123#13063436—21no—
124#13063436—21no—
125C23×C173—21no—
126C23×C173C2×D411yescharacteristic
127Q8×C173—24no—
128Q8×C173—21no—
129Q8×C173—21no—
130Q8×C173#4112yes—
131Q8×C173#4112yes—
132D4×C173—21no—
133D4×C173—21no—
134D4×C173—81no—
135C2×C4×C173—24no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2411yescharacteristic
143C8×C173—21no—
144C8×C173—21no—
145C22×C173—21no—
146C22×C173—21no—
147C22×C173—21no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173C22×D411yescharacteristic
152C2×D173—41no—
153C2×D173—41no—
154C2×D173—41no—
155C4×C173—22no—
156C4×C173—22no—
157C4×C173—21no—
158C4×C173—41no—
159C4×C173C2×#4111yescharacteristic
160C4×C173C2×#4111yescharacteristic
161Dic173—21no—
162Dic173—21no—
163Dic173—21no—
164Dic173—21no—
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168C2×C173—41no—
169C2×C173—81no—
170C2×C173#52511yescharacteristic
171C2×C173#57711yescharacteristic
172C2×C173#57711yescharacteristic
173D173—81no—
174C173#312611yescharacteristic, complemented, Sylow
175#3126—1731nomaximal, Sylow
176#540—1731no—
177#519—1731no—
178#482—1731no—
179#482—1731no—
180#479—1731no—
181#479—1731no—
182#460—1731no—
183#460—1731no—
184#448—1731no—
185#448D17311yescharacteristic, complemented
186#440—1731no—
187#440—1731no—
188C4×SD16—1731no—
189C4×SD16—1731no—
190#433—1731no—
191C22×Q8C2×D17311yescharacteristic
192C2×SD16—1731no—
193C2×SD16—1731no—
194C2×SD16C2×D17311yescharacteristic
195C2×SD16C2×D17311yescharacteristic
196#127—1731no—
197#125—1731no—
198#123—3461no—
199#123—3461no—
200#121—3461no—
201#121—3461no—
202#121—1731no—
203#121C2×D17311yescharacteristic
204C4×Q8—3461no—
205C4×Q8—3461no—
206C4×Q8—1731no—
207C4×Q8—1731no—
208C4×Q8—1731no—
209C4×Q8—1731no—
210C4×D4—1731no—
211C4×D4—1731no—
212#117—3461no—
213#117—3461no—
214#117—1731no—
215#117—1731no—
216#107—1731no—
217#107—1731no—
218#106—1731no—
219#106—1731no—
220C4⋊C8—1731no—
221C4⋊C8—1731no—
222#103—1731no—
223#103—1731no—
224#103C2×D17311yescharacteristic
225#103C2×D17311yescharacteristic
226#102—1731no—
227#102—1731no—
228#102—1731no—
229#102—1731no—
230C22⋊C8—1731no—
231C22⋊C8C2×D17311yescharacteristic
232#97—1731no—
233C4×C8—1731no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D17311yescharacteristic
238C2×Q8C22×D17311yescharacteristic
239C2×D4—21no—
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—1731no—
244C2×D4C22×D17311yescharacteristic
245C22×C4—21no—
246C22×C4—3461no—
247C22×C4—3461no—
248C22×C4C22×D17311yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—3462no—
252SD16—3462no—
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⋊C4—1731no—
268C4⋊C4C22×D17311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—3461no—
271C22⋊C4—3461no—
272C22⋊C4—3461no—
273C22⋊C4—3461no—
274C22⋊C4—1731no—
275C22⋊C4—1731no—
276C42—3461no—
277C42—3461no—
278C42—3461no—
279C42—3461no—
280C42—3461no—
281C42—1731no—
282C42—1731no—
283C23—21no—
284C23—3461no—
285C23D4×D17311yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1458382512yes—
290Q8#1458382512yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—3461no—
295D4—3461no—
296D4—13841no—
297D4—13841no—
298D4—13841no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
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×C4—1731no—
314C2×C4D4×D17311yescharacteristic
315C2×C4C23×D17311yescharacteristic
316C8—21no—
317C8—21no—
318C8—3461no—
319C8—3461no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—6921no—
327C22—6921no—
328C22—6921no—
329C22C2×D4×D17311yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—3461no—
335C4—3461no—
336C4—3461no—
337C4—3461no—
338C4—6921no—
339C4—6921no—
340C4—6921no—
341C4C2×#1458382511yescharacteristic
342C4C2×#1458382511yescharacteristic
343C2—41no—
344C2—81no—
345C2—13841no—
346C2#2601199711yescharacteristic
347C2#2601229711yescharacteristic
348C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
prime radical#26012094C2
hypercenter#448D173
hypocenterC173#3126
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #26011969C2semidirect
3. #26011816C2semidirect
4. #26011812C2semidirect
5. #26011800C2semidirect
6. #26011796C2semidirect
7. #26011689C2semidirect
8. #26011689C2semidirect
9. #26011630C2semidirect
10. #26011594C2semidirect
11. #26011592C2semidirect
12. SD16×Dic173C2semidirect
13. #26011576C2semidirect
14. #26011549C2semidirect
15. #448×C173C2semidirect
174. C173#3126semidirect
185. #448D173semidirect

Maximal subgroups

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

subgroupindex
#260120942
#260119692
#260118162
#260118122
#260118002
#260117962
#260116892
#260116302
#260115942
#260115922
SD16×Dic1732
#260115762
#260115492
#448×C1732
#3126173