Finite Group Databasebeta

← #36365457

Subgroups — #36365457

Order 22144.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012182C211yesmaximal, characteristic, complemented
2#26012149C211yesmaximal, characteristic
3#26011845C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011803C211yesmaximal, characteristic, complemented
6#26011799C211yesmaximal, characteristic, complemented
7#26011729C211yesmaximal, characteristic, complemented
8#26011717C211yesmaximal, characteristic, complemented
9#26011650C211yesmaximal, characteristic, complemented
10#26011645C211yesmaximal, characteristic, complemented
11C4×#14583816C211yesmaximal, characteristic, complemented
12#26011556C211yesmaximal, characteristic, complemented
13#26011484C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#464×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149673—21no—
22#19149671C2211yescharacteristic
23#19149669—21no—
24#19149665—21no—
25#19149664—21no—
26#19149656—21no—
27#19149655C2211yescharacteristic
28#19149655C2211yescharacteristic
29#19149655C2211yescharacteristic
30#19149652—21no—
31#19149650C2211yescharacteristic
32C4×Dic346C2211yescharacteristic
33C4×D692C2211yescharacteristic
34#19149639—21no—
35#19149637C2211yescharacteristic
36#19149636—21no—
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51C4×#13063436C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54#124×C173C2211yescharacteristic
55C4×Q8×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173—41no—
65C2×D692D411yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C22×Dic173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173C2311yescharacteristic
72C2×C4×D173C2311yescharacteristic
73#14583816—22no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C8×D173—41no—
77C8×D173—41no—
78C4⋊Dic173D411yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803D411yescharacteristic
95#14583803D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173—21no—
98C2×Q8×C173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173C2311yescharacteristic
101Q16×C173—41no—
102SD16×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173C2×D411yescharacteristic
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—42no—
115Dic346#4112yes—
116D692—22no—
117C22⋊D173—42no—
118C22⋊D173—42no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—42no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C4×D173—21no—
127C4×D173—21no—
128C4×D173—21no—
129C4×D173—21no—
130C4×D173—42no—
131C4×D173—42no—
132#13063436—21no—
133#13063436—21no—
134C23×C173—21no—
135Q8×C173—21no—
136Q8×C173—21no—
137Q8×C173—21no—
138Q8×C173—21no—
139D4×C173—21no—
140D4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173C22×D411yescharacteristic
153C2×D173—21no—
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—21no—
157C2×D173—21no—
158C2×D173#13512yes—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173—41no—
164C4×C173C2×#4111yescharacteristic
165C4×C173H(5,𝔽2)11yescharacteristic
166Dic173—21no—
167Dic173—21no—
168Dic173—22no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—81no—
173C2×C173#54611yescharacteristic
174C2×C173C2×#13511yescharacteristic
175C2×C173#57811yescharacteristic
176D173—22no—
177D173—41no—
178C173#330011yescharacteristic, complemented, Sylow
179#3300—1731nomaximal, Sylow
180#552—1731no—
181#548—1731no—
182#488—1731no—
183#485—1731no—
184#480—1731no—
185#479—1731no—
186#467—1731no—
187#465—1731no—
188#464D17311yescharacteristic, complemented
189#452—1731no—
190#451—1731no—
191C4×Q16—1731no—
192C8×D4—1731no—
193#416—1731no—
194C2×#103—1731no—
195C2×#41—1731no—
196C22×Q8—1731no—
197C2×Q16—1731no—
198C2×Q16C2×D17311yescharacteristic
199C2×SD16C2×D17311yescharacteristic
200C22×C8—1731no—
201C22×C8—1731no—
202#126—3461no—
203#125—3461no—
204#125—1731no—
205#124—3461no—
206#124C2×D17311yescharacteristic
207#123—3461no—
208C22⋊Q8—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8—1731no—
212#121—3461no—
213#121—1731no—
214C4×Q8—1731no—
215C4×Q8C2×D17311yescharacteristic
216C4×D4—3461no—
217C4×D4—1731no—
218#117—3461no—
219#117—1731no—
220C2×C4⋊C4—1731no—
221#107—1731no—
222#107—1731no—
223#106—1731no—
224C4⋊C8C2×D17311yescharacteristic
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103—1731no—
231#103C2×D17311yescharacteristic
232#102—1731no—
233#102—1731no—
234#102C2×D17311yescharacteristic
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237C4×C8—1731no—
238#41—3462no—
239#41—6921no—
240C2×Q8—3461no—
241C2×Q8—3462no—
242C2×Q8—1731no—
243C2×Q8—6921no—
244C2×Q8C22×D17311yescharacteristic
245C2×Q8C22×D17311yescharacteristic
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4C22×D17311yescharacteristic
250C22×C4—3461no—
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4—1731no—
254C22×C4—1731no—
255Q16—41no—
256Q16—3462no—
257SD16—41no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8—6921no—
261C2×C8—6921no—
262C2×C8C22×D17311yescharacteristic
263C2×C8C22×D17311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—3461no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4—1731no—
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—1731no—
285C22⋊C4—1731no—
286C42—21no—
287C42—3461no—
288C42—3461no—
289C42C22×D17311yescharacteristic
290C23—21no—
291C23—1731no—
292C23—1731no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—3461no—
298Q8—1732no—
299Q8—6922no—
300D4—21no—
301D4—21no—
302D4—3462no—
303D4—6922no—
304D4—6922no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—1731no—
317C2×C4—1731no—
318C2×C4—6922no—
319C2×C4—6922no—
320C2×C4—6922no—
321C2×C4D4×D17311yescharacteristic
322C2×C4D4×D17311yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C8—21no—
325C8—21no—
326C8—3461no—
327C8—3461no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—3461no—
332C22—3461no—
333C22—3461no—
334C22—3461no—
335C22—3461no—
336C22—1732no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3461no—
344C4—3461no—
345C4—3462no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4#41×D17311yescharacteristic
350C4#1914972411yescharacteristic
351C2—81no—
352C2—3462no—
353C2—6921no—
354C2#2601213811yescharacteristic
355C2#135×D17311yescharacteristic
356C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#464×C173C2
socleC22×C173C22×D4
prime radical#26012149C2
hypercenter#464D173
hypocenterC173#3300
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #26012182C2semidirect
3. #26011845C2semidirect
4. #26011830C2semidirect
5. #26011803C2semidirect
6. #26011799C2semidirect
7. #26011729C2semidirect
8. #26011717C2semidirect
9. #26011650C2semidirect
10. #26011645C2semidirect
11. C4×#14583816C2semidirect
12. #26011556C2semidirect
13. #26011484C2semidirect
14. #103×D173C2semidirect
15. #464×C173C2semidirect
178. C173#3300semidirect
188. #464D173semidirect

Maximal subgroups

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

subgroupindex
#260121822
#260121492
#260118452
#260118302
#260118032
#260117992
#260117292
#260117172
#260116502
#260116452
C4×#145838162
#260115562
#260114842
#103×D1732
#464×C1732
#3300173