Finite Group Databasebeta

← #36365455

Subgroups — #36365455

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#26012153C211yesmaximal, characteristic
3#26011846C211yesmaximal, characteristic, complemented
4#26011834C211yesmaximal, characteristic, complemented
5#26011806C211yesmaximal, characteristic, complemented
6#26011798C211yesmaximal, characteristic, complemented
7#26011731C211yesmaximal, characteristic, complemented
8#26011723C211yesmaximal, characteristic, complemented
9#26011713C211yesmaximal, characteristic, complemented
10#26011652C211yesmaximal, characteristic, complemented
11#26011645C211yesmaximal, characteristic, complemented
12C8×D692C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15C4×Q16×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149673—21no—
23#19149671C2211yescharacteristic
24#19149669—21no—
25#19149667—21no—
26#19149667C2211yescharacteristic
27#19149664—21no—
28#19149659—21no—
29#19149659C2211yescharacteristic
30#19149655C2211yescharacteristic
31#19149655C2211yescharacteristic
32#19149654—21no—
33#19149654C2211yescharacteristic
34C4×D692—21no—
35C4×D692C2211yescharacteristic
36#19149637C2211yescharacteristic
37#19149636—21no—
38C4⋊C4×D173C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173—41no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692D411yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73C2×C4×D173C2311yescharacteristic
74#14583816—41no—
75#14583813—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C8×D173—41no—
79C8×D173—41no—
80C4⋊Dic173—21no—
81C4⋊Dic173D411yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803D411yescharacteristic
97#14583803D411yescharacteristic
98C4×Dic173—21no—
99C2×Q8×C173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101Q16×C173—22no—
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173—21no—
105C4⋊C4×C173—21no—
106C4⋊C4×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C42×C173—21no—
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173C2×D411yescharacteristic
113C22×D173C2×D411yescharacteristic
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117D692—22no—
118D692—21no—
119D692—21no—
120D692—42no—
121D692—42no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—21no—
128C4×D173—21no—
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—42no—
132C4×D173—42no—
133C4×D173—42no—
134#13063436—21no—
135#13063436—21no—
136Q8×C173—21no—
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×C173#4112yes—
140C2×C4×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×C173C22×D411yescharacteristic
150C2×D173—21no—
151C2×D173—21no—
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—21no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173#13512yes—
159C4×C173—22no—
160C4×C173—21no—
161C4×C173—21no—
162C4×C173—41no—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173C2×#4111yescharacteristic
166C4×C173H(5,𝔽2)11yescharacteristic
167Dic173—21no—
168Dic173—21no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173#54611yescharacteristic
173C2×C173C2×#13511yescharacteristic
174C2×C173#57811yescharacteristic
175D173—22no—
176D173—41no—
177D173—81no—
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#464—1731no—
189#452—1731no—
190#451—1731no—
191C4×Q16D17311yescharacteristic, complemented
192C8×D4—1731no—
193#416—1731no—
194C2×#103—1731no—
195C2×#41—1731no—
196C22×Q8—1731no—
197C2×Q16—1731no—
198C2×Q16C2×D17311yescharacteristic
199C2×SD16—1731no—
200C22×C8—1731no—
201C22×C8—1731no—
202#126—3461no—
203#125—3461no—
204#125—1731no—
205#124—3461no—
206#124—1731no—
207#123—3461no—
208C22⋊Q8—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8—1731no—
212#121—3461no—
213#121—1731no—
214C4×Q8C2×D17311yescharacteristic
215C4×Q8C2×D17311yescharacteristic
216C4×D4—3461no—
217C4×D4—1731no—
218#117—3461no—
219#117—1731no—
220C2×C4⋊C4—1731no—
221#107—1731no—
222#107C2×D17311yescharacteristic
223#106—1731no—
224C4⋊C8—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103C2×D17311yescharacteristic
231#103C2×D17311yescharacteristic
232#102—1731no—
233#102—1731no—
234#102—1731no—
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237C4×C8C2×D17311yescharacteristic
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×D4—1731no—
250C22×C4—3461no—
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4—1731no—
254C22×C4—1731no—
255Q16—22no—
256Q16—6921no—
257SD16—6921no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8—6921no—
261C2×C8—6921no—
262C2×C8C22×D17311yescharacteristic
263C2×C8C22×D17311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
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⋊C4C22×D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—1731no—
285C22⋊C4—1731no—
286C42—21no—
287C42—21no—
288C42—3461no—
289C42C22×D17311yescharacteristic
290C23—3461no—
291C23—1731no—
292C23—1731no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—3461no—
297Q8—3461no—
298Q8—6922no—
299Q8#1458382612yes—
300D4—3462no—
301D4—3461no—
302D4—3461no—
303D4—6922no—
304D4—6922no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
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—3461no—
329C22—3461no—
330C22—3461no—
331C22—3461no—
332C22—3461no—
333C22—1732no—
334C22—6921no—
335C22—6921no—
336C22—6921no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—3461no—
345C4—3461no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4C2×#1458382611yescharacteristic
350C4#1914972511yescharacteristic
351C2—3462no—
352C2—13841no—
353C2—6921no—
354C2#2601213711yescharacteristic
355C2#135×D17311yescharacteristic
356C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×Q16×C173C2
socleC22×C173C22×D4
prime radical#26012153C2
hypercenterC4×Q16D173
hypocenterC173#3300
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #26012182C2semidirect
3. #26011846C2semidirect
4. #26011834C2semidirect
5. #26011806C2semidirect
6. #26011798C2semidirect
7. #26011731C2semidirect
8. #26011723C2semidirect
9. #26011713C2semidirect
10. #26011652C2semidirect
11. #26011645C2semidirect
12. C8×D692C2semidirect
13. #26011486C2semidirect
14. #103×D173C2semidirect
15. C4×Q16×C173C2semidirect
178. C173#3300semidirect
191. C4×Q16D173semidirect

Maximal subgroups

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

subgroupindex
#260121822
#260121532
#260118462
#260118342
#260118062
#260117982
#260117312
#260117232
#260117132
#260116522
#260116452
C8×D6922
#260114862
#103×D1732
C4×Q16×C1732
#3300173