Finite Group Databasebeta

← #36364540

Subgroups — #36364540

Order 22144.

400 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012040C211yesmaximal, characteristic, complemented
2#26012026C211yesmaximal, characteristic, complemented
3#26011875C211yesmaximal, characteristic, complemented
4#26011873C211yesmaximal, characteristic, complemented
5#26011848C211yesmaximal, characteristic, complemented
6#26011700C211yesmaximal, characteristic, complemented
7#26011695C211yesmaximal, characteristic, complemented
8#26011690C211yesmaximal, characteristic, complemented
9#26011685C211yesmaximal, characteristic, complemented
10#26011656C211yesmaximal, characteristic, complemented
11#26011654C211yesmaximal, characteristic, complemented
12#26011639C211yesmaximal, characteristic, complemented
13#26011637C211yesmaximal, characteristic, complemented
14#26011542C211yesmaximal, characteristic, complemented
15#488×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583815C2211yescharacteristic
22C2×#14583814C2211yescharacteristic
23C2×#14583813C2211yescharacteristic
24C2×D1384C2211yescharacteristic
25C2×#14583811C2211yescharacteristic
26#19149677C2211yescharacteristic
27#19149675C2211yescharacteristic
28#19149674—21no—
29#19149667C2211yescharacteristic
30#19149667C2211yescharacteristic
31#19149664—21no—
32#19149659C2211yescharacteristic
33#19149654C2211yescharacteristic
34#19149652—21no—
35#19149652—21no—
36#19149644C2211yescharacteristic
37#19149641—21no—
38#19149641C2211yescharacteristic
39#19149639—21no—
40#19149639—21no—
41#19149637—21no—
42C42×D173C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51C4×#13063436C2211yescharacteristic
52#125×C173C2211yescharacteristic
53#124×C173C2211yescharacteristic
54#103×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59#14583827—41no—
60#14583827—41no—
61#14583825—41no—
62#14583825—41no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692—21no—
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C22×Dic173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173D411yescharacteristic
76C2×C4×D173D411yescharacteristic
77C2×C4×D173C2311yescharacteristic
78Dic692—41no—
79#14583816—41no—
80#14583815—41no—
81#14583815—41no—
82#14583814—41no—
83#14583813—41no—
84D1384—41no—
85#14583811—41no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C4⋊Dic173—21no—
89#14583805—21no—
90#14583805—21no—
91#14583805—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99C4×Dic173—21no—
100C4×Dic173D411yescharacteristic
101C4×Dic173D411yescharacteristic
102C4×Dic173C2311yescharacteristic
103C2×Q8×C173C2311yescharacteristic
104C2×D4×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173—21no—
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C4⋊C4×C173C2311yescharacteristic
111C22⋊C4×C173—21no—
112C22⋊C4×C173—21no—
113C42×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—21no—
116C22×D173C2×D411yescharacteristic
117Dic346—21no—
118Dic346—21no—
119Dic346—21no—
120Dic346—21no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—21no—
125D692—42no—
126D692—42no—
127C22⋊D173—42no—
128C22⋊D173—42no—
129C22⋊D173—42no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—42no—
133C2×Dic173C2×D411yescharacteristic
134C2×Dic173C2×D411yescharacteristic
135C2×Dic173C2×D411yescharacteristic
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—22no—
139C4×D173—41no—
140C4×D173—42no—
141C4×D173—42no—
142#13063436—21no—
143#13063436—21no—
144C23×C173—21no—
145Q8×C173—21no—
146Q8×C173—21no—
147D4×C173—21no—
148D4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2×D411yescharacteristic
154C2×C4×C173C2411yescharacteristic
155C8×C173—21no—
156C8×C173—21no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173C22×D411yescharacteristic
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—21no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C2×D173—41no—
169C2×D173—41no—
170C4×C173—21no—
171C4×C173—21no—
172C4×C173—41no—
173C4×C173—41no—
174C4×C173—41no—
175C4×C173H(5,𝔽2)11yescharacteristic
176C4×C173H(5,𝔽2)11yescharacteristic
177Dic173—21no—
178Dic173—21no—
179Dic173—41no—
180Dic173—41no—
181Dic173#13512yes—
182C2×C173—81no—
183C2×C173#53511yescharacteristic
184C2×C173C2×#13511yescharacteristic
185C2×C173#57511yescharacteristic
186D173—41no—
187D173—81no—
188D173—81no—
189C173#321211yescharacteristic, complemented, Sylow
190#3212—1731nomaximal, Sylow
191#532—1731no—
192#529—1731no—
193#495—1731no—
194#495—1731no—
195#488—1731no—
196#488D17311yescharacteristic, complemented
197#462—1731no—
198#461—1731no—
199#460—1731no—
200#459—1731no—
201#452—1731no—
202#452—1731no—
203#449—1731no—
204#449—1731no—
205#431—1731no—
206C2×#41—1731no—
207C2×#41—1731no—
208C2×Q16—1731no—
209C2×Q16—1731no—
210C2×SD16—1731no—
211C2×SD16—1731no—
212C2×SD16—1731no—
213C2×SD16—1731no—
214C2×D8—1731no—
215C2×D8—1731no—
216C4⋊Q8—1731no—
217#127—1731no—
218#126—3461no—
219#125C2×D17311yescharacteristic
220#124—1731no—
221#124—1731no—
222#124C2×D17311yescharacteristic
223#123—3461no—
224C22⋊Q8—1731no—
225#121—3461no—
226#121—3461no—
227#121—1731no—
228C4×Q8—1731no—
229C4×D4—3461no—
230C4×D4—3461no—
231C4×D4—3461no—
232C4×D4—1731no—
233#117—3461no—
234C2×C42—1731no—
235C4⋊C8—1731no—
236C4⋊C8—1731no—
237#103—1731no—
238#103—1731no—
239#103C2×D17311yescharacteristic
240#103C2×D17311yescharacteristic
241#102—1731no—
242#102—1731no—
243#102C2×D17311yescharacteristic
244#102C2×D17311yescharacteristic
245C22⋊C8—1731no—
246C22⋊C8—1731no—
247C4×C8—1731no—
248C4×C8C2×D17311yescharacteristic
249#41—6921no—
250#41—6921no—
251#41—6921no—
252#41—6921no—
253C2×Q8—1731no—
254C2×Q8—1731no—
255C2×Q8C22×D17311yescharacteristic
256C2×D4—3461no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—3461no—
261C2×D4—1731no—
262C2×D4—1731no—
263C2×D4C22×D17311yescharacteristic
264C22×C4—3461no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—1731no—
268C22×C4—1731no—
269C22×C4—1731no—
270Q16—6921no—
271Q16—6921no—
272SD16—6921no—
273SD16—6921no—
274SD16—6921no—
275SD16—6921no—
276D8—6921no—
277D8—6921no—
278C2×C8—1731no—
279C2×C8—1731no—
280C2×C8C22×D17311yescharacteristic
281C2×C8C22×D17311yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4C22×D17311yescharacteristic
289C4⋊C4C22×D17311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C42—3461no—
300C42—1731no—
301C42—1731no—
302C42—1731no—
303C42C22×D17311yescharacteristic
304C23—21no—
305C23—3461no—
306C23—3461no—
307C23—1731no—
308Q8—21no—
309Q8—21no—
310Q8—3461no—
311Q8—3461no—
312Q8—3461no—
313Q8—3461no—
314D4—21no—
315D4—21no—
316D4—3461no—
317D4—3461no—
318D4—3461no—
319D4—3461no—
320D4—6922no—
321D4—6922no—
322D4—6922no—
323D4—6922no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—3462no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—1731no—
336C2×C4—6921no—
337C2×C4—6922no—
338C2×C4—6922no—
339C2×C4—6922no—
340C2×C4D4×D17311yescharacteristic
341C2×C4D4×D17311yescharacteristic
342C2×C4C23×D17311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3461no—
346C8—3461no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—3461no—
351C22—3461no—
352C22—3461no—
353C22—6921no—
354C22—6921no—
355C22—6921no—
356C22—6921no—
357C22—6921no—
358C22—6921no—
359C22C2×D4×D17311yescharacteristic
360C4—21no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—3461no—
366C4—3461no—
367C4—1732no—
368C4—6921no—
369C4—6921no—
370C4#1914972511yescharacteristic
371C4#1914972511yescharacteristic
372C2—81no—
373C2—13841no—
374C2—13841no—
375C2—6921no—
376C2#2601205911yescharacteristic
377C2#135×D17311yescharacteristic
378C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#488×C173C2
socleC22×C173C22×D4
hypercenter#488D173
hypocenterC173#3212
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #26012040C2semidirect
2. #26012026C2semidirect
3. #26011875C2semidirect
4. #26011873C2semidirect
5. #26011848C2semidirect
6. #26011700C2semidirect
7. #26011695C2semidirect
8. #26011690C2semidirect
9. #26011685C2semidirect
10. #26011656C2semidirect
11. #26011654C2semidirect
12. #26011639C2semidirect
13. #26011637C2semidirect
14. #26011542C2semidirect
15. #488×C173C2semidirect
189. C173#3212semidirect
196. #488D173semidirect

Maximal subgroups

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

subgroupindex
#260120402
#260120262
#260118752
#260118732
#260118482
#260117002
#260116952
#260116902
#260116852
#260116562
#260116542
#260116392
#260116372
#260115422
#488×C1732
#3212173