Finite Group Databasebeta

← #36365536

Subgroups — #36365536

Order 22144.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012194C211yesmaximal, characteristic, complemented
2#26012145C211yesmaximal, characteristic, complemented
3#26011861C211yesmaximal, characteristic, complemented
4#26011834C211yesmaximal, characteristic, complemented
5#26011823C211yesmaximal, characteristic, complemented
6#26011738C211yesmaximal, characteristic, complemented
7#26011721C211yesmaximal, characteristic, complemented
8#26011697C211yesmaximal, characteristic, complemented
9#26011637C211yesmaximal, characteristic, complemented
10#26011637C211yesmaximal, characteristic, complemented
11#26011601C211yesmaximal, characteristic, complemented
12#26011561C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#478×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21C2×#14583809C2211yescharacteristic
22C2×C8×D173C2211yescharacteristic
23#19149679C2211yescharacteristic
24#19149674—21no—
25#19149664—21no—
26#19149663—21no—
27#19149663—21no—
28#19149659C2211yescharacteristic
29#19149659C2211yescharacteristic
30#19149654—21no—
31#19149654C2211yescharacteristic
32#19149654C2211yescharacteristic
33#19149653—21no—
34C4×D692C2211yescharacteristic
35C4×D692C2211yescharacteristic
36#19149641—21no—
37#19149637C2211yescharacteristic
38C2×C4⋊Dic173—21no—
39C4⋊C4×D173C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic
52#125×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—22no—
61#14583825—41no—
62#14583825—41no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692—21no—
66C2×D692—21no—
67C2×D692D411yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C22×Dic173—21no—
72C22×Dic173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76C2×C4×D173C2311yescharacteristic
77#14583815—41no—
78#14583815—41no—
79#14583811—22no—
80#14583809—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C8×D173—41no—
84C4⋊Dic173—21no—
85C4⋊Dic173—21no—
86C4⋊Dic173—21no—
87C4⋊Dic173—21no—
88C4⋊Dic173D411yescharacteristic
89C4⋊Dic173C2311yescharacteristic
90#14583805—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803D411yescharacteristic
98#14583803D411yescharacteristic
99C4×Dic173—21no—
100C2×D4×C173C2311yescharacteristic
101C22×C4×C173—21no—
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173—21no—
105C4⋊C4×C173—21no—
106C4⋊C4×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173C2×D411yescharacteristic
113C22×D173C2×D411yescharacteristic
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118D692—21no—
119D692—21no—
120D692—41no—
121D692—42no—
122D692—81no—
123C22⋊D173—42no—
124C22⋊D173—42no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—42no—
129C2×Dic173—42no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—21no—
136C4×D173—81no—
137C4×D173—81no—
138#13063436—21no—
139#13063436—21no—
140C23×C173—21no—
141D4×C173—21no—
142D4×C173#4112yes—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—42no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C22×C173—22no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173C22×D411yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—21no—
159C2×D173—21no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173C2×#4111yescharacteristic
171C4×C173H(5,𝔽2)11yescharacteristic
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—42no—
178C2×C173#54611yescharacteristic
179C2×C173#57511yescharacteristic
180C2×C173#57711yescharacteristic
181D173—41no—
182D173—41no—
183D173—81no—
184C173#330711yescharacteristic, complemented, Sylow
185#3307—1731nomaximal, Sylow
186#553—1731no—
187#547—1731no—
188#491—1731no—
189#485—1731no—
190#483—1731no—
191#478D17311yescharacteristic, complemented
192#468—1731no—
193#465—1731no—
194#461—1731no—
195#449—1731no—
196#449—1731no—
197#442—1731no—
198#435—1731no—
199#417—1731no—
200#416—1731no—
201C2×#41—1731no—
202C2×#41—1731no—
203C2×SD16—1731no—
204C2×SD16—1731no—
205C2×D8—1731no—
206C2×M16—1731no—
207C22×C8—1731no—
208C4⋊Q8—1731no—
209#126—3461no—
210#125C2×D17311yescharacteristic
211#123—3461no—
212#123—3461no—
213#123—3461no—
214C22⋊Q8—1731no—
215C22⋊Q8—1731no—
216#121—3461no—
217#121—3461no—
218#121—1731no—
219#121—1731no—
220C4×D4—3461no—
221C4×D4—1731no—
222C4×D4—1731no—
223C4×D4C2×D17311yescharacteristic
224#117—1731no—
225C2×C4⋊C4—3461no—
226C2×C4⋊C4—1731no—
227#107C2×D17311yescharacteristic
228#106—1731no—
229#106C2×D17311yescharacteristic
230C4⋊C8C2×D17311yescharacteristic
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#102—1731no—
238#102—1731no—
239#102C2×D17311yescharacteristic
240#102C2×D17311yescharacteristic
241C22⋊C8—1731no—
242C22⋊C8—1731no—
243#97—1731no—
244#41—3462no—
245#41—3462no—
246#41—6921no—
247#41—6921no—
248C2×Q8—1731no—
249C2×Q8—1731no—
250C2×D4—3461no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—3461no—
254C2×D4—1731no—
255C2×D4—1731no—
256C2×D4C22×D17311yescharacteristic
257C22×C4—21no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4—1731no—
263C22×C4—1731no—
264SD16—6921no—
265SD16—6921no—
266D8—3462no—
267M16—6921no—
268C2×C8—1731no—
269C2×C8—1731no—
270C2×C8—6921no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4C22×D17311yescharacteristic
283C4⋊C4C22×D17311yescharacteristic
284C4⋊C4C22×D17311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—1731no—
293C22⋊C4—1731no—
294C42—3461no—
295C42C22×D17311yescharacteristic
296C23—21no—
297C23—3461no—
298C23—1731no—
299C23—1731no—
300Q8—3461no—
301Q8—3461no—
302Q8—3461no—
303Q8—3461no—
304D4—21no—
305D4—3461no—
306D4—3461no—
307D4—13841no—
308D4—6921no—
309D4—6922no—
310D4—6922no—
311D4—6922no—
312D4#1458382712yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—13841no—
325C2×C4—13841no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4—6922no—
329C2×C4—6922no—
330C2×C4D4×D17311yescharacteristic
331C2×C4D4×D17311yescharacteristic
332C2×C4C23×D17311yescharacteristic
333C8—21no—
334C8—21no—
335C8—3461no—
336C8—3461no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—3461no—
341C22—3461no—
342C22—3461no—
343C22—3461no—
344C22—3461no—
345C22—3461no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3461no—
356C4—3461no—
357C4—6921no—
358C4—6921no—
359C4—6921no—
360C4C2×#1458382711yescharacteristic
361C4#1914972511yescharacteristic
362C2—42no—
363C2—13841no—
364C2—6921no—
365C2—6921no—
366C2#2601213211yescharacteristic
367C2#2601228711yescharacteristic
368C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#478×C173C2
socleC22×C173C22×D4
hypercenter#478D173
hypocenterC173#3307
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #26012194C2semidirect
2. #26012145C2semidirect
3. #26011861C2semidirect
4. #26011834C2semidirect
5. #26011823C2semidirect
6. #26011738C2semidirect
7. #26011721C2semidirect
8. #26011697C2semidirect
9. #26011637C2semidirect
10. #26011637C2semidirect
11. #26011601C2semidirect
12. #26011561C2semidirect
13. #26011489C2semidirect
14. #26011484C2semidirect
15. #478×C173C2semidirect
184. C173#3307semidirect
191. #478D173semidirect

Maximal subgroups

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

subgroupindex
#260121942
#260121452
#260118612
#260118342
#260118232
#260117382
#260117212
#260116972
#260116372
#260116012
#260115612
#260114892
#260114842
#478×C1732
#3307173