Finite Group Databasebeta

← #36363767

Subgroups — #36363767

Order 22144.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012097C211yesmaximal, characteristic, complemented
2#26011970C211yesmaximal, characteristic, complemented
3#26011811C211yesmaximal, characteristic, complemented
4#26011806C211yesmaximal, characteristic, complemented
5#26011795C211yesmaximal, characteristic, complemented
6#26011692C211yesmaximal, characteristic, complemented
7#26011686C211yesmaximal, characteristic, complemented
8#26011637C211yesmaximal, characteristic, complemented
9#26011637C211yesmaximal, characteristic, complemented
10#26011600C211yesmaximal, characteristic, complemented
11#26011590C211yesmaximal, characteristic, complemented
12#26011578C211yesmaximal, characteristic, complemented
13#26011571C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic, complemented
15#478×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583811C2211yescharacteristic
21#19149676C2211yescharacteristic
22#19149664—21no—
23#19149664—21no—
24#19149654—21no—
25#19149654—21no—
26#19149654C2211yescharacteristic
27#19149654C2211yescharacteristic
28C4×Dic346C2211yescharacteristic
29#19149644C2211yescharacteristic
30D4×Dic173C2211yescharacteristic
31#19149641C2211yescharacteristic
32#19149641C2211yescharacteristic
33C4×C22⋊D173—21no—
34#19149639—21no—
35#19149636C2211yescharacteristic
36#19149635—21no—
37C2×#14583805—21no—
38C4⋊C4×D173C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149602C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
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—
61C2×Dic346C2311yescharacteristic
62C2×D692—21no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C22×Dic173—21no—
70C22×Dic173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173C2311yescharacteristic
75#14583815—22no—
76#14583815—22no—
77#14583811—22no—
78#14583811—22no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—21no—
110C22×D173C2×D411yescharacteristic
111Dic346—21no—
112Dic346#4112yes—
113D692—21no—
114D692—21no—
115D692—21no—
116D692—21no—
117D692—81no—
118D692—81no—
119D692—81no—
120D692—81no—
121C22⋊D173—24no—
122C2×Dic173—24no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—81no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—81no—
132C4×D173—81no—
133C4×D173—81no—
134#13063436—21no—
135#13063436—21no—
136C23×C173—21no—
137D4×C173—21no—
138D4×C173#4112yes—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—81no—
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173—21no—
148C22×C173—22no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173C22×D411yescharacteristic
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—21no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C4×C173—21no—
162C4×C173—21no—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173C2×#4111yescharacteristic
167C4×C173C2×#4111yescharacteristic
168Dic173—21no—
169Dic173—22no—
170Dic173—21no—
171Dic173—21no—
172Dic173—41no—
173C2×C173—42no—
174C2×C173#52511yescharacteristic
175C2×C173#57611yescharacteristic
176C2×C173#57711yescharacteristic
177D173—41no—
178D173—81no—
179D173—81no—
180C173#311911yescharacteristic, complemented, Sylow
181#3119—1731nomaximal, Sylow
182#540—1731no—
183#520—1731no—
184#481—1731no—
185#480—1731no—
186#479—1731no—
187#478D17311yescharacteristic, complemented
188#460—1731no—
189#459—1731no—
190#449—1731no—
191#449—1731no—
192#442—1731no—
193#440—1731no—
194C4×SD16—1731no—
195C4×D8—1731no—
196#433—1731no—
197C2×#41—1731no—
198C2×SD16—1731no—
199C2×SD16—1731no—
200C2×D8—1731no—
201C2×D8—1731no—
202#127—1731no—
203#125C2×D17311yescharacteristic
204#123—3461no—
205#123—3461no—
206#121—3461no—
207#121—3461no—
208#121—1731no—
209#121—1731no—
210C4×Q8—1731no—
211C4×Q8—1731no—
212C4×D4—3461no—
213C4×D4—3461no—
214C4×D4—1731no—
215C4×D4—1731no—
216C4×D4—1731no—
217C4×D4C2×D17311yescharacteristic
218#117—3461no—
219#117—1731no—
220C2×C4⋊C4—3461no—
221C2×C4⋊C4—1731no—
222#107—1731no—
223#107C2×D17311yescharacteristic
224#106—1731no—
225#106C2×D17311yescharacteristic
226C4⋊C8—1731no—
227C4⋊C8C2×D17311yescharacteristic
228#103—1731no—
229#103—1731no—
230#102—1731no—
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102C2×D17311yescharacteristic
235#102C2×D17311yescharacteristic
236C22⋊C8—1731no—
237C22⋊C8—1731no—
238#97—1731no—
239C4×C8—1731no—
240#41—3462no—
241#41—3462no—
242C2×Q8—1731no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—3461no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4—1731no—
250C2×D4C22×D17311yescharacteristic
251C22×C4—21no—
252C22×C4—3461no—
253C22×C4—3461no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4—1731no—
258SD16—3462no—
259SD16—3462no—
260D8—3462no—
261D8—3462no—
262C2×C8—1731no—
263C2×C8—1731no—
264C2×C8C22×D17311yescharacteristic
265C2×C8C22×D17311yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4C22×D17311yescharacteristic
275C4⋊C4C22×D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—1731no—
284C22⋊C4—1731no—
285C42—3461no—
286C42—3461no—
287C42—1731no—
288C42C22×D17311yescharacteristic
289C23—21no—
290C23—3461no—
291C23—3461no—
292C23—1731no—
293Q8—3461no—
294Q8—1732no—
295D4—21no—
296D4—3464no—
297D4—3461no—
298D4—3461no—
299D4—3461no—
300D4—3461no—
301D4—13841no—
302D4—13841no—
303D4—13841no—
304D4—13841no—
305D4#1458382712yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—3464no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—13841no—
316C2×C4—13841no—
317C2×C4—13841no—
318C2×C4—13841no—
319C2×C4—1731no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4D4×D17311yescharacteristic
323C2×C4D4×D17311yescharacteristic
324C2×C4C23×D17311yescharacteristic
325C8—21no—
326C8—21no—
327C8—3461no—
328C8—3461no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—3461no—
333C22—3461no—
334C22—3461no—
335C22—6921no—
336C22—6921no—
337C22—6921no—
338C22—6921no—
339C22—6921no—
340C22—6921no—
341C22C2×D4×D17311yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—3461no—
348C4—3462no—
349C4—3461no—
350C4—3461no—
351C4—6921no—
352C4#41×D17311yescharacteristic
353C4C2×#1458382711yescharacteristic
354C2—42no—
355C2—13841no—
356C2—13841no—
357C2—6921no—
358C2#2601200411yescharacteristic
359C2#2601229511yescharacteristic
360C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#478×C173C2
socleC22×C173C22×D4
hypercenter#478D173
hypocenterC173#3119
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. #26012097C2semidirect
2. #26011970C2semidirect
3. #26011811C2semidirect
4. #26011806C2semidirect
5. #26011795C2semidirect
6. #26011692C2semidirect
7. #26011686C2semidirect
8. #26011637C2semidirect
9. #26011637C2semidirect
10. #26011600C2semidirect
11. #26011590C2semidirect
12. #26011578C2semidirect
13. #26011571C2semidirect
14. #26011551C2semidirect
15. #478×C173C2semidirect
180. C173#3119semidirect
187. #478D173semidirect

Maximal subgroups

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

subgroupindex
#260120972
#260119702
#260118112
#260118062
#260117952
#260116922
#260116862
#260116372
#260116002
#260115902
#260115782
#260115712
#260115512
#478×C1732
#3119173