Finite Group Databasebeta

← #36365656

Subgroups — #36365656

Order 22144.

382 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#26012179C211yesmaximal, characteristic, complemented
2#26012145C211yesmaximal, characteristic
3#26011861C211yesmaximal, characteristic, complemented
4#26011835C211yesmaximal, characteristic, complemented
5#26011828C211yesmaximal, characteristic, complemented
6#26011750C211yesmaximal, characteristic, complemented
7#26011733C211yesmaximal, characteristic, complemented
8#26011702C211yesmaximal, characteristic, complemented
9#26011654C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12#26011561C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#478×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×Dic692C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583809C2211yescharacteristic
22C2×C8×D173C2211yescharacteristic
23#19149679C2211yescharacteristic
24#19149674—21no—
25#19149670—21no—
26#19149664—21no—
27#19149663—21no—
28#19149663—21no—
29#19149659C2211yescharacteristic
30#19149659C2211yescharacteristic
31#19149659C2211yescharacteristic
32#19149659C2211yescharacteristic
33#19149653—21no—
34C4×Dic346C2211yescharacteristic
35C4×D692C2211yescharacteristic
36#19149637—21no—
37#19149637C2211yescharacteristic
38C2×C4⋊Dic173—21no—
39C4⋊C4×D173C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149600C2211yescharacteristic
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×Dic346C2311yescharacteristic
66C2×D692D411yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C22×Dic173—21no—
70C22×Dic173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73C2×C4×D173C2311yescharacteristic
74Dic692—41no—
75Dic692—41no—
76#14583814—22no—
77#14583809—41no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—41no—
81C4⋊Dic173—21no—
82C4⋊Dic173—21no—
83C4⋊Dic173—21no—
84C4⋊Dic173—21no—
85C4⋊Dic173—21no—
86C4⋊Dic173—21no—
87C4⋊Dic173D411yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89#14583805—21no—
90#14583805—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803D411yescharacteristic
97#14583803D411yescharacteristic
98C4×Dic173—21no—
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×D173C2×D411yescharacteristic
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119D692—41no—
120C22⋊D173—42no—
121C22⋊D173—42no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—42no—
127C2×Dic173—42no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C4×D173—21no—
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—42no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138D4×C173—21no—
139D4×C173#4112yes—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—42no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—22no—
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—21no—
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—41no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—42no—
173C2×C173#54611yescharacteristic
174C2×C173#57511yescharacteristic
175C2×C173#57811yescharacteristic
176D173—41no—
177D173—41no—
178C173#331511yescharacteristic, complemented, Sylow
179#3315—1731nomaximal, Sylow
180#552—1731no—
181#547—1731no—
182#491—1731no—
183#485—1731no—
184#484—1731no—
185#478D17311yescharacteristic, complemented
186#470—1731no—
187#467—1731no—
188#462—1731no—
189#452—1731no—
190#452—1731no—
191#440—1731no—
192#435—1731no—
193#417—1731no—
194#416—1731no—
195C2×#41—1731no—
196C2×#41—1731no—
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×SD16—1731no—
200C2×M16—1731no—
201C22×C8—1731no—
202C4⋊Q8—1731no—
203#126—3461no—
204#125—3461no—
205#125C2×D17311yescharacteristic
206#123—3461no—
207#123—3461no—
208#123—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8—1731no—
212C22⋊Q8—1731no—
213#121—3461no—
214C4×Q8—1731no—
215C4×D4—1731no—
216C4×D4C2×D17311yescharacteristic
217#117—3461no—
218#117—1731no—
219C2×C4⋊C4—3461no—
220C2×C4⋊C4—1731no—
221#107—1731no—
222#107C2×D17311yescharacteristic
223#106C2×D17311yescharacteristic
224C4⋊C8C2×D17311yescharacteristic
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103—1731no—
231#103—1731no—
232#102—1731no—
233#102C2×D17311yescharacteristic
234#102C2×D17311yescharacteristic
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237#97—1731no—
238#41—3462no—
239#41—3462no—
240#41—6921no—
241#41—6921no—
242C2×Q8—1731no—
243C2×Q8—1731no—
244C2×Q8—1731no—
245C2×D4—3461no—
246C2×D4—3461no—
247C2×D4—1731no—
248C2×D4C22×D17311yescharacteristic
249C22×C4—21no—
250C22×C4—3461no—
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4—1731no—
254C22×C4—1731no—
255Q16—6921no—
256Q16—6921no—
257SD16—3462no—
258M16—6921no—
259C2×C8—1731no—
260C2×C8—1731no—
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—3461no—
273C4⋊C4—3461no—
274C4⋊C4—1731no—
275C4⋊C4—1731no—
276C4⋊C4C22×D17311yescharacteristic
277C4⋊C4C22×D17311yescharacteristic
278C4⋊C4C22×D17311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—1731no—
286C22⋊C4—1731no—
287C42—3461no—
288C42—3461no—
289C42C22×D17311yescharacteristic
290C23—21no—
291C23—1731no—
292C23—1731no—
293Q8—3461no—
294Q8—3461no—
295Q8—3461no—
296Q8—3461no—
297Q8—3461no—
298Q8—3461no—
299D4—21no—
300D4—6921no—
301D4—6922no—
302D4—6922no—
303D4#1458382712yes—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
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—22no—
329C22—41no—
330C22—41no—
331C22—3461no—
332C22—3461no—
333C22—3461no—
334C22—3461no—
335C22—3461no—
336C22—3461no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3461no—
344C4—3461no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4C2×#1458382711yescharacteristic
350C4#1914972511yescharacteristic
351C2—42no—
352C2—6921no—
353C2—6921no—
354C2#2601213211yescharacteristic
355C2#2601228711yescharacteristic
356C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#478×C173C2
socleC22×C173C22×D4
prime radical#26012145C2
hypercenter#478D173
hypocenterC173#3315
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. #26012179C2semidirect
3. #26011861C2semidirect
4. #26011835C2semidirect
5. #26011828C2semidirect
6. #26011750C2semidirect
7. #26011733C2semidirect
8. #26011702C2semidirect
9. #26011654C2semidirect
10. #26011654C2semidirect
11. #26011591C2semidirect
12. #26011561C2semidirect
13. #26011489C2semidirect
14. #26011484C2semidirect
15. #478×C173C2semidirect
178. C173#3315semidirect
185. #478D173semidirect

Maximal subgroups

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

subgroupindex
#260121792
#260121452
#260118612
#260118352
#260118282
#260117502
#260117332
#260117022
#260116542
#260115912
#260115612
#260114892
#260114842
#478×C1732
#3315173