Finite Group Databasebeta

← #36365478

Subgroups — #36365478

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#26012209C211yesmaximal, characteristic, complemented
2#26012143C211yesmaximal, characteristic
3#26011840C211yesmaximal, characteristic, complemented
4#26011835C211yesmaximal, characteristic, complemented
5#26011770C211yesmaximal, characteristic, complemented
6#26011733C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011700C211yesmaximal, characteristic, complemented
9#26011656C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011578C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14#26011485C211yesmaximal, characteristic, complemented
15#485×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C22×#13063436C2211yescharacteristic
22C2×C8×D173C2211yescharacteristic
23#19149678C2211yescharacteristic
24#19149677C2211yescharacteristic
25#19149673—21no—
26#19149672—21no—
27#19149665—21no—
28#19149664—21no—
29#19149661—21no—
30#19149659C2211yescharacteristic
31#19149659C2211yescharacteristic
32#19149658C2211yescharacteristic
33#19149652—21no—
34C4×Dic346—21no—
35#19149644C2211yescharacteristic
36#19149641C2211yescharacteristic
37C4×C22⋊D173C2211yescharacteristic
38#19149637C2211yescharacteristic
39C4⋊C4×D173—21no—
40#19149612C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53#117×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—41no—
61#14583826—22no—
62#14583826—41no—
63C2×Dic346—21no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173D411yescharacteristic
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74Dic692—41no—
75#14583816—41no—
76#14583815—22no—
77C2×#13063436—41no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—41no—
81C4⋊Dic173—21no—
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803D411yescharacteristic
97C4×Dic173—21no—
98C4×Dic173—21no—
99C4×Dic173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C4⋊C4×C173—21no—
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C22⋊C4×C173—21no—
109C22⋊C4×C173—21no—
110C42×C173—21no—
111C22×D173—21no—
112C22×D173C2×D411yescharacteristic
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—42no—
118D692—21no—
119D692—42no—
120D692#4112yes—
121C22⋊D173—22no—
122C22⋊D173—42no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—42no—
132C4×D173—42no—
133C4×D173—42no—
134#13063436—21no—
135#13063436—21no—
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173C22×D411yescharacteristic
153C2×D173—21no—
154C2×D173—22no—
155C2×D173—21no—
156C2×D173—21no—
157C2×D173—41no—
158C2×D173—41no—
159C4×C173—21no—
160C4×C173—41no—
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#13512yes—
172C2×C173—41no—
173C2×C173#54611yescharacteristic
174C2×C173C2×#13511yescharacteristic
175C2×C173#57811yescharacteristic
176D173—41no—
177D173—42no—
178C173#330211yescharacteristic, complemented, Sylow
179#3302—1731nomaximal, Sylow
180#556—1731no—
181#547—1731no—
182#487—1731no—
183#485—1731no—
184#485D17311yescharacteristic, complemented
185#474—1731no—
186#467—1731no—
187#467—1731no—
188#462—1731no—
189#452—1731no—
190#452—1731no—
191C4×SD16—1731no—
192C8×D4—1731no—
193#416—1731no—
194#416—1731no—
195C2×#41—1731no—
196C2×#41—1731no—
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×SD16—1731no—
200C22×C8—1731no—
201C22×C8—1731no—
202C4⋊Q8—1731no—
203C4⋊Q8—1731no—
204#126—3461no—
205#126—3461no—
206#124—3461no—
207#123—3461no—
208#123—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8—1731no—
212C22⋊Q8C2×D17311yescharacteristic
213#121—3461no—
214C4×Q8—3461no—
215C4×Q8—1731no—
216C4×D4—1731no—
217C4×D4—1731no—
218#117—1731no—
219#117C2×D17311yescharacteristic
220C2×C4⋊C4—3461no—
221#107—1731no—
222#107C2×D17311yescharacteristic
223#106C2×D17311yescharacteristic
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⋊C8C2×D17311yescharacteristic
237C4×C8—1731no—
238#41—3462no—
239#41—3462no—
240#41—6921no—
241#41—6921no—
242C2×Q8—3461no—
243C2×Q8—1731no—
244C2×Q8—1731no—
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×C4C22×D17311yescharacteristic
255Q16—6921no—
256Q16—6921no—
257SD16—3462no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8—6921no—
261C2×C8—6921no—
262C2×C8C22×D17311yescharacteristic
263C2×C8C22×D17311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—3461no—
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⋊C4C22×D17311yescharacteristic
275C4⋊C4C22×D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—1731no—
285C22⋊C4—1731no—
286C42—21no—
287C42—3461no—
288C42—3461no—
289C42—1731no—
290C23—3461no—
291C23—1731no—
292C23D4×D17311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3461no—
296Q8—3461no—
297Q8—3461no—
298Q8—3461no—
299Q8—6922no—
300D4—3462no—
301D4—3461no—
302D4—1732no—
303D4—6922no—
304D4—6922no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—1731no—
317C2×C4—1731no—
318C2×C4—1731no—
319C2×C4—6922no—
320C2×C4—6922no—
321C2×C4—6922no—
322C2×C4D4×D17311yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C8—21no—
325C8—21no—
326C8—3461no—
327C8—3461no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—3461no—
332C22—3461no—
333C22—3461no—
334C22—3462no—
335C22—6921no—
336C22—6921no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3461no—
344C4—3461no—
345C4—1732no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4#41×D17311yescharacteristic
350C4#1914972511yescharacteristic
351C2—41no—
352C2—6921no—
353C2—6922no—
354C2#2601213311yescharacteristic
355C2#135×D17311yescharacteristic
356C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#485×C173C2
socleC22×C173C22×D4
prime radical#26012143C2
hypercenter#485D173
hypocenterC173#3302
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #26012209C2semidirect
3. #26011840C2semidirect
4. #26011835C2semidirect
5. #26011770C2semidirect
6. #26011733C2semidirect
7. #26011730C2semidirect
8. #26011700C2semidirect
9. #26011656C2semidirect
10. #26011655C2semidirect
11. #26011578C2semidirect
12. #26011554C2semidirect
13. #26011486C2semidirect
14. #26011485C2semidirect
15. #485×C173C2semidirect
178. C173#3302semidirect
184. #485D173semidirect

Maximal subgroups

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

subgroupindex
#260122092
#260121432
#260118402
#260118352
#260117702
#260117332
#260117302
#260117002
#260116562
#260116552
#260115782
#260115542
#260114862
#260114852
#485×C1732
#3302173