Finite Group Databasebeta

← #36365476

Subgroups — #36365476

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#26012140C211yesmaximal, characteristic
3#26011840C211yesmaximal, characteristic, complemented
4#26011832C211yesmaximal, characteristic, complemented
5#26011831C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011732C211yesmaximal, characteristic, complemented
8#26011731C211yesmaximal, characteristic, complemented
9#26011700C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011653C211yesmaximal, characteristic, complemented
12#26011582C211yesmaximal, characteristic, complemented
13C8×C22⋊D173C211yesmaximal, characteristic, complemented
14#26011484C211yesmaximal, characteristic, complemented
15#416×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583814C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149677C2211yescharacteristic
23#19149673—21no—
24#19149672—21no—
25#19149665—21no—
26#19149662—21no—
27#19149660—21no—
28#19149659C2211yescharacteristic
29#19149658C2211yescharacteristic
30#19149658C2211yescharacteristic
31#19149655C2211yescharacteristic
32#19149651—21no—
33C4×Dic346—21no—
34#19149644C2211yescharacteristic
35D4×Dic173C2211yescharacteristic
36C4×C22⋊D173C2211yescharacteristic
37#19149637C2211yescharacteristic
38C2×#14583805—21no—
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54#117×C173C2211yescharacteristic
55#103×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#14583825—22no—
60#14583825—41no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×Dic346C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173D411yescharacteristic
66C22×Dic173—21no—
67C22×Dic173—21no—
68C2×C4×D173C2311yescharacteristic
69Dic692—41no—
70#14583816—41no—
71#14583814—22no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C8×D173—41no—
75C4⋊Dic173—21no—
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805D411yescharacteristic
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173D411yescharacteristic
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97#41×C173—22no—
98#41×C173—41no—
99C2×Q8×C173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104C2×C8×C173—41no—
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C42×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—42no—
117C22⋊D173—22no—
118C22⋊D173—42no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—42no—
124C2×Dic173—42no—
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—21no—
129C4×D173—21no—
130#13063436—21no—
131#13063436—21no—
132C23×C173—21no—
133C23×C173C2×D411yescharacteristic
134Q8×C173—21no—
135Q8×C173—21no—
136D4×C173—21no—
137D4×C173—42no—
138D4×C173#4112yes—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—42no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—21no—
150C22×C173—22no—
151C22×C173—21no—
152C22×C173—21no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173C22×D411yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—21no—
159C4×C173—21no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173C2×#4111yescharacteristic
164C4×C173H(5,𝔽2)11yescharacteristic
165Dic173—21no—
166Dic173—21no—
167Dic173—41no—
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171Dic173#13512yes—
172C2×C173—41no—
173C2×C173—42no—
174C2×C173#54611yescharacteristic
175C2×C173C2×#13511yescharacteristic
176C2×C173#57811yescharacteristic
177D173—41no—
178C173#330211yescharacteristic, complemented, Sylow
179#3302—1731nomaximal, Sylow
180#556—1731no—
181#547—1731no—
182#487—1731no—
183#485—1731no—
184#485—1731no—
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#416D17311yescharacteristic, complemented
195C2×#41—1731no—
196C2×#41C2×D17311yescharacteristic
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×SD16—1731no—
200C22×C8—1731no—
201C22×C8C2×D17311yescharacteristic
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⋊Q8—1731no—
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#107—1731no—
223#106—1731no—
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#102C2×D17311yescharacteristic
234#102C2×D17311yescharacteristic
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237C4×C8—1731no—
238#41—22no—
239#41—41no—
240#41—3462no—
241#41—6921no—
242C2×Q8—3461no—
243C2×Q8—1731no—
244C2×Q8—1731no—
245C2×Q8C22×D17311yescharacteristic
246C2×D4—21no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4C22×D17311yescharacteristic
250C22×C4—21no—
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4—1731no—
254C22×C4C22×D17311yescharacteristic
255Q16—6921no—
256Q16—6921no—
257SD16—3462no—
258C2×C8—41no—
259C2×C8—1731no—
260C2×C8—1731no—
261C2×C8—6921no—
262C2×C8C22×D17311yescharacteristic
263C2×C8C22×D17311yescharacteristic
264C4⋊C4—3461no—
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⋊C4—1731no—
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—3461no—
284C22⋊C4—1731no—
285C22⋊C4—1731no—
286C42—21no—
287C42—3461no—
288C42—3461no—
289C42—1731no—
290C23—21no—
291C23—1731no—
292C23D4×D17311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3461no—
296Q8—3461no—
297Q8—3461no—
298Q8—3461no—
299Q8—6922no—
300D4—21no—
301D4—42no—
302D4—3462no—
303D4—6922no—
304D4#1458382612yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—1731no—
318C2×C4—1731no—
319C2×C4—1731no—
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—22no—
332C22—41no—
333C22—41no—
334C22—3461no—
335C22—3461no—
336C22—3461no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—3461no—
343C4—3461no—
344C4—1732no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4C2×#1458382611yescharacteristic
350C4#1914972411yescharacteristic
351C2—41no—
352C2—42no—
353C2—6921no—
354C2#2601213111yescharacteristic
355C2#135×D17311yescharacteristic
356C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#416×C173C2
socleC22×C173C22×D4
prime radical#26012140C2
hypercenter#416D173
hypocenterC173#3302
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #26012209C2semidirect
3. #26011840C2semidirect
4. #26011832C2semidirect
5. #26011831C2semidirect
6. #26011769C2semidirect
7. #26011732C2semidirect
8. #26011731C2semidirect
9. #26011700C2semidirect
10. #26011654C2semidirect
11. #26011653C2semidirect
12. #26011582C2semidirect
13. C8×C22⋊D173C2semidirect
14. #26011484C2semidirect
15. #416×C173C2semidirect
178. C173#3302semidirect
194. #416D173semidirect

Maximal subgroups

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

subgroupindex
#260122092
#260121402
#260118402
#260118322
#260118312
#260117692
#260117322
#260117312
#260117002
#260116542
#260116532
#260115822
C8×C22⋊D1732
#260114842
#416×C1732
#3302173