Finite Group Databasebeta

← #36365623

Subgroups — #36365623

Order 22144.

380 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#26012150C211yesmaximal, characteristic
2#26011853C211yesmaximal, characteristic, complemented
3#26011832C211yesmaximal, characteristic, complemented
4#26011815C211yesmaximal, characteristic, complemented
5#26011783C211yesmaximal, characteristic, complemented
6#26011739C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011713C211yesmaximal, characteristic, complemented
9#26011651C211yesmaximal, characteristic, complemented
10#26011646C211yesmaximal, characteristic, complemented
11#26011596C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011499C211yesmaximal, characteristic, complemented
14#26011485C211yesmaximal, characteristic, complemented
15#556×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583810C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149667C2211yescharacteristic
24#19149665—21no—
25#19149658C2211yescharacteristic
26#19149657—21no—
27#19149653C2211yescharacteristic
28#19149651—21no—
29C4×Dic346C2211yescharacteristic
30#19149639—21no—
31#19149612C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149603C2211yescharacteristic
36#19149602C2211yescharacteristic
37#19149602C2211yescharacteristic
38#19149602C2211yescharacteristic
39#19149602C2211yescharacteristic
40#19149602C2211yescharacteristic
41#19149602C2211yescharacteristic
42#19149599C2211yescharacteristic
43#19149599C2211yescharacteristic
44#19149599C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47#19149590C2211yescharacteristic
48C4⋊Q8×C173C2211yescharacteristic
49#126×C173—21no—
50#126×C173—21no—
51#124×C173—21no—
52C22⋊Q8×C173C2211yescharacteristic
53C22⋊Q8×C173C2211yescharacteristic
54C4×Q8×C173—21no—
55C4×Q8×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic
57#117×C173C2211yescharacteristic
58#117×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—41no—
61C2×Dic346—22no—
62C2×Dic346—21no—
63C2×Dic346—41no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71#14583816—22no—
72#14583816—41no—
73#14583813—41no—
74#14583810—41no—
75C2×#13063436—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83C22⋊Dic173—21no—
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87C4×Dic173—21no—
88C2×Q8×C173—21no—
89C2×Q8×C173C2311yescharacteristic
90C2×D4×C173D411yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C4⋊C4×C173—21no—
94C4⋊C4×C173—21no—
95C4⋊C4×C173—21no—
96C4⋊C4×C173—21no—
97C4⋊C4×C173D411yescharacteristic
98C4⋊C4×C173C2311yescharacteristic
99C4⋊C4×C173C2311yescharacteristic
100C4⋊C4×C173C2311yescharacteristic
101C22⋊C4×C173—21no—
102C22⋊C4×C173—21no—
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C22⋊C4×C173D411yescharacteristic
106C22⋊C4×C173D411yescharacteristic
107C42×C173—21no—
108C42×C173—21no—
109C42×C173—21no—
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—42no—
116Dic346#4112yes—
117D692—21no—
118D692—21no—
119C22⋊D173—42no—
120C22⋊D173—42no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—42no—
125C4×D173—42no—
126#13063436—21no—
127#13063436—21no—
128#13063436—21no—
129#13063436—21no—
130C23×C173C2×D411yescharacteristic
131C23×C173C2×D411yescharacteristic
132Q8×C173—21no—
133Q8×C173—21no—
134Q8×C173—42no—
135D4×C173—41no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173C22×D411yescharacteristic
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—21no—
162C4×C173—21no—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173—41no—
167C4×C173C2×#4111yescharacteristic
168C4×C173H(5,𝔽2)11yescharacteristic
169Dic173—22no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—41no—
173C2×C173—41no—
174C2×C173#54611yescharacteristic
175C2×C173#57511yescharacteristic
176C2×C173#57811yescharacteristic
177D173—81no—
178C173#331311yescharacteristic, complemented, Sylow
179#3313—1731nomaximal, Sylow
180#556D17311yescharacteristic, complemented
181#548—1731no—
182#489—1731no—
183#485—1731no—
184#482—1731no—
185#477—1731no—
186#468—1731no—
187#467—1731no—
188#464—1731no—
189#452—1731no—
190#451—1731no—
191#441—1731no—
192#435—1731no—
193#419—1731no—
194#416—1731no—
195C2×#41—1731no—
196C22×Q8—1731no—
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×SD16—1731no—
200C2×M16—1731no—
201C22×C8—1731no—
202C4⋊Q8C2×D17311yescharacteristic
203#126—21no—
204#126—21no—
205#124—21no—
206#124—3461no—
207#124—1731no—
208C22⋊Q8—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8C2×D17311yescharacteristic
211C22⋊Q8C2×D17311yescharacteristic
212#121—3461no—
213#121—1731no—
214C4×Q8—21no—
215C4×Q8—1731no—
216C4×Q8C2×D17311yescharacteristic
217C4×D4—3461no—
218C4×D4C2×D17311yescharacteristic
219#117C2×D17311yescharacteristic
220#117C2×D17311yescharacteristic
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#103—1731no—
231#103—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237#97—1731no—
238#41—3462no—
239#41—6921no—
240C2×Q8—21no—
241C2×Q8—3462no—
242C2×Q8—3461no—
243C2×Q8—1731no—
244C2×Q8—1731no—
245C2×Q8—6921no—
246C2×Q8C22×D17311yescharacteristic
247C2×D4—3461no—
248C2×D4—3461no—
249C2×D4—1731no—
250C2×D4C22⋊D17311yescharacteristic
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4C22×D17311yescharacteristic
254C22×C4C22×D17311yescharacteristic
255Q16—3462no—
256Q16—6921no—
257SD16—6921no—
258M16—6921no—
259C2×C8—1731no—
260C2×C8—1731no—
261C2×C8—1731no—
262C2×C8—1731no—
263C2×C8—6921no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—1731no—
271C4⋊C4C22⋊D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
273C4⋊C4C22×D17311yescharacteristic
274C4⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4C22⋊D17311yescharacteristic
284C22⋊C4C22⋊D17311yescharacteristic
285C42—21no—
286C42—21no—
287C42—21no—
288C42—3461no—
289C42C22×D17311yescharacteristic
290C23—3461no—
291C23C2×C22⋊D17311yescharacteristic
292C23C2×C22⋊D17311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—3461no—
297Q8—3461no—
298Q8—3461no—
299Q8—1732no—
300Q8—6922no—
301D4—41no—
302D4—3461no—
303D4—3461no—
304D4—6922no—
305D4—6922no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—6922no—
318C2×C4—6922no—
319C2×C4C2×C22⋊D17311yescharacteristic
320C2×C4C2×C22⋊D17311yescharacteristic
321C2×C4C2×C22⋊D17311yescharacteristic
322C2×C4C2×C22⋊D17311yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C8—3461no—
325C8—3461no—
326C8—3461no—
327C8—3461no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—6921no—
335C22—6921no—
336C22—6921no—
337C22C22×C22⋊D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3462no—
347C4—6921no—
348C4—6921no—
349C4#41×D17311yescharacteristic
350C4#1914972511yescharacteristic
351C2—41no—
352C2—41no—
353C2—13841no—
354C2#2601212911yescharacteristic
355C2#2601228911yescharacteristic
356C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#556×C173C2
socleC22×C173C22×D4
prime radical#26012150C2
hypercenter#556D173
hypocenterC173#3313
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #26011853C2semidirect
3. #26011832C2semidirect
4. #26011815C2semidirect
5. #26011783C2semidirect
6. #26011739C2semidirect
7. #26011730C2semidirect
8. #26011713C2semidirect
9. #26011651C2semidirect
10. #26011646C2semidirect
11. #26011596C2semidirect
12. #26011563C2semidirect
13. #26011499C2semidirect
14. #26011485C2semidirect
15. #556×C173C2semidirect
178. C173#3313semidirect
180. #556D173semidirect

Maximal subgroups

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

subgroupindex
#260121502
#260118532
#260118322
#260118152
#260117832
#260117392
#260117302
#260117132
#260116512
#260116462
#260115962
#260115632
#260114992
#260114852
#556×C1732
#3313173