Finite Group Databasebeta

← #36365642

Subgroups — #36365642

Order 22144.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012196C211yesmaximal, characteristic, complemented
2#26012132C211yesmaximal, characteristic
3#26011852C211yesmaximal, characteristic, complemented
4#26011806C211yesmaximal, characteristic, complemented
5#26011770C211yesmaximal, characteristic, complemented
6#26011738C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011716C211yesmaximal, characteristic, complemented
9#26011656C211yesmaximal, characteristic, complemented
10#26011644C211yesmaximal, characteristic, complemented
11#26011590C211yesmaximal, characteristic, complemented
12#26011559C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14C2×#19149599C211yesmaximal, characteristic, complemented
15#484×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic, complemented
19C2×#14583815C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583809C2211yescharacteristic, complemented
22C22×#13063436C2211yescharacteristic
23#19149678C2211yescharacteristic, complemented
24#19149669—21no—
25#19149668C2211yescharacteristic
26#19149664—21no—
27#19149659C2211yescharacteristic, complemented
28#19149658—21no—
29#19149658C2211yescharacteristic
30#19149657—21no—
31#19149654—21no—
32#19149654C2211yescharacteristic
33#19149649—21no—
34#19149644C2211yescharacteristic, complemented
35#19149641C2211yescharacteristic
36C4×C22⋊D173—21no—
37C4×C22⋊D173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic, complemented
39C22⋊C4×D173—21no—
40#19149612C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C22⋊Q8×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic, complemented
55#107×C173C2211yescharacteristic, complemented
56#103×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic
59C23×D173—21no—
60#14583827—22no—
61#14583827—41no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×D692—22no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—21no—
68C2×D692—41no—
69C2×D692C2311yescharacteristic
70C2×D692C2311yescharacteristic
71C2×C22⋊D173D411yescharacteristic
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76#14583816—41no—
77#14583815—22no—
78#14583815—41no—
79#14583809—41no—
80C2×#13063436—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C4⋊Dic173—21no—
84C4⋊Dic173C2311yescharacteristic
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#14583803—21no—
97#14583803D411yescharacteristic
98C4×Dic173—21no—
99C4×Dic173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101C22×C4×C173—21no—
102C22×C4×C173C2311yescharacteristic
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116C22×D173C2×D411yescharacteristic
117Dic346—21no—
118Dic346—21no—
119Dic346—81no—
120D692—21no—
121D692—21no—
122D692—21no—
123D692—42no—
124D692—42no—
125D692—81no—
126D692#4112yes—
127C22⋊D173—22no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C2×Dic173C2×D411yescharacteristic
133C4×D173—21no—
134C4×D173—21no—
135C4×D173—42no—
136C4×D173—81no—
137C4×D173—81no—
138#13063436—21no—
139#13063436—21no—
140C23×C173C2×D411yescharacteristic
141Q8×C173—21no—
142Q8×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—81no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173#13712yes—
157C22×C173C22×D411yescharacteristic
158C2×D173—21no—
159C2×D173—22no—
160C2×D173—21no—
161C2×D173—21no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—84no—
168C4×C173—21no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173—41no—
173C4×C173C2×#4111yescharacteristic
174C4×C173H(5,𝔽2)11yescharacteristic
175Dic173—21no—
176Dic173—21no—
177Dic173—21no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—22no—
181C2×C173#54611yescharacteristic
182C2×C173C2×#13711yescharacteristic
183C2×C173#57611yescharacteristic
184D173—41no—
185D173—42no—
186D173—81no—
187C173#331411yescharacteristic, complemented, Sylow
188#3314—1731nomaximal, Sylow
189#554—1731no—
190#546—1731no—
191#489—1731no—
192#484D17311yescharacteristic, complemented
193#480—1731no—
194#474—1731no—
195#468—1731no—
196#467—1731no—
197#464—1731no—
198#452—1731no—
199#450—1731no—
200#440—1731no—
201#435—1731no—
202#417—1731no—
203C2×#102—1731no—
204C2×#41—1731no—
205C22×D4—1731no—
206C2×Q16—1731no—
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×M16—1731no—
210C22×C8—1731no—
211C4⋊Q8—1731no—
212#125—3461no—
213#124—1731no—
214#123—3461no—
215C22⋊Q8—3461no—
216C22⋊Q8—3461no—
217C22⋊Q8—1731no—
218C22⋊Q8—1731no—
219C22⋊Q8C2×D17311yescharacteristic
220#121—3461no—
221#121—1731no—
222#120—3461no—
223C4×Q8—1731no—
224C4×D4—3461no—
225C4×D4—1731no—
226C4×D4—1731no—
227C2×C4⋊C4—1731no—
228C2×C4⋊C4C2×D17311yescharacteristic
229C2×C22⋊C4—3461no—
230#107—1731no—
231#107C2×D17311yescharacteristic, complemented
232#107C2×D17311yescharacteristic, complemented
233C4⋊C8—1731no—
234#103—1731no—
235#103—1731no—
236#103C2×D17311yescharacteristic, complemented
237#103C2×D17311yescharacteristic, complemented
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244C22⋊C8—1731no—
245C22⋊C8C2×D17311yescharacteristic
246#97—1731no—
247C24—3461no—
248#41—3462no—
249#41—6921no—
250C2×Q8—3461no—
251C2×Q8—1731no—
252C2×Q8C22×D17311yescharacteristic
253C2×D4—3462no—
254C2×D4—3461no—
255C2×D4—3461no—
256C2×D4—3461no—
257C2×D4—1731no—
258C2×D4—1731no—
259C2×D4—1731no—
260C2×D4—6921no—
261C22×C4—21no—
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—1731no—
266C22×C4C22×D17311yescharacteristic
267Q16—6921no—
268SD16—3462no—
269SD16—6921no—
270M16—6921no—
271C2×C8—1731no—
272C2×C8—1731no—
273C2×C8—6921no—
274C2×C8C22×D17311yescharacteristic
275C2×C8C22×D17311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—1731no—
284C4⋊C4—1731no—
285C4⋊C4C22×D17311yescharacteristic
286C4⋊C4C22×D17311yescharacteristic
287C4⋊C4C22×D17311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—1731no—
296C22⋊C4—1731no—
297C42—3461no—
298C42—1731no—
299C23—3461no—
300C23—3461no—
301C23—13841no—
302C23—1731no—
303C23—6922no—
304C23—6922no—
305C23D4×D17311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—3461no—
309Q8—3461no—
310Q8—13841no—
311D4—3462no—
312D4—3461no—
313D4—3461no—
314D4—3461no—
315D4—13841no—
316D4—1732no—
317D4—6922no—
318D4—6922no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—81no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—13841no—
331C2×C4—13841no—
332C2×C4—1731no—
333C2×C4—1731no—
334C2×C4—1731no—
335C2×C4—6922no—
336C2×C4D4×D17311yescharacteristic
337C2×C4C23×D17311yescharacteristic
338C8—21no—
339C8—21no—
340C8—3461no—
341C8—3461no—
342C22—21no—
343C22—21no—
344C22—3461no—
345C22—3461no—
346C22—3461no—
347C22—3462no—
348C22—13844no—
349C22—6921no—
350C22—6921no—
351C22—6921no—
352C22—6921no—
353C22—6921no—
354C22#1914971112yes—
355C22C2×D4×D17311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—3461no—
362C4—3461no—
363C4—3461no—
364C4—6921no—
365C4—6921no—
366C4#41×D17311yescharacteristic
367C4#1914972511yescharacteristic
368C2—22no—
369C2—13841no—
370C2—6921no—
371C2—6922no—
372C2#2601213311yescharacteristic
373C2C2×#1914971111yescharacteristic
374C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#484×C173C2
socleC22×C173C22×D4
prime radical#26012132C2
hypercenter#484D173
hypocenterC173#3314
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #26012196C2semidirect
3. #26011852C2semidirect
4. #26011806C2semidirect
5. #26011770C2semidirect
6. #26011738C2semidirect
7. #26011730C2semidirect
8. #26011716C2semidirect
9. #26011656C2semidirect
10. #26011644C2semidirect
11. #26011590C2semidirect
12. #26011559C2semidirect
13. #26011491C2semidirect
14. C2×#19149599C2semidirect
15. #484×C173C2semidirect
18. C2×#14583816C22semidirect
21. C2×#14583809C22semidirect
23. #19149678C22semidirect
27. #19149659C22semidirect
34. #19149644C22semidirect
38. C4⋊C4×D173C22semidirect
40. #19149612C22semidirect
41. #19149608C22semidirect
42. #19149603C22semidirect
43. #19149602C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
54. #107×C173C22semidirect
55. #107×C173C22semidirect
56. #103×C173C22semidirect
57. #103×C173C22semidirect
187. C173#3314semidirect
192. #484D173semidirect
231. #107C2×D173semidirect
232. #107C2×D173semidirect
236. #103C2×D173semidirect
237. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121962
#260121322
#260118522
#260118062
#260117702
#260117382
#260117302
#260117162
#260116562
#260116442
#260115902
#260115592
#260114912
C2×#191495992
#484×C1732
#3314173