Finite Group Databasebeta

← #36365650

Subgroups — #36365650

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#26012199C211yesmaximal, characteristic, complemented
2#26012132C211yesmaximal, characteristic
3#26011852C211yesmaximal, characteristic, complemented
4#26011824C211yesmaximal, characteristic, complemented
5#26011806C211yesmaximal, characteristic, complemented
6#26011770C211yesmaximal, characteristic, complemented
7#26011740C211yesmaximal, characteristic, complemented
8#26011715C211yesmaximal, characteristic, complemented
9#26011652C211yesmaximal, characteristic, complemented
10#26011641C211yesmaximal, characteristic, complemented
11#26011594C211yesmaximal, characteristic, complemented
12#26011562C211yesmaximal, characteristic, complemented
13#26011491C211yesmaximal, characteristic, complemented
14C2×#19149601C211yesmaximal, characteristic, complemented
15#467×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×#14583809C2211yescharacteristic, complemented
21#19149678C2211yescharacteristic, complemented
22#19149669—21no—
23#19149668C2211yescharacteristic
24#19149664—21no—
25#19149659—21no—
26#19149656—21no—
27#19149655C2211yescharacteristic, complemented
28#19149654—21no—
29#19149654C2211yescharacteristic
30#19149649—21no—
31Q8×Dic173C2211yescharacteristic, complemented
32#19149641C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149639—21no—
35C2×C4⋊Dic173C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic, complemented
37C22⋊C4×D173—21no—
38#19149612C2211yescharacteristic, complemented
39#19149612C2211yescharacteristic, complemented
40#19149608C2211yescharacteristic, complemented
41#19149603C2211yescharacteristic, complemented
42#19149603C2211yescharacteristic, complemented
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149589C2211yescharacteristic, complemented
52C2×Q16×C173C2211yescharacteristic, complemented
53C22×C8×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#103×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583827—22no—
61#14583827—41no—
62C2×Dic346—21no—
63C2×D692—22no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692—41no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173D411yescharacteristic
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76#14583813—22no—
77#14583813—41no—
78#14583809—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C4⋊Dic173—21no—
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805D411yescharacteristic
89C22⋊Dic173D411yescharacteristic
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803D411yescharacteristic
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C2×Q8×C173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101Q16×C173—41no—
102C2×C8×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116C22×D173C2×D411yescharacteristic
117Dic346—81no—
118D692—21no—
119D692—21no—
120D692—21no—
121D692—42no—
122D692—42no—
123D692—81no—
124D692#4112yes—
125C22⋊D173—22no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—81no—
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—42no—
135C4×D173—81no—
136C4×D173—81no—
137#13063436—21no—
138#13063436—21no—
139C23×C173C2×D411yescharacteristic
140Q8×C173—21no—
141Q8×C173—21no—
142Q8×C173—21no—
143Q8×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—21no—
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#484—1731no—
193#480—1731no—
194#474—1731no—
195#468—1731no—
196#467D17311yescharacteristic, complemented
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×Q16C2×D17311yescharacteristic, complemented
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×M16—1731no—
210C22×C8C2×D17311yescharacteristic
211C4⋊Q8—1731no—
212#125—3461no—
213#124—1731no—
214#123—3461no—
215C22⋊Q8—3461no—
216C22⋊Q8—3461no—
217C22⋊Q8—1731no—
218C22⋊Q8C2×D17311yescharacteristic
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⋊C4—1731no—
229C2×C22⋊C4—3461no—
230#107—1731no—
231#107—1731no—
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⋊C8—1731no—
246#97—1731no—
247C24—3461no—
248#41—3462no—
249#41—6921no—
250C2×Q8—3461no—
251C2×Q8C22×D17311yescharacteristic
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—3461no—
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—1731no—
266C22×C4C22×D17311yescharacteristic
267Q16—41no—
268SD16—3462no—
269SD16—6921no—
270M16—6921no—
271C2×C8—41no—
272C2×C8—1731no—
273C2×C8—1731no—
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⋊C4—1731no—
286C4⋊C4C22×D17311yescharacteristic
287C4⋊C4C22×D17311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
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—21no—
309Q8—21no—
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—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—13841no—
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#1914972411yescharacteristic
368C2—22no—
369C2—13841no—
370C2—6921no—
371C2—6922no—
372C2#2601213011yescharacteristic
373C2C2×#1914971111yescharacteristic
374C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#467×C173C2
socleC22×C173C22×D4
prime radical#26012132C2
hypercenter#467D173
hypocenterC173#3314
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #26012199C2semidirect
3. #26011852C2semidirect
4. #26011824C2semidirect
5. #26011806C2semidirect
6. #26011770C2semidirect
7. #26011740C2semidirect
8. #26011715C2semidirect
9. #26011652C2semidirect
10. #26011641C2semidirect
11. #26011594C2semidirect
12. #26011562C2semidirect
13. #26011491C2semidirect
14. C2×#19149601C2semidirect
15. #467×C173C2semidirect
20. C2×#14583809C22semidirect
21. #19149678C22semidirect
27. #19149655C22semidirect
31. Q8×Dic173C22semidirect
36. C4⋊C4×D173C22semidirect
38. #19149612C22semidirect
39. #19149612C22semidirect
40. #19149608C22semidirect
41. #19149603C22semidirect
42. #19149603C22semidirect
50. #19149591C22semidirect
51. #19149589C22semidirect
52. C2×Q16×C173C22semidirect
56. #107×C173C22semidirect
57. #103×C173C22semidirect
58. #103×C173C22semidirect
187. C173#3314semidirect
196. #467D173semidirect
206. C2×Q16C2×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
#260121992
#260121322
#260118522
#260118242
#260118062
#260117702
#260117402
#260117152
#260116522
#260116412
#260115942
#260115622
#260114912
C2×#191496012
#467×C1732
#3314173