Finite Group Databasebeta

← #36365647

Subgroups — #36365647

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#26012138C211yesmaximal, characteristic
3#26011849C211yesmaximal, characteristic, complemented
4#26011828C211yesmaximal, characteristic, complemented
5#26011770C211yesmaximal, characteristic, complemented
6#26011735C211yesmaximal, characteristic, complemented
7#26011733C211yesmaximal, characteristic, complemented
8#26011712C211yesmaximal, characteristic, complemented
9#26011655C211yesmaximal, characteristic, complemented
10#26011643C211yesmaximal, characteristic, complemented
11#26011590C211yesmaximal, characteristic, complemented
12#26011565C211yesmaximal, characteristic, complemented
13#26011490C211yesmaximal, characteristic, complemented
14#102×D173C211yesmaximal, characteristic, complemented
15#480×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18C2×Dic692C2211yescharacteristic, complemented
19C2×#14583815C2211yescharacteristic
20C2×#14583814C2211yescharacteristic
21C2×#14583810C2211yescharacteristic, complemented
22C2×C8×D173C2211yescharacteristic
23#19149678C2211yescharacteristic, complemented
24#19149669—21no—
25#19149666C2211yescharacteristic
26#19149661—21no—
27#19149659C2211yescharacteristic
28#19149659C2211yescharacteristic
29#19149658—21no—
30#19149658C2211yescharacteristic, complemented
31#19149657—21no—
32#19149650—21no—
33#19149647—21no—
34#19149644C2211yescharacteristic, complemented
35#19149641C2211yescharacteristic
36C4×C22⋊D173—21no—
37C4×C22⋊D173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39C22⋊C4×D173—21no—
40#19149612C2211yescharacteristic, complemented
41#19149608C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149602C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149589C2211yescharacteristic, complemented
52#121×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic, complemented
54#107×C173C2211yescharacteristic, complemented
55#107×C173C2211yescharacteristic, complemented
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—22no—
61#14583826—41no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173D411yescharacteristic
69D4×D173—22no—
70D4×D173—41no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173C2311yescharacteristic
75Dic692—41no—
76#14583815—22no—
77#14583814—41no—
78#14583810—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C8×D173—41no—
82C4⋊Dic173—21no—
83C4⋊Dic173—21no—
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173D411yescharacteristic
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803D411yescharacteristic
98C4×Dic173—21no—
99C4×Dic173C2311yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173—21no—
112C22×D173—42no—
113C22×D173—42no—
114C22×D173—81no—
115C22×D173C2×D411yescharacteristic
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—21no—
120Dic346—81no—
121D692—21no—
122D692#4112yes—
123C22⋊D173—22no—
124C22⋊D173—42no—
125C22⋊D173—42no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
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×C173—21no—
140C23×C173C2×D411yescharacteristic
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—81no—
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—21no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173C22×D411yescharacteristic
161C2×D173—21no—
162C2×D173—21no—
163C2×D173—22no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—84no—
167C2×D173#13712yes—
168C4×C173—21no—
169C4×C173—41no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173C2×#4111yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—21no—
175Dic173—21no—
176Dic173—21no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—41no—
181C2×C173—81no—
182C2×C173#54611yescharacteristic
183C2×C173C2×#13711yescharacteristic
184C2×C173#57611yescharacteristic
185D173—22no—
186D173—42no—
187C173#331411yescharacteristic, complemented, Sylow
188#3314—1731nomaximal, Sylow
189#554—1731no—
190#546—1731no—
191#489—1731no—
192#484—1731no—
193#480D17311yescharacteristic, complemented
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⋊Q8—1731no—
220#121—3461no—
221#121C2×D17311yescharacteristic
222#120—3461no—
223C4×Q8—1731no—
224C4×D4—3461no—
225C4×D4—1731no—
226C4×D4—1731no—
227C2×C4⋊C4—1731no—
228C2×C4⋊C4C2×D17311yescharacteristic, complemented
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#103—1731no—
237#103—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242#102C2×D17311yescharacteristic
243#102C2×D17311yescharacteristic
244C22⋊C8—1731no—
245C22⋊C8C2×D17311yescharacteristic, complemented
246#97—1731no—
247C24—3461no—
248#41—3462no—
249#41—6921no—
250C2×Q8—3461no—
251C2×Q8—1731no—
252C2×Q8—1731no—
253C2×D4—21no—
254C2×D4—3462no—
255C2×D4—3461no—
256C2×D4—3461no—
257C2×D4—1731no—
258C2×D4—1731no—
259C2×D4—6921no—
260C2×D4C22×D17311yescharacteristic
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—3461no—
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—21no—
300C23—3461no—
301C23—13841no—
302C23—1731no—
303C23—6922no—
304C23—6922no—
305C23D4×D17311yescharacteristic
306Q8—3461no—
307Q8—3461no—
308Q8—3461no—
309Q8—3461no—
310Q8—13841no—
311D4—21no—
312D4—21no—
313D4—81no—
314D4—3462no—
315D4—3461no—
316D4—1732no—
317D4—6922no—
318D4—6922no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—81no—
325C2×C4—3461no—
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—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—3461no—
349C22—3461no—
350C22—3462no—
351C22—13844no—
352C22—1732no—
353C22—6921no—
354C22—6921no—
355C22C2×D4×D17311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3461no—
361C4—3461no—
362C4—3461no—
363C4—6921no—
364C4—6921no—
365C4—6921no—
366C4#41×D17311yescharacteristic
367C4#1914972511yescharacteristic
368C2—41no—
369C2—81no—
370C2—3462no—
371C2—6922no—
372C2#2601213311yescharacteristic
373C2#137×D17311yescharacteristic
374C2#2601229211yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#480×C173C2
socleC22×C173C22×D4
prime radical#26012138C2
hypercenter#480D173
hypocenterC173#3314
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #26012196C2semidirect
3. #26011849C2semidirect
4. #26011828C2semidirect
5. #26011770C2semidirect
6. #26011735C2semidirect
7. #26011733C2semidirect
8. #26011712C2semidirect
9. #26011655C2semidirect
10. #26011643C2semidirect
11. #26011590C2semidirect
12. #26011565C2semidirect
13. #26011490C2semidirect
14. #102×D173C2semidirect
15. #480×C173C2semidirect
18. C2×Dic692C22semidirect
21. C2×#14583810C22semidirect
23. #19149678C22semidirect
30. #19149658C22semidirect
34. #19149644C22semidirect
40. #19149612C22semidirect
41. #19149608C22semidirect
42. #19149604C22semidirect
43. #19149602C22semidirect
44. #19149602C22semidirect
45. #19149602C22semidirect
51. #19149589C22semidirect
53. C2×C4⋊C4×C173C22semidirect
54. #107×C173C22semidirect
55. #107×C173C22semidirect
58. C22⋊C8×C173C22semidirect
187. C173#3314semidirect
193. #480D173semidirect
228. C2×C4⋊C4C2×D173semidirect
231. #107C2×D173semidirect
232. #107C2×D173semidirect
245. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121962
#260121382
#260118492
#260118282
#260117702
#260117352
#260117332
#260117122
#260116552
#260116432
#260115902
#260115652
#260114902
#102×D1732
#480×C1732
#3314173