Finite Group Databasebeta

← #36365077

Subgroups — #36365077

Order 22144.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149708C211yesmaximal, characteristic, complemented
2C22×#14583814C211yesmaximal, characteristic, complemented
3#26012154C211yesmaximal, characteristic, complemented
4#26012135C211yesmaximal, characteristic
5#26011882C211yesmaximal, characteristic, complemented
6#26011736C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011712C211yesmaximal, characteristic, complemented
9#26011694C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011649C211yesmaximal, characteristic, complemented
12#26011628C211yesmaximal, characteristic, complemented
13#26011588C211yesmaximal, characteristic, complemented
14#26011559C211yesmaximal, characteristic, complemented
15#450×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic, complemented
19#19149708—24no—
20C2×Dic692C2211yescharacteristic, complemented
21C2×Dic692C2211yescharacteristic, complemented
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583814—24no—
24C2×#14583814C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583814C2211yescharacteristic
27C2×#14583814C2211yescharacteristic
28C2×#14583809C2211yescharacteristic, complemented
29C22×#13063436C2211yescharacteristic
30#19149678—21no—
31#19149677C2211yescharacteristic, complemented
32#19149666C2211yescharacteristic
33#19149660—21no—
34#19149659C2211yescharacteristic, complemented
35#19149658C2211yescharacteristic
36#19149657—21no—
37#19149656—21no—
38#19149651—21no—
39#19149650C2211yescharacteristic
40#19149648—21no—
41#19149644C2211yescharacteristic, complemented
42D4×Dic173C2211yescharacteristic
43C4×C22⋊D173C2211yescharacteristic
44#19149639—21no—
45C2×C22⋊Dic173—21no—
46#19149612C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149604C2211yescharacteristic, complemented
50#19149603C2211yescharacteristic, complemented
51#19149602C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic, complemented
53#19149589C2211yescharacteristic, complemented
54C22×D4×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C2×SD16×C173C2211yescharacteristic
57C22⋊Q8×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59#102×C173C2211yescharacteristic
60C22⋊C8×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—41no—
63C2×Dic346—21no—
64C2×Dic346—22no—
65C2×Dic346—21no—
66C2×Dic346—41no—
67C2×Dic346C2311yescharacteristic
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70Q8×D173—22no—
71Q8×D173—41no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173D411yescharacteristic
75C22×Dic173—21no—
76C22×Dic173—21no—
77C22×Dic173—21no—
78C2×C4×D173—21no—
79C2×C4×D173C2311yescharacteristic
80Dic692—22no—
81Dic692—41no—
82#14583816—22no—
83#14583814—22no—
84#14583814—22no—
85#14583814—22no—
86#14583814D414yescomplemented
87#14583809—22no—
88C2×#13063436—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C4⋊Dic173—21no—
92C4⋊Dic173C2311yescharacteristic
93#14583805—21no—
94#14583805—21no—
95#14583805—21no—
96#14583805—21no—
97#14583805D411yescharacteristic
98C22⋊Dic173—21no—
99C22⋊Dic173—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173D411yescharacteristic
104#14583803—21no—
105#14583803D411yescharacteristic
106C4×Dic173—21no—
107C4×Dic173C2311yescharacteristic
108C24×C173—21no—
109C2×Q8×C173C2311yescharacteristic
110C2×D4×C173—22no—
111C2×D4×C173—21no—
112C2×D4×C173—41no—
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173C2311yescharacteristic
116SD16×C173—22no—
117SD16×C173—41no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173—21no—
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—21no—
129Dic346—42no—
130Dic346—42no—
131Dic346C2×D412yes—
132C22⋊D173—22no—
133C22⋊D173—42no—
134C22⋊D173—81no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—42no—
140C2×Dic173—42no—
141C2×Dic173—81no—
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C4×D173—21no—
146C4×D173—21no—
147C4×D173—42no—
148#13063436—21no—
149#13063436C2×D412yes—
150C23×C173—21no—
151C23×C173—21no—
152C23×C173—42no—
153C23×C173—42no—
154C23×C173—81no—
155C23×C173C2×D411yescharacteristic
156Q8×C173—21no—
157Q8×C173—21no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—42no—
162D4×C173C2×D412yes—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2411yescharacteristic
169C8×C173—21no—
170C8×C173—21no—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173—22no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—84no—
180C22×C173#13712yescomplemented
181C22×C173C22×D411yescharacteristic
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—21no—
185C4×C173—21no—
186C4×C173—41no—
187C4×C173—41no—
188C4×C173C22×D411yescharacteristic
189C4×C173H(5,𝔽2)11yescharacteristic
190Dic173—21no—
191Dic173—22no—
192Dic173—21no—
193Dic173—21no—
194Dic173—41no—
195Dic173—41no—
196Dic173—41no—
197C2×C173—22no—
198C2×C173—42no—
199C2×C173—81no—
200C2×C173D4211yescharacteristic
201C2×C173C2×#13711yescharacteristic
202C2×C173#57711yescharacteristic
203D173—41no—
204C173#326611yescharacteristic, complemented, Sylow
205#3266—1731nomaximal, Sylow
206C2×#137—1731no—
207C22×SD16—1731no—
208D4×Q8—1731no—
209#546—1731no—
210#497—1731no—
211#468—1731no—
212#467—1731no—
213#464—1731no—
214#461—1731no—
215#452—1731no—
216#451—1731no—
217#450D17311yescharacteristic, complemented
218#448—1731no—
219#440—1731no—
220#435—1731no—
221C2×#41—1731no—
222C22×Q8—1731no—
223C22×Q8—1731no—
224C22×D4C2×D17311yescharacteristic
225#137—3464no—
226C2×Q16—1731no—
227C2×Q16—1731no—
228C2×Q16—1731no—
229C2×SD16—3464no—
230C2×SD16—1731no—
231C2×SD16—1731no—
232C2×SD16—1731no—
233C2×SD16—1731no—
234C2×SD16C2×D17311yescharacteristic
235C2×SD16C2×D17311yescharacteristic
236C2×M16—1731no—
237C22×C8—1731no—
238C4⋊Q8—3461no—
239C4⋊Q8—1731no—
240#124—1731no—
241#123—3461no—
242C22⋊Q8—3461no—
243C22⋊Q8—3461no—
244C22⋊Q8—1731no—
245C22⋊Q8—1731no—
246C22⋊Q8C2×D17311yescharacteristic
247#121—3461no—
248#121—1731no—
249#120—3461no—
250C4×Q8—1731no—
251C4×D4—3461no—
252C4×D4—1731no—
253C4×D4—1731no—
254C2×C22⋊C4—3461no—
255#107—1731no—
256C4⋊C8—1731no—
257#103—1731no—
258#103—1731no—
259#103—1731no—
260#103—1731no—
261#102C2×D17311yescharacteristic
262#102C2×D17311yescharacteristic
263C22⋊C8—1731no—
264C22⋊C8C2×D17311yescharacteristic
265#97—1731no—
266C24—21no—
267#41—3462no—
268#41—6921no—
269C2×Q8—3461no—
270C2×Q8—3462no—
271C2×Q8—3461no—
272C2×Q8—3462no—
273C2×Q8—1731no—
274C2×Q8—1731no—
275C2×Q8—1731no—
276C2×Q8—6921no—
277C2×Q8—6921no—
278C2×Q8C22×D17311yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—3461no—
283C2×D4—3461no—
284C2×D4—1731no—
285C2×D4C22×D17311yescharacteristic
286C2×D4C22×D17311yescharacteristic
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—1731no—
292C22×C4C22×D17311yescharacteristic
293Q16—3462no—
294Q16—3462no—
295Q16—6921no—
296SD16—22no—
297SD16—41no—
298SD16—3462no—
299SD16—3462no—
300SD16—3462no—
301SD16—1734no—
302M16—3462no—
303C2×C8—3462no—
304C2×C8—1731no—
305C2×C8—1731no—
306C2×C8C22×D17311yescharacteristic
307C2×C8C22×D17311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3461no—
310C4⋊C4—3461no—
311C4⋊C4—3461no—
312C4⋊C4—3461no—
313C4⋊C4—3461no—
314C4⋊C4—1731no—
315C4⋊C4—1731no—
316C4⋊C4C22×D17311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—1731no—
325C22⋊C4—1731no—
326C42—3461no—
327C42—1731no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1731no—
334C23D4×D17311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—3461no—
338Q8—3461no—
339Q8—3461no—
340Q8—3461no—
341Q8—3461no—
342Q8—1732no—
343Q8—6922no—
344Q8—6922no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—3462no—
350D4—13841no—
351D4—6922no—
352D4D4×D17312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—13841no—
364C2×C4—1731no—
365C2×C4—1731no—
366C2×C4—1731no—
367C2×C4—6922no—
368C2×C4—6922no—
369C2×C4—6922no—
370C2×C4D4×D17311yescharacteristic
371C2×C4C23×D17311yescharacteristic
372C8—21no—
373C8—21no—
374C8—3461no—
375C8—1732no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—3461no—
386C22—3461no—
387C22—3461no—
388C22#1914970812yescomplemented
389C22C2×D4×D17311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—3461no—
394C4—3461no—
395C4—3461no—
396C4—3462no—
397C4—6921no—
398C4—6921no—
399C4—6921no—
400C4C2×D4×D17311yescharacteristic
401C4#1914972411yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—6921no—
406C2#2601212311yescharacteristic
407C2C2×#1914970811yescharacteristic
408C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radical#26012135C2
hypercenter#450D173
hypocenterC173#3266
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
2. C22×#14583814C2semidirect
3. #26012154C2semidirect
5. #26011882C2semidirect
6. #26011736C2semidirect
7. #26011730C2semidirect
8. #26011712C2semidirect
9. #26011694C2semidirect
10. #26011654C2semidirect
11. #26011649C2semidirect
12. #26011628C2semidirect
13. #26011588C2semidirect
14. #26011559C2semidirect
15. #450×C173C2semidirect
18. C2×Q8×D173C22semidirect
20. C2×Dic692C22semidirect
21. C2×Dic692C22semidirect
22. C2×#14583816C22semidirect
28. C2×#14583809C22semidirect
31. #19149677C22semidirect
34. #19149659C22semidirect
41. #19149644C22semidirect
46. #19149612C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
49. #19149604C22semidirect
50. #19149603C22semidirect
51. #19149602C22semidirect
52. #19149591C22semidirect
53. #19149589C22semidirect
86. #14583814D4semidirect
180. C22×C173#137semidirect
204. C173#3266semidirect
217. #450D173semidirect
388. C22#19149708semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
C22×#145838142
#260121542
#260121352
#260118822
#260117362
#260117302
#260117122
#260116942
#260116542
#260116492
#260116282
#260115882
#260115592
#450×C1732
#3266173