Finite Group Databasebeta

← #36365079

Subgroups — #36365079

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
2C2×SD16×D173C211yesmaximal, characteristic, complemented
3#26012156C211yesmaximal, characteristic, complemented
4#26012136C211yesmaximal, characteristic
5#26011883C211yesmaximal, characteristic, complemented
6#26011740C211yesmaximal, characteristic, complemented
7#26011731C211yesmaximal, characteristic, complemented
8#26011714C211yesmaximal, characteristic, complemented
9#26011650C211yesmaximal, characteristic, complemented
10#26011648C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011631C211yesmaximal, characteristic, complemented
13#26011591C211yesmaximal, characteristic, complemented
14#26011561C211yesmaximal, characteristic, complemented
15#461×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic, complemented
19C2×D4×D173C2211yescharacteristic
20#19149708—24no—
21C2×Dic692C2211yescharacteristic, complemented
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583816C2211yescharacteristic, complemented
24C2×#14583815C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583813C2211yescharacteristic
27C2×#14583813C2211yescharacteristic
28SD16×D173—24no—
29C2×#14583809C2211yescharacteristic, complemented
30C2×C8×D173C2211yescharacteristic
31#19149678—21no—
32#19149667C2211yescharacteristic
33#19149663—21no—
34#19149659C2211yescharacteristic
35#19149657—21no—
36#19149657—21no—
37#19149655C2211yescharacteristic
38#19149655C2211yescharacteristic, complemented
39#19149654C2211yescharacteristic
40#19149653—21no—
41#19149649—21no—
42C4×Dic346C2211yescharacteristic, complemented
43C4×D692C2211yescharacteristic
44#19149641—21no—
45C2×#14583803—21no—
46#19149612C2211yescharacteristic, complemented
47#19149604C2211yescharacteristic, complemented
48#19149602C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic
50#19149600C2211yescharacteristic
51#19149591C2211yescharacteristic
52#19149591C2211yescharacteristic, complemented
53#19149590C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic
55C2×SD16×C173C2211yescharacteristic
56C4⋊Q8×C173C2211yescharacteristic, complemented
57C4×D4×C173C2211yescharacteristic
58C4⋊C8×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60#103×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583825—22no—
63#14583825—41no—
64C2×Dic346—21no—
65C2×Dic346—21no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68Q8×D173—22no—
69Q8×D173—22no—
70Q8×D173—41no—
71Q8×D173—41no—
72C2×D692—21no—
73C2×D692D411yescharacteristic
74C2×D692C2311yescharacteristic
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77D4×D173—22no—
78D4×D173—41no—
79C22×Dic173—21no—
80C2×C4×D173—21no—
81C2×C4×D173—21no—
82C2×C4×D173C2311yescharacteristic
83C2×C4×D173C2311yescharacteristic
84Dic692—22no—
85#14583816—22no—
86#14583816—41no—
87#14583815—22no—
88#14583814D414yescomplemented
89#14583813—22no—
90#14583813—41no—
91#14583809—22no—
92C2×#13063436C2311yescharacteristic
93C2×#13063436C2311yescharacteristic
94C8×D173—22no—
95C4⋊Dic173—21no—
96C4⋊Dic173D411yescharacteristic
97C4⋊Dic173C2311yescharacteristic
98#14583805—21no—
99#14583805—21no—
100#14583805—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803D411yescharacteristic
108#14583803D411yescharacteristic
109C4×Dic173—21no—
110C2×Q8×C173C2311yescharacteristic
111C2×Q8×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173—21no—
114SD16×C173—22no—
115SD16×C173—22no—
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C4⋊C4×C173—21no—
119C4⋊C4×C173—21no—
120C4⋊C4×C173C2311yescharacteristic
121C22⋊C4×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—42no—
125C22×D173—42no—
126C22×D173—81no—
127C22×D173C2×D411yescharacteristic
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346—21no—
131Dic346—21no—
132Dic346—42no—
133Dic346—42no—
134Dic346C2×D412yes—
135D692—22no—
136D692—21no—
137D692—21no—
138D692—81no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—42no—
145C2×Dic173C2×D411yescharacteristic
146C2×Dic173C2×D411yescharacteristic
147C4×D173—21no—
148C4×D173—21no—
149C4×D173—21no—
150C4×D173—21no—
151C4×D173—42no—
152C4×D173—42no—
153#13063436—21no—
154#13063436C2×D412yes—
155C23×C173—21no—
156Q8×C173—21no—
157Q8×C173—21no—
158Q8×C173—21no—
159Q8×C173—21no—
160D4×C173—21no—
161D4×C173C2×D412yes—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—81no—
166C2×C4×C173C2×D411yescharacteristic
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2411yescharacteristic
169C8×C173—21no—
170C8×C173—21no—
171C22×C173—22no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173C22×D411yescharacteristic
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—21no—
179C2×D173—21no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—84no—
184C2×D173#13712yescomplemented
185C4×C173—21no—
186C4×C173—21no—
187C4×C173—41no—
188C4×C173—41no—
189C4×C173—41no—
190C4×C173C22×D411yescharacteristic
191C4×C173H(5,𝔽2)11yescharacteristic
192Dic173—21no—
193Dic173—21no—
194Dic173—22no—
195Dic173—41no—
196Dic173—41no—
197C2×C173—42no—
198C2×C173D4211yescharacteristic
199C2×C173C2×#13711yescharacteristic
200C2×C173#57711yescharacteristic
201D173—22no—
202D173—41no—
203D173—81no—
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#461D17311yescharacteristic, complemented
215#452—1731no—
216#451—1731no—
217#450—1731no—
218#448—1731no—
219#440—1731no—
220#435—1731no—
221C2×#41—1731no—
222C22×Q8—1731no—
223C22×Q8—1731no—
224C22×D4—1731no—
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⋊Q8C2×D17311yescharacteristic, complemented
240#124—1731no—
241#123—3461no—
242C22⋊Q8—3461no—
243C22⋊Q8—3461no—
244C22⋊Q8—1731no—
245C22⋊Q8—1731no—
246C22⋊Q8—1731no—
247#121—3461no—
248#121—1731no—
249#120—3461no—
250C4×Q8—1731no—
251C4×D4—3461no—
252C4×D4—1731no—
253C4×D4C2×D17311yescharacteristic
254C2×C22⋊C4—3461no—
255#107—1731no—
256C4⋊C8C2×D17311yescharacteristic, complemented
257#103—1731no—
258#103—1731no—
259#103C2×D17311yescharacteristic, complemented
260#103C2×D17311yescharacteristic, complemented
261#102—1731no—
262#102—1731no—
263C22⋊C8—1731no—
264C22⋊C8—1731no—
265#97—1731no—
266C24—3461no—
267#41—3462no—
268#41—6921no—
269C2×Q8—3461no—
270C2×Q8—3461no—
271C2×Q8—3462no—
272C2×Q8—3462no—
273C2×Q8—1731no—
274C2×Q8—1731no—
275C2×Q8—6921no—
276C2×Q8—6921no—
277C2×Q8C22×D17311yescharacteristic
278C2×Q8C22×D17311yescharacteristic
279C2×D4—3461no—
280C2×D4—3462no—
281C2×D4—3461no—
282C2×D4—3461no—
283C2×D4—1731no—
284C2×D4—1731no—
285C2×D4—6921no—
286C2×D4C22×D17311yescharacteristic
287C22×C4—21no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—1731no—
292C22×C4—1731no—
293Q16—3462no—
294Q16—3462no—
295Q16—6921no—
296SD16—22no—
297SD16—22no—
298SD16—3462no—
299SD16—3462no—
300SD16—1734no—
301SD16—6921no—
302M16—3462no—
303C2×C8—3462no—
304C2×C8—1731no—
305C2×C8—1731no—
306C2×C8C22×D17311yescharacteristic
307C2×C8C22×D17311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
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—
327C42C22×D17311yescharacteristic
328C23—21no—
329C23—3461no—
330C23—13841no—
331C23—1731no—
332C23—1731no—
333C23—6922no—
334C23—6922no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—3461no—
340Q8—3461no—
341Q8—3461no—
342Q8—1732no—
343Q8—6922no—
344Q8—6922no—
345D4—21no—
346D4—3462no—
347D4—3461no—
348D4—3461no—
349D4—13841no—
350D4—6922no—
351D4—6922no—
352D4D4×D17312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—1731no—
365C2×C4—1731no—
366C2×C4—6922no—
367C2×C4—6922no—
368C2×C4—6922no—
369C2×C4D4×D17311yescharacteristic
370C2×C4D4×D17311yescharacteristic
371C2×C4C23×D17311yescharacteristic
372C8—21no—
373C8—21no—
374C8—3461no—
375C8—1732no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—3461no—
380C22—3461no—
381C22—3461no—
382C22—3461no—
383C22—3461no—
384C22—13844no—
385C22—1732no—
386C22—6921no—
387C22—6921no—
388C22—6921no—
389C22C2×D4×D17311yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3461no—
396C4—3461no—
397C4—3462no—
398C4—6921no—
399C4—6921no—
400C4C2×D4×D17311yescharacteristic
401C4#1914972511yescharacteristic
402C2—42no—
403C2—3462no—
404C2—13841no—
405C2—6921no—
406C2#2601212511yescharacteristic
407C2#137×D17311yescharacteristic
408C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#461×C173C2
socleC22×C173C22×D4
prime radical#26012136C2
hypercenter#461D173
hypocenterC173#3266
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
2. C2×SD16×D173C2semidirect
3. #26012156C2semidirect
5. #26011883C2semidirect
6. #26011740C2semidirect
7. #26011731C2semidirect
8. #26011714C2semidirect
9. #26011650C2semidirect
10. #26011648C2semidirect
11. #26011640C2semidirect
12. #26011631C2semidirect
13. #26011591C2semidirect
14. #26011561C2semidirect
15. #461×C173C2semidirect
18. C2×Q8×D173C22semidirect
21. C2×Dic692C22semidirect
22. C2×#14583816C22semidirect
23. C2×#14583816C22semidirect
29. C2×#14583809C22semidirect
38. #19149655C22semidirect
42. C4×Dic346C22semidirect
46. #19149612C22semidirect
47. #19149604C22semidirect
48. #19149602C22semidirect
52. #19149591C22semidirect
53. #19149590C22semidirect
56. C4⋊Q8×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #103×C173C22semidirect
60. #103×C173C22semidirect
88. #14583814D4semidirect
184. C2×D173#137semidirect
204. C173#3266semidirect
214. #461D173semidirect
239. C4⋊Q8C2×D173semidirect
256. C4⋊C8C2×D173semidirect
259. #103C2×D173semidirect
260. #103C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
C2×SD16×D1732
#260121562
#260121362
#260118832
#260117402
#260117312
#260117142
#260116502
#260116482
#260116402
#260116312
#260115912
#260115612
#461×C1732
#3266173