Finite Group Databasebeta

← #49170635

Subgroups — #49170635

Order 37504.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036825C211yesmaximal, characteristic, complemented
2#34036483C211yesmaximal, characteristic, complemented
3#34036462C211yesmaximal, characteristic, complemented
4#34036450C211yesmaximal, characteristic, complemented
5#34036429C211yesmaximal, characteristic, complemented
6#34036363C211yesmaximal, characteristic, complemented
7#34036360C211yesmaximal, characteristic, complemented
8#34036341C211yesmaximal, characteristic, complemented
9#34036274C211yesmaximal, characteristic, complemented
10#34036263C211yesmaximal, characteristic, complemented
11C4×#17989787C211yesmaximal, characteristic, complemented
12D4×#14556873C211yesmaximal, characteristic, complemented
13#34036125C211yesmaximal, characteristic, complemented
14C2×#22627545C211yesmaximal, characteristic, complemented
15#546×C293C211yesmaximal, characteristic, complemented
16C2×#17989789C2211yescharacteristic, complemented
17C2×#17989787C2211yescharacteristic
18C2×#17989787C2211yescharacteristic
19C22×#14556873C2211yescharacteristic
20C22×#14556873C2211yescharacteristic, complemented
21#22627619—21no—
22#22627616C2211yescharacteristic, complemented
23#22627606—21no—
24#22627598C2211yescharacteristic
25#22627598C2211yescharacteristic, complemented
26#22627596—21no—
27C4×D1172C2211yescharacteristic, complemented
28#22627584—21no—
29#22627583C2211yescharacteristic, complemented
30C2×C4⋊Dic293C2211yescharacteristic
31#22627557C2211yescharacteristic, complemented
32#22627557C2211yescharacteristic, complemented
33#22627555C2211yescharacteristic, complemented
34C4⋊#14556873C2211yescharacteristic, complemented
35#22627548C2211yescharacteristic, complemented
36#22627548C2211yescharacteristic, complemented
37#22627548C2211yescharacteristic, complemented
38#22627545C2211yescharacteristic
39#22627545C2211yescharacteristic
40#22627545C2211yescharacteristic
41#22627545C2211yescharacteristic
42#22627545C2211yescharacteristic
43#22627545C2211yescharacteristic
44#22627544C2211yescharacteristic, complemented
45C22⋊#14556873C2211yescharacteristic
46C22⋊#14556873C2211yescharacteristic, complemented
47C4×#14556873C2211yescharacteristic, complemented
48C2×#41×C293C2211yescharacteristic
49C22×D4×C293C2211yescharacteristic
50#124×C293C2211yescharacteristic
51#123×C293—21no—
52C22⋊Q8×C293C2211yescharacteristic
53#121×C293—21no—
54#121×C293C2211yescharacteristic
55#120×C293—21no—
56C4×D4×C293C2211yescharacteristic
57C4×D4×C293C2211yescharacteristic
58C2×C22⋊C4×C293—21no—
59C2×D1172C2311yescharacteristic
60C2×C22⋊D293—21no—
61C2×C22⋊D293—21no—
62C22×Dic293—21no—
63C2×C4×D293—21no—
64#17989789—41no—
65#17989787—22no—
66#17989787—41no—
67C2×#14556873—41no—
68C2×#14556873—41no—
69C2×#14556873C2311yescharacteristic
70C2×#14556873C2311yescharacteristic
71C2×#14556873C2311yescharacteristic
72C2×#14556873C2311yescharacteristic
73C4⋊Dic293—21no—
74C4⋊Dic293C2311yescharacteristic
75C4⋊Dic293C2311yescharacteristic
76C4⋊Dic293C2311yescharacteristic
77#17989781—21no—
78#17989781—21no—
79C22⋊Dic293—21no—
80#17989779—21no—
81#17989779—21no—
82#17989779—21no—
83C4×Dic293—21no—
84C24×C293—21no—
85#41×C293—22no—
86#41×C293—41no—
87C2×Q8×C293C2311yescharacteristic
88C2×D4×C293—22no—
89C2×D4×C293—21no—
90C2×D4×C293—21no—
91C2×D4×C293—21no—
92C2×D4×C293—41no—
93C2×D4×C293D411yescharacteristic
94C2×D4×C293C2311yescharacteristic
95C2×D4×C293C2311yescharacteristic
96C22×C4×C293—21no—
97C22×C4×C293—21no—
98C22×C4×C293C2311yescharacteristic
99C22×C4×C293C2311yescharacteristic
100C4⋊C4×C293—21no—
101C4⋊C4×C293D411yescharacteristic
102C4⋊C4×C293C2311yescharacteristic
103C22⋊C4×C293—21no—
104C22⋊C4×C293—21no—
105C22⋊C4×C293—21no—
106C22⋊C4×C293—21no—
107C22⋊C4×C293—21no—
108C22⋊C4×C293D411yescharacteristic
109C22⋊C4×C293D411yescharacteristic
110C42×C293C2311yescharacteristic
111C22×D293—21no—
112D1172—21no—
113D1172—21no—
114C22⋊D293—42no—
115C22⋊D293—81no—
116C2×Dic293—21no—
117C2×Dic293—21no—
118C2×Dic293—21no—
119C2×Dic293—81no—
120C4×D293—81no—
121#14556873—21no—
122#14556873—21no—
123#14556873—21no—
124#14556873—21no—
125C23×C293—21no—
126C23×C293—21no—
127C23×C293—42no—
128C23×C293—42no—
129C23×C293—81no—
130C23×C293C2×D411yescharacteristic
131C23×C293C2×D411yescharacteristic
132Q8×C293—21no—
133Q8×C293—21no—
134D4×C293—22no—
135D4×C293—21no—
136D4×C293—21no—
137D4×C293—21no—
138D4×C293—42no—
139D4×C293—42no—
140D4×C293—81no—
141D4×C293#4112yes—
142C2×C4×C293—21no—
143C2×C4×C293—21no—
144C2×C4×C293—21no—
145C2×C4×C293—21no—
146C2×C4×C293—21no—
147C2×C4×C293—21no—
148C2×C4×C293—42no—
149C2×C4×C293—81no—
150C2×C4×C293C2×D411yescharacteristic
151C2×C4×C293C2×D411yescharacteristic
152C2×C4×C293C2×D411yescharacteristic
153C2×C4×C293C2×D411yescharacteristic
154C2×C4×C293C2411yescharacteristic
155C22×C293—21no—
156C22×C293—21no—
157C22×C293—21no—
158C22×C293—22no—
159C22×C293—21no—
160C22×C293—21no—
161C22×C293—41no—
162C22×C293—41no—
163C22×C293—41no—
164C22×C293—41no—
165C22×C293—41no—
166C22×C293—84no—
167C22×C293#13512yes—
168C22×C293C22×D411yescharacteristic
169C2×D293—41no—
170C2×D293—41no—
171C2×D293—41no—
172C4×C293—21no—
173C4×C293—21no—
174C4×C293—21no—
175C4×C293—21no—
176C4×C293—41no—
177C4×C293—41no—
178C4×C293C2×#4111yescharacteristic
179C4×C293H(5,𝔽2)11yescharacteristic
180Dic293—41no—
181Dic293—41no—
182Dic293—41no—
183C2×C293—22no—
184C2×C293—41no—
185C2×C293—42no—
186C2×C293—81no—
187C2×C293#54611yescharacteristic
188C2×C293C2×#13511yescharacteristic
189C2×C293#57611yescharacteristic
190D293—81no—
191C293#329711yescharacteristic, complemented, Sylow
192#3297—2931nomaximal, Sylow
193#553—2931no—
194#546D29311yescharacteristic, complemented
195#488—2931no—
196#484—2931no—
197#482—2931no—
198#478—2931no—
199#466—2931no—
200#465—2931no—
201#463—2931no—
202#449—2931no—
203#447—2931no—
204C4×D8—2931no—
205C8×D4—2931no—
206#416—2931no—
207C2×#102—2931no—
208C2×#41C2×D29311yescharacteristic
209C22×D4C2×D29311yescharacteristic
210C2×SD16—2931no—
211C2×D8—2931no—
212C2×D8—2931no—
213C22×C8—2931no—
214C22×C8—2931no—
215#126—5861no—
216#125—2931no—
217#124C2×D29311yescharacteristic
218#123—21no—
219#123—5861no—
220C22⋊Q8C2×D29311yescharacteristic
221#121—21no—
222#121—2931no—
223#121—2931no—
224#121—5861no—
225#121C2×D29311yescharacteristic
226#120—21no—
227C4×D4—2931no—
228C4×D4—5861no—
229C4×D4C2×D29311yescharacteristic
230C4×D4C2×D29311yescharacteristic
231#117—2931no—
232C2×C4⋊C4—2931no—
233C2×C22⋊C4—21no—
234#107—2931no—
235#107—2931no—
236#106—2931no—
237C4⋊C8—2931no—
238#103—2931no—
239#103—2931no—
240#103—2931no—
241#102—2931no—
242#102—2931no—
243#102—2931no—
244#102—2931no—
245#102—2931no—
246#102—2931no—
247#102—2931no—
248C22⋊C8—2931no—
249C22⋊C8—2931no—
250C4×C8—2931no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D29311yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—2931no—
261C2×D4—5861no—
262C2×D4—5861no—
263C2×D4C22⋊D29311yescharacteristic
264C2×D4C22×D29311yescharacteristic
265C2×D4C22×D29311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—5861no—
269C22×C4—5861no—
270C22×C4C22×D29311yescharacteristic
271C22×C4C22×D29311yescharacteristic
272SD16—11721no—
273D8—5862no—
274D8—11721no—
275C2×C8—2931no—
276C2×C8—2931no—
277C2×C8—2931no—
278C2×C8—2931no—
279C2×C8—11721no—
280C2×C8—11721no—
281C4⋊C4—21no—
282C4⋊C4—2931no—
283C4⋊C4—2931no—
284C4⋊C4—2931no—
285C4⋊C4—5861no—
286C4⋊C4—5861no—
287C4⋊C4—5861no—
288C4⋊C4C22⋊D29311yescharacteristic
289C4⋊C4C22×D29311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—5861no—
296C22⋊C4—5861no—
297C22⋊C4—5861no—
298C22⋊C4—5861no—
299C22⋊C4C22⋊D29311yescharacteristic
300C22⋊C4C22⋊D29311yescharacteristic
301C42—5861no—
302C42C22×D29311yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—5861no—
309C23C2×C22⋊D29311yescharacteristic
310C23C2×C22⋊D29311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—23441no—
321D4—5861no—
322D4—5861no—
323D4—11722no—
324D4#1798980112yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—23441no—
334C2×C4—23441no—
335C2×C4—5861no—
336C2×C4—5861no—
337C2×C4—5861no—
338C2×C4C2×C22⋊D29311yescharacteristic
339C2×C4C2×C22⋊D29311yescharacteristic
340C2×C4C2×C22⋊D29311yescharacteristic
341C2×C4C2×C22⋊D29311yescharacteristic
342C2×C4C23×D29311yescharacteristic
343C8—5861no—
344C8—5861no—
345C8—5861no—
346C8—5861no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—11721no—
360C22—11721no—
361C22—11721no—
362C22#2262764412yes—
363C22C22×C22⋊D29311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—11721no—
371C4—11721no—
372C4—11721no—
373C4C2×#1798980111yescharacteristic
374C4#2262766911yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—23441no—
380C2#3403677311yescharacteristic
381C2C2×#2262764411yescharacteristic
382C2#3403693211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#546×C293C2
socleC22×C293C22×D4
hypercenter#546D293
hypocenterC293#3297
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #34036825C2semidirect
2. #34036483C2semidirect
3. #34036462C2semidirect
4. #34036450C2semidirect
5. #34036429C2semidirect
6. #34036363C2semidirect
7. #34036360C2semidirect
8. #34036341C2semidirect
9. #34036274C2semidirect
10. #34036263C2semidirect
11. C4×#17989787C2semidirect
12. D4×#14556873C2semidirect
13. #34036125C2semidirect
14. C2×#22627545C2semidirect
15. #546×C293C2semidirect
16. C2×#17989789C22semidirect
20. C22×#14556873C22semidirect
22. #22627616C22semidirect
25. #22627598C22semidirect
27. C4×D1172C22semidirect
29. #22627583C22semidirect
31. #22627557C22semidirect
32. #22627557C22semidirect
33. #22627555C22semidirect
34. C4⋊#14556873C22semidirect
35. #22627548C22semidirect
36. #22627548C22semidirect
37. #22627548C22semidirect
44. #22627544C22semidirect
46. C22⋊#14556873C22semidirect
47. C4×#14556873C22semidirect
191. C293#3297semidirect
194. #546D293semidirect

Maximal subgroups

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

subgroupindex
#340368252
#340364832
#340364622
#340364502
#340364292
#340363632
#340363602
#340363412
#340362742
#340362632
C4×#179897872
D4×#145568732
#340361252
C2×#226275452
#546×C2932
#3297293