Finite Group Databasebeta

← #49168927

Subgroups — #49168927

Order 37504.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036666C211yesmaximal, characteristic, complemented
2C4×D4×D293C211yesmaximal, characteristic, complemented
3#34036458C211yesmaximal, characteristic, complemented
4#34036444C211yesmaximal, characteristic, complemented
5#34036432C211yesmaximal, characteristic, complemented
6#34036431C211yesmaximal, characteristic, complemented
7#34036347C211yesmaximal, characteristic, complemented
8#34036346C211yesmaximal, characteristic, complemented
9#34036278C211yesmaximal, characteristic, complemented
10#34036264C211yesmaximal, characteristic
11#34036242C211yesmaximal, characteristic, complemented
12#34036239C211yesmaximal, characteristic, complemented
13#34036227C211yesmaximal, characteristic, complemented
14#34036184C211yesmaximal, characteristic, complemented
15#440×C293C211yesmaximal, characteristic, complemented
16C2×D4×D293C2211yescharacteristic
17C22×C4×D293—21no—
18C2×#17989789C2211yescharacteristic
19C2×D2344C2211yescharacteristic
20C2×#17989787C2211yescharacteristic
21#22627619—21no—
22#22627615C2211yescharacteristic, complemented
23#22627613C2211yescharacteristic, complemented
24#22627609—21no—
25#22627600C2211yescharacteristic
26#22627599C2211yescharacteristic
27C4×D1172—21no—
28C4×D1172C2211yescharacteristic
29C4×D1172C2211yescharacteristic, complemented
30D4×Dic293C2211yescharacteristic
31#22627586C2211yescharacteristic
32C4×C22⋊D293—21no—
33#22627584—21no—
34#22627581—21no—
35C4⋊C4×D293C2211yescharacteristic
36C22⋊C4×D293—21no—
37C42×D293C2211yescharacteristic
38#22627557C2211yescharacteristic, complemented
39#22627556C2211yescharacteristic, complemented
40#22627555C2211yescharacteristic, complemented
41#22627553C2211yescharacteristic, complemented
42#22627553C2211yescharacteristic, complemented
43#22627548C2211yescharacteristic, complemented
44#22627546C2211yescharacteristic
45#22627546C2211yescharacteristic, complemented
46#22627545C2211yescharacteristic
47#22627544C2211yescharacteristic
48#22627544C2211yescharacteristic, complemented
49#22627536C2211yescharacteristic
50#22627536C2211yescharacteristic
51#22627535C2211yescharacteristic, complemented
52C2×SD16×C293C2211yescharacteristic
53C4×Q8×C293C2211yescharacteristic, complemented
54C4×D4×C293C2211yescharacteristic
55#107×C293C2211yescharacteristic, complemented
56#103×C293C2211yescharacteristic, complemented
57#102×C293C2211yescharacteristic
58#97×C293C2211yescharacteristic, complemented
59C23×D293—21no—
60C2×D1172—21no—
61C2×D1172C2311yescharacteristic
62C2×D1172C2311yescharacteristic
63C2×C22⋊D293—21no—
64D4×D293—22no—
65D4×D293—22no—
66C22×Dic293—21no—
67C2×C4×D293—21no—
68C2×C4×D293—21no—
69C2×C4×D293—21no—
70C2×C4×D293—21no—
71C2×C4×D293—44no—
72C2×C4×D293D411yescharacteristic
73C2×C4×D293D411yescharacteristic
74C2×C4×D293C2311yescharacteristic
75#17989789—22no—
76D2344—22no—
77#17989787—22no—
78C2×#14556873C2311yescharacteristic
79C2×#14556873C2311yescharacteristic
80C4⋊Dic293—21no—
81C4⋊Dic293C2311yescharacteristic
82C4⋊Dic293C2311yescharacteristic
83#17989781—21no—
84#17989781—21no—
85#17989781—21no—
86C22⋊Dic293—21no—
87#17989779—21no—
88#17989779—21no—
89#17989779—21no—
90#17989779—21no—
91#17989779—21no—
92#17989779—21no—
93#17989779—21no—
94C4×Dic293D411yescharacteristic
95C4×Dic293D411yescharacteristic
96C4×Dic293C2311yescharacteristic
97C2×Q8×C293C2311yescharacteristic
98C2×D4×C293C2311yescharacteristic
99C22×C4×C293—21no—
100SD16×C293—22no—
101C2×C8×C293C2311yescharacteristic
102C2×C8×C293C2311yescharacteristic
103C4⋊C4×C293—21no—
104C4⋊C4×C293C2311yescharacteristic
105C4⋊C4×C293C2311yescharacteristic
106C22⋊C4×C293—21no—
107C42×C293—21no—
108C42×C293C2311yescharacteristic
109C22×D293—24no—
110C22×D293—21no—
111C22×D293—21no—
112C22×D293—42no—
113C22×D293—42no—
114C22×D293C2×D411yescharacteristic
115D1172—21no—
116D1172—21no—
117D1172—21no—
118D1172—42no—
119D1172#4112yes—
120C22⋊D293—24no—
121C2×Dic293—21no—
122C2×Dic293—21no—
123C2×Dic293—42no—
124C2×Dic293C2×D411yescharacteristic
125C2×Dic293C2×D411yescharacteristic
126C2×Dic293C2×D411yescharacteristic
127C4×D293—22no—
128C4×D293—21no—
129C4×D293—21no—
130C4×D293—41no—
131C4×D293—42no—
132C4×D293—42no—
133C4×D293—42no—
134C4×D293—42no—
135#14556873—21no—
136#14556873—21no—
137C23×C293—21no—
138Q8×C293—21no—
139Q8×C293—21no—
140D4×C293—21no—
141D4×C293#4112yes—
142C2×C4×C293—21no—
143C2×C4×C293—21no—
144C2×C4×C293—21no—
145C2×C4×C293—42no—
146C2×C4×C293C2×D411yescharacteristic
147C2×C4×C293C2×D411yescharacteristic
148C2×C4×C293C2411yescharacteristic
149C8×C293—21no—
150C8×C293—21no—
151C22×C293—22no—
152C22×C293—41no—
153C22×C293—41no—
154C22×C293C22×D411yescharacteristic
155C2×D293—21no—
156C2×D293—21no—
157C2×D293—22no—
158C2×D293—48no—
159C2×D293—41no—
160C2×D293—41no—
161C2×D293—41no—
162C2×D293—41no—
163C2×D293—41no—
164C2×D293AGL(1,ℤ/8)12yes—
165C4×C293—21no—
166C4×C293—21no—
167C4×C293—41no—
168C4×C293—41no—
169C4×C293—41no—
170C4×C293C2×#4111yescharacteristic
171C4×C293C2×#4111yescharacteristic
172Dic293—21no—
173Dic293—21no—
174Dic293—21no—
175Dic293—41no—
176Dic293—41no—
177C2×C293—42no—
178C2×C293#52511yescharacteristic
179C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C293#57511yescharacteristic
181D293—22no—
182D293—42no—
183D293—81no—
184C293#311411yescharacteristic, complemented, Sylow
185#3114—2931nomaximal, Sylow
186#529—2931no—
187C2×C4×D4—2931no—
188#483—2931no—
189#480—2931no—
190#478—2931no—
191#478—2931no—
192#463—2931no—
193#463—2931no—
194#450—2931no—
195#447—2931no—
196#442—2931no—
197#442—2931no—
198#440—2931no—
199#440D29311yescharacteristic, complemented
200#432—2931no—
201C22×D4—2931no—
202C23×C4—5861no—
203C2×SD16—2931no—
204C2×SD16C2×D29311yescharacteristic
205C2×D8—2931no—
206C2×D8—2931no—
207#126—5861no—
208#125—2931no—
209#124—2931no—
210#123—5861no—
211C22⋊Q8—2931no—
212#121—2931no—
213C4×Q8C2×D29311yescharacteristic, complemented
214C4×D4—2931no—
215C4×D4—2931no—
216C4×D4—2931no—
217C4×D4—2931no—
218C4×D4—5861no—
219C4×D4—5861no—
220C4×D4—5861no—
221C4×D4C2×D29311yescharacteristic
222#117—5861no—
223C2×C4⋊C4—2931no—
224C2×C22⋊C4—5861no—
225C2×C42—2931no—
226#107—2931no—
227#107C2×D29311yescharacteristic, complemented
228#106—2931no—
229#106—2931no—
230C4⋊C8—2931no—
231C4⋊C8—2931no—
232#103—2931no—
233#103C2×D29311yescharacteristic, complemented
234#102—2931no—
235#102—2931no—
236#102—2931no—
237#102—2931no—
238#102—2931no—
239#102C2×D29311yescharacteristic
240C22⋊C8—2931no—
241C22⋊C8—2931no—
242#97—2931no—
243#97C2×D29311yescharacteristic, complemented
244C24—5861no—
245C2×Q8C22×D29311yescharacteristic
246C2×D4—2931no—
247C2×D4—2931no—
248C2×D4—5861no—
249C2×D4—5861no—
250C2×D4—5862no—
251C2×D4—5862no—
252C2×D4C22×D29311yescharacteristic
253C22×C4—21no—
254C22×C4—2931no—
255C22×C4—2931no—
256C22×C4—2931no—
257C22×C4—5861no—
258C22×C4—5861no—
259C22×C4—5861no—
260C22×C4—5861no—
261C22×C4—5861no—
262C22×C4—11724no—
263SD16—22no—
264SD16—5862no—
265D8—5862no—
266D8—5862no—
267C2×C8—2931no—
268C2×C8—2931no—
269C2×C8C22×D29311yescharacteristic
270C2×C8C22×D29311yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—2931no—
273C4⋊C4—2931no—
274C4⋊C4—5861no—
275C4⋊C4—5861no—
276C4⋊C4—5861no—
277C4⋊C4—5861no—
278C4⋊C4C22×D29311yescharacteristic
279C4⋊C4C22×D29311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—5861no—
282C22⋊C4—5861no—
283C22⋊C4—5861no—
284C22⋊C4—5861no—
285C22⋊C4—5861no—
286C22⋊C4—5861no—
287C22⋊C4—5861no—
288C22⋊C4—5861no—
289C42—21no—
290C42—2931no—
291C42—2931no—
292C42—2931no—
293C42C22×D29311yescharacteristic
294C23—21no—
295C23—2931no—
296C23—5864no—
297C23—5861no—
298C23—5861no—
299C23—11722no—
300C23—11722no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—2932no—
305D4—5864no—
306D4—5861no—
307D4—5861no—
308D4—5861no—
309D4—11722no—
310D4#1798980212yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—2931no—
316C2×C4—2931no—
317C2×C4—2931no—
318C2×C4—5862no—
319C2×C4—5861no—
320C2×C4—5861no—
321C2×C4—5861no—
322C2×C4—5861no—
323C2×C4—11721no—
324C2×C4—11722no—
325C2×C4—11722no—
326C2×C4—11722no—
327C2×C4—11722no—
328C2×C4—11722no—
329C2×C4D4×D29311yescharacteristic
330C2×C4D4×D29311yescharacteristic
331C2×C4C23×D29311yescharacteristic
332C8—21no—
333C8—21no—
334C8—5861no—
335C8—5861no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—2932no—
340C22—5861no—
341C22—5861no—
342C22—5862no—
343C22—11728no—
344C22—11721no—
345C22—11721no—
346C22—11721no—
347C22—11721no—
348C22—11721no—
349C22C2×D4×D29311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—5861no—
356C4—5861no—
357C4—5861no—
358C4—11721no—
359C4—11721no—
360C4#41×D29311yescharacteristic
361C4C2×#1798980211yescharacteristic
362C2—42no—
363C2—23441no—
364C2—5862no—
365C2—11722no—
366C2#3403663811yescharacteristic
367C2AGL(1,ℤ/8)×D29311yescharacteristic
368C2#3403692811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#440×C293C2
socleC22×C293C22×D4
prime radical#34036264C2
hypercenter#440D293
hypocenterC293#3114
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #34036666C2semidirect
2. C4×D4×D293C2semidirect
3. #34036458C2semidirect
4. #34036444C2semidirect
5. #34036432C2semidirect
6. #34036431C2semidirect
7. #34036347C2semidirect
8. #34036346C2semidirect
9. #34036278C2semidirect
11. #34036242C2semidirect
12. #34036239C2semidirect
13. #34036227C2semidirect
14. #34036184C2semidirect
15. #440×C293C2semidirect
22. #22627615C22semidirect
23. #22627613C22semidirect
29. C4×D1172C22semidirect
38. #22627557C22semidirect
39. #22627556C22semidirect
40. #22627555C22semidirect
41. #22627553C22semidirect
42. #22627553C22semidirect
43. #22627548C22semidirect
45. #22627546C22semidirect
48. #22627544C22semidirect
51. #22627535C22semidirect
53. C4×Q8×C293C22semidirect
55. #107×C293C22semidirect
56. #103×C293C22semidirect
58. #97×C293C22semidirect
184. C293#3114semidirect
199. #440D293semidirect
213. C4×Q8C2×D293semidirect
227. #107C2×D293semidirect
233. #103C2×D293semidirect
243. #97C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340366662
C4×D4×D2932
#340364582
#340364442
#340364322
#340364312
#340363472
#340363462
#340362782
#340362642
#340362422
#340362392
#340362272
#340361842
#440×C2932
#3114293