Finite Group Databasebeta

← #49170752

Subgroups — #49170752

Order 37504.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036779C211yesmaximal, characteristic, complemented
2#34036770C211yesmaximal, characteristic, complemented
3#34036490C211yesmaximal, characteristic, complemented
4#34036457C211yesmaximal, characteristic, complemented
5#34036449C211yesmaximal, characteristic, complemented
6#34036418C211yesmaximal, characteristic, complemented
7#34036381C211yesmaximal, characteristic, complemented
8#34036346C211yesmaximal, characteristic, complemented
9#34036273C211yesmaximal, characteristic, complemented
10#34036262C211yesmaximal, characteristic, complemented
11#34036238C211yesmaximal, characteristic, complemented
12#34036200C211yesmaximal, characteristic, complemented
13#34036129C211yesmaximal, characteristic, complemented
14C2×#22627546C211yesmaximal, characteristic, complemented
15#465×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989801—21no—
18C22×D1172C2211yescharacteristic
19C2×#17989787C2211yescharacteristic
20C2×#17989787C2211yescharacteristic
21C2×#17989785C2211yescharacteristic, complemented
22#22627623C2211yescharacteristic, complemented
23#22627613C2211yescharacteristic
24#22627609—21no—
25#22627606—21no—
26#22627605—21no—
27#22627601—21no—
28#22627599—21no—
29#22627599C2211yescharacteristic
30#22627595C2211yescharacteristic, complemented
31#22627594—21no—
32D4×Dic293C2211yescharacteristic, complemented
33#22627586C2211yescharacteristic
34C4×C22⋊D293C2211yescharacteristic
35C2×C4⋊Dic293C2211yescharacteristic
36C4⋊C4×D293C2211yescharacteristic, complemented
37C22⋊C4×D293—21no—
38#22627555C2211yescharacteristic, complemented
39#22627555C2211yescharacteristic, complemented
40#22627553C2211yescharacteristic, complemented
41#22627548C2211yescharacteristic, complemented
42#22627548C2211yescharacteristic, complemented
43#22627546C2211yescharacteristic
44#22627546C2211yescharacteristic
45#22627546C2211yescharacteristic
46#22627546C2211yescharacteristic
47#22627544C2211yescharacteristic
48#22627544C2211yescharacteristic
49C22⋊#14556873C2211yescharacteristic
50#22627536C2211yescharacteristic, complemented
51#22627534C2211yescharacteristic, complemented
52C2×SD16×C293C2211yescharacteristic, complemented
53C22×C8×C293C2211yescharacteristic
54C22⋊Q8×C293C2211yescharacteristic
55#121×C293C2211yescharacteristic
56#106×C293C2211yescharacteristic, complemented
57#103×C293C2211yescharacteristic, complemented
58#102×C293C2211yescharacteristic, complemented
59C23×D293—21no—
60#17989803—22no—
61#17989803—41no—
62#17989801—44no—
63C2×Dic586—21no—
64C2×D1172—22no—
65C2×D1172—21no—
66C2×D1172—21no—
67C2×D1172—21no—
68C2×D1172—41no—
69C2×D1172C2311yescharacteristic
70C2×D1172C2311yescharacteristic
71C2×C22⋊D293—21no—
72C2×C22⋊D293D411yescharacteristic
73C22×Dic293—21no—
74C22×Dic293—21no—
75C2×C4×D293—21no—
76C2×C4×D293—21no—
77C2×C4×D293—21no—
78C2×C4×D293C2311yescharacteristic
79#17989787—22no—
80#17989787—41no—
81#17989785—41no—
82C2×#14556873C2311yescharacteristic
83C2×#14556873C2311yescharacteristic
84C4⋊Dic293—21no—
85C4⋊Dic293C2311yescharacteristic
86C4⋊Dic293C2311yescharacteristic
87#17989781—21no—
88#17989781—21no—
89#17989781D411yescharacteristic
90C22⋊Dic293—21no—
91C22⋊Dic293—21no—
92C22⋊Dic293D411yescharacteristic
93#17989779—21no—
94#17989779—21no—
95#17989779—21no—
96#17989779—21no—
97#17989779D411yescharacteristic
98C4×Dic293C2311yescharacteristic
99C2×Q8×C293C2311yescharacteristic
100C2×D4×C293—21no—
101C2×D4×C293C2311yescharacteristic
102C22×C4×C293C2311yescharacteristic
103SD16×C293—41no—
104C2×C8×C293—41no—
105C2×C8×C293C2311yescharacteristic
106C2×C8×C293C2311yescharacteristic
107C4⋊C4×C293—21no—
108C4⋊C4×C293C2311yescharacteristic
109C4⋊C4×C293C2311yescharacteristic
110C22⋊C4×C293—21no—
111C22⋊C4×C293—21no—
112C22×D293—21no—
113C22×D293—21no—
114C22×D293—42no—
115C22×D293—42no—
116C22×D293—81no—
117C22×D293C2×D411yescharacteristic
118Dic586—42no—
119D1172—21no—
120D1172—21no—
121D1172—21no—
122D1172—42no—
123D1172—42no—
124D1172—81no—
125D1172#4112yescomplemented
126C22⋊D293—22no—
127C22⋊D293—42no—
128C2×Dic293—21no—
129C2×Dic293—21no—
130C2×Dic293—42no—
131C2×Dic293—42no—
132C2×Dic293C2×D411yescharacteristic
133C2×Dic293C2×D411yescharacteristic
134C2×Dic293C2×D411yescharacteristic
135C4×D293—21no—
136C4×D293—21no—
137C4×D293—42no—
138C4×D293—42no—
139C4×D293—81no—
140#14556873—21no—
141#14556873—21no—
142C23×C293—21no—
143C23×C293C2×D411yescharacteristic
144Q8×C293—21no—
145Q8×C293—21no—
146D4×C293—21no—
147D4×C293—21no—
148D4×C293—42no—
149C2×C4×C293—21no—
150C2×C4×C293—21no—
151C2×C4×C293—21no—
152C2×C4×C293—21no—
153C2×C4×C293—21no—
154C2×C4×C293C2×D411yescharacteristic
155C2×C4×C293C2411yescharacteristic
156C8×C293—21no—
157C8×C293—21no—
158C22×C293—21no—
159C22×C293—21no—
160C22×C293—41no—
161C22×C293—41no—
162C22×C293—41no—
163C22×C293AGL(1,ℤ/8)12yes—
164C22×C293C22×D411yescharacteristic
165C2×D293—21no—
166C2×D293—22no—
167C2×D293—21no—
168C2×D293—21no—
169C2×D293—41no—
170C2×D293—41no—
171C2×D293—41no—
172C2×D293—41no—
173C2×D293—41no—
174C2×D293—84no—
175C4×C293—21no—
176C4×C293—41no—
177C4×C293—41no—
178C4×C293—41no—
179C4×C293C2×#4111yescharacteristic
180C4×C293H(5,𝔽2)11yescharacteristic
181Dic293—21no—
182Dic293—21no—
183Dic293—21no—
184Dic293—41no—
185Dic293—41no—
186C2×C293—22no—
187C2×C293—81no—
188C2×C293#54611yescharacteristic
189C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C293#57711yescharacteristic
191D293—41no—
192D293—42no—
193D293—81no—
194C293#330611yescharacteristic, complemented, Sylow
195#3306—2931nomaximal, Sylow
196#547—2931no—
197#546—2931no—
198#489—2931no—
199#483—2931no—
200#481—2931no—
201#476—2931no—
202#469—2931no—
203#465D29311yescharacteristic, complemented
204#463—2931no—
205#449—2931no—
206#447—2931no—
207#442—2931no—
208#435—2931no—
209#417—2931no—
210C2×#102—2931no—
211C2×#41—2931no—
212C2×#41—5861no—
213C22×D4—2931no—
214C2×SD16C2×D29311yescharacteristic, complemented
215C2×D8—2931no—
216C2×D8—2931no—
217C2×M16—2931no—
218C22×C8C2×D29311yescharacteristic
219C4⋊Q8—2931no—
220#124—2931no—
221#123—5861no—
222#123—5861no—
223#123—5861no—
224C22⋊Q8—5861no—
225C22⋊Q8C2×D29311yescharacteristic
226#121—2931no—
227#121—2931no—
228#121—5861no—
229#121C2×D29311yescharacteristic
230#120—5861no—
231C4×D4—2931no—
232C4×D4—2931no—
233C4×D4—2931no—
234C2×C4⋊C4—2931no—
235C2×C4⋊C4—2931no—
236C2×C22⋊C4—5861no—
237#106—2931no—
238#106—2931no—
239#106C2×D29311yescharacteristic, complemented
240C4⋊C8—2931no—
241#103—2931no—
242#103—2931no—
243#103C2×D29311yescharacteristic, complemented
244#102—2931no—
245#102—2931no—
246#102—2931no—
247#102—2931no—
248#102—2931no—
249#102—2931no—
250#102C2×D29311yescharacteristic, complemented
251C22⋊C8—2931no—
252C22⋊C8—2931no—
253#97—2931no—
254C24—5861no—
255#41—5862no—
256#41—11721no—
257#41—11724no—
258C2×Q8—5861no—
259C2×Q8C22×D29311yescharacteristic
260C2×D4—21no—
261C2×D4—2931no—
262C2×D4—2931no—
263C2×D4—2931no—
264C2×D4—5862no—
265C2×D4—5861no—
266C2×D4—5861no—
267C2×D4—5861no—
268C2×D4—5861no—
269C2×D4—11721no—
270C2×D4C22×D29311yescharacteristic
271C22×C4—2931no—
272C22×C4—5861no—
273C22×C4—5861no—
274C22×C4—5861no—
275C22×C4—5861no—
276C22×C4—5861no—
277C22×C4C22×D29311yescharacteristic
278SD16—41no—
279D8—5862no—
280D8—11721no—
281M16—11721no—
282C2×C8—41no—
283C2×C8—2931no—
284C2×C8—2931no—
285C2×C8C22×D29311yescharacteristic
286C2×C8C22×D29311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—2931no—
289C4⋊C4—2931no—
290C4⋊C4—2931no—
291C4⋊C4—5861no—
292C4⋊C4—5861no—
293C4⋊C4—5861no—
294C4⋊C4C22×D29311yescharacteristic
295C4⋊C4C22×D29311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—2931no—
299C22⋊C4—2931no—
300C22⋊C4—5861no—
301C22⋊C4—5861no—
302C22⋊C4—5861no—
303C22⋊C4—5861no—
304C22⋊C4—5861no—
305C22⋊C4—5861no—
306C42—2931no—
307C23—21no—
308C23—2931no—
309C23—23441no—
310C23—5861no—
311C23—5861no—
312C23—11722no—
313C23—11722no—
314C23D4×D29311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—11722no—
318D4—21no—
319D4—21no—
320D4—42no—
321D4—2932no—
322D4—23441no—
323D4—5862no—
324D4—5861no—
325D4—5861no—
326D4—5861no—
327D4—11722no—
328D4—11722no—
329D4—11722no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—2931no—
336C2×C4—2931no—
337C2×C4—2931no—
338C2×C4—23441no—
339C2×C4—5861no—
340C2×C4—5861no—
341C2×C4—5861no—
342C2×C4—5861no—
343C2×C4—11722no—
344C2×C4—11722no—
345C2×C4—11722no—
346C2×C4—11722no—
347C2×C4D4×D29311yescharacteristic
348C2×C4C23×D29311yescharacteristic
349C8—21no—
350C8—21no—
351C8—5861no—
352C8—5861no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—23444no—
359C22—5861no—
360C22—5862no—
361C22—5861no—
362C22—5861no—
363C22—11721no—
364C22—11721no—
365C22—11721no—
366C22—11721no—
367C22—11721no—
368C22#2262765112yes—
369C22C2×D4×D29311yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—5861no—
375C4—5861no—
376C4—5861no—
377C4—11721no—
378C4—11721no—
379C4#41×D29311yescharacteristic
380C4#2262766911yescharacteristic
381C2—22no—
382C2—81no—
383C2—23441no—
384C2—11722no—
385C2—11721no—
386C2#3403676811yescharacteristic
387C2C2×#2262765111yescharacteristic
388C2#3403693611yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#465×C293C2
socleC22×C293C22×D4
hypercenter#465D293
hypocenterC293#3306
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #34036779C2semidirect
2. #34036770C2semidirect
3. #34036490C2semidirect
4. #34036457C2semidirect
5. #34036449C2semidirect
6. #34036418C2semidirect
7. #34036381C2semidirect
8. #34036346C2semidirect
9. #34036273C2semidirect
10. #34036262C2semidirect
11. #34036238C2semidirect
12. #34036200C2semidirect
13. #34036129C2semidirect
14. C2×#22627546C2semidirect
15. #465×C293C2semidirect
21. C2×#17989785C22semidirect
22. #22627623C22semidirect
30. #22627595C22semidirect
32. D4×Dic293C22semidirect
36. C4⋊C4×D293C22semidirect
38. #22627555C22semidirect
39. #22627555C22semidirect
40. #22627553C22semidirect
41. #22627548C22semidirect
42. #22627548C22semidirect
50. #22627536C22semidirect
51. #22627534C22semidirect
52. C2×SD16×C293C22semidirect
56. #106×C293C22semidirect
57. #103×C293C22semidirect
58. #102×C293C22semidirect
125. D1172#41semidirect
194. C293#3306semidirect
203. #465D293semidirect
214. C2×SD16C2×D293semidirect
239. #106C2×D293semidirect
243. #103C2×D293semidirect
250. #102C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340367792
#340367702
#340364902
#340364572
#340364492
#340364182
#340363812
#340363462
#340362732
#340362622
#340362382
#340362002
#340361292
C2×#226275462
#465×C2932
#3306293