Finite Group Databasebeta

← #49170641

Subgroups — #49170641

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#34036831C211yesmaximal, characteristic, complemented
2#34036770C211yesmaximal, characteristic, complemented
3#34036485C211yesmaximal, characteristic, complemented
4#34036462C211yesmaximal, characteristic, complemented
5#34036455C211yesmaximal, characteristic, complemented
6#34036428C211yesmaximal, characteristic, complemented
7#34036364C211yesmaximal, characteristic, complemented
8#34036346C211yesmaximal, characteristic, complemented
9#34036273C211yesmaximal, characteristic, complemented
10#34036262C211yesmaximal, characteristic, complemented
11#34036209C211yesmaximal, characteristic, complemented
12C8×C22⋊D293C211yesmaximal, characteristic, complemented
13#34036124C211yesmaximal, characteristic, complemented
14C2×#22627546C211yesmaximal, characteristic, complemented
15#465×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18C2×#17989787C2211yescharacteristic
19C2×#17989787C2211yescharacteristic
20C2×C8×D293C2211yescharacteristic, complemented
21#22627617—21no—
22#22627615C2211yescharacteristic, complemented
23#22627613C2211yescharacteristic
24#22627609—21no—
25#22627608—21no—
26#22627599—21no—
27#22627599C2211yescharacteristic
28#22627596—21no—
29#22627595C2211yescharacteristic, complemented
30#22627594—21no—
31D4×Dic293—21no—
32D4×Dic293C2211yescharacteristic, complemented
33#22627586C2211yescharacteristic
34C4×C22⋊D293C2211yescharacteristic
35#22627582C2211yescharacteristic, complemented
36C2×C4⋊Dic293C2211yescharacteristic
37C22⋊C4×D293—21no—
38#22627557C2211yescharacteristic, complemented
39#22627557C2211yescharacteristic, 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
51C8×Dic293C2211yescharacteristic, 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—
62C2×D1172—22no—
63C2×D1172—21no—
64C2×D1172—21no—
65C2×D1172—21no—
66C2×D1172—41no—
67C2×D1172C2311yescharacteristic
68C2×D1172C2311yescharacteristic
69C2×C22⋊D293—21no—
70C2×C22⋊D293D411yescharacteristic
71C22×Dic293—21no—
72C22×Dic293—21no—
73C2×C4×D293—21no—
74C2×C4×D293—21no—
75C2×C4×D293C2311yescharacteristic
76#17989787—22no—
77#17989787—41no—
78C2×#14556873C2311yescharacteristic
79C2×#14556873C2311yescharacteristic
80C8×D293—41no—
81C4⋊Dic293—21no—
82C4⋊Dic293—21no—
83C4⋊Dic293C2311yescharacteristic
84C4⋊Dic293C2311yescharacteristic
85#17989781—21no—
86#17989781D411yescharacteristic
87C22⋊Dic293—21no—
88C22⋊Dic293—21no—
89C22⋊Dic293D411yescharacteristic
90#17989779—21no—
91#17989779—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779—21no—
95#17989779D411yescharacteristic
96C4×Dic293—21no—
97C4×Dic293C2311yescharacteristic
98C2×Q8×C293C2311yescharacteristic
99C2×D4×C293—21no—
100C2×D4×C293C2311yescharacteristic
101C22×C4×C293C2311yescharacteristic
102SD16×C293—41no—
103C2×C8×C293—41no—
104C2×C8×C293C2311yescharacteristic
105C2×C8×C293C2311yescharacteristic
106C4⋊C4×C293—21no—
107C4⋊C4×C293C2311yescharacteristic
108C4⋊C4×C293C2311yescharacteristic
109C22⋊C4×C293—21no—
110C22⋊C4×C293—21no—
111C22×D293—21no—
112C22×D293—21no—
113C22×D293—42no—
114C22×D293—42no—
115C22×D293—81no—
116C22×D293C2×D411yescharacteristic
117D1172—21no—
118D1172—21no—
119D1172—21no—
120D1172—42no—
121D1172—42no—
122D1172—81no—
123D1172#4112yes—
124C22⋊D293—22no—
125C22⋊D293—81no—
126C2×Dic293—21no—
127C2×Dic293—21no—
128C2×Dic293—81no—
129C2×Dic293—81no—
130C2×Dic293C2×D411yescharacteristic
131C2×Dic293C2×D411yescharacteristic
132C2×Dic293C2×D411yescharacteristic
133C4×D293—21no—
134C4×D293—21no—
135C4×D293—42no—
136C4×D293—81no—
137#14556873—21no—
138#14556873—21no—
139C23×C293—21no—
140C23×C293C2×D411yescharacteristic
141Q8×C293—21no—
142Q8×C293—21no—
143D4×C293—21no—
144D4×C293—21no—
145D4×C293—42no—
146C2×C4×C293—21no—
147C2×C4×C293—21no—
148C2×C4×C293—21no—
149C2×C4×C293—21no—
150C2×C4×C293—21no—
151C2×C4×C293C2×D411yescharacteristic
152C2×C4×C293C2411yescharacteristic
153C8×C293—21no—
154C8×C293—21no—
155C22×C293—21no—
156C22×C293—21no—
157C22×C293—41no—
158C22×C293—41no—
159C22×C293—41no—
160C22×C293#13512yes—
161C22×C293C22×D411yescharacteristic
162C2×D293—21no—
163C2×D293—22no—
164C2×D293—21no—
165C2×D293—21no—
166C2×D293—41no—
167C2×D293—41no—
168C2×D293—41no—
169C2×D293—41no—
170C2×D293—41no—
171C2×D293—84no—
172C4×C293—21no—
173C4×C293—41no—
174C4×C293—41no—
175C4×C293—41no—
176C4×C293C2×#4111yescharacteristic
177C4×C293H(5,𝔽2)11yescharacteristic
178Dic293—21no—
179Dic293—21no—
180Dic293—21no—
181Dic293—41no—
182Dic293—41no—
183C2×C293—22no—
184C2×C293—81no—
185C2×C293#54611yescharacteristic
186C2×C293C2×#13511yescharacteristic
187C2×C293#57611yescharacteristic
188D293—41no—
189D293—42no—
190D293—81no—
191C293#329711yescharacteristic, complemented, Sylow
192#3297—2931nomaximal, Sylow
193#553—2931no—
194#546—2931no—
195#488—2931no—
196#484—2931no—
197#482—2931no—
198#478—2931no—
199#466—2931no—
200#465D29311yescharacteristic, complemented
201#463—2931no—
202#449—2931no—
203#447—2931no—
204C4×D8—2931no—
205C8×D4—2931no—
206#416—2931no—
207C2×#102—2931no—
208C2×#41—2931no—
209C22×D4—2931no—
210C2×SD16C2×D29311yescharacteristic, complemented
211C2×D8—2931no—
212C2×D8—2931no—
213C22×C8—2931no—
214C22×C8C2×D29311yescharacteristic
215#126—5861no—
216#125—2931no—
217#124—2931no—
218#123—5861no—
219#123—5861no—
220C22⋊Q8C2×D29311yescharacteristic
221#121—2931no—
222#121—2931no—
223#121—5861no—
224#121—5861no—
225#121C2×D29311yescharacteristic
226#120—5861no—
227C4×D4—2931no—
228C4×D4—2931no—
229C4×D4—2931no—
230C4×D4—5861no—
231#117—2931no—
232C2×C4⋊C4—2931no—
233C2×C22⋊C4—5861no—
234#107—2931no—
235#107—2931no—
236#106C2×D29311yescharacteristic, complemented
237C4⋊C8—2931no—
238#103—2931no—
239#103—2931no—
240#103C2×D29311yescharacteristic, complemented
241#102—2931no—
242#102—2931no—
243#102—2931no—
244#102—2931no—
245#102—2931no—
246#102—2931no—
247#102C2×D29311yescharacteristic, complemented
248C22⋊C8—2931no—
249C22⋊C8—2931no—
250C4×C8—2931no—
251C24—5861no—
252#41—5862no—
253#41—11721no—
254C2×Q8C22×D29311yescharacteristic
255C2×D4—21no—
256C2×D4—2931no—
257C2×D4—2931no—
258C2×D4—2931no—
259C2×D4—5862no—
260C2×D4—5861no—
261C2×D4—5861no—
262C2×D4—5861no—
263C2×D4—5861no—
264C2×D4—11721no—
265C2×D4C22×D29311yescharacteristic
266C22×C4—2931no—
267C22×C4—5861no—
268C22×C4—5861no—
269C22×C4—5861no—
270C22×C4—5861no—
271C22×C4C22×D29311yescharacteristic
272SD16—41no—
273D8—5862no—
274D8—11721no—
275C2×C8—41no—
276C2×C8—2931no—
277C2×C8—2931no—
278C2×C8—11721no—
279C2×C8C22×D29311yescharacteristic
280C2×C8C22×D29311yescharacteristic
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—2931no—
293C22⋊C4—2931no—
294C22⋊C4—5861no—
295C22⋊C4—5861no—
296C22⋊C4—5861no—
297C22⋊C4—5861no—
298C22⋊C4—5861no—
299C22⋊C4—5861no—
300C22⋊C4—5861no—
301C42—2931no—
302C42—5861no—
303C23—21no—
304C23—2931no—
305C23—23441no—
306C23—5861no—
307C23—5861no—
308C23—11722no—
309C23—11722no—
310C23D4×D29311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—2932no—
317D4—23441no—
318D4—23441no—
319D4—5862no—
320D4—5861no—
321D4—5861no—
322D4—5861no—
323D4—11722no—
324D4—11722no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—2931no—
331C2×C4—2931no—
332C2×C4—2931no—
333C2×C4—23441no—
334C2×C4—23441no—
335C2×C4—23441no—
336C2×C4—5861no—
337C2×C4—5861no—
338C2×C4—5861no—
339C2×C4—5861no—
340C2×C4—11722no—
341C2×C4D4×D29311yescharacteristic
342C2×C4C23×D29311yescharacteristic
343C8—21no—
344C8—21no—
345C8—5861no—
346C8—5861no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—23444no—
353C22—5861no—
354C22—5862no—
355C22—5861no—
356C22—5861no—
357C22—11721no—
358C22—11721no—
359C22—11721no—
360C22—11721no—
361C22—11721no—
362C22#2262764512yes—
363C22C2×D4×D29311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—5861no—
369C4—5861no—
370C4—5861no—
371C4—11721no—
372C4—11721no—
373C4#41×D29311yescharacteristic
374C4#2262766911yescharacteristic
375C2—22no—
376C2—81no—
377C2—23441no—
378C2—11722no—
379C2—11721no—
380C2#3403676811yescharacteristic
381C2C2×#2262764511yescharacteristic
382C2#3403693111yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#465×C293C2
socleC22×C293C22×D4
hypercenter#465D293
hypocenterC293#3297
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. #34036831C2semidirect
2. #34036770C2semidirect
3. #34036485C2semidirect
4. #34036462C2semidirect
5. #34036455C2semidirect
6. #34036428C2semidirect
7. #34036364C2semidirect
8. #34036346C2semidirect
9. #34036273C2semidirect
10. #34036262C2semidirect
11. #34036209C2semidirect
12. C8×C22⋊D293C2semidirect
13. #34036124C2semidirect
14. C2×#22627546C2semidirect
15. #465×C293C2semidirect
20. C2×C8×D293C22semidirect
22. #22627615C22semidirect
29. #22627595C22semidirect
32. D4×Dic293C22semidirect
35. #22627582C22semidirect
38. #22627557C22semidirect
39. #22627557C22semidirect
40. #22627553C22semidirect
41. #22627548C22semidirect
42. #22627548C22semidirect
50. #22627536C22semidirect
51. C8×Dic293C22semidirect
52. C2×SD16×C293C22semidirect
56. #106×C293C22semidirect
57. #103×C293C22semidirect
58. #102×C293C22semidirect
191. C293#3297semidirect
200. #465D293semidirect
210. C2×SD16C2×D293semidirect
236. #106C2×D293semidirect
240. #103C2×D293semidirect
247. #102C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340368312
#340367702
#340364852
#340364622
#340364552
#340364282
#340363642
#340363462
#340362732
#340362622
#340362092
C8×C22⋊D2932
#340361242
C2×#226275462
#465×C2932
#3297293