Finite Group Databasebeta

← #18003951

Subgroups — #18003951

Order 4736.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924582C211yesmaximal, characteristic, complemented
2#14558331C211yesmaximal, characteristic, complemented
3#14558097C211yesmaximal, characteristic, complemented
4#14557709C214yesmaximal, complemented
5#14557699C214yesmaximal, complemented
6#14557550C211yesmaximal, characteristic, complemented
7#14557549C211yesmaximal, characteristic, complemented
8C2×#11924452C211yesmaximal, characteristic, complemented
9C37×#575C211yesmaximal, characteristic, complemented
10#11924582—22no—
11#11924582—22no—
12#11924582—22no—
13#11924582C2214yes—
14C2×#10596966—21no—
15C2×#10596962—21no—
16C2×#10596962C2211yescharacteristic
17C22×D148—21no—
18C22×D148C2211yescharacteristic
19C2×D4×D37—22no—
20C2×D4×D37—21no—
21#11924567C2212yescomplemented
22#11924566—21no—
23#11924562C2212yescomplemented
24#11924561—21no—
25#11924556—22no—
26C2×#10596934C2211yescharacteristic, complemented
27#11924532C2212yescomplemented
28#11924525C2212yescomplemented
29#11924510—22no—
30#11924504—22no—
31#11924495C2211yescharacteristic, complemented
32#11924463C2214yescomplemented
33#11924462C2214yescomplemented
34#11924452C2214yescomplemented
35C2×#41×C37C2211yescharacteristic
36C2×#41×C37C2211yescharacteristic, complemented
37#137×C37—21no—
38#137×C37C2212yescomplemented
39AGL(1,ℤ/8)×C37—21no—
40AGL(1,ℤ/8)×C37C2212yescomplemented
41#135×C37—22no—
42C2×M16×C37C2211yescharacteristic, complemented
43C23×D37—22no—
44C23×D37D412yes—
45#41×C37—21no—
46#41×C37—21no—
47#41×C37—21no—
48#41×C37—21no—
49#41×C37—21no—
50#41×C37—21no—
51#41×C37D412yescomplemented
52#41×C37C2312yes—
53C2×Q8×C37D411yescharacteristic, complemented
54C2×Q8×C37C2311yescharacteristic, complemented
55C2×D4×C37—21no—
56C2×D4×C37—21no—
57C2×D4×C37D411yescharacteristic, complemented
58C2×D4×C37C2311yescharacteristic, complemented
59C22×C4×C37—21no—
60C22×C4×C37—21no—
61C22×C4×C37C2311yescharacteristic
62C2×C22⋊D37—22no—
63C2×C22⋊D37—21no—
64C2×C22⋊D37—24no—
65#10596966—24no—
66#10596966—24no—
67D4×D37—28no—
68D4×D37—24no—
69D4×D37—28no—
70D4×D37—24no—
71#10596962—24no—
72#10596962—21no—
73#10596962—21no—
74#10596962—24no—
75#10596962D412yescomplemented
76#10596962C2312yes—
77C2×D148—22no—
78C2×D148—24no—
79C2×D148—21no—
80C2×D148—21no—
81C2×D148—21no—
82C2×D148—21no—
83C2×D148—24no—
84C2×D148D412yes—
85C2×D148D411yescharacteristic, complemented
86C2×D148C2312yes—
87C2×C4×D37—22no—
88C2×C4×D37—21no—
89C2×Dic74D411yescharacteristic, complemented
90Q16×C37—22no—
91SD16×C37—22no—
92SD16×C37—22no—
93D8×C37—22no—
94M16×C37—21no—
95M16×C37C2312yescomplemented
96C2×C8×C37—21no—
97C22⋊Dic37—21no—
98#10596942—22no—
99#10596941—22no—
100#10596940—22no—
101#10596939—22no—
102#10596938—24no—
103C4⋊Dic37—21no—
104C4×Dic37C2312yescomplemented
105#10596934—21no—
106#10596934C2312yescomplemented
107C2×#64436—21no—
108C23×C37—21no—
109C23×C37—21no—
110C23×C37C2×D411yescharacteristic
111C22×D37—22no—
112C22×D37—21no—
113C22×D37—28no—
114C22×D37—22no—
115C22×D37—82no—
116C22×D37—84no—
117C22×D37C2×D414yes—
118Q8×C37—21no—
119Q8×C37—21no—
120Q8×C37—21no—
121Q8×C37C2×D412yes—
122D4×C37—22no—
123D4×C37—22no—
124D4×C37—21no—
125D4×C37—21no—
126D4×C37—21no—
127D4×C37—41no—
128D4×C37—41no—
129D4×C37C2×D412yes—
130C2×C4×C37—21no—
131C2×C4×C37—22no—
132C2×C4×C37—22no—
133C2×C4×C37—21no—
134C2×C4×C37—41no—
135C2×C4×C37—41no—
136C2×C4×C37C2×D411yescharacteristic
137C2×C4×C37C2×D411yescharacteristic
138C2×C4×C37C2×D411yescharacteristic
139C2×C4×C37C2×D411yescharacteristic
140C2×C4×C37C2×D411yescharacteristic
141C2×C4×C37C2411yescharacteristic
142C22⋊D37—28no—
143C22⋊D37—22no—
144C22⋊D37—22no—
145C22⋊D37—84no—
146C2×Dic37—21no—
147C2×Dic37—22no—
148D148—28no—
149D148—22no—
150D148—21no—
151D148—22no—
152D148—22no—
153D148—22no—
154D148C2×D412yes—
155C4×D37—28no—
156C4×D37—22no—
157C4×D37—22no—
158Dic74—21no—
159Dic74C2×D412yes—
160C8×C37—22no—
161#64436—22no—
162C22×C37—22no—
163C22×C37—41no—
164C22×C37—42no—
165C22×C37—42no—
166C22×C37—41no—
167C22×C37#12011yescharacteristic
168C22×C37#12011yescharacteristic
169C22×C37C22×D411yescharacteristic
170C2×D37—24no—
171C2×D37—22no—
172C2×D37—44no—
173C2×D37—42no—
174C2×D37—44no—
175C2×D37—82no—
176C2×D37—88no—
177C4×C37—22no—
178C4×C37—41no—
179C4×C37#12011yescharacteristic
180C4×C37#12011yescharacteristic
181C4×C37C22×D411yescharacteristic
182C4×C37C22×D411yescharacteristic
183Dic37—22no—
184Dic37—42no—
185#3019—371nomaximal, Sylow
186C2×C37—21no—
187C2×C37—21no—
188C2×C37—21no—
189C2×C37—42no—
190C2×C37—81no—
191C2×C37C2×#12011yescharacteristic
192D37—44no—
193D37—42no—
194C2×H(5,𝔽2)—371no—
195#575—371no—
196#575D3711yescharacteristic, complemented
197#535—371no—
198#455—374no—
199#453—374no—
200#421—371no—
201#421—371no—
202C2×#100—371no—
203C37#301911yescharacteristic, complemented, Sylow
204H(5,𝔽2)—374no—
205H(5,𝔽2)—742no—
206H(5,𝔽2)—742no—
207H(5,𝔽2)—742no—
208C2×#41—371no—
209C2×#41—741no—
210C2×#41—741no—
211C2×#41C2×D3711yescharacteristic
212C2×#41C2×D3711yescharacteristic, complemented
213C22×D4—371no—
214C22×D4—741no—
215C22×D4—742no—
216C22×D4—741no—
217#137—21no—
218#137—372no—
219#137—741no—
220#137C2×D3712yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—372no—
223AGL(1,ℤ/8)—741no—
224AGL(1,ℤ/8)C2×D3712yescomplemented
225#135—22no—
226#135—742no—
227C2×M16—371no—
228C2×M16C2×D3711yescharacteristic, complemented
229#127—372no—
230#124—372no—
231#121—742no—
232#120—742no—
233#117—371no—
234#104—374no—
235#104—374no—
236#100—374no—
237C24—372no—
238C24—742no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—372no—
246#41—372no—
247#41—744no—
248#41—744no—
249#41—744no—
250#41—744no—
251#41—741no—
252#41—741no—
253#41C22⋊D3712yescomplemented
254#41C22×D3712yes—
255C2×Q8—371no—
256C2×Q8C22⋊D3711yescharacteristic, complemented
257C2×Q8C22×D3711yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—371no—
261C2×D4—372no—
262C2×D4—372no—
263C2×D4—744no—
264C2×D4—742no—
265C2×D4—744no—
266C2×D4—748no—
267C2×D4—748no—
268C2×D4—744no—
269C2×D4—744no—
270C2×D4—742no—
271C2×D4—741no—
272C2×D4—744no—
273C2×D4—741no—
274C2×D4—741no—
275C2×D4—741no—
276C2×D4—741no—
277C2×D4C22⋊D3711yescharacteristic, complemented
278C2×D4C22×D3711yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—742no—
282C22×C4—741no—
283C22×C4C22×D3711yescharacteristic
284Q16—22no—
285Q16—742no—
286SD16—22no—
287SD16—22no—
288SD16—742no—
289SD16—742no—
290D8—22no—
291D8—742no—
292M16—21no—
293M16—372no—
294M16—741no—
295M16C22×D3712yescomplemented
296C2×C8—21no—
297C2×C8—741no—
298C4⋊C4—741no—
299C22⋊C4—744no—
300C22⋊C4—741no—
301C42—372no—
302C23—21no—
303C23—21no—
304C23—374no—
305C23—2964no—
306C23—2962no—
307C23—748no—
308C23—742no—
309C23—741no—
310C23—742no—
311C23D4×D3711yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—372no—
316Q8—741no—
317Q8C2×C22⋊D3712yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—372no—
326D4—2964no—
327D4—748no—
328D4—748no—
329D4—742no—
330D4—742no—
331D4—741no—
332D4—742no—
333D4—742no—
334D4—742no—
335D4—742no—
336D4C2×C22⋊D3712yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—748no—
344C2×C4—741no—
345C2×C4—742no—
346C2×C4—742no—
347C2×C4—742no—
348C2×C4D4×D3711yescharacteristic
349C2×C4D4×D3711yescharacteristic
350C2×C4D4×D3711yescharacteristic
351C2×C4C2×C22⋊D3711yescharacteristic
352C2×C4C2×C22⋊D3711yescharacteristic
353C2×C4C23×D3711yescharacteristic
354C8—22no—
355C8—742no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—2962no—
362C22—2968no—
363C22—742no—
364C22—744no—
365C22—1484no—
366C22—1482no—
367C22—1484no—
368C22#1192450411yescharacteristic
369C22#1192450411yescharacteristic
370C22C2×D4×D3711yescharacteristic
371C4—22no—
372C4—41no—
373C4—742no—
374C4—1482no—
375C4#1192450411yescharacteristic
376C4#1192450411yescharacteristic
377C4C2×D4×D3711yescharacteristic
378C4C22×C22⋊D3711yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—1482no—
385C2—1484no—
386C2C2×#1192450411yescharacteristic
subgroupG/H
centerC2C2×#11924504
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#575C2
socleC2×C37C2×#120
hypercenter#575D37
hypocenterC37#3019
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#11924582C2semidirect
2. #14558331C2semidirect
3. #14558097C2semidirect
4. #14557709C2semidirect
5. #14557699C2semidirect
6. #14557550C2semidirect
7. #14557549C2semidirect
8. C2×#11924452C2semidirect
9. C37×#575C2semidirect
21. #11924567C22semidirect
23. #11924562C22semidirect
26. C2×#10596934C22semidirect
27. #11924532C22semidirect
28. #11924525C22semidirect
31. #11924495C22semidirect
32. #11924463C22semidirect
33. #11924462C22semidirect
34. #11924452C22semidirect
36. C2×#41×C37C22semidirect
38. #137×C37C22semidirect
40. AGL(1,ℤ/8)×C37C22semidirect
42. C2×M16×C37C22semidirect
51. #41×C37D4semidirect
53. C2×Q8×C37D4semidirect
54. C2×Q8×C37C23semidirect
57. C2×D4×C37D4semidirect
58. C2×D4×C37C23semidirect
75. #10596962D4semidirect
85. C2×D148D4semidirect
89. C2×Dic74D4semidirect
95. M16×C37C23semidirect
104. C4×Dic37C23semidirect
106. #10596934C23semidirect
196. #575D37semidirect
203. C37#3019semidirect
212. C2×#41C2×D37semidirect
220. #137C2×D37semidirect
224. AGL(1,ℤ/8)C2×D37semidirect
228. C2×M16C2×D37semidirect
253. #41C22⋊D37semidirect
256. C2×Q8C22⋊D37semidirect
257. C2×Q8C22×D37semidirect
277. C2×D4C22⋊D37semidirect
278. C2×D4C22×D37semidirect
295. M16C22×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245822
#145583312
#145580972
#145577092
#145576992
#145575502
#145575492
C2×#119244522
C37×#5752
#301937