Finite Group Databasebeta

← #18004059

Subgroups — #18004059

Order 4736.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#11924564C211yesmaximal, characteristic, complemented
2C2×#11924561C211yesmaximal, characteristic, complemented
3#14558079C211yesmaximal, characteristic, complemented
4C2×#11924498C211yesmaximal, characteristic, complemented
5#14557751C212yesmaximal, complemented
6#14557747C212yesmaximal, complemented
7#14557737C212yesmaximal, complemented
8#14557727C212yesmaximal, complemented
9#14557556C211yesmaximal, characteristic, complemented
10#14557532C211yesmaximal, characteristic, complemented
11C37×#417C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13C22×D148C2211yescharacteristic
14C22×C4×D37—21no—
15#11924564—24no—
16#11924561—24no—
17C2×#10596930C2212yes—
18C2×#10596941C2212yes—
19C2×D296C2212yes—
20C2×#10596939C2212yes—
21C2×#10596934C2211yescharacteristic
22#11924534C2211yescharacteristic, complemented
23#11924532C2211yescharacteristic, complemented
24#11924525C2211yescharacteristic, complemented
25#11924523C2211yescharacteristic, complemented
26#11924508—21no—
27#11924508—21no—
28#11924498—21no—
29#11924498—21no—
30#11924498C2212yes—
31#11924498C2212yes—
32C4×C22⋊D37—21no—
33C4×C22⋊D37—21no—
34C2×#10596938—21no—
35C2×C4×Dic37C2211yescharacteristic
36#11924465C2212yescomplemented
37#11924465C2212yescomplemented
38#11924459C2212yescomplemented
39#11924456C2212yescomplemented
40C2×#41×C37C2211yescharacteristic
41C2×M16×C37C2211yescharacteristic
42C2×C4⋊C4×C37C2211yescharacteristic
43#103×C37C2212yescomplemented
44#102×C37C2212yescomplemented
45C23×D37—21no—
46#41×C37—22no—
47#41×C37—22no—
48C2×Q8×C37C2311yescharacteristic
49C2×D4×C37—21no—
50C2×D4×C37C2311yescharacteristic
51C22×C4×C37—21no—
52C22×C4×C37—21no—
53C22×C4×C37C2311yescharacteristic
54C2×C22⋊D37—21no—
55C2×C22⋊D37—21no—
56C2×C22⋊D37—21no—
57C22×Dic37D411yescharacteristic
58C22×Dic37D411yescharacteristic
59#10596962—22no—
60#10596962—22no—
61C2×D148—21no—
62C2×D148—22no—
63C2×D148D414yescomplemented
64C2×D148C2311yescharacteristic
65C2×D148C2312yes—
66C2×C4×D37—21no—
67C2×C4×D37—22no—
68C2×C4×D37—22no—
69C2×C4×D37—44no—
70C2×Dic74C2311yescharacteristic
71M16×C37—22no—
72C2×C8×C37C2312yes—
73C4⋊C4×C37—21no—
74C4⋊C4×C37C2312yes—
75C22⋊Dic37—21no—
76C22⋊Dic37—21no—
77#10596941—24no—
78#10596939—24no—
79#10596938—22no—
80#10596938—22no—
81#10596938—21no—
82#10596938—21no—
83#10596936—21no—
84#10596936—21no—
85C4×Dic37D411yescharacteristic
86C4×Dic37D411yescharacteristic
87C4×Dic37C2311yescharacteristic
88C4×Dic37C2311yescharacteristic
89#10596934—22no—
90C2×#64436C2312yes—
91D296—24no—
92#10596930—24no—
93C23×C37—21no—
94C23×C37C2×D411yescharacteristic
95C22×D37—24no—
96C22×D37—22no—
97C22×D37—21no—
98C22×D37—42no—
99C22×D37—42no—
100Q8×C37—21no—
101Q8×C37—21no—
102D4×C37—21no—
103D4×C37—21no—
104D4×C37—42no—
105C2×C4×C37—21no—
106C2×C4×C37—22no—
107C2×C4×C37—42no—
108C2×C4×C37—42no—
109C2×C4×C37C2×D411yescharacteristic
110C2×C4×C37C2×D412yes—
111C2×C4×C37#4112yes—
112C2×C4×C37C2411yescharacteristic
113C22⋊D37—42no—
114C22⋊D37—42no—
115C22⋊D37—42no—
116C2×Dic37—22no—
117C2×Dic37—21no—
118C2×Dic37—41no—
119C2×Dic37C2×D411yescharacteristic
120C2×Dic37C2×D411yescharacteristic
121C2×Dic37C2×D411yescharacteristic
122C2×Dic37C2×D411yescharacteristic
123D148—24no—
124D148—21no—
125D148—22no—
126D148—21no—
127D148C2×D414yes—
128C4×D37—44no—
129C4×D37—42no—
130C4×D37—44no—
131Dic74—21no—
132Dic74—21no—
133C8×C37—22no—
134#64436—22no—
135C22×C37—21no—
136C22×C37—21no—
137C22×C37—41no—
138C22×C37—41no—
139C22×C37—41no—
140C22×C37#12112yes—
141C22×C37C22×D411yescharacteristic
142C2×D37—24no—
143C2×D37—48no—
144C2×D37—42no—
145C2×D37—42no—
146C2×D37—41no—
147C2×D37—41no—
148C2×D37—41no—
149C4×C37—41no—
150C4×C37—42no—
151C4×C37#12112yes—
152C4×C37C22×D411yescharacteristic
153C4×C37C2×#4111yescharacteristic
154Dic37—21no—
155Dic37—21no—
156Dic37—21no—
157Dic37—41no—
158Dic37AGL(1,ℤ/8)12yescomplemented
159#3036—371nomaximal, Sylow
160C2×C37—22no—
161C2×C37—81no—
162C2×C37C2×#12111yescharacteristic
163C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C37#57511yescharacteristic
165D37—44no—
166D37—81no—
167C2×AGL(1,ℤ/8)—371no—
168C2×AGL(1,ℤ/8)—371no—
169#532—371no—
170C2×C4×D4—371no—
171#463—372no—
172#463—372no—
173#461—372no—
174#459—372no—
175#423—371no—
176#417—371no—
177#417D3711yescharacteristic, complemented
178C37#303611yescharacteristic, complemented, Sylow
179C2×#41—371no—
180C2×#41C2×D3711yescharacteristic
181C22×D4—371no—
182C23×C4—741no—
183AGL(1,ℤ/8)—744no—
184AGL(1,ℤ/8)—744no—
185C2×SD16—372no—
186C2×SD16—372no—
187C2×D8—372no—
188C2×D8—372no—
189C2×M16—371no—
190C2×M16C2×D3711yescharacteristic
191C4⋊Q8—371no—
192#127—371no—
193#124—371no—
194#124—371no—
195#121—741no—
196#121—741no—
197C4×D4—372no—
198C4×D4—372no—
199C4×D4—741no—
200C4×D4—741no—
201C4×D4—741no—
202C4×D4—741no—
203C2×C4⋊C4C2×D3711yescharacteristic
204C2×C22⋊C4—741no—
205C2×C42—371no—
206C4⋊C8—372no—
207C4⋊C8—372no—
208#103—372no—
209#103C2×D3712yescomplemented
210#102—372no—
211#102C2×D3712yescomplemented
212C24—741no—
213#41—22no—
214#41—22no—
215#41—742no—
216#41—742no—
217C2×Q8—371no—
218C2×Q8C22×D3711yescharacteristic
219C2×D4—21no—
220C2×D4—374no—
221C2×D4—372no—
222C2×D4—371no—
223C2×D4—741no—
224C2×D4—741no—
225C2×D4—741no—
226C2×D4—742no—
227C2×D4—741no—
228C2×D4C22×D3711yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—371no—
232C22×C4—371no—
233C22×C4—742no—
234C22×C4—742no—
235C22×C4—741no—
236C22×C4—1484no—
237C22×C4C22×D3711yescharacteristic
238SD16—744no—
239SD16—744no—
240D8—744no—
241D8—744no—
242M16—22no—
243M16—742no—
244C2×C8—372no—
245C2×C8C22×D3712yes—
246C4⋊C4—21no—
247C4⋊C4—741no—
248C4⋊C4—741no—
249C4⋊C4C22×D3712yes—
250C22⋊C4—742no—
251C22⋊C4—742no—
252C22⋊C4—741no—
253C22⋊C4—741no—
254C22⋊C4—741no—
255C22⋊C4—741no—
256C42—371no—
257C42—371no—
258C42—371no—
259C42—371no—
260C23—21no—
261C23—744no—
262C23—742no—
263C23—741no—
264C23—1482no—
265C23—1482no—
266C23D4×D3711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—741no—
270Q8—741no—
271D4—21no—
272D4—21no—
273D4—42no—
274D4—374no—
275D4—744no—
276D4—741no—
277D4—742no—
278D4—741no—
279D4—1482no—
280D4—1482no—
281D4—1482no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—371no—
287C2×C4—371no—
288C2×C4—371no—
289C2×C4—371no—
290C2×C4—742no—
291C2×C4—741no—
292C2×C4—1484no—
293C2×C4—1484no—
294C2×C4—1481no—
295C2×C4—1482no—
296C2×C4#1059696212yes—
297C2×C4D4×D3711yescharacteristic
298C2×C4D4×D3712yes—
299C2×C4C23×D3711yescharacteristic
300C8—22no—
301C8—742no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—744no—
308C22—1488no—
309C22—1481no—
310C22—1481no—
311C22—1481no—
312C22—1482no—
313C22—1482no—
314C22#1192450912yes—
315C22C2×D4×D3711yescharacteristic
316C4—41no—
317C4—42no—
318C4—372no—
319C4—741no—
320C4—741no—
321C4—741no—
322C4—1481no—
323C4#1192450912yes—
324C4C2×D4×D3711yescharacteristic
325C4C2×#1059696211yescharacteristic
326C2—22no—
327C2—81no—
328C2—2961no—
329C2—1484no—
330C2C2×#1192450911yescharacteristic
331C2AGL(1,ℤ/8)×D3711yescharacteristic
332C2#1455833211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#417C2
socleC22×C37C22×D4
hypercenter#417D37
hypocenterC37#3036
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#11924564C2semidirect
2. C2×#11924561C2semidirect
3. #14558079C2semidirect
4. C2×#11924498C2semidirect
5. #14557751C2semidirect
6. #14557747C2semidirect
7. #14557737C2semidirect
8. #14557727C2semidirect
9. #14557556C2semidirect
10. #14557532C2semidirect
11. C37×#417C2semidirect
22. #11924534C22semidirect
23. #11924532C22semidirect
24. #11924525C22semidirect
25. #11924523C22semidirect
36. #11924465C22semidirect
37. #11924465C22semidirect
38. #11924459C22semidirect
39. #11924456C22semidirect
43. #103×C37C22semidirect
44. #102×C37C22semidirect
63. C2×D148D4semidirect
158. Dic37AGL(1,ℤ/8)semidirect
177. #417D37semidirect
178. C37#3036semidirect
209. #103C2×D37semidirect
211. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245642
C2×#119245612
#145580792
C2×#119244982
#145577512
#145577472
#145577372
#145577272
#145575562
#145575322
C37×#4172
#303637