Finite Group Databasebeta

← #18005370

Subgroups — #18005370

Order 4736.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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×#11924516C211yesmaximal, characteristic
2C22⋊Q8×D37C211yesmaximal, characteristic
3#14557870C212yesmaximal, complemented
4#14557868C211yesmaximal, characteristic, complemented
5#14557775C212yesmaximal, complemented
6#14557773C212yesmaximal, complemented
7#14557691C211yesmaximal, characteristic, complemented
8#14557690C212yesmaximal, complemented
9#14557688C211yesmaximal, characteristic, complemented
10C22⋊C8×D37C211yesmaximal, characteristic
11C37×#484C211yesmaximal, characteristic, complemented
12C22×Dic74C2211yescharacteristic
13C2×Q8×D37C2211yescharacteristic
14C22×C4×D37C2211yescharacteristic
15C2×Dic148C2212yescomplemented
16C2×#10596942C2212yescomplemented
17C2×C8×D37C2212yes—
18#11924516—21no—
19#11924516—21no—
20#11924516C2211yescharacteristic
21#11924516C2212yes—
22#11924516C2212yes—
23#11924515C2211yescharacteristic
24#11924514—21no—
25#11924513—21no—
26#11924512C2211yescharacteristic
27C2×C4⋊Dic37—21no—
28C4⋊C4×D37—21no—
29C4⋊C4×D37C2211yescharacteristic
30C2×#10596938—21no—
31C22⋊C4×D37—21no—
32#11924469C2212yescomplemented
33#11924461C2212yescomplemented
34#11924459C2212yescomplemented
35#11924459C2212yescomplemented
36C22⋊#64436C2211yescharacteristic
37#11924448C2212yes—
38C22⋊Q8×C37C2211yescharacteristic
39C2×C4⋊C4×C37C2211yescharacteristic
40#107×C37C2212yescomplemented
41#103×C37C2212yescomplemented
42C22⋊C8×C37C2211yescharacteristic
43C23×D37D411yescharacteristic
44C2×Q8×C37C2311yescharacteristic
45C22×C4×C37—21no—
46C22×C4×C37C2311yescharacteristic
47C22×Dic37—21no—
48C22×Dic37D411yescharacteristic
49Q8×D37—42no—
50C2×C4×D37—21no—
51C2×C4×D37—21no—
52C2×C4×D37—42no—
53C2×C4×D37D411yescharacteristic
54C2×C4×D37D411yescharacteristic
55C2×C4×D37C2311yescharacteristic
56C2×C4×D37C2311yescharacteristic
57C2×Dic74—21no—
58C2×Dic74—21no—
59C2×Dic74—42no—
60C2×Dic74C2311yescharacteristic
61C2×Dic74C2312yes—
62C2×C8×C37C2312yes—
63C4⋊C4×C37—21no—
64C4⋊C4×C37—21no—
65C4⋊C4×C37C2311yescharacteristic
66C4⋊C4×C37C2312yes—
67C22⋊C4×C37—21no—
68C22⋊Dic37—21no—
69#10596942—42no—
70#10596938—21no—
71#10596938—21no—
72#10596938—22no—
73#10596938—22no—
74C4⋊Dic37—21no—
75C4⋊Dic37—22no—
76C4⋊Dic37—22no—
77C4⋊Dic37C2311yescharacteristic
78#10596936—21no—
79#10596936—21no—
80#10596936—21no—
81C2×#64436C2312yes—
82Dic148—42no—
83C8×D37—42no—
84C23×C37C2×D411yescharacteristic
85C22×D37—41no—
86C22×D37—41no—
87C22×D37Q1614yescomplemented
88C22×D37C2×D411yescharacteristic
89C22×D37C2×D411yescharacteristic
90Q8×C37—22no—
91C2×C4×C37—22no—
92C2×C4×C37—21no—
93C2×C4×C37—21no—
94C2×C4×C37—22no—
95C2×C4×C37—42no—
96C2×C4×C37C2×D411yescharacteristic
97C2×C4×C37C2411yescharacteristic
98C2×Dic37—21no—
99C2×Dic37—21no—
100C2×Dic37—22no—
101C2×Dic37—41no—
102C2×Dic37—42no—
103C2×Dic37C2×D411yescharacteristic
104C2×Dic37C2×D411yescharacteristic
105C4×D37—22no—
106C4×D37—22no—
107C4×D37—41no—
108C4×D37—41no—
109C4×D37—42no—
110C4×D37—42no—
111Dic74—22no—
112Dic74—22no—
113Dic74—22no—
114Dic74—42no—
115Dic74—42no—
116C8×C37—22no—
117#64436—22no—
118C22×C37—21no—
119C22×C37—21no—
120C22×C37C2×Q1612yes—
121C22×C37C22×D411yescharacteristic
122C2×D37—21no—
123C2×D37—21no—
124C2×D37—21no—
125C2×D37—21no—
126C2×D37—44no—
127C2×D37C2×Q1612yes—
128C2×D37C2×Q1612yes—
129C4×C37—21no—
130C4×C37—41no—
131C4×C37—41no—
132C4×C37—42no—
133C4×C37H(5,𝔽2)12yes—
134Dic37—21no—
135Dic37—21no—
136Dic37—41no—
137Dic37—42no—
138Dic37—41no—
139#3189—371nomaximal, Sylow
140C2×C37—22no—
141C2×C37#53411yescharacteristic
142C2×C37C22×Q1611yescharacteristic
143C2×C37#57511yescharacteristic
144D37—22no—
145D37—22no—
146C2×C22⋊Q8—371no—
147C2×C22⋊Q8—371no—
148#484—371no—
149#484—372no—
150#484D3711yescharacteristic, complemented
151#467—372no—
152#467—372no—
153#451—371no—
154#451—372no—
155#451—371no—
156C2×C22⋊C8—371no—
157C37#318911yescharacteristic, complemented, Sylow
158C22×Q8—371no—
159C22×Q8—371no—
160C23×C4—371no—
161C2×Q16—372no—
162C2×Q16—372no—
163C22×C8—372no—
164C22⋊Q8—371no—
165C22⋊Q8—371no—
166C22⋊Q8—372no—
167C22⋊Q8—371no—
168C22⋊Q8—372no—
169C22⋊Q8—741no—
170C22⋊Q8—741no—
171C22⋊Q8—741no—
172C22⋊Q8—741no—
173C22⋊Q8C2×D3711yescharacteristic
174C2×C4⋊C4—371no—
175C2×C4⋊C4—741no—
176C2×C4⋊C4—741no—
177C2×C4⋊C4C2×D3711yescharacteristic
178C2×C22⋊C4—741no—
179C2×C22⋊C4—741no—
180#107—372no—
181#107C2×D3712yescomplemented
182#103—372no—
183#103—372no—
184#103—372no—
185#103C2×D3712yescomplemented
186C22⋊C8—371no—
187C22⋊C8—372no—
188C22⋊C8C2×D3711yescharacteristic
189C24—371no—
190C2×Q8—371no—
191C2×Q8—372no—
192C2×Q8—741no—
193C2×Q8—741no—
194C2×Q8—1482no—
195C2×Q8—1482no—
196C2×Q8C22×D3711yescharacteristic
197C22×C4—21no—
198C22×C4—371no—
199C22×C4—371no—
200C22×C4—371no—
201C22×C4—371no—
202C22×C4—371no—
203C22×C4—741no—
204C22×C4—741no—
205C22×C4—741no—
206C22×C4—1482no—
207C22×C4C22×D3711yescharacteristic
208Q16—1482no—
209Q16—1482no—
210C2×C8—372no—
211C2×C8—1482no—
212C2×C8C22×D3712yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—371no—
216C4⋊C4—741no—
217C4⋊C4—741no—
218C4⋊C4—741no—
219C4⋊C4—742no—
220C4⋊C4—742no—
221C4⋊C4—741no—
222C4⋊C4C22×D3711yescharacteristic
223C4⋊C4C22×D3712yes—
224C22⋊C4—21no—
225C22⋊C4—741no—
226C22⋊C4—741no—
227C22⋊C4—742no—
228C22⋊C4—742no—
229C22⋊C4—741no—
230C23—371no—
231C23—371no—
232C23—374no—
233C23—1481no—
234C23—1481no—
235C23D4×D3711yescharacteristic
236Q8—22no—
237Q8—742no—
238Q8—742no—
239Q8—742no—
240Q8—1482no—
241Q8—1482no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—371no—
248C2×C4—371no—
249C2×C4—742no—
250C2×C4—742no—
251C2×C4—741no—
252C2×C4—742no—
253C2×C4—741no—
254C2×C4—1481no—
255C2×C4—1481no—
256C2×C4—1482no—
257C2×C4—1482no—
258C2×C4—1481no—
259C2×C4—1482no—
260C2×C4D4×D3711yescharacteristic
261C2×C4C23×D3711yescharacteristic
262C8—22no—
263C8—742no—
264C22—21no—
265C22—21no—
266C22—372no—
267C22—372no—
268C22—741no—
269C22—741no—
270C22—741no—
271C22—741no—
272C22—1484no—
273C22Q16×D3712yes—
274C22C2×D4×D3711yescharacteristic
275C4—21no—
276C4—42no—
277C4—41no—
278C4—41no—
279C4—741no—
280C4—741no—
281C4—1481no—
282C4—1481no—
283C4—1482no—
284C4#1192458212yes—
285C2—22no—
286C2—742no—
287C2—742no—
288C2#1455809411yescharacteristic
289C2C2×Q16×D3711yescharacteristic
290C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#484C2
socleC22×C37C22×D4
prime radicalC23×D37D4
hypercenter#484D37
hypocenterC37#3189
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #14557870C2semidirect
4. #14557868C2semidirect
5. #14557775C2semidirect
6. #14557773C2semidirect
7. #14557691C2semidirect
8. #14557690C2semidirect
9. #14557688C2semidirect
11. C37×#484C2semidirect
15. C2×Dic148C22semidirect
16. C2×#10596942C22semidirect
32. #11924469C22semidirect
33. #11924461C22semidirect
34. #11924459C22semidirect
35. #11924459C22semidirect
40. #107×C37C22semidirect
41. #103×C37C22semidirect
87. C22×D37Q16semidirect
150. #484D37semidirect
157. C37#3189semidirect
181. #107C2×D37semidirect
185. #103C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245162
C22⋊Q8×D372
#145578702
#145578682
#145577752
#145577732
#145576912
#145576902
#145576882
C22⋊C8×D372
C37×#4842
#318937