Finite Group Databasebeta

← #18004175

Subgroups — #18004175

Order 4736.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11924555C211yesmaximal, characteristic, complemented
2C22×C8×D37C211yesmaximal, characteristic, complemented
3#14558043C212yesmaximal, complemented
4#14557775C211yesmaximal, characteristic, complemented
5#14557774C211yesmaximal, characteristic, complemented
6#14557766C211yesmaximal, characteristic, complemented
7#14557761C212yesmaximal, complemented
8#14557760C212yesmaximal, complemented
9#14557570C211yesmaximal, characteristic, complemented
10#14557526C212yesmaximal, complemented
11C37×#466C211yesmaximal, characteristic, complemented
12C2×#10596964C2212yes—
13C22×C4×D37C2211yescharacteristic
14#11924555—24no—
15C2×Dic148C2211yescharacteristic, complemented
16C2×#10596940C2212yescomplemented
17C22×#64436C2211yescharacteristic
18C2×C8×D37—24no—
19C2×C8×D37C2211yescharacteristic
20C2×C8×D37C2211yescharacteristic
21C2×C8×D37C2211yescharacteristic
22C2×C8×D37C2211yescharacteristic
23#11924518—22no—
24#11924516C2212yes—
25#11924515C2212yes—
26#11924507C2212yes—
27#11924504—22no—
28D4×Dic37—22no—
29#11924496—22no—
30#11924494C2212yes—
31#11924470C2211yescharacteristic, complemented
32#11924467C2212yescomplemented
33#11924461C2212yescomplemented
34#11924459C2212yescomplemented
35#11924457C2212yescomplemented
36C2×D8×C37C2211yescharacteristic, complemented
37C22×C8×C37C2211yescharacteristic
38#121×C37C2212yes—
39#107×C37C2211yescharacteristic, complemented
40#102×C37C2212yescomplemented
41C23×D37D411yescharacteristic
42C2×D4×C37—22no—
43C2×D4×C37C2312yes—
44C22×C4×C37C2311yescharacteristic
45C2×C22⋊D37—22no—
46C2×C22⋊D37—22no—
47C22×Dic37—22no—
48C22×Dic37D411yescharacteristic
49#10596964—24no—
50#10596964—24no—
51C2×C4×D37—22no—
52C2×C4×D37—22no—
53C2×C4×D37D411yescharacteristic
54C2×C4×D37D411yescharacteristic
55C2×C4×D37C2311yescharacteristic
56C2×C4×D37C2311yescharacteristic
57C2×Dic74C2312yes—
58D8×C37—22no—
59C2×C8×C37—22no—
60C2×C8×C37C2311yescharacteristic
61C2×C8×C37C2311yescharacteristic
62C4⋊C4×C37C2312yes—
63C22⋊C4×C37—22no—
64C22⋊Dic37—22no—
65C22⋊Dic37—22no—
66#10596940—24no—
67#10596938—22no—
68#10596938—22no—
69C4⋊Dic37—22no—
70C4⋊Dic37C2312yes—
71#10596936—22no—
72C4×Dic37—22no—
73C2×#64436—22no—
74C2×#64436C2311yescharacteristic
75C2×#64436C2311yescharacteristic
76Dic148—22no—
77C8×D37—22no—
78C8×D37—22no—
79C8×D37—22no—
80C8×D37D414yescomplemented
81C23×C37—22no—
82C23×C37C2×D411yescharacteristic
83C22×D37—22no—
84C22×D37—22no—
85C22×D37—22no—
86C22×D37C2×D411yescharacteristic
87C22×D37C2×D411yescharacteristic
88D4×C37—22no—
89D4×C37—22no—
90D4×C37—44no—
91C2×C4×C37—21no—
92C2×C4×C37—21no—
93C2×C4×C37—22no—
94C2×C4×C37C2×D411yescharacteristic
95C2×C4×C37C2411yescharacteristic
96C22⋊D37—44no—
97C22⋊D37—44no—
98C2×Dic37—22no—
99C2×Dic37—22no—
100C2×Dic37—22no—
101C2×Dic37—44no—
102C2×Dic37C2×D411yescharacteristic
103C2×Dic37C2×D411yescharacteristic
104C4×D37—22no—
105C4×D37—21no—
106C4×D37—21no—
107C4×D37—22no—
108C4×D37C2×D412yes—
109C4×D37#4112yes—
110Dic74—22no—
111Dic74—22no—
112C8×C37—21no—
113C8×C37C2×D412yes—
114#64436—21no—
115#64436C2×D412yes—
116C22×C37—21no—
117C22×C37—21no—
118C22×C37—42no—
119C22×C37—42no—
120C22×C37—42no—
121C22×C37#13512yescomplemented
122C22×C37C22×D411yescharacteristic
123C2×D37—28no—
124C2×D37—21no—
125C2×D37—21no—
126C2×D37—21no—
127C2×D37—21no—
128C2×D37#12112yes—
129C2×D37#13512yescomplemented
130C4×C37—21no—
131C4×C37—42no—
132C4×C37C22×D411yescharacteristic
133C4×C37C2×#4111yescharacteristic
134Dic37—21no—
135Dic37—42no—
136Dic37—42no—
137Dic37#12112yes—
138#3050—371nomaximal, Sylow
139C2×C37—22no—
140C2×C37—82no—
141C2×C37C2×#12111yescharacteristic
142C2×C37C2×#13511yescharacteristic
143C2×C37C2×#13511yescharacteristic
144D37—22no—
145D37—22no—
146C2×#135—371no—
147C23×C8—371no—
148#525—372no—
149#467—371no—
150#467—371no—
151#466—371no—
152#466D3711yescharacteristic, complemented
153#465—372no—
154#465—372no—
155#427—371no—
156#416—372no—
157C37#305011yescharacteristic, complemented, Sylow
158C2×#41—372no—
159C23×C4—371no—
160#135—744no—
161C2×Q16—371no—
162C2×SD16—372no—
163C2×D8C2×D3711yescharacteristic, complemented
164C22×C8—371no—
165C22×C8—371no—
166C22×C8—371no—
167C22×C8—371no—
168C22×C8—371no—
169C22×C8—744no—
170C22×C8C2×D3711yescharacteristic
171#123—742no—
172C22⋊Q8—372no—
173C22⋊Q8—372no—
174#121—372no—
175#121C2×D3712yes—
176#120—742no—
177C4×D4—742no—
178C4×D4—742no—
179#117—372no—
180#107—371no—
181#107C2×D3711yescharacteristic, complemented
182#106—372no—
183#103—372no—
184#103—372no—
185#102—372no—
186#102C2×D3712yescomplemented
187C24—371no—
188#41—744no—
189#41—744no—
190C2×Q8—372no—
191C2×D4—22no—
192C2×D4—742no—
193C2×D4—742no—
194C2×D4C22×D3712yes—
195C22×C4—371no—
196C22×C4—371no—
197C22×C4—371no—
198C22×C4—371no—
199C22×C4—371no—
200C22×C4—742no—
201C22×C4—742no—
202C22×C4—742no—
203C22×C4C22×D3711yescharacteristic
204Q16—742no—
205SD16—744no—
206D8—22no—
207C2×C8—22no—
208C2×C8—371no—
209C2×C8—374no—
210C2×C8—371no—
211C2×C8—742no—
212C2×C8—742no—
213C2×C8—742no—
214C2×C8—742no—
215C2×C8C22×D3711yescharacteristic
216C2×C8C22×D3711yescharacteristic
217C4⋊C4—372no—
218C4⋊C4—742no—
219C4⋊C4—742no—
220C4⋊C4C22×D3712yes—
221C22⋊C4—22no—
222C22⋊C4—742no—
223C22⋊C4—742no—
224C22⋊C4—742no—
225C22⋊C4—742no—
226C42—742no—
227C23—22no—
228C23—371no—
229C23—371no—
230C23—742no—
231C23—742no—
232C23—742no—
233C23D4×D3711yescharacteristic
234Q8—742no—
235Q8—742no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—1484no—
240D4—1484no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—371no—
245C2×C4—372no—
246C2×C4—372no—
247C2×C4—371no—
248C2×C4—742no—
249C2×C4—742no—
250C2×C4—742no—
251C2×C4—741no—
252C2×C4—741no—
253C2×C4—742no—
254C2×C4—742no—
255C2×C4—1484no—
256C2×C4D4×D3711yescharacteristic
257C2×C4C23×D3711yescharacteristic
258C8—21no—
259C8—372no—
260C8—741no—
261C8D4×D3712yes—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—372no—
268C22—372no—
269C22—748no—
270C22—741no—
271C22—741no—
272C22—741no—
273C22—741no—
274C22#1192455512yescomplemented
275C22C2×D4×D3711yescharacteristic
276C4—21no—
277C4—42no—
278C4—372no—
279C4—741no—
280C4—1482no—
281C4—1482no—
282C4C2×D4×D3711yescharacteristic
283C4#41×D3711yescharacteristic
284C2—22no—
285C2—82no—
286C2—742no—
287C2—742no—
288C2#121×D3711yescharacteristic
289C2#135×D3711yescharacteristic
290C2C2×#1192455511yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#466C2
socleC22×C37C22×D4
hypercenter#466D37
hypocenterC37#3050
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#11924555C2semidirect
2. C22×C8×D37C2semidirect
3. #14558043C2semidirect
4. #14557775C2semidirect
5. #14557774C2semidirect
6. #14557766C2semidirect
7. #14557761C2semidirect
8. #14557760C2semidirect
9. #14557570C2semidirect
10. #14557526C2semidirect
11. C37×#466C2semidirect
15. C2×Dic148C22semidirect
16. C2×#10596940C22semidirect
31. #11924470C22semidirect
32. #11924467C22semidirect
33. #11924461C22semidirect
34. #11924459C22semidirect
35. #11924457C22semidirect
36. C2×D8×C37C22semidirect
39. #107×C37C22semidirect
40. #102×C37C22semidirect
80. C8×D37D4semidirect
121. C22×C37#135semidirect
129. C2×D37#135semidirect
152. #466D37semidirect
157. C37#3050semidirect
163. C2×D8C2×D37semidirect
181. #107C2×D37semidirect
186. #102C2×D37semidirect
274. C22#11924555semidirect

Maximal subgroups

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

subgroupindex
C2×#119245552
C22×C8×D372
#145580432
#145577752
#145577742
#145577662
#145577612
#145577602
#145575702
#145575262
C37×#4662
#305037