Finite Group Databasebeta

← #17994561

Subgroups — #17994561

Order 4736.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#14558369C211yesmaximal, characteristic
2#14557551C211yesmaximal, characteristic, complemented
3#14557550C211yesmaximal, characteristic, complemented
4#14557530C211yesmaximal, characteristic, complemented
5#14557506C211yesmaximal, characteristic, complemented
6#14557499C211yesmaximal, characteristic, complemented
7C37×#416C211yesmaximal, characteristic, complemented
8#11924584—21no—
9#11924583—21no—
10#11924583C412yescomplemented
11#11924582—21no—
12#11924582C412yescomplemented
13#11924581—21no—
14#11924581—22no—
15C2×#10596962—22no—
16C2×#10596962C2211yescharacteristic
17C2×#10596962C2211yescharacteristic
18#41×D37—22no—
19#41×D37—22no—
20C2×#10596934C2211yescharacteristic, complemented
21#11924495C2211yescharacteristic, complemented
22#11924464—22no—
23#11924463—22no—
24#11924461—21no—
25#11924458—21no—
26#11924450—22no—
27#11924448—22no—
28C2×#41×C37C2211yescharacteristic
29C22×C8×C37C2211yescharacteristic, complemented
30#117×C37C2211yescharacteristic, complemented
31#103×C37—21no—
32#102×C37—21no—
33#41×C37—21no—
34#41×C37—21no—
35#41×C37C2×C412yes—
36#41×C37D412yes—
37C2×Q8×C37D411yescharacteristic
38C2×D4×C37—21no—
39C2×D4×C37D411yescharacteristic
40C22×C4×C37—21no—
41C22×C4×C37C2311yescharacteristic
42C2×C22⋊D37—24no—
43C2×C22⋊D37—21no—
44C2×C22⋊D37D412yes—
45#10596966—22no—
46#10596966—22no—
47Q8×D37—22no—
48Q8×D37—22no—
49#10596964—24no—
50#10596964—22no—
51#10596964—22no—
52#10596964—24no—
53D4×D37—24no—
54D4×D37—22no—
55D4×D37—24no—
56D4×D37—22no—
57#10596962—24no—
58#10596962—24no—
59#10596962—21no—
60#10596962—21no—
61#10596962—21no—
62#10596962—21no—
63#10596962—21no—
64#10596962—21no—
65#10596962—22no—
66#10596962—22no—
67#10596962C2×C412yes—
68#10596962D412yes—
69C2×D148—22no—
70C2×D148C2×C411yescharacteristic
71C2×D148D411yescharacteristic
72C2×C4×D37—24no—
73C2×C4×D37—21no—
74C2×C4×D37D412yes—
75C2×Dic74—22no—
76C2×Dic74C2×C411yescharacteristic
77C2×Dic74D411yescharacteristic
78C2×C8×C37—21no—
79C2×C8×C37—21no—
80C2×C8×C37—21no—
81C4⋊C4×C37—21no—
82C22⋊C4×C37—21no—
83C42×C37—21no—
84C22⋊Dic37—21no—
85C4⋊Dic37—21no—
86C4×Dic37—21no—
87#10596934—21no—
88#10596934—21no—
89C2×#64436—21no—
90C23×C37—21no—
91C23×C37C2×D411yescharacteristic
92C22×D37—24no—
93C22×D37—21no—
94C22×D37C22⋊C412yes—
95Q8×C37—21no—
96Q8×C37C22⋊C412yes—
97D4×C37—22no—
98D4×C37—22no—
99D4×C37—21no—
100D4×C37C22⋊C412yes—
101C2×C4×C37—21no—
102C2×C4×C37—22no—
103C2×C4×C37—22no—
104C2×C4×C37—41no—
105C2×C4×C37C22×C411yescharacteristic
106C2×C4×C37C2×D411yescharacteristic
107C2×C4×C37C2×D411yescharacteristic
108C2×C4×C37C2×D411yescharacteristic
109C2×C4×C37C2×D411yescharacteristic
110C2×C4×C37C2×D411yescharacteristic
111C22⋊D37—24no—
112C22⋊D37—24no—
113C22⋊D37—22no—
114C22⋊D37—22no—
115C22⋊D37—22no—
116C22⋊D37—22no—
117C2×Dic37—24no—
118C2×Dic37—21no—
119C2×Dic37—41no—
120C2×Dic37C22⋊C412yes—
121D148—24no—
122D148—21no—
123D148—21no—
124D148—21no—
125D148C22⋊C412yes—
126C4×D37—24no—
127C4×D37—24no—
128C4×D37—22no—
129C4×D37—22no—
130C4×D37—22no—
131C4×D37—22no—
132Dic74—24no—
133Dic74—21no—
134Dic74—21no—
135Dic74—21no—
136Dic74C22⋊C412yes—
137C8×C37—41no—
138#64436—41no—
139C22×C37—22no—
140C22×C37—41no—
141C22×C37—81no—
142C22×C37C2×C22⋊C411yescharacteristic
143C22×C37#12011yescharacteristic
144C22×C37#12011yescharacteristic
145C2×D37—22no—
146C2×D37—22no—
147C2×D37—42no—
148C2×D37—81no—
149C2×D37—84no—
150C4×C37—22no—
151C4×C37—81no—
152C4×C37C2×C22⋊C411yescharacteristic
153C4×C37C2×C22⋊C411yescharacteristic
154C4×C37#12011yescharacteristic
155C4×C37#12011yescharacteristic
156Dic37—22no—
157Dic37—22no—
158Dic37—81no—
159#1794—371nomaximal, Sylow
160C2×C37—21no—
161C2×C37—21no—
162C2×C37—21no—
163C2×C37—42no—
164C2×C37#37911yescharacteristic
165D37—42no—
166D37—42no—
167#585—371no—
168#421—371no—
169#421—371no—
170#416—371no—
171#416D3711yescharacteristic, complemented
172#410—371no—
173#408—371no—
174C37#179411yescharacteristic, complemented, Sylow
1752−1+4—372no—
1762−1+4—741no—
1772−1+4—741no—
178H(5,𝔽2)—372no—
179H(5,𝔽2)—742no—
180H(5,𝔽2)—741no—
181H(5,𝔽2)—741no—
182C2×#41—371no—
183C2×#41—371no—
184C2×#41—742no—
185C2×#41—742no—
186C2×#41—742no—
187C2×#41C2×D3711yescharacteristic
188C2×M16—371no—
189C22×C8C2×D3711yescharacteristic, complemented
190#117—371no—
191#117C2×D3711yescharacteristic, complemented
192#104—742no—
193#104—742no—
194#103—21no—
195#103—741no—
196#102—21no—
197#102—741no—
198#99—742no—
199C22⋊C8—742no—
200#41—21no—
201#41—21no—
202#41—372no—
203#41—372no—
204#41—742no—
205#41—742no—
206#41—744no—
207#41—744no—
208#41—744no—
209#41—744no—
210#41—742no—
211#41—742no—
212#41—742no—
213#41—742no—
214#41—741no—
215#41—741no—
216#41—741no—
217#41—741no—
218#41—741no—
219#41—741no—
220#41C4×D3712yes—
221#41D14812yes—
222C2×Q8—371no—
223C2×Q8—371no—
224C2×Q8—742no—
225C2×Q8—742no—
226C2×Q8—742no—
227C2×Q8C22⋊D3711yescharacteristic
228C2×D4—21no—
229C2×D4—372no—
230C2×D4—371no—
231C2×D4—371no—
232C2×D4—742no—
233C2×D4—744no—
234C2×D4—744no—
235C2×D4—742no—
236C2×D4—744no—
237C2×D4—741no—
238C2×D4—742no—
239C2×D4C22⋊D3711yescharacteristic
240C22×C4—21no—
241C22×C4—372no—
242C22×C4—744no—
243C22×C4—741no—
244C22×C4C22×D3711yescharacteristic
245M16—741no—
246M16—741no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—741no—
251C4⋊C4—21no—
252C4⋊C4—741no—
253C22⋊C4—21no—
254C22⋊C4—741no—
255C42—21no—
256C42—741no—
257C23—21no—
258C23—372no—
259C23—744no—
260C23—741no—
261C23D4×D3711yescharacteristic
262Q8—21no—
263Q8—372no—
264Q8—744no—
265Q8—741no—
266Q8—741no—
267Q8—741no—
268Q8#1059693812yes—
269D4—22no—
270D4—22no—
271D4—21no—
272D4—372no—
273D4—744no—
274D4—744no—
275D4—744no—
276D4—742no—
277D4—742no—
278D4—742no—
279D4—742no—
280D4—741no—
281D4—741no—
282D4—741no—
283D4#1059693812yes—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—372no—
289C2×C4—744no—
290C2×C4—744no—
291C2×C4—744no—
292C2×C4—741no—
293C2×C4—742no—
294C2×C4—742no—
295C2×C4—742no—
296C2×C4—742no—
297C2×C4—1481no—
298C2×C4C2×C4×D3711yescharacteristic
299C2×C4C2×D14811yescharacteristic
300C2×C4D4×D3711yescharacteristic
301C2×C4D4×D3711yescharacteristic
302C2×C4D4×D3711yescharacteristic
303C2×C4C2×C22⋊D3711yescharacteristic
304C8—41no—
305C8—1481no—
306C22—22no—
307C22—41no—
308C22—81no—
309C22—2961no—
310C22—2964no—
311C22—742no—
312C22—742no—
313C22—1482no—
314C22C22⋊C4×D3711yescharacteristic
315C22#1192450411yescharacteristic
316C22#1192450611yescharacteristic
317C4—22no—
318C4—81no—
319C4—2961no—
320C4—742no—
321C4—742no—
322C4C22⋊C4×D3711yescharacteristic
323C4C2×#1059693811yescharacteristic
324C4#1192450411yescharacteristic
325C4#1192450611yescharacteristic
326C2—21no—
327C2—21no—
328C2—21no—
329C2—42no—
330C2—1482no—
331C2—1482no—
332C2#1455738811yescharacteristic
subgroupG/H
centerC4#11924506
commutatorC2×C4×C37C22×C4
FrattiniC22×C4C22×D37
FittingC37×#416C2
socleC2×C37#379
prime radical#14558369C2
cosocleC22×C4×C37C23
hypercenter#416D37
hypocenterC37#1794
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #14557551C2semidirect
3. #14557550C2semidirect
4. #14557530C2semidirect
5. #14557506C2semidirect
6. #14557499C2semidirect
7. C37×#416C2semidirect
10. #11924583C4semidirect
12. #11924582C4semidirect
20. C2×#10596934C22semidirect
21. #11924495C22semidirect
29. C22×C8×C37C22semidirect
30. #117×C37C22semidirect
171. #416D37semidirect
174. C37#1794semidirect
189. C22×C8C2×D37semidirect
191. #117C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145583692
#145575512
#145575502
#145575302
#145575062
#145574992
C37×#4162
#179437