Finite Group Databasebeta

← #17995600

Subgroups — #17995600

Order 4736.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×#10596940C211yesmaximal, characteristic, complemented
2C2×#11924516C211yesmaximal, characteristic, complemented
3C2×#11924507C211yesmaximal, characteristic
4C2×#11924461C211yesmaximal, characteristic, complemented
5C2×#11924448C211yesmaximal, characteristic, complemented
6#14557295C211yesmaximal, characteristic, complemented
7C2×#102×C37C211yesmaximal, characteristic, complemented
8C22×Dic74C2211yescharacteristic
9C22×C22⋊D37—21no—
10C22×C4×D37C2211yescharacteristic
11C2×#10596940—22no—
12C2×#10596940—22no—
13C2×#10596940—22no—
14C22×#64436C2211yescharacteristic
15#11924516—22no—
16#11924516—22no—
17#11924507—22no—
18#11924507—22no—
19C2×C4⋊Dic37—21no—
20C2×C4⋊Dic37C2211yescharacteristic
21C2×C22⋊Dic37—21no—
22C2×#10596938—21no—
23#11924461—22no—
24#11924448—22no—
25C22×D4×C37C2211yescharacteristic
26C22×C8×C37C2211yescharacteristic
27C2×C4⋊C4×C37C2211yescharacteristic
28#102×C37—22no—
29C24×C37—21no—
30C23×D37D411yescharacteristic
31C2×D4×C37—21no—
32C2×D4×C37—21no—
33C2×D4×C37—21no—
34C2×D4×C37—21no—
35C2×D4×C37—21no—
36C2×D4×C37D412yes—
37C22×C4×C37—21no—
38C22×C4×C37C2311yescharacteristic
39C2×C22⋊D37—24no—
40C2×C22⋊D37—42no—
41C2×C22⋊D37—42no—
42C22×Dic37—21no—
43C22×Dic37—21no—
44C22×Dic37D411yescharacteristic
45C2×C4×D37—21no—
46C2×C4×D37—21no—
47C2×C4×D37—21no—
48C2×C4×D37—21no—
49C2×C4×D37—21no—
50C2×C4×D37—21no—
51C2×Dic74—21no—
52C2×Dic74—21no—
53C2×Dic74—21no—
54C2×Dic74—21no—
55C2×Dic74—21no—
56C2×Dic74D412yescomplemented
57C2×C8×C37—21no—
58C2×C8×C37—21no—
59C2×C8×C37—21no—
60C4⋊C4×C37—21no—
61C4⋊C4×C37D412yescomplemented
62C22⋊Dic37—24no—
63#10596940—28no—
64#10596938—24no—
65C4⋊Dic37—24no—
66C4⋊Dic37—21no—
67C4⋊Dic37D412yes—
68C2×#64436—21no—
69C2×#64436—21no—
70C2×#64436D412yescomplemented
71C23×C37—22no—
72C23×C37—41no—
73C23×C37—41no—
74C23×C37—41no—
75C23×C37—41no—
76C23×C37—41no—
77C23×C37—41no—
78C23×C37C2×D411yescharacteristic
79C22×D37—21no—
80C22×D37—21no—
81C22×D37—21no—
82C22×D37—21no—
83C22×D37—21no—
84C22×D37—21no—
85C22×D37SD1612yescomplemented
86D4×C37—22no—
87D4×C37—22no—
88D4×C37—41no—
89D4×C37—41no—
90C2×C4×C37—22no—
91C2×C4×C37—41no—
92C2×C4×C37—41no—
93C2×C4×C37C2×D411yescharacteristic
94C2×C4×C37C2×D411yescharacteristic
95C2×C4×C37C2×D411yescharacteristic
96C2×C4×C37C2×D411yescharacteristic
97C2×C4×C37C2×D411yescharacteristic
98C2×C4×C37#4111yescharacteristic
99C22⋊D37—48no—
100C2×Dic37—21no—
101C2×Dic37—22no—
102C2×Dic37—22no—
103C2×Dic37—21no—
104C2×Dic37—21no—
105C2×Dic37—41no—
106C2×Dic37—41no—
107C2×Dic37—41no—
108C2×Dic37—41no—
109C4×D37—41no—
110C4×D37—41no—
111C4×D37—41no—
112C4×D37—41no—
113Dic74—22no—
114Dic74—22no—
115Dic74—41no—
116Dic74—41no—
117C8×C37—41no—
118#64436—22no—
119C22×C37—42no—
120C22×C37—42no—
121C22×C37—42no—
122C22×C37—81no—
123C22×C37—81no—
124C22×C37—81no—
125C22×C37—81no—
126C22×C37#12011yescharacteristic
127C22×C37#12011yescharacteristic
128C22×C37#12111yescharacteristic
129C22×C37C2×SD1611yescharacteristic
130C22×C37#13511yescharacteristic
131C22×C37AGL(1,ℤ/8)11yescharacteristic
132C22×C37#13711yescharacteristic
133C2×D37—22no—
134C2×D37—22no—
135C2×D37—22no—
136C2×D37—41no—
137C2×D37—41no—
138C2×D37—41no—
139C2×D37—41no—
140C4×C37—42no—
141C4×C37#12011yescharacteristic
142C4×C37#12111yescharacteristic
143C4×C37#12111yescharacteristic
144C4×C37#12711yescharacteristic
145Dic37—42no—
146Dic37—41no—
147Dic37—42no—
148#2000—371nomaximal, Sylow
149C2×C37—82no—
150C2×C37#39211yescharacteristic
151C2×C37#44811yescharacteristic
152C2×C37#44911yescharacteristic
153C2×C37#45011yescharacteristic
154C2×C37#45211yescharacteristic
155C2×C37#46511yescharacteristic
156C2×C37#46811yescharacteristic
157D37—42no—
158C22×SD16—371no—
159C2×C22⋊Q8—371no—
160C2×#121—371no—
161C2×#103—371no—
162C2×#102D3711yescharacteristic, complemented
163C2×C22⋊C8—371no—
164#340—371no—
165C37#200011yescharacteristic, complemented, Sylow
166C22×Q8—371no—
167C22×D4—741no—
168C22×D4C2×D3711yescharacteristic
169C23×C4—371no—
170C2×SD16—742no—
171C2×SD16—742no—
172C2×SD16—742no—
173C22×C8—371no—
174C22×C8C2×D3711yescharacteristic
175C22⋊Q8—742no—
176C22⋊Q8—742no—
177#121—742no—
178#121—742no—
179C2×C4⋊C4—371no—
180C2×C4⋊C4—741no—
181C2×C4⋊C4C2×D3711yescharacteristic
182C2×C22⋊C4—741no—
183C2×C22⋊C4—741no—
184#103—742no—
185#102—22no—
186C22⋊C8—742no—
187C24—21no—
188C24—371no—
189C2×Q8—372no—
190C2×Q8—741no—
191C2×Q8—741no—
192C2×Q8—741no—
193C2×Q8—741no—
194C2×Q8—741no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—744no—
201C2×D4—1482no—
202C2×D4—1482no—
203C2×D4D14812yes—
204C22×C4—21no—
205C22×C4—371no—
206C22×C4—741no—
207C22×C4—741no—
208C22×C4—741no—
209C22×C4—741no—
210C22×C4—741no—
211C22×C4—741no—
212C22×C4—741no—
213C22×C4—741no—
214C22×C4C22×D3711yescharacteristic
215SD16—748no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—372no—
220C2×C8—741no—
221C2×C8—741no—
222C4⋊C4—21no—
223C4⋊C4—372no—
224C4⋊C4—744no—
225C4⋊C4—741no—
226C4⋊C4C22⋊D3712yescomplemented
227C22⋊C4—744no—
228C22⋊C4—744no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—372no—
237C23—741no—
238C23—741no—
239C23—741no—
240C23—741no—
241C23—741no—
242C23—741no—
243C23D4×D3711yescharacteristic
244Q8—742no—
245Q8—742no—
246Q8—1481no—
247Q8—1481no—
248D4—22no—
249D4—22no—
250D4—41no—
251D4—41no—
252D4—1488no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—742no—
257C2×C4—741no—
258C2×C4—741no—
259C2×C4—742no—
260C2×C4—741no—
261C2×C4—1481no—
262C2×C4—1481no—
263C2×C4—1481no—
264C2×C4—1481no—
265C2×C4—1481no—
266C2×C4—1481no—
267C2×C4—1481no—
268C2×C4—1481no—
269C2×C4C2×D14811yescharacteristic
270C2×C4#1059696211yescharacteristic
271C2×C4D4×D3711yescharacteristic
272C2×C4D4×D3711yescharacteristic
273C2×C4D4×D3711yescharacteristic
274C2×C4C2×C22⋊D3711yescharacteristic
275C8—41no—
276C8—742no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—742no—
285C22—742no—
286C22—742no—
287C22—1481no—
288C22—1481no—
289C22—1481no—
290C22—1481no—
291C22#1192450411yescharacteristic
292C22#1192450611yescharacteristic
293C22#1192450911yescharacteristic
294C22SD16×D3711yescharacteristic
295C22#1192455511yescharacteristic
296C22#1192456011yescharacteristic
297C22#1192456511yescharacteristic
298C4—42no—
299C4—1481no—
300C4—1482no—
301C4—1482no—
302C4#1192450611yescharacteristic
303C4#1192450911yescharacteristic
304C4#1192451011yescharacteristic
305C4#1192453211yescharacteristic
306C2—82no—
307C2—1482no—
308C2#1455744211yescharacteristic
309C2#1455767011yescharacteristic
310C2#1455767511yescharacteristic
311C2#1455768511yescharacteristic
312C2#1455769611yescharacteristic
313C2#1455776111yescharacteristic
314C2#1455777811yescharacteristic
subgroupG/H
centerC23D4×D37
commutator, cosocleC22×C4×C37C23
FrattiniC22×C4C22×D37
FittingC2×#102×C37C2
socleC23×C37C2×D4
prime radicalC2×#11924507C2
hypercenterC2×#102D37
hypocenterC37#2000
LCSG ▷ C22×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#10596940C2semidirect
2. C2×#11924516C2semidirect
4. C2×#11924461C2semidirect
5. C2×#11924448C2semidirect
6. #14557295C2semidirect
7. C2×#102×C37C2semidirect
56. C2×Dic74D4semidirect
61. C4⋊C4×C37D4semidirect
70. C2×#64436D4semidirect
85. C22×D37SD16semidirect
162. C2×#102D37semidirect
165. C37#2000semidirect
226. C4⋊C4C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C22×#105969402
C2×#119245162
C2×#119245072
C2×#119244612
C2×#119244482
#145572952
C2×#102×C372
#200037