Finite Group Databasebeta

← #18004065

Subgroups — #18004065

Order 4736.

484 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 143 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×#11924567C211yesmaximal, characteristic, complemented
2#14558079C211yesmaximal, characteristic
3C2×Q8×Dic37C211yesmaximal, characteristic, complemented
4#14557757C212yesmaximal, complemented
5#14557753C212yesmaximal, complemented
6#14557741C212yesmaximal, complemented
7#14557731C212yesmaximal, complemented
8#14557556C211yesmaximal, characteristic, complemented
9#14557535C211yesmaximal, characteristic, complemented
10#14557534C211yesmaximal, characteristic, complemented
11C2×#137×C37C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13#11924567—24no—
14C2×#10596942C2212yes—
15C2×#10596941C2212yes—
16C2×#10596934C2211yescharacteristic
17#11924534C2211yescharacteristic
18#11924532C2211yescharacteristic
19#11924525C2211yescharacteristic
20#11924523C2211yescharacteristic
21#11924508—21no—
22#11924508—21no—
23Q8×Dic37—21no—
24Q8×Dic37—21no—
25Q8×Dic37C2212yes—
26Q8×Dic37C2212yes—
27C4×C22⋊D37—21no—
28C4×C22⋊D37—21no—
29C2×C4⋊Dic37—21no—
30C2×C4⋊Dic37C2211yescharacteristic
31C2×C4×Dic37—21no—
32C2×C4×Dic37C2211yescharacteristic
33#11924465C2212yes—
34#11924465C2212yes—
35#11924461C2212yes—
36#11924460C2212yes—
37#11924458C2212yes—
38#11924457C2212yes—
39C2×#41×C37C2211yescharacteristic
40C22×Q8×C37C2211yescharacteristic
41#137×C37—24no—
42C2×Q16×C37C2212yes—
43C2×SD16×C37C2212yes—
44C2×M16×C37C2211yescharacteristic
45#41×C37—22no—
46#41×C37—22no—
47C2×Q8×C37—21no—
48C2×Q8×C37—22no—
49C2×Q8×C37D414yescomplemented
50C2×Q8×C37C2311yescharacteristic
51C2×Q8×C37C2312yes—
52C2×D4×C37—21no—
53C2×D4×C37C2311yescharacteristic
54C22×C4×C37—21no—
55C22×C4×C37—21no—
56C22×C4×C37C2311yescharacteristic
57C2×C22⋊D37—21no—
58C2×C22⋊D37—21no—
59C2×C22⋊D37—21no—
60C22×Dic37—21no—
61C22×Dic37D411yescharacteristic
62C22×Dic37D411yescharacteristic
63#10596962—22no—
64#10596962—22no—
65C2×D148C2311yescharacteristic
66C2×C4×D37—21no—
67C2×Dic74C2311yescharacteristic
68Q16×C37—24no—
69SD16×C37—24no—
70M16×C37—22no—
71C2×C8×C37C2312yes—
72C22⋊Dic37—21no—
73C22⋊Dic37—21no—
74#10596942—24no—
75#10596941—24no—
76#10596938—21no—
77#10596938—21no—
78C4⋊Dic37—21no—
79C4⋊Dic37—22no—
80C4⋊Dic37—22no—
81C4⋊Dic37C2312yes—
82#10596936—21no—
83#10596936—21no—
84C4×Dic37—22no—
85C4×Dic37—22no—
86C4×Dic37D411yescharacteristic
87C4×Dic37D411yescharacteristic
88C4×Dic37C2311yescharacteristic
89C4×Dic37C2311yescharacteristic
90#10596934—22no—
91C2×#64436C2312yes—
92C23×C37—21no—
93C23×C37C2×D411yescharacteristic
94C22×D37—21no—
95Q8×C37—24no—
96Q8×C37—22no—
97Q8×C37—21no—
98Q8×C37—21no—
99Q8×C37C2×D414yes—
100D4×C37—21no—
101D4×C37—21no—
102D4×C37—42no—
103C2×C4×C37—24no—
104C2×C4×C37—22no—
105C2×C4×C37—21no—
106C2×C4×C37—42no—
107C2×C4×C37C2×D411yescharacteristic
108C2×C4×C37C2×D412yes—
109C2×C4×C37#4112yes—
110C2×C4×C37C2411yescharacteristic
111C22⋊D37—42no—
112C22⋊D37—42no—
113C22⋊D37—42no—
114C2×Dic37—22no—
115C2×Dic37—22no—
116C2×Dic37—21no—
117C2×Dic37—41no—
118C2×Dic37—42no—
119C2×Dic37C2×D411yescharacteristic
120C2×Dic37C2×D411yescharacteristic
121C2×Dic37C2×D411yescharacteristic
122C2×Dic37C2×D411yescharacteristic
123D148—21no—
124D148—21no—
125C4×D37—42no—
126Dic74—21no—
127Dic74—21no—
128C8×C37—22no—
129#64436—22no—
130C22×C37—21no—
131C22×C37—21no—
132C22×C37—41no—
133C22×C37—41no—
134C22×C37—41no—
135C22×C37#12112yes—
136C22×C37C22×D411yescharacteristic
137C2×D37—41no—
138C2×D37—41no—
139C2×D37—41no—
140C4×C37—24no—
141C4×C37—41no—
142C4×C37#12112yes—
143C4×C37C22×D411yescharacteristic
144C4×C37C2×#4111yescharacteristic
145Dic37—21no—
146Dic37—21no—
147Dic37—21no—
148Dic37—41no—
149Dic37—42no—
150Dic37#13712yescomplemented
151#3037—371nomaximal, Sylow
152C2×C37—22no—
153C2×C37—81no—
154C2×C37C2×#12111yescharacteristic
155C2×C37C2×#13711yescharacteristic
156C2×C37#57511yescharacteristic
157D37—81no—
158C2×#137—371no—
159C2×#137D3711yescharacteristic, complemented
160#532—371no—
161C2×C4×Q8—371no—
162#464—372no—
163#464—372no—
164#462—372no—
165#460—372no—
166#423—371no—
167#417—371no—
168#417—371no—
169C37#303711yescharacteristic, complemented, Sylow
170C2×#41—371no—
171C2×#41C2×D3711yescharacteristic
172C22×Q8C2×D3711yescharacteristic
173#137—24no—
174#137—744no—
175C2×Q16—372no—
176C2×Q16C2×D3712yes—
177C2×SD16—372no—
178C2×SD16C2×D3712yes—
179C2×M16—371no—
180C2×M16C2×D3711yescharacteristic
181C4⋊Q8—371no—
182#127—371no—
183#124—371no—
184#124—371no—
185#121—741no—
186#121—741no—
187C4×Q8—372no—
188C4×Q8—372no—
189C4×Q8—741no—
190C4×Q8—741no—
191C4×D4—741no—
192C4×D4—741no—
193C2×C4⋊C4—371no—
194C2×C4⋊C4—741no—
195C2×C42—371no—
196C2×C42—741no—
197C4⋊C8—372no—
198C4⋊C8—372no—
199#103—372no—
200#103—372no—
201#102—372no—
202#102—372no—
203#41—22no—
204#41—22no—
205#41—742no—
206#41—742no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×Q8—371no—
210C2×Q8C22⋊D3714yescomplemented
211C2×Q8C22×D3711yescharacteristic
212C2×Q8C22×D3712yes—
213C2×D4—21no—
214C2×D4—371no—
215C2×D4—741no—
216C2×D4—741no—
217C2×D4—741no—
218C2×D4C22×D3711yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—371no—
222C22×C4—371no—
223C22×C4—741no—
224C22×C4—741no—
225C22×C4C22×D3711yescharacteristic
226Q16—24no—
227Q16—744no—
228SD16—24no—
229SD16—744no—
230M16—22no—
231M16—742no—
232C2×C8—372no—
233C2×C8C22×D3712yes—
234C4⋊C4—372no—
235C4⋊C4—741no—
236C4⋊C4—741no—
237C4⋊C4—742no—
238C4⋊C4—742no—
239C4⋊C4—741no—
240C22⋊C4—741no—
241C22⋊C4—741no—
242C22⋊C4—741no—
243C22⋊C4—741no—
244C42—371no—
245C42—371no—
246C42—371no—
247C42—371no—
248C42—742no—
249C42—742no—
250C23—21no—
251C23—741no—
252C23D4×D3711yescharacteristic
253Q8—24no—
254Q8—22no—
255Q8—21no—
256Q8—21no—
257Q8—741no—
258Q8—741no—
259Q8C2×C22⋊D3714yes—
260D4—21no—
261D4—21no—
262D4—42no—
263D4—741no—
264D4—741no—
265D4—1482no—
266D4—1482no—
267D4—1482no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—371no—
273C2×C4—371no—
274C2×C4—371no—
275C2×C4—371no—
276C2×C4—742no—
277C2×C4—742no—
278C2×C4—741no—
279C2×C4—1481no—
280C2×C4—1482no—
281C2×C4—1482no—
282C2×C4D4×D3711yescharacteristic
283C2×C4#1059696412yes—
284C2×C4C2×C22⋊D3712yes—
285C2×C4C23×D3711yescharacteristic
286C8—22no—
287C8—742no—
288C22—21no—
289C22—21no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—1481no—
294C22—1481no—
295C22—1481no—
296C22#1192450812yes—
297C22C2×D4×D3711yescharacteristic
298C4—24no—
299C4—41no—
300C4—372no—
301C4—741no—
302C4—741no—
303C4—741no—
304C4—1481no—
305C4—1482no—
306C4#1192450812yes—
307C4C22×C22⋊D3711yescharacteristic
308C4C2×#1059696411yescharacteristic
309C2—22no—
310C2—81no—
311C2—2961no—
312C2C2×#1192450811yescharacteristic
313C2#137×D3711yescharacteristic
314C2#1455833211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×#137×C37C2
socleC22×C37C22×D4
prime radical#14558079C2
hypercenterC2×#137D37
hypocenterC37#3037
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#11924567C2semidirect
3. C2×Q8×Dic37C2semidirect
4. #14557757C2semidirect
5. #14557753C2semidirect
6. #14557741C2semidirect
7. #14557731C2semidirect
8. #14557556C2semidirect
9. #14557535C2semidirect
10. #14557534C2semidirect
11. C2×#137×C37C2semidirect
49. C2×Q8×C37D4semidirect
150. Dic37#137semidirect
159. C2×#137D37semidirect
169. C37#3037semidirect
210. C2×Q8C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245672
#145580792
C2×Q8×Dic372
#145577572
#145577532
#145577412
#145577312
#145575562
#145575352
#145575342
C2×#137×C372
#303737