Finite Group Databasebeta

← #13125312

Subgroups — #13125312

Order 1664.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#88163C211yesmaximal, characteristic, complemented
2C2×#88162C211yesmaximal, characteristic, complemented
3C2×#88158C211yesmaximal, characteristic, complemented
4C22×#6095C211yesmaximal, characteristic, complemented
5#11719231C211yesmaximal, characteristic, complemented
6#11719230C211yesmaximal, characteristic, complemented
7C13×#405C211yesmaximal, characteristic, complemented
8#11718376C212yesmaximal, complemented
9#11718373C212yesmaximal, complemented
10#11718357C212yesmaximal, complemented
11#11718354C212yesmaximal, complemented
12C2×#6127C2211yescharacteristic
13C2×#6127C2211yescharacteristic
14C22×D52C2211yescharacteristic
15C22×Dic26C2211yescharacteristic
16C2×C13×M16C2211yescharacteristic
17C22×C8×C13C2211yescharacteristic
18C13×#117C2211yescharacteristic
19#88163—24no—
20#88162—24no—
21C2×Dic52C2211yescharacteristic, complemented
22C2×Dic52C2212yescomplemented
23#88158—24no—
24C2×D104C2212yes—
25C2×D104C2211yescharacteristic, complemented
26C2×#6095—24no—
27C2×#6095C2212yes—
28C2×#6095C2212yes—
29C2×#6095C2212yes—
30C2×#6095C2211yescharacteristic, complemented
31C2×#6095C2211yescharacteristic, complemented
32C2×#6095C2212yescomplemented
33#88151—22no—
34#88149—22no—
35#88140—22no—
36#88136—22no—
37#88129C2212yes—
38#88129C2212yescomplemented
39#88128C2212yes—
40#88123C2212yescomplemented
41C13×#97C2212yescomplemented
42C4×C8×C13C2212yescomplemented
43C23×D13—21no—
44C22×C4×C13C2311yescharacteristic
45C2×C22⋊D13—21no—
46C2×C22⋊D13—21no—
47C22×Dic13—21no—
48#6127—22no—
49#6127—22no—
50#6127—22no—
51#6127—22no—
52C2×D52—22no—
53C2×D52—22no—
54C2×D52—21no—
55C2×D52—24no—
56C2×D52C2311yescharacteristic
57C2×D52C2312yes—
58C2×D52C2311yescharacteristic
59C2×C4×D13—21no—
60C2×C4×D13—21no—
61C2×Dic26—22no—
62C2×Dic26—22no—
63C2×Dic26—21no—
64C2×Dic26C2311yescharacteristic
65C2×Dic26C2312yes—
66C2×Dic26C2311yescharacteristic
67C13×M16D414yescomplemented
68C2×C8×C13D414yescomplemented
69C2×C8×C13C2311yescharacteristic
70C2×C8×C13C2312yes—
71C2×C8×C13C2311yescharacteristic
72C13×C4⋊C4D412yes—
73C13×C22⋊C4D412yes—
74C42×C13C2312yes—
75#6103—24no—
76#6103—24no—
77C4⋊Dic13—24no—
78Dic52—22no—
79Dic52—24no—
80D104—24no—
81D104—22no—
82#6095—24no—
83#6095—24no—
84#6095—24no—
85#6095—22no—
86#6095—22no—
87#6095—24no—
88#3154—131nomaximal, Sylow
89C23×C13C2×D411yescharacteristic
90C22×D13—21no—
91C22×D13—22no—
92C22×D13—21no—
93C22×D13—42no—
94C22×D13—82no—
95C2×C4×C13C2×D414yes—
96C2×C4×C13C2×D411yescharacteristic
97C2×C4×C13C2×D412yes—
98C2×C4×C13C2×D412yes—
99C2×C4×C13C2411yescharacteristic
100C22⋊D13—42no—
101C22⋊D13—42no—
102C2×Dic13—21no—
103C2×Dic13—22no—
104C2×Dic13—21no—
105C2×Dic13—42no—
106D52—22no—
107D52—21no—
108D52—22no—
109D52—21no—
110D52—21no—
111D52—21no—
112D52—42no—
113D52—84no—
114C4×D13—42no—
115C4×D13—42no—
116Dic26—22no—
117Dic26—21no—
118Dic26—22no—
119Dic26—21no—
120Dic26—21no—
121Dic26—21no—
122Dic26—42no—
123C8×C13C2×D412yes—
124C8×C13C2×D412yes—
125C8×C13C2×D414yes—
126C2×#137—131no—
127C2×AGL(1,ℤ/8)—131no—
128C2×#135—131no—
129C22×SD16—131no—
130#536—131no—
131#535—131no—
132#497—132no—
133#496—132no—
134#495—132no—
135#492—132no—
136#405D1311yescharacteristic, complemented
137C22×C13—22no—
138C22×C13#12712yes—
139C22×C13C22×D411yescharacteristic
140C2×D13—41no—
141C2×D13—41no—
142C2×D13—42no—
143C2×D13—42no—
144C2×D13—44no—
145C2×D13—42no—
146C2×D13—84no—
147C4×C13—24no—
148C4×C13#12712yes—
149C4×C13C22×D411yescharacteristic
150C4×C13C22×D411yescharacteristic
151Dic13—42no—
152Dic13—41no—
153Dic13—41no—
154C2×#41—131no—
155C2×#41—131no—
156C22×Q8—131no—
157C22×D4—131no—
158#137—264no—
159AGL(1,ℤ/8)—264no—
160#135—264no—
161C2×Q16—131no—
162C2×Q16—132no—
163C2×SD16—132no—
164C2×SD16—132no—
165C2×SD16—131no—
166C2×SD16—131no—
167C2×SD16—132no—
168C2×SD16—132no—
169C2×SD16—264no—
170C2×D8—131no—
171C2×D8—132no—
172C2×M16C2×D1311yescharacteristic
173C22×C8C2×D1311yescharacteristic
174C4⋊Q8—132no—
175#127—132no—
176#124—132no—
177#124—132no—
178#123—262no—
179C22⋊Q8—262no—
180#121—262no—
181#120—262no—
182#117C2×D1311yescharacteristic
183#97C2×D1312yescomplemented
184C4×C8C2×D1312yescomplemented
185C2×C13—22no—
186C2×C13C2×#12711yescharacteristic
187C2×C13#57712yes—
188D13—81no—
189D13—81no—
190D13—82no—
191C24—261no—
192#41—262no—
193#41—262no—
194#41—262no—
195#41—262no—
196C2×Q8—131no—
197C2×Q8—132no—
198C2×Q8—131no—
199C2×Q8—261no—
200C2×Q8—262no—
201C2×Q8—262no—
202C2×D4—131no—
203C2×D4—132no—
204C2×D4—131no—
205C2×D4—261no—
206C2×D4—261no—
207C2×D4—262no—
208C2×D4—262no—
209C2×D4—261no—
210C2×D4—264no—
211C22×C4—261no—
212C22×C4—261no—
213C22×C4—261no—
214C22×C4C22×D1311yescharacteristic
215Q16—262no—
216Q16—264no—
217SD16—264no—
218SD16—264no—
219SD16—262no—
220SD16—262no—
221SD16—264no—
222SD16—264no—
223D8—262no—
224D8—264no—
225M16D5214yescomplemented
226C2×C8D5214yescomplemented
227C2×C8C22×D1311yescharacteristic
228C2×C8C22×D1312yes—
229C2×C8C22×D1311yescharacteristic
230C4⋊C4—264no—
231C4⋊C4D5212yes—
232C22⋊C4—264no—
233C22⋊C4—264no—
234C22⋊C4D5212yes—
235C42C22×D1312yes—
236C13#315411yescharacteristic, complemented, Sylow
237C23—261no—
238C23—262no—
239C23—261no—
240C23—522no—
241C23—1042no—
242C23C2×D5211yescharacteristic
243Q8—262no—
244Q8—261no—
245Q8—262no—
246Q8—261no—
247Q8—261no—
248Q8—261no—
249Q8—522no—
250D4—261no—
251D4—262no—
252D4—261no—
253D4—261no—
254D4—262no—
255D4—261no—
256D4—522no—
257D4—522no—
258D4—522no—
259D4—1044no—
260C2×C4—261no—
261C2×C4—262no—
262C2×C4—261no—
263C2×C4—522no—
264C2×C4—522no—
265C2×C4—522no—
266C2×C4C2×D5214yes—
267C2×C4C2×D5211yescharacteristic
268C2×C4C2×D5212yes—
269C2×C4C2×D5212yes—
270C2×C4C23×D1311yescharacteristic
271C8C2×D5212yes—
272C8C2×D5212yes—
273C8C2×D5214yes—
274C22—22no—
275C22—521no—
276C22—521no—
277C22—522no—
278C22—522no—
279C22—524no—
280C22—522no—
281C22—1044no—
282C22#8812812yes—
283C22C22×D5211yescharacteristic
284C4—24no—
285C4—521no—
286C4—521no—
287C4—522no—
288C4#8812812yes—
289C4C22×D5211yescharacteristic
290C4C22×D5211yescharacteristic
291C2—22no—
292C2—1042no—
293C2—1041no—
294C2—1041no—
295C2C2×#8812811yescharacteristic
296C2#1171945612yes—
subgroupG/H
centerC22C22×D52
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#405C2
socleC22×C13C22×D4
hypercenter#405D13
hypocenterC13#3154
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#88163C2semidirect
2. C2×#88162C2semidirect
3. C2×#88158C2semidirect
4. C22×#6095C2semidirect
5. #11719231C2semidirect
6. #11719230C2semidirect
7. C13×#405C2semidirect
8. #11718376C2semidirect
9. #11718373C2semidirect
10. #11718357C2semidirect
11. #11718354C2semidirect
21. C2×Dic52C22semidirect
22. C2×Dic52C22semidirect
25. C2×D104C22semidirect
30. C2×#6095C22semidirect
31. C2×#6095C22semidirect
32. C2×#6095C22semidirect
38. #88129C22semidirect
40. #88123C22semidirect
41. C13×#97C22semidirect
42. C4×C8×C13C22semidirect
67. C13×M16D4semidirect
68. C2×C8×C13D4semidirect
136. #405D13semidirect
183. #97C2×D13semidirect
184. C4×C8C2×D13semidirect
225. M16D52semidirect
226. C2×C8D52semidirect
236. C13#3154semidirect

Maximal subgroups

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

subgroupindex
C2×#881632
C2×#881622
C2×#881582
C22×#60952
#117192312
#117192302
C13×#4052
#117183762
#117183732
#117183572
#117183542
#315413