Finite Group Databasebeta

← #13119729

Subgroups — #13119729

Order 1664.

800 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 157 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×#88207C211yesmaximal, characteristic
2C22×C22⋊Dic13C211yesmaximal, characteristic, complemented
3C2×#88140C211yesmaximal, characteristic, complemented
4C2×#88135C211yesmaximal, characteristic, complemented
5C13×#379C211yesmaximal, characteristic, complemented
6#11718620C212yesmaximal, complemented
7#11718617C212yesmaximal, complemented
8#11718612C212yesmaximal, complemented
9#11718609C212yesmaximal, complemented
10#11718604C211yesmaximal, characteristic, complemented
11#11718603C211yesmaximal, characteristic, complemented
12C25×C13C2211yescharacteristic
13C22×C22⋊D13C2212yes—
14C22×C22⋊D13C2211yescharacteristic
15C23×Dic13C2211yescharacteristic
16C23×Dic13C2211yescharacteristic
17C22×C4×D13C2211yescharacteristic, complemented
18C2×C13×C22⋊C4C2211yescharacteristic
19C2×C13×C22⋊C4C2211yescharacteristic
20C2×C13×C22⋊C4C2212yes—
21C2×C13×C22⋊C4C2212yes—
22#88207—24no—
23C2×C22⋊Dic13—24no—
24C2×C22⋊Dic13C2212yes—
25C2×C22⋊Dic13C2212yes—
26C2×C22⋊Dic13C2211yescharacteristic
27C2×C22⋊Dic13C2212yes—
28C2×#6103C2211yescharacteristic, complemented
29C2×#6103C2212yescomplemented
30C2×C4⋊Dic13C2212yescomplemented
31C2×#6101C2211yescharacteristic, complemented
32C2×#6101C2212yescomplemented
33C2×C4×Dic13C2211yescharacteristic, complemented
34#88140—24no—
35#88135—24no—
36#88071C2212yescomplemented
37#88071C2212yescomplemented
38#88071C2212yescomplemented
39C24×C13—22no—
40C24×C13—24no—
41C24×C13—22no—
42C24×C13—22no—
43C24×C13—42no—
44C24×C13C2312yes—
45C24×C13C2311yescharacteristic
46C23×D13C2311yescharacteristic
47C22×C4×C13C2311yescharacteristic
48C22×C4×C13C2311yescharacteristic
49C22×C4×C13C2312yes—
50C2×C22⋊D13—22no—
51C2×C22⋊D13—22no—
52C2×C22⋊D13—24no—
53C2×C22⋊D13—24no—
54C2×C22⋊D13—42no—
55C2×C22⋊D13D414yes—
56C22×Dic13—22no—
57C22×Dic13—22no—
58C22×Dic13—22no—
59C22×Dic13—42no—
60C22×Dic13D414yescomplemented
61C22×Dic13C2312yes—
62C22×Dic13C2312yes—
63C22×Dic13C2311yescharacteristic
64C22×Dic13C2311yescharacteristic
65C22×Dic13C2311yescharacteristic
66C2×C4×D13—22no—
67C2×C4×D13—42no—
68C13×C22⋊C4—22no—
69C13×C22⋊C4—22no—
70C13×C22⋊C4—24no—
71C13×C22⋊C4—42no—
72C22⋊Dic13—22no—
73C22⋊Dic13—24no—
74C22⋊Dic13—24no—
75C22⋊Dic13—24no—
76#6103—24no—
77#6103—22no—
78C4⋊Dic13—24no—
79#6101—22no—
80#6101—42no—
81C4×Dic13—22no—
82#2534—131nomaximal, Sylow
83C23×C13—22no—
84C23×C13—22no—
85C23×C13—28no—
86C23×C13—22no—
87C23×C13—22no—
88C23×C13—22no—
89C23×C13—22no—
90C23×C13—22no—
91C23×C13—22no—
92C23×C13—48no—
93C23×C13—48no—
94C23×C13—44no—
95C23×C13—44no—
96C23×C13C2×D412yes—
97C23×C13C2×D412yes—
98C23×C13#4114yes—
99C23×C13#4112yes—
100C23×C13C2411yescharacteristic
101C22×D13—21no—
102C22×D13—21no—
103C22×D13—22no—
104C22×D13—22no—
105C22×D13C2×D412yes—
106C2×C4×C13—21no—
107C2×C4×C13—21no—
108C2×C4×C13—22no—
109C2×C4×C13—22no—
110C2×C4×C13—22no—
111C2×C4×C13—22no—
112C2×C4×C13—22no—
113C22⋊D13—28no—
114C22⋊D13—48no—
115C2×Dic13—21no—
116C2×Dic13—21no—
117C2×Dic13—22no—
118C2×Dic13—22no—
119C2×Dic13—22no—
120C2×Dic13—22no—
121C2×Dic13—22no—
122C2×Dic13—22no—
123C2×Dic13—22no—
124C2×Dic13—22no—
125C2×Dic13—44no—
126C2×Dic13—44no—
127C2×Dic13C2×D412yes—
128C2×Dic13C2×D412yes—
129C2×Dic13C2×D412yes—
130C4×D13—44no—
131C2×#123—131no—
132C2×#120—131no—
133C2×C4×D4—131no—
134C22×C22⋊C4—131no—
135#396—132no—
136#394—132no—
137#388—132no—
138#386—132no—
139#381—131no—
140#380—131no—
141#379D1311yescharacteristic, complemented
142C22×C13—22no—
143C22×C13—22no—
144C22×C13—24no—
145C22×C13—24no—
146C22×C13—24no—
147C22×C13—24no—
148C22×C13—24no—
149C22×C13—416no—
150C22×C13—42no—
151C22×C13—42no—
152C22×C13—42no—
153C22×C13—42no—
154C22×C13#12114yes—
155C22×C13C22×D411yescharacteristic
156C22×C13C22×D411yescharacteristic
157C22×C13C2×#4111yescharacteristic
158C22×C13C2×#4112yes—
159C22×C13H(5,𝔽2)12yes—
160C2×D13—22no—
161C2×D13—24no—
162C2×D13—41no—
163C2×D13—41no—
164C2×D13—42no—
165C4×C13—41no—
166C4×C13—41no—
167C4×C13—42no—
168Dic13—22no—
169Dic13—22no—
170Dic13—22no—
171Dic13—42no—
172Dic13—42no—
173C25C2×D1311yescharacteristic
174C22×D4—131no—
175C22×D4—132no—
176C23×C4—131no—
177C23×C4—131no—
178C23×C4—131no—
179#123—264no—
180#120—264no—
181C4×D4—264no—
182C2×C4⋊C4—131no—
183C2×C4⋊C4—132no—
184C2×C4⋊C4—132no—
185C2×C22⋊C4—132no—
186C2×C22⋊C4—132no—
187C2×C22⋊C4—131no—
188C2×C22⋊C4—132no—
189C2×C22⋊C4—131no—
190C2×C22⋊C4—132no—
191C2×C22⋊C4—264no—
192C2×C22⋊C4C2×D1311yescharacteristic
193C2×C22⋊C4C2×D1311yescharacteristic
194C2×C22⋊C4C2×D1312yes—
195C2×C22⋊C4C2×D1312yes—
196C2×C42—131no—
197#95—132no—
198#95—132no—
199#95—132no—
200C2×C13—24no—
201C2×C13—44no—
202C2×C13C2×#12111yescharacteristic
203C2×C13#52511yescharacteristic
204C2×C13#53411yescharacteristic
205C2×C13#54612yes—
206C2×C13#55112yes—
207D13—42no—
208C24—22no—
209C24—24no—
210C24—22no—
211C24—22no—
212C24—42no—
213C24—131no—
214C24C22×D1312yes—
215C24C22×D1311yescharacteristic
216C2×D4—134no—
217C2×D4—262no—
218C2×D4—262no—
219C2×D4—264no—
220C2×D4—264no—
221C2×D4—522no—
222C22×C4—131no—
223C22×C4—131no—
224C22×C4—131no—
225C22×C4—134no—
226C22×C4—132no—
227C22×C4—132no—
228C22×C4—262no—
229C22×C4—262no—
230C22×C4—262no—
231C22×C4—262no—
232C22×C4—522no—
233C22×C4—522no—
234C22×C4C22×D1311yescharacteristic
235C22×C4C22×D1311yescharacteristic
236C22×C4C22×D1312yes—
237C4⋊C4—262no—
238C4⋊C4—264no—
239C4⋊C4—522no—
240C22⋊C4—22no—
241C22⋊C4—22no—
242C22⋊C4—24no—
243C22⋊C4—42no—
244C22⋊C4—262no—
245C22⋊C4—264no—
246C22⋊C4—264no—
247C22⋊C4—264no—
248C22⋊C4—264no—
249C22⋊C4—262no—
250C42—262no—
251C13#253411yescharacteristic, complemented, Sylow
252C23—22no—
253C23—22no—
254C23—28no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—48no—
262C23—48no—
263C23—44no—
264C23—44no—
265C23—132no—
266C23—261no—
267C23—261no—
268C23—262no—
269C23—262no—
270C23#612712yes—
271C23D4×D1312yes—
272C23D4×D1312yes—
273C23#612914yes—
274C23C23×D1311yescharacteristic
275D4—268no—
276D4—528no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—132no—
285C2×C4—132no—
286C2×C4—132no—
287C2×C4—261no—
288C2×C4—261no—
289C2×C4—262no—
290C2×C4—262no—
291C2×C4—262no—
292C2×C4—262no—
293C2×C4—262no—
294C2×C4—262no—
295C2×C4—262no—
296C2×C4—262no—
297C2×C4—524no—
298C2×C4—524no—
299C2×C4—524no—
300C22—24no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—416no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—262no—
313C22—264no—
314C22—522no—
315C22—521no—
316C22—521no—
317C22#8813814yes—
318C22C2×#612711yescharacteristic
319C22C2×D4×D1311yescharacteristic
320C22C2×D4×D1311yescharacteristic
321C22C2×#612912yes—
322C22#8825112yes—
323C4—41no—
324C4—41no—
325C4—42no—
326C4—262no—
327C4—262no—
328C4—262no—
329C4—522no—
330C4—522no—
331C2—24no—
332C2—44no—
333C2—522no—
334C2C2×#8813811yescharacteristic
335C2#1171920012yes—
336C2#1171929111yescharacteristic
337C2#1171929411yescharacteristic
338C2#1171929612yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#379C2
prime radicalC2×#88207C2
hypercenter#379D13
hypocenterC13#2534
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊Dic13C2semidirect
3. C2×#88140C2semidirect
4. C2×#88135C2semidirect
5. C13×#379C2semidirect
6. #11718620C2semidirect
7. #11718617C2semidirect
8. #11718612C2semidirect
9. #11718609C2semidirect
10. #11718604C2semidirect
11. #11718603C2semidirect
17. C22×C4×D13C22semidirect
28. C2×#6103C22semidirect
29. C2×#6103C22semidirect
30. C2×C4⋊Dic13C22semidirect
31. C2×#6101C22semidirect
32. C2×#6101C22semidirect
33. C2×C4×Dic13C22semidirect
36. #88071C22semidirect
37. #88071C22semidirect
38. #88071C22semidirect
60. C22×Dic13D4semidirect
141. #379D13semidirect
251. C13#2534semidirect

Maximal subgroups

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

subgroupindex
C2×#882072
C22×C22⋊Dic132
C2×#881402
C2×#881352
C13×#3792
#117186202
#117186172
#117186122
#117186092
#117186042
#117186032
#253413