Finite Group Databasebeta

← #13120720

Subgroups — #13120720

Order 1664.

620 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 141 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×C4×D4×C13C211yesmaximal, characteristic, complemented
2C2×#88184C211yesmaximal, characteristic, complemented
3#11718884C212yesmaximal, complemented
4#11718874C212yesmaximal, complemented
5#11718872C211yesmaximal, characteristic, complemented
6#11718652C211yesmaximal, characteristic, complemented
7#11718647C211yesmaximal, characteristic, complemented
8#11718621C211yesmaximal, characteristic, complemented
9#11718620C212yesmaximal, complemented
10#11718610C211yesmaximal, characteristic, complemented
11#11718601C212yesmaximal, complemented
12C22×D4×C13C2211yescharacteristic
13C23×C4×C13C2211yescharacteristic
14C23×C4×C13C2211yescharacteristic
15C22×C22⋊D13C2212yes—
16C22×C22⋊D13C2212yes—
17C22×D52C2211yescharacteristic
18C4×D4×C13—24no—
19C2×C13×C4⋊C4C2211yescharacteristic
20C2×C13×C22⋊C4C2211yescharacteristic
21C2×C13×C22⋊C4C2211yescharacteristic
22C2×C42×C13C2211yescharacteristic
23C2×C22⋊Dic13C2212yes—
24C2×C22⋊Dic13C2212yes—
25#88184—24no—
26C2×#6103C2212yes—
27C2×#6103C2212yes—
28C2×#6103C2212yes—
29C2×#6103C2212yes—
30C2×C4⋊Dic13C2211yescharacteristic
31C2×#6101C2212yes—
32#88071C2212yes—
33#88071C2212yes—
34#88071C2211yescharacteristic
35#88071C2211yescharacteristic
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C24×C13C2311yescharacteristic
39C23×D13C2312yes—
40C2×D4×C13—22no—
41C2×D4×C13—22no—
42C2×D4×C13—22no—
43C22×C4×C13—22no—
44C22×C4×C13—42no—
45C22×C4×C13—42no—
46C22×C4×C13D414yescomplemented
47C22×C4×C13C2311yescharacteristic
48C22×C4×C13C2311yescharacteristic
49C22×C4×C13C2311yescharacteristic
50C22×C4×C13C2311yescharacteristic
51C22×C4×C13C2311yescharacteristic
52C2×C22⋊D13—24no—
53C2×C22⋊D13—24no—
54C2×C22⋊D13—42no—
55C2×C22⋊D13—42no—
56C2×C22⋊D13—42no—
57C2×C22⋊D13—42no—
58C22×Dic13C2312yes—
59C22×Dic13C2312yes—
60C22×Dic13C2312yes—
61C2×D52—22no—
62C2×D52—42no—
63C13×C4⋊C4—22no—
64C13×C22⋊C4—22no—
65C13×C22⋊C4—22no—
66C42×C13—22no—
67C22⋊Dic13—42no—
68C22⋊Dic13—42no—
69#6103—24no—
70#6103—24no—
71#6103—42no—
72#6103—42no—
73C4⋊Dic13—22no—
74#6101—42no—
75#2635—131nomaximal, Sylow
76C23×C13—21no—
77C23×C13—21no—
78C23×C13—21no—
79C23×C13—21no—
80C23×C13—22no—
81C23×C13—22no—
82C23×C13—22no—
83C23×C13—22no—
84C23×C13C2×D412yes—
85C23×C13#4112yes—
86C23×C13C2411yescharacteristic
87C22×D13—22no—
88C22×D13—22no—
89C22×D13—22no—
90C22×D13—22no—
91C22×D13—22no—
92C22×D13—22no—
93C22×D13—22no—
94D4×C13—28no—
95C2×C4×C13—21no—
96C2×C4×C13—21no—
97C2×C4×C13—22no—
98C2×C4×C13—22no—
99C2×C4×C13—21no—
100C2×C4×C13—21no—
101C2×C4×C13—22no—
102C2×C4×C13—22no—
103C2×C4×C13—44no—
104C2×C4×C13—44no—
105C2×C4×C13C2×D412yes—
106C2×C4×C13C2×D412yes—
107C2×C4×C13#4112yes—
108C22⋊D13—48no—
109C22⋊D13—48no—
110C2×Dic13—22no—
111C2×Dic13—22no—
112C2×Dic13—22no—
113C2×Dic13—22no—
114C2×Dic13—22no—
115C2×Dic13—22no—
116C2×Dic13—22no—
117C2×Dic13—22no—
118C2×Dic13—22no—
119D52—44no—
120C2×#121—131no—
121C2×C4×D4D1311yescharacteristic, complemented
122#400—131no—
123#399—131no—
124#394—131no—
125#394—132no—
126#392—132no—
127#392—131no—
128#388—132no—
129#386—132no—
130#381—131no—
131C22×C13—22no—
132C22×C13—22no—
133C22×C13—22no—
134C22×C13—22no—
135C22×C13—24no—
136C22×C13—41no—
137C22×C13—41no—
138C22×C13—42no—
139C22×C13—42no—
140C22×C13—42no—
141C22×C13C22×D411yescharacteristic
142C22×C13C2×#4111yescharacteristic
143C22×C13C2×#4111yescharacteristic
144C22×C13H(5,𝔽2)12yes—
145C22×C13H(5,𝔽2)12yes—
146C2×D13—42no—
147C2×D13—42no—
148C2×D13—42no—
149C2×D13—42no—
150C2×D13—42no—
151C2×D13—42no—
152C2×D13—42no—
153C4×C13—22no—
154C4×C13—22no—
155C4×C13—22no—
156C4×C13—41no—
157C4×C13—41no—
158Dic13—42no—
159Dic13—42no—
160Dic13—42no—
161C22×D4—132no—
162C22×D4—132no—
163C22×D4—131no—
164C22×D4C2×D1311yescharacteristic
165C23×C4C2×D1311yescharacteristic
166C23×C4C2×D1311yescharacteristic
167#121—264no—
168C4×D4—24no—
169C2×C4⋊C4—131no—
170C2×C4⋊C4—132no—
171C2×C4⋊C4C2×D1311yescharacteristic
172C2×C22⋊C4—132no—
173C2×C22⋊C4—132no—
174C2×C22⋊C4—132no—
175C2×C22⋊C4—132no—
176C2×C22⋊C4—132no—
177C2×C22⋊C4—132no—
178C2×C22⋊C4C2×D1311yescharacteristic
179C2×C22⋊C4C2×D1311yescharacteristic
180C2×C42C2×D1311yescharacteristic
181#95—132no—
182#95—131no—
183#95—131no—
184#95—132no—
185#95—132no—
186C2×C13—42no—
187C2×C13—42no—
188C2×C13#52511yescharacteristic
189C2×C13#53411yescharacteristic
190C2×C13#53511yescharacteristic
191C2×C13#53812yes—
192C2×C13#54012yes—
193D13—82no—
194C24—132no—
195C24C22×D1311yescharacteristic
196C24C22×D1311yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—264no—
201C2×D4—264no—
202C2×D4—262no—
203C2×D4—522no—
204C2×D4—522no—
205C2×D4—522no—
206C2×D4—522no—
207C2×D4—522no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—132no—
212C22×C4—132no—
213C22×C4—132no—
214C22×C4C22⋊D1314yescomplemented
215C22×C4C22×D1311yescharacteristic
216C22×C4C22×D1311yescharacteristic
217C22×C4C22×D1311yescharacteristic
218C22×C4C22×D1311yescharacteristic
219C22×C4C22×D1311yescharacteristic
220C4⋊C4—22no—
221C4⋊C4—262no—
222C4⋊C4—522no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—264no—
226C22⋊C4—264no—
227C22⋊C4—522no—
228C22⋊C4—522no—
229C22⋊C4—522no—
230C22⋊C4—522no—
231C42—22no—
232C13#263511yescharacteristic, complemented, Sylow
233C23—21no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—262no—
242C23—262no—
243C23—262no—
244C23—262no—
245C23—262no—
246C23—262no—
247C23—262no—
248C23#612712yes—
249C23C2×C22⋊D1312yes—
250C23C23×D1311yescharacteristic
251D4—28no—
252D4—524no—
253D4—528no—
254D4—528no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—262no—
266C2×C4—262no—
267C2×C4—262no—
268C2×C4—262no—
269C2×C4—262no—
270C2×C4—262no—
271C2×C4—262no—
272C2×C4—262no—
273C2×C4—262no—
274C2×C4#612712yes—
275C2×C4C2×C22⋊D1312yes—
276C2×C4C2×C22⋊D1312yes—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—24no—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—522no—
288C22—522no—
289C22—522no—
290C22—522no—
291C22—522no—
292C22—522no—
293C22—522no—
294C22C2×#612711yescharacteristic
295C22C2×#612711yescharacteristic
296C22#8825112yes—
297C22#8825612yes—
298C22C22×C22⋊D1311yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—522no—
305C4—522no—
306C4—522no—
307C2—42no—
308C2—42no—
309C2—1042no—
310C2#1171926512yes—
311C2#1171926712yes—
312C2#1171949611yescharacteristic
313C2#1171950911yescharacteristic
314C2#1171953511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC2×C4×D4×C13C2
hypercenterC2×C4×D4D13
hypocenterC13#2635
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D4×C13C2semidirect
2. C2×#88184C2semidirect
3. #11718884C2semidirect
4. #11718874C2semidirect
5. #11718872C2semidirect
6. #11718652C2semidirect
7. #11718647C2semidirect
8. #11718621C2semidirect
9. #11718620C2semidirect
10. #11718610C2semidirect
11. #11718601C2semidirect
46. C22×C4×C13D4semidirect
121. C2×C4×D4D13semidirect
214. C22×C4C22⋊D13semidirect
232. C13#2635semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×C132
C2×#881842
#117188842
#117188742
#117188722
#117186522
#117186472
#117186212
#117186202
#117186102
#117186012
#263513