Finite Group Databasebeta

← #13120584

Subgroups — #13120584

Order 1664.

632 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 149 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×#88193C211yesmaximal, characteristic
2C2×#88189C211yesmaximal, characteristic, complemented
3C22×#6101C211yesmaximal, characteristic, complemented
4C2×#88140C211yesmaximal, characteristic, complemented
5C13×#386C211yesmaximal, characteristic, complemented
6#11718874C211yesmaximal, characteristic, complemented
7#11718620C212yesmaximal, complemented
8#11718609C212yesmaximal, complemented
9#11718603C211yesmaximal, characteristic, complemented
10#11718337C212yesmaximal, complemented
11#11718318C212yesmaximal, complemented
12C22×D4×C13C2211yescharacteristic
13C23×C4×C13C2211yescharacteristic
14C22×C22⋊D13C2211yescharacteristic
15C22×C22⋊D13C2211yescharacteristic
16C23×Dic13C2211yescharacteristic
17C23×Dic13C2211yescharacteristic
18C2×C13×C22⋊C4C2211yescharacteristic
19C2×C13×C22⋊C4C2212yes—
20C2×C22⋊Dic13C2211yescharacteristic
21C2×C22⋊Dic13C2212yes—
22C2×C22⋊Dic13C2211yescharacteristic
23#88193—24no—
24#88189—24no—
25C2×#6103C2212yes—
26C2×#6103C2211yescharacteristic
27C2×#6103C2212yes—
28C2×C4⋊Dic13C2212yes—
29C2×#6101—24no—
30C2×#6101C2212yes—
31C2×#6101C2212yes—
32C2×#6101C2212yes—
33C2×#6101C2211yescharacteristic
34C2×#6101C2212yes—
35#88140—24no—
36C13×#95C2212yes—
37#88071C2212yes—
38#88071C2212yes—
39C24×C13C2311yescharacteristic
40C24×C13C2311yescharacteristic
41C23×D13C2311yescharacteristic
42C2×D4×C13—22no—
43C2×D4×C13—22no—
44C2×D4×C13—22no—
45C22×C4×C13—22no—
46C22×C4×C13—42no—
47C22×C4×C13C2312yes—
48C22×C4×C13C2312yes—
49C22×C4×C13C2311yescharacteristic
50C2×C22⋊D13—22no—
51C2×C22⋊D13—22no—
52C2×C22⋊D13—22no—
53C2×C22⋊D13—22no—
54C2×C22⋊D13—42no—
55C22×Dic13—22no—
56C22×Dic13—22no—
57C22×Dic13—42no—
58C22×Dic13D414yes—
59C22×Dic13D414yescomplemented
60C22×Dic13C2312yes—
61C22×Dic13C2311yescharacteristic
62C22×Dic13C2311yescharacteristic
63C22×Dic13C2311yescharacteristic
64C22×Dic13C2312yes—
65C13×C22⋊C4—22no—
66C13×C22⋊C4—42no—
67C22⋊Dic13—22no—
68C22⋊Dic13—24no—
69C22⋊Dic13—22no—
70#6103—24no—
71#6103—22no—
72#6103—42no—
73C4⋊Dic13—24no—
74#6101—24no—
75#6101—24no—
76#6101—24no—
77#6101—22no—
78#6101—42no—
79#2618—131nomaximal, Sylow
80C23×C13—22no—
81C23×C13—22no—
82C23×C13—21no—
83C23×C13—21no—
84C23×C13—21no—
85C23×C13—22no—
86C23×C13—22no—
87C23×C13C2×D412yes—
88C23×C13C2×D412yes—
89C23×C13#4112yes—
90C23×C13C2411yescharacteristic
91C22×D13—21no—
92C22×D13—21no—
93C22×D13—21no—
94C22×D13—22no—
95C22×D13—22no—
96D4×C13—44no—
97C2×C4×C13—22no—
98C2×C4×C13—21no—
99C2×C4×C13—21no—
100C2×C4×C13—22no—
101C2×C4×C13—21no—
102C2×C4×C13—22no—
103C2×C4×C13—22no—
104C2×C4×C13—22no—
105C2×C4×C13—22no—
106C2×C4×C13—44no—
107C22⋊D13—44no—
108C22⋊D13—44no—
109C2×Dic13—28no—
110C2×Dic13—22no—
111C2×Dic13—22no—
112C2×Dic13—22no—
113C2×Dic13—22no—
114C2×Dic13—22no—
115C2×Dic13—21no—
116C2×Dic13—21no—
117C2×Dic13—21no—
118C2×Dic13—22no—
119C2×Dic13—22no—
120C2×Dic13—44no—
121C2×Dic13C2×D412yes—
122C2×Dic13C2×D412yes—
123C2×Dic13C2×D414yes—
124C2×#123—131no—
125C2×#123—131no—
126C2×#121—131no—
127C22×C4⋊C4—131no—
128#398—132no—
129#396—132no—
130#396—132no—
131#394—132no—
132#386—131no—
133#386D1311yescharacteristic, complemented
134#380—131no—
135C22×C13—22no—
136C22×C13—22no—
137C22×C13—24no—
138C22×C13—24no—
139C22×C13—41no—
140C22×C13—41no—
141C22×C13—41no—
142C22×C13—42no—
143C22×C13—42no—
144C22×C13#12114yes—
145C22×C13C22×D411yescharacteristic
146C22×C13C22×D411yescharacteristic
147C22×C13C2×#4111yescharacteristic
148C22×C13H(5,𝔽2)12yes—
149C22×C132−1+412yes—
150C2×D13—41no—
151C2×D13—41no—
152C2×D13—41no—
153C2×D13—42no—
154C2×D13—42no—
155C4×C13—41no—
156C4×C13—42no—
157C4×C13—42no—
158Dic13—24no—
159Dic13—22no—
160Dic13—22no—
161Dic13—41no—
162Dic13—42no—
163C22×D4—131no—
164C22×D4—131no—
165C22×D4C2×D1311yescharacteristic
166C23×C4—131no—
167C23×C4—131no—
168C23×C4C2×D1311yescharacteristic
169#123—264no—
170#123—264no—
171#121—264no—
172C2×C4⋊C4—132no—
173C2×C4⋊C4—132no—
174C2×C4⋊C4—131no—
175C2×C4⋊C4—132no—
176C2×C4⋊C4—132no—
177C2×C4⋊C4—132no—
178C2×C4⋊C4—264no—
179C2×C22⋊C4—131no—
180C2×C22⋊C4—131no—
181C2×C22⋊C4—132no—
182C2×C22⋊C4—131no—
183C2×C22⋊C4—132no—
184C2×C22⋊C4—132no—
185C2×C22⋊C4C2×D1311yescharacteristic
186C2×C22⋊C4C2×D1312yes—
187#95—132no—
188#95—132no—
189#95C2×D1312yes—
190C2×C13—24no—
191C2×C13—81no—
192C2×C13C2×#12111yescharacteristic
193C2×C13#53411yescharacteristic
194C2×C13#53611yescharacteristic
195C2×C13#53712yes—
196C2×C13#53912yes—
197D13—81no—
198C24—131no—
199C24C22×D1311yescharacteristic
200C24C22×D1311yescharacteristic
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—262no—
205C2×D4—262no—
206C2×D4—262no—
207C2×D4—262no—
208C2×D4—522no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—134no—
212C22×C4—131no—
213C22×C4—132no—
214C22×C4—131no—
215C22×C4—134no—
216C22×C4—132no—
217C22×C4—131no—
218C22×C4—262no—
219C22×C4—262no—
220C22×C4—522no—
221C22×C4C22×D1312yes—
222C22×C4C22×D1312yes—
223C22×C4C22×D1311yescharacteristic
224C4⋊C4—264no—
225C4⋊C4—264no—
226C4⋊C4—262no—
227C4⋊C4—264no—
228C4⋊C4—264no—
229C4⋊C4—522no—
230C22⋊C4—22no—
231C22⋊C4—42no—
232C22⋊C4—262no—
233C22⋊C4—264no—
234C22⋊C4—262no—
235C22⋊C4—264no—
236C22⋊C4—262no—
237C22⋊C4—522no—
238C13#261811yescharacteristic, complemented, Sylow
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—261no—
247C23—261no—
248C23—261no—
249C23—262no—
250C23—262no—
251C23#612712yes—
252C23D4×D1312yes—
253C23D4×D1312yes—
254C23C23×D1311yescharacteristic
255D4—44no—
256D4—524no—
257D4—524no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—44no—
268C2×C4—134no—
269C2×C4—132no—
270C2×C4—132no—
271C2×C4—268no—
272C2×C4—262no—
273C2×C4—262no—
274C2×C4—262no—
275C2×C4—262no—
276C2×C4—261no—
277C2×C4—262no—
278C2×C4—261no—
279C2×C4—261no—
280C2×C4—262no—
281C2×C4—262no—
282C2×C4—524no—
283C22—24no—
284C22—22no—
285C22—24no—
286C22—22no—
287C22—41no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22—521no—
293C22—522no—
294C22—521no—
295C22—522no—
296C22—521no—
297C22#8813814yes—
298C22C2×#612711yescharacteristic
299C22C2×D4×D1311yescharacteristic
300C22C2×D4×D1311yescharacteristic
301C22#8825112yes—
302C22#8825712yes—
303C4—41no—
304C4—42no—
305C4—42no—
306C4—264no—
307C4—262no—
308C4—262no—
309C4—521no—
310C4—522no—
311C2—24no—
312C2—81no—
313C2—1041no—
314C2C2×#8813811yescharacteristic
315C2#1171926912yes—
316C2#1171929411yescharacteristic
317C2#1171930212yes—
318C2#1171935311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#386C2
prime radicalC2×#88193C2
hypercenter#386D13
hypocenterC13#2618
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#88189C2semidirect
3. C22×#6101C2semidirect
4. C2×#88140C2semidirect
5. C13×#386C2semidirect
6. #11718874C2semidirect
7. #11718620C2semidirect
8. #11718609C2semidirect
9. #11718603C2semidirect
10. #11718337C2semidirect
11. #11718318C2semidirect
59. C22×Dic13D4semidirect
133. #386D13semidirect
238. C13#2618semidirect

Maximal subgroups

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

subgroupindex
C2×#881932
C2×#881892
C22×#61012
C2×#881402
C13×#3862
#117188742
#117186202
#117186092
#117186032
#117183372
#117183182
#261813