Finite Group Databasebeta

← #13120967

Subgroups — #13120967

Order 1664.

598 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 133 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×#88192C211yesmaximal, characteristic, complemented
2C2×#88137C211yesmaximal, characteristic, complemented
3C13×#392C211yesmaximal, characteristic, complemented
4#11718884C212yesmaximal, complemented
5#11718883C211yesmaximal, characteristic, complemented
6#11718882C212yesmaximal, complemented
7#11718622C212yesmaximal, complemented
8#11718614C211yesmaximal, characteristic, complemented
9#11718612C212yesmaximal, complemented
10#11718327C211yesmaximal, characteristic, complemented
11#11718322C211yesmaximal, characteristic, complemented
12C22×D4×C13C2211yescharacteristic
13C22×D4×C13C2212yes—
14C22×C22⋊D13C2211yescharacteristic
15C22×C22⋊D13C2212yes—
16C22×C4×D13C2211yescharacteristic
17C2×C13×C22⋊C4C2211yescharacteristic
18C2×C13×C22⋊C4C2212yes—
19C2×C22⋊Dic13C2211yescharacteristic
20C2×C22⋊Dic13C2212yes—
21C2×C22⋊Dic13C2212yes—
22C2×C22⋊Dic13C2212yes—
23C2×C22⋊Dic13C2212yes—
24#88192—24no—
25C2×#6103C2211yescharacteristic
26C2×#6103C2212yes—
27C2×C4⋊Dic13C2211yescharacteristic
28C2×C4⋊Dic13C2211yescharacteristic
29C2×C4⋊Dic13C2212yes—
30C2×#6101C2211yescharacteristic
31C2×C4×Dic13C2212yes—
32#88137—24no—
33C13×#95C2211yescharacteristic
34#88071C2212yes—
35#88071C2211yescharacteristic
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C24×C13C2312yes—
39C23×D13C2311yescharacteristic
40C2×D4×C13—22no—
41C2×D4×C13—24no—
42C2×D4×C13—42no—
43C2×D4×C13—42no—
44C2×D4×C13—42no—
45C22×C4×C13C2311yescharacteristic
46C22×C4×C13C2311yescharacteristic
47C22×C4×C13C2312yes—
48C2×C22⋊D13—22no—
49C2×C22⋊D13—24no—
50C2×C22⋊D13—42no—
51C2×C22⋊D13—42no—
52C2×C22⋊D13—42no—
53C22×Dic13C2312yes—
54C22×Dic13C2311yescharacteristic
55C22×Dic13C2312yes—
56C22×Dic13C2311yescharacteristic
57C22×Dic13C2311yescharacteristic
58C2×C4×D13—42no—
59C2×C4×D13D414yescomplemented
60C13×C22⋊C4—22no—
61C13×C22⋊C4—24no—
62C22⋊Dic13—22no—
63C22⋊Dic13—24no—
64C22⋊Dic13—42no—
65C22⋊Dic13—42no—
66C22⋊Dic13—42no—
67#6103—22no—
68#6103—24no—
69C4⋊Dic13—22no—
70C4⋊Dic13—22no—
71C4⋊Dic13—42no—
72#6101—22no—
73C4×Dic13—42no—
74#2661—131nomaximal, Sylow
75C23×C13—21no—
76C23×C13—22no—
77C23×C13—22no—
78C23×C13—22no—
79C23×C13—22no—
80C23×C13—21no—
81C23×C13—22no—
82C23×C13—22no—
83C23×C13—22no—
84C23×C13—22no—
85C23×C13—22no—
86C23×C13—21no—
87C23×C13C2411yescharacteristic
88C22×D13—22no—
89C22×D13—22no—
90C22×D13—21no—
91C22×D13—21no—
92C22×D13C2×D412yes—
93D4×C13—44no—
94D4×C13—48no—
95C2×C4×C13—22no—
96C2×C4×C13—22no—
97C2×C4×C13—22no—
98C2×C4×C13—21no—
99C2×C4×C13—22no—
100C2×C4×C13C2×D412yes—
101C2×C4×C13#4114yes—
102C22⋊D13—44no—
103C22⋊D13—48no—
104C2×Dic13—22no—
105C2×Dic13—21no—
106C2×Dic13—21no—
107C2×Dic13—22no—
108C2×Dic13—22no—
109C2×Dic13—22no—
110C2×Dic13—21no—
111C2×Dic13—22no—
112C2×Dic13—21no—
113C2×Dic13—22no—
114C2×Dic13—22no—
115C2×Dic13—22no—
116C2×Dic13C2×D412yes—
117C4×D13—44no—
118C2×#123—131no—
119C2×#121—131no—
120#398—131no—
121#397—131no—
122#394—131no—
123#394—132no—
124#392—132no—
125#392D1311yescharacteristic, complemented
126#390—132no—
127#388—132no—
128C4×C22⋊C4—131no—
129C22×C13—41no—
130C22×C13—42no—
131C22×C13—41no—
132C22×C13—42no—
133C22×C13—42no—
134C22×C13—41no—
135C22×C13—42no—
136C22×C13—42no—
137C22×C13—42no—
138C22×C13—42no—
139C22×C13—42no—
140C22×C13—42no—
141C22×C13C22×D411yescharacteristic
142C22×C13C2×#4112yes—
143C22×C13H(5,𝔽2)12yes—
144C22×C13H(5,𝔽2)11yescharacteristic
145C22×C13H(5,𝔽2)11yescharacteristic
146C2×D13—22no—
147C2×D13—22no—
148C2×D13—22no—
149C2×D13—42no—
150C2×D13—42no—
151C4×C13—22no—
152C4×C13—24no—
153C4×C13—41no—
154Dic13—22no—
155Dic13—41no—
156Dic13—41no—
157Dic13—42no—
158Dic13—42no—
159C22×D4—132no—
160C22×D4—131no—
161C22×D4C2×D1311yescharacteristic
162C22×D4C2×D1312yes—
163C23×C4—131no—
164#123—264no—
165#121—264no—
166C2×C4⋊C4—131no—
167C2×C4⋊C4—131no—
168C2×C4⋊C4—131no—
169C2×C4⋊C4—132no—
170C2×C22⋊C4—132no—
171C2×C22⋊C4—131no—
172C2×C22⋊C4—131no—
173C2×C22⋊C4—132no—
174C2×C22⋊C4—132no—
175C2×C22⋊C4—132no—
176C2×C22⋊C4—132no—
177C2×C22⋊C4C2×D1311yescharacteristic
178C2×C22⋊C4C2×D1312yes—
179C2×C42—132no—
180#95—131no—
181#95—132no—
182#95—132no—
183#95C2×D1311yescharacteristic
184C2×C13—81no—
185C2×C13—82no—
186C2×C13#53211yescharacteristic
187C2×C13#53411yescharacteristic
188C2×C13#53511yescharacteristic
189C2×C13#53812yes—
190C2×C13#54012yes—
191D13—42no—
192C24—131no—
193C24C22×D1311yescharacteristic
194C24C22×D1312yes—
195C2×D4—22no—
196C2×D4—24no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—264no—
201C2×D4—262no—
202C2×D4—522no—
203C2×D4—522no—
204C2×D4—522no—
205C22×C4—131no—
206C22×C4—134no—
207C22×C4—132no—
208C22×C4—132no—
209C22×C4—131no—
210C22×C4—131no—
211C22×C4—522no—
212C22×C4C22×D1311yescharacteristic
213C22×C4C22×D1311yescharacteristic
214C22×C4C22×D1312yes—
215C4⋊C4—262no—
216C4⋊C4—262no—
217C4⋊C4—262no—
218C4⋊C4—522no—
219C22⋊C4—22no—
220C22⋊C4—24no—
221C22⋊C4—262no—
222C22⋊C4—264no—
223C22⋊C4—262no—
224C22⋊C4—264no—
225C22⋊C4—522no—
226C22⋊C4—522no—
227C22⋊C4—522no—
228C42—522no—
229C13#266111yescharacteristic, complemented, Sylow
230C23—21no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—22no—
242C23—132no—
243C23—262no—
244C23—262no—
245C23—261no—
246C23—261no—
247C23C23×D1311yescharacteristic
248D4—44no—
249D4—48no—
250D4—524no—
251D4—528no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—132no—
258C2×C4—262no—
259C2×C4—261no—
260C2×C4—262no—
261C2×C4—261no—
262C2×C4—262no—
263C2×C4—262no—
264C2×C4—262no—
265C2×C4—261no—
266C2×C4—262no—
267C2×C4—261no—
268C2×C4—262no—
269C2×C4—262no—
270C2×C4—524no—
271C2×C4D4×D1312yes—
272C2×C4#612914yes—
273C22—42no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—42no—
285C22—262no—
286C22—262no—
287C22—262no—
288C22—522no—
289C22—522no—
290C22C2×D4×D1311yescharacteristic
291C22C2×#612912yes—
292C22#8825112yes—
293C22#8825111yescharacteristic
294C22#8825611yescharacteristic
295C4—22no—
296C4—24no—
297C4—41no—
298C4—262no—
299C4—521no—
300C4—521no—
301C4—522no—
302C4—522no—
303C2—82no—
304C2—81no—
305C2—522no—
306C2#1171929411yescharacteristic
307C2#1171929812yes—
308C2#1171931211yescharacteristic
309C2#1171932412yes—
310C2#1171942111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#392C2
hypercenter#392D13
hypocenterC13#2661
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#88192C2semidirect
2. C2×#88137C2semidirect
3. C13×#392C2semidirect
4. #11718884C2semidirect
5. #11718883C2semidirect
6. #11718882C2semidirect
7. #11718622C2semidirect
8. #11718614C2semidirect
9. #11718612C2semidirect
10. #11718327C2semidirect
11. #11718322C2semidirect
59. C2×C4×D13D4semidirect
125. #392D13semidirect
229. C13#2661semidirect

Maximal subgroups

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

subgroupindex
C2×#881922
C2×#881372
C13×#3922
#117188842
#117188832
#117188822
#117186222
#117186142
#117186122
#117183272
#117183222
#266113