Finite Group Databasebeta

← #13120620

Subgroups — #13120620

Order 1664.

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

normal subgroups only

This group has 526 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×#88189C211yesmaximal, characteristic
2C22×#6101C211yesmaximal, characteristic, complemented
3C13×#386C211yesmaximal, characteristic
4#11718881C211yesmaximal, characteristic
5#11718871C211yesmaximal, characteristic, complemented
6#11718613C212yesmaximal, complemented
7#11718611C211yesmaximal, characteristic, complemented
8#11718610C212yesmaximal, complemented
9#11718604C211yesmaximal, characteristic, complemented
10#11718319C212yesmaximal, complemented
11#11718318C212yesmaximal, complemented
12C22×D4×C13C2211yescharacteristic
13C23×C4×C13C2211yescharacteristic
14C23×Dic13C2211yescharacteristic
15C23×Dic13C2211yescharacteristic
16C2×C13×C22⋊C4C2211yescharacteristic
17C2×C13×C22⋊C4C2212yes—
18C2×C22⋊Dic13C2211yescharacteristic
19C2×C22⋊Dic13C2211yescharacteristic
20C2×C22⋊Dic13C2211yescharacteristic
21C2×C22⋊Dic13C2211yescharacteristic
22C2×C22⋊Dic13C2212yes—
23#88189—24no—
24C2×C4⋊Dic13C2212yes—
25C2×#6101—24no—
26C2×#6101C2212yes—
27C2×#6101C2212yes—
28C2×#6101C2211yescharacteristic
29C2×#6101C2211yescharacteristic
30C2×#6101C2212yes—
31C13×#95C2212yes—
32#88071C2212yes—
33#88071C2212yes—
34#88071C2212yes—
35#88071C2212yes—
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C24×C13C2311yescharacteristic
39C2×D4×C13—22no—
40C2×D4×C13—22no—
41C2×D4×C13—22no—
42C22×C4×C13—22no—
43C22×C4×C13—42no—
44C22×C4×C13C2312yes—
45C22×C4×C13C2311yescharacteristic
46C22×C4×C13C2312yes—
47C22×Dic13—22no—
48C22×Dic13—22no—
49C22×Dic13—22no—
50C22×Dic13—42no—
51C22×Dic13D414yes—
52C22×Dic13C2311yescharacteristic
53C22×Dic13C2312yes—
54C22×Dic13C2312yes—
55C22×Dic13C2311yescharacteristic
56C22×Dic13C2311yescharacteristic
57C22×Dic13C2311yescharacteristic
58C13×C22⋊C4—22no—
59C13×C22⋊C4—42no—
60C22⋊Dic13—22no—
61C22⋊Dic13—22no—
62C22⋊Dic13—22no—
63C22⋊Dic13—22no—
64C22⋊Dic13—42no—
65C4⋊Dic13—42no—
66#6101—24no—
67#6101—24no—
68#6101—22no—
69#6101—22no—
70#6101—42no—
71#2623—131nomaximal, Sylow
72C23×C13—22no—
73C23×C13—22no—
74C23×C13—21no—
75C23×C13—21no—
76C23×C13—21no—
77C23×C13—22no—
78C23×C13—22no—
79C23×C13C2×D412yes—
80C23×C13#4112yes—
81C23×C13#4112yes—
82C23×C13C2411yescharacteristic
83D4×C13—44no—
84C2×C4×C13—22no—
85C2×C4×C13—21no—
86C2×C4×C13—22no—
87C2×C4×C13—21no—
88C2×C4×C13—21no—
89C2×C4×C13—22no—
90C2×C4×C13—22no—
91C2×C4×C13—22no—
92C2×C4×C13—22no—
93C2×C4×C13—44no—
94C2×Dic13—28no—
95C2×Dic13—21no—
96C2×Dic13—21no—
97C2×Dic13—21no—
98C2×Dic13—21no—
99C2×Dic13—22no—
100C2×Dic13—21no—
101C2×Dic13—21no—
102C2×Dic13—22no—
103C2×Dic13—22no—
104C2×Dic13—22no—
105C2×Dic13—22no—
106C2×Dic13—22no—
107C2×Dic13—22no—
108C2×Dic13—22no—
109C2×Dic13—44no—
110C2×Dic13C2×D412yes—
111C2×Dic13C2×D412yes—
112C2×#123—131no—
113C22×C4⋊C4—131no—
114#400—132no—
115#399—132no—
116#398—132no—
117#397—132no—
118#386—131no—
119#386D1311yescharacteristic
120#385—131no—
121#385—131no—
122#381—131no—
123C22×C13—22no—
124C22×C13—22no—
125C22×C13—24no—
126C22×C13—24no—
127C22×C13—41no—
128C22×C13—41no—
129C22×C13—41no—
130C22×C13—42no—
131C22×C13—42no—
132C22×C13#12314yes—
133C22×C13C22×D411yescharacteristic
134C22×C13C2×#4111yescharacteristic
135C22×C13C2×#4111yescharacteristic
136C22×C13H(5,𝔽2)12yes—
137C22×C132−1+412yes—
138C4×C13—41no—
139C4×C13—42no—
140C4×C13—42no—
141Dic13—22no—
142Dic13—22no—
143Dic13—41no—
144Dic13—42no—
145Dic13—41no—
146Dic13—42no—
147C22×D4C2×D1311yescharacteristic
148C23×C4—131no—
149C23×C4—131no—
150C23×C4C2×D1311yescharacteristic
151#123—264no—
152C2×C4⋊C4—132no—
153C2×C4⋊C4—132no—
154C2×C4⋊C4—131no—
155C2×C4⋊C4—131no—
156C2×C4⋊C4—132no—
157C2×C4⋊C4—132no—
158C2×C4⋊C4—264no—
159C2×C22⋊C4—131no—
160C2×C22⋊C4—131no—
161C2×C22⋊C4—131no—
162C2×C22⋊C4—131no—
163C2×C22⋊C4—132no—
164C2×C22⋊C4C2×D1311yescharacteristic
165C2×C22⋊C4C2×D1312yes—
166#95—132no—
167#95—132no—
168#95—132no—
169#95—132no—
170#95—132no—
171#95C2×D1312yes—
172C2×C13—24no—
173C2×C13—81no—
174C2×C13C2×#12311yescharacteristic
175C2×C13#53511yescharacteristic
176C2×C13#53611yescharacteristic
177C2×C13#53912yes—
178C2×C13#53912yes—
179C24C22×D1311yescharacteristic
180C24C22×D1311yescharacteristic
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—134no—
187C22×C4—131no—
188C22×C4—131no—
189C22×C4—132no—
190C22×C4—131no—
191C22×C4—132no—
192C22×C4—131no—
193C22×C4—262no—
194C22×C4—262no—
195C22×C4—262no—
196C22×C4—522no—
197C22×C4C22×D1312yes—
198C22×C4C22×D1311yescharacteristic
199C22×C4C22×D1312yes—
200C4⋊C4—264no—
201C4⋊C4—264no—
202C4⋊C4—262no—
203C4⋊C4—262no—
204C4⋊C4—522no—
205C4⋊C4—522no—
206C22⋊C4—22no—
207C22⋊C4—42no—
208C22⋊C4—262no—
209C22⋊C4—262no—
210C22⋊C4—262no—
211C22⋊C4—262no—
212C22⋊C4—522no—
213C13#262311yescharacteristic, complemented, Sylow
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23#612712yes—
222C23D4×D1312yes—
223C23#612912yes—
224C23C23×D1311yescharacteristic
225D4—44no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—132no—
237C2×C4—132no—
238C2×C4—268no—
239C2×C4—261no—
240C2×C4—261no—
241C2×C4—261no—
242C2×C4—261no—
243C2×C4—262no—
244C2×C4—261no—
245C2×C4—261no—
246C2×C4—262no—
247C2×C4—262no—
248C2×C4—262no—
249C2×C4—262no—
250C2×C4—262no—
251C2×C4—262no—
252C2×C4—262no—
253C2×C4—524no—
254C22—24no—
255C22—22no—
256C22—24no—
257C22—22no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22#8813714yes—
264C22C2×#612711yescharacteristic
265C22C2×D4×D1311yescharacteristic
266C22C2×#612911yescharacteristic
267C22#8825112yes—
268C22#8825712yes—
269C4—41no—
270C4—42no—
271C4—42no—
272C4—262no—
273C4—262no—
274C4—522no—
275C4—521no—
276C4—522no—
277C4—521no—
278C2—24no—
279C2—81no—
280C2C2×#8813711yescharacteristic
281C2#1171926912yes—
282C2#1171929711yescharacteristic
283C2#1171930712yes—
284C2#1171935311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#386C2
prime radicalC22×D4×C13C22
hypercenter#386D13
hypocenterC13#2623
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×#6101C2semidirect
5. #11718871C2semidirect
6. #11718613C2semidirect
7. #11718611C2semidirect
8. #11718610C2semidirect
9. #11718604C2semidirect
10. #11718319C2semidirect
11. #11718318C2semidirect
213. C13#2623semidirect

Maximal subgroups

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

subgroupindex
C2×#881892
C22×#61012
C13×#3862
#117188812
#117188712
#117186132
#117186112
#117186102
#117186042
#117183192
#117183182
#262313