Finite Group Databasebeta

← #13120004

Subgroups — #13120004

Order 1664.

564 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 151 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×#88198C211yesmaximal, characteristic, complemented
2C2×#88135C211yesmaximal, characteristic
3C13×#393C211yesmaximal, characteristic, complemented
4#11718892C211yesmaximal, characteristic, complemented
5#11718618C212yesmaximal
6#11718608C211yesmaximal, characteristic, complemented
7#11718603C212yesmaximal, complemented
8#11718334C211yesmaximal, characteristic, complemented
9#11718328C212yesmaximal, complemented
10#11718311C212yesmaximal, complemented
11#11718310C211yesmaximal, characteristic, complemented
12C22×Q8×C13C2211yescharacteristic
13C22×C22⋊D13C2211yescharacteristic
14C23×Dic13C2211yescharacteristic
15C22×C4×D13C2211yescharacteristic
16C22×Dic26C2211yescharacteristic
17C2×C13×C22⋊C4C2211yescharacteristic
18C2×C13×C22⋊C4C2212yes—
19C2×C22⋊Dic13C2212yes—
20#88198—24no—
21C2×#6103C2211yescharacteristic
22C2×#6103C2212yes—
23C2×#6103C2212yes—
24C2×#6101C2211yescharacteristic
25C2×#6101C2212yes—
26C2×#6101C2212yes—
27C2×#6101C2212yes—
28C2×C4×Dic13C2212yes—
29C2×C4×Dic13C2211yescharacteristic
30#88135—24no—
31C13×#95C2211yescharacteristic
32C13×#95C2212yes—
33#88071C2211yescharacteristic
34#88071C2212yes—
35#88071C2212yes—
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C23×D13C2311yescharacteristic
39C2×Q8×C13—22no—
40C2×Q8×C13—42no—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2312yes—
43C22×C4×C13C2312yes—
44C22×C4×C13C2311yescharacteristic
45C2×C22⋊D13—22no—
46C2×C22⋊D13—22no—
47C2×C22⋊D13Q814yescomplemented
48C22×Dic13—22no—
49C22×Dic13—24no—
50C22×Dic13C2311yescharacteristic
51C22×Dic13C2312yes—
52C22×Dic13C2312yes—
53C22×Dic13C2311yescharacteristic
54C22×Dic13C2311yescharacteristic
55C2×C4×D13—22no—
56C2×C4×D13—24no—
57C2×Dic26—22no—
58C2×Dic26—42no—
59C13×C22⋊C4—22no—
60C13×C22⋊C4—42no—
61C22⋊Dic13—42no—
62#6103—22no—
63#6103—42no—
64#6103—42no—
65#6101—22no—
66#6101—24no—
67#6101—24no—
68#6101—24no—
69C4×Dic13—24no—
70C4×Dic13D414yes—
71#2563—131nomaximal, Sylow
72C23×C13—21no—
73C23×C13—22no—
74C23×C13—21no—
75C23×C13—22no—
76C23×C13C2×Q812yes—
77C23×C13C2411yescharacteristic
78C22×D13—21no—
79C22×D13—22no—
80C22×D13—21no—
81C22×D13—22no—
82C22×D13C2×Q812yes—
83Q8×C13—44no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13—21no—
87C2×C4×C13—22no—
88C2×C4×C13—22no—
89C2×C4×C13—22no—
90C2×C4×C13—22no—
91C2×C4×C13—22no—
92C2×C4×C13—22no—
93C2×C4×C13C2×D412yes—
94C22⋊D13—28no—
95C2×Dic13—22no—
96C2×Dic13—21no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13—22no—
102C2×Dic13—22no—
103C2×Dic13—44no—
104C2×Dic13C2×D412yes—
105C2×Dic13C2×D412yes—
106C2×Dic13C2×Q812yes—
107C2×Dic13#4112yes—
108C2×Dic13#4114yes—
109C4×D13—44no—
110Dic26—44no—
111C2×C4⋊Q8—131no—
112C2×C4×D4—131no—
113#393—131no—
114#393—131no—
115#393—131no—
116#393D1311yescharacteristic, complemented
117#390—132no—
118#382—132no—
119#380—132no—
120#380—132no—
121C4×C4⋊C4—131no—
122C22×C13—22no—
123C22×C13—24no—
124C22×C13—41no—
125C22×C13—41no—
126C22×C13—42no—
127C22×C13C22×D411yescharacteristic
128C22×C13C22×Q811yescharacteristic
129C22×C13C2×#4111yescharacteristic
130C22×C13C2×#4112yes—
131C22×C13H(5,𝔽2)12yes—
132C2×D13—22no—
133C2×D13—24no—
134C2×D13—41no—
135C2×D13—41no—
136C2×D13—42no—
137C4×C13—22no—
138C4×C13—41no—
139C4×C13—42no—
140C4×C13—42no—
141Dic13—41no—
142Dic13—42no—
143Dic13—42no—
144Dic13C22⋊Q814yes—
145Dic13#12414yes—
146C22×Q8—131no—
147C22×Q8C2×D1311yescharacteristic
148C22×D4—131no—
149C23×C4—131no—
150C23×C4—131no—
151C4⋊Q8—264no—
152C4×D4—264no—
153C2×C4⋊C4—131no—
154C2×C4⋊C4—132no—
155C2×C4⋊C4—132no—
156C2×C4⋊C4—132no—
157C2×C22⋊C4—131no—
158C2×C22⋊C4—132no—
159C2×C22⋊C4—132no—
160C2×C22⋊C4—132no—
161C2×C22⋊C4C2×D1311yescharacteristic
162C2×C22⋊C4C2×D1312yes—
163C2×C42—131no—
164C2×C42—132no—
165#95—131no—
166#95—132no—
167#95—132no—
168#95—132no—
169#95C2×D1311yescharacteristic
170#95C2×D1312yes—
171C2×C13—42no—
172C2×C13C2×C22⋊Q811yescharacteristic
173C2×C13C2×#12411yescharacteristic
174C2×C13#53511yescharacteristic
175C2×C13#55412yes—
176C2×C13#55512yes—
177D13—42no—
178C24—131no—
179C24C22×D1311yescharacteristic
180C2×Q8—22no—
181C2×Q8—42no—
182C2×Q8—262no—
183C2×Q8—522no—
184C2×D4—134no—
185C2×D4—262no—
186C2×D4—262no—
187C22×C4—131no—
188C22×C4—131no—
189C22×C4—131no—
190C22×C4—132no—
191C22×C4—132no—
192C22×C4—262no—
193C22×C4—264no—
194C22×C4—262no—
195C22×C4—264no—
196C22×C4C22×D1311yescharacteristic
197C22×C4C22×D1312yes—
198C22×C4C22×D1312yes—
199C22×C4C22×D1311yescharacteristic
200C4⋊C4—262no—
201C4⋊C4—264no—
202C4⋊C4—264no—
203C4⋊C4—264no—
204C22⋊C4—22no—
205C22⋊C4—42no—
206C22⋊C4—262no—
207C22⋊C4—522no—
208C22⋊C4—522no—
209C22⋊C4—522no—
210C42—134no—
211C42—264no—
212C13#256311yescharacteristic, complemented, Sylow
213C23—21no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—132no—
218C23—261no—
219C23—262no—
220C23—262no—
221C23—261no—
222C23Q8×D1312yes—
223C23C23×D1311yescharacteristic
224Q8—44no—
225Q8—524no—
226D4—268no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—132no—
237C2×C4—132no—
238C2×C4—132no—
239C2×C4—132no—
240C2×C4—134no—
241C2×C4—261no—
242C2×C4—262no—
243C2×C4—262no—
244C2×C4—262no—
245C2×C4—262no—
246C2×C4—262no—
247C2×C4—262no—
248C2×C4—262no—
249C2×C4—524no—
250C2×C4—524no—
251C2×C4D4×D1312yes—
252C22—22no—
253C22—24no—
254C22—41no—
255C22—42no—
256C22—41no—
257C22—262no—
258C22—264no—
259C22—521no—
260C22—522no—
261C22—521no—
262C22C2×D4×D1311yescharacteristic
263C22#8825112yes—
264C22C2×Q8×D1311yescharacteristic
265C22#41×D1311yescharacteristic
266C22#41×D1312yes—
267C4—22no—
268C4—42no—
269C4—42no—
270C4—41no—
271C4—134no—
272C4—134no—
273C4—522no—
274C4—522no—
275C4—521no—
276C2—42no—
277C2—522no—
278C2#1171929711yescharacteristic
279C2D13×C22⋊Q811yescharacteristic
280C2#1171934112yes—
281C2#1171937512yes—
282C2D13×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#393C2
prime radicalC22×C22⋊D13C22
hypercenter#393D13
hypocenterC13#2563
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#88198C2semidirect
3. C13×#393C2semidirect
4. #11718892C2semidirect
6. #11718608C2semidirect
7. #11718603C2semidirect
8. #11718334C2semidirect
9. #11718328C2semidirect
10. #11718311C2semidirect
11. #11718310C2semidirect
47. C2×C22⋊D13Q8semidirect
116. #393D13semidirect
212. C13#2563semidirect

Maximal subgroups

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

subgroupindex
C2×#881982
C2×#881352
C13×#3932
#117188922
#117186182
#117186082
#117186032
#117183342
#117183282
#117183112
#117183102
#256313