Finite Group Databasebeta

← #13124657

Subgroups — #13124657

Order 1664.

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

normal subgroups only

This group has 384 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
1#11719227C211yesmaximal, characteristic
2C2×#88144C211yesmaximal, characteristic, complemented
3C13×#428C211yesmaximal, characteristic, complemented
4#11718794C211yesmaximal, characteristic
5#11718779C211yesmaximal, characteristic, complemented
6#11718656C211yesmaximal, characteristic, complemented
7#11718627C211yesmaximal, characteristic
8#11718552C212yesmaximal, complemented
9#11718538C212yesmaximal, complemented
10#11718532C212yesmaximal, complemented
11#11718518C212yesmaximal, complemented
12C2×#6127C2211yescharacteristic
13C2×C13×M16C2211yescharacteristic
14C13×#117C2211yescharacteristic
15C2×C13×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic13—21no—
17C2×C4⋊Dic13C2211yescharacteristic
18C2×#6101—21no—
19C2×C4×Dic13C2211yescharacteristic
20C2×#6099C2211yescharacteristic
21#88147C2211yescharacteristic
22#88147C2211yescharacteristic
23#88144—21no—
24#88144—21no—
25#88144C2212yes—
26#88144C2212yes—
27#88141C2211yescharacteristic
28#88141C2211yescharacteristic
29#88135—21no—
30#88135—21no—
31#88126—21no—
32C42×D13—21no—
33C13×#107C2212yes—
34C13×#106C2212yes—
35#88061C2212yes—
36#88056C2212yes—
37#88054C2212yes—
38#88054C2212yes—
39#88050C2212yes—
40#88049C2212yes—
41#88048C2212yes—
42#88047C2212yes—
43C22×C4×C13—21no—
44C22×C4×C13C2311yescharacteristic
45C2×C22⋊D13—21no—
46C22×Dic13—21no—
47C22×Dic13D411yescharacteristic
48C22×Dic13D411yescharacteristic
49#6127—22no—
50#6127Q814yes—
51C2×D52C2311yescharacteristic
52C2×C4×D13—21no—
53C2×C4×D13—21no—
54C2×C4×D13—21no—
55C2×Dic26C2311yescharacteristic
56C13×M16—22no—
57C2×C8×C13C2312yes—
58C13×C4⋊C4—21no—
59C13×C4⋊C4C2311yescharacteristic
60C13×C4⋊C4C2312yes—
61C13×C4⋊C4C2311yescharacteristic
62C13×C22⋊C4—21no—
63C42×C13—21no—
64#6103—21no—
65#6103—21no—
66C4⋊Dic13—21no—
67C4⋊Dic13—22no—
68C4⋊Dic13—22no—
69C4⋊Dic13C2312yes—
70#6101—21no—
71#6101—21no—
72#6101—22no—
73#6101—22no—
74C4×Dic13—21no—
75C4×Dic13—21no—
76C4×Dic13D411yescharacteristic
77C4×Dic13D411yescharacteristic
78C4×Dic13C2311yescharacteristic
79C4×Dic13C2311yescharacteristic
80#6099—22no—
81C2×#1055C2312yes—
82#3080—131nomaximal, Sylow
83C23×C13C2×D411yescharacteristic
84C22×D13—21no—
85C2×C4×C13—21no—
86C2×C4×C13—21no—
87C2×C4×C13—22no—
88C2×C4×C13—42no—
89C2×C4×C13C2×D411yescharacteristic
90C2×C4×C13C2×Q812yes—
91C2×C4×C13#4112yes—
92C2×C4×C13C2411yescharacteristic
93C22⋊D13—24no—
94C2×Dic13—21no—
95C2×Dic13—22no—
96C2×Dic13—41no—
97C2×Dic13—41no—
98C2×Dic13—42no—
99C2×Dic13C2×D411yescharacteristic
100C2×Dic13C2×D411yescharacteristic
101C2×Dic13C2×D411yescharacteristic
102C2×Dic13C2×D411yescharacteristic
103D52—21no—
104D52C2×Q812yes—
105C4×D13—24no—
106C4×D13—42no—
107C4×D13—42no—
108Dic26—21no—
109Dic26C2×Q812yes—
110C8×C13—22no—
111#1055—22no—
112C2×#125—131no—
113C4×#41—131no—
114#479—132no—
115#479—132no—
116#478—132no—
117#478—132no—
118#428—131no—
119#428D1311yescharacteristic, complemented
120#423—131no—
121#417—131no—
122#417—131no—
123C22×C13—21no—
124C22×C13—21no—
125C22×C13C22⋊Q812yes—
126C22×C13C22×D411yescharacteristic
127C2×D13—22no—
128C2×D13—41no—
129C2×D13—41no—
130C4×C13—41no—
131C4×C13—41no—
132C4×C13—42no—
133C4×C13C22⋊Q812yes—
134C4×C13C22×Q811yescharacteristic
135C4×C13C2×#4111yescharacteristic
136Dic13—22no—
137Dic13—21no—
138Dic13—21no—
139Dic13—21no—
140Dic13—21no—
141Dic13—42no—
142C2×#41—131no—
143C2×M16—131no—
144C2×M16C2×D1311yescharacteristic
145#125—132no—
146#125—132no—
147#125—261no—
148#125—261no—
149C4×Q8—131no—
150C4×Q8—131no—
151C4×D4—131no—
152C4×D4—131no—
153C4×D4—261no—
154C4×D4—261no—
155#117—261no—
156#117C2×D1311yescharacteristic
157C2×C4⋊C4—131no—
158C2×C4⋊C4—261no—
159C2×C4⋊C4—261no—
160C2×C4⋊C4C2×D1311yescharacteristic
161C2×C42—131no—
162C2×C42—261no—
163#107—132no—
164#107C2×D1312yes—
165#106—132no—
166#106C2×D1312yes—
167C4⋊C8—132no—
168C4⋊C8—132no—
169#103—132no—
170#103—132no—
171#102—132no—
172#102—132no—
173C2×C13—22no—
174C2×C13C2×C22⋊Q811yescharacteristic
175C2×C13#57511yescharacteristic
176C2×C13#57511yescharacteristic
177D13—42no—
178#41—134no—
179#41—262no—
180C2×Q8—131no—
181C2×D4—131no—
182C2×D4—261no—
183C22×C4—21no—
184C22×C4—131no—
185C22×C4—131no—
186C22×C4—261no—
187C22×C4—261no—
188C22×C4—261no—
189C22×C4—261no—
190C22×C4C22×D1311yescharacteristic
191M16—22no—
192M16—262no—
193C2×C8—132no—
194C2×C8C22×D1312yes—
195C4⋊C4—21no—
196C4⋊C4—132no—
197C4⋊C4—262no—
198C4⋊C4—262no—
199C4⋊C4—262no—
200C4⋊C4—262no—
201C4⋊C4—261no—
202C4⋊C4—261no—
203C4⋊C4—261no—
204C4⋊C4C22×D1312yes—
205C4⋊C4C22×D1311yescharacteristic
206C4⋊C4C22×D1311yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—261no—
209C22⋊C4—261no—
210C42—21no—
211C42—131no—
212C42—131no—
213C42—131no—
214C42—131no—
215C42—261no—
216C42—261no—
217C13#308011yescharacteristic, complemented, Sylow
218C23—261no—
219C23D4×D1311yescharacteristic
220Q8—132no—
221Q8—261no—
222D4—132no—
223D4—264no—
224D4—261no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—131no—
230C2×C4—131no—
231C2×C4—131no—
232C2×C4—131no—
233C2×C4—261no—
234C2×C4—264no—
235C2×C4—262no—
236C2×C4—522no—
237C2×C4—522no—
238C2×C4—521no—
239C2×C4—521no—
240C2×C4—522no—
241C2×C4#612712yes—
242C2×C4D4×D1311yescharacteristic
243C2×C4Q8×D1312yes—
244C2×C4C23×D1311yescharacteristic
245C8—22no—
246C8—262no—
247C22—21no—
248C22—21no—
249C22—262no—
250C22—521no—
251C22—521no—
252C22#8815012yes—
253C22C2×D4×D1311yescharacteristic
254C4—42no—
255C4—41no—
256C4—41no—
257C4—262no—
258C4—261no—
259C4—261no—
260C4—261no—
261C4—261no—
262C4—522no—
263C4#8815012yes—
264C4C2×#612711yescharacteristic
265C4C2×Q8×D1311yescharacteristic
266C2—22no—
267C2—522no—
268C2C2×#8815011yescharacteristic
269C2#1171947811yescharacteristic
270C2#1171948211yescharacteristic
subgroupG/H
centerC22C2×D4×D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#428C2
socleC22×C13C22×D4
prime radicalC2×#6127C22
hypercenter#428D13
hypocenterC13#3080
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#88144C2semidirect
3. C13×#428C2semidirect
5. #11718779C2semidirect
6. #11718656C2semidirect
8. #11718552C2semidirect
9. #11718538C2semidirect
10. #11718532C2semidirect
11. #11718518C2semidirect
119. #428D13semidirect
217. C13#3080semidirect

Maximal subgroups

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

subgroupindex
#117192272
C2×#881442
C13×#4282
#117187942
#117187792
#117186562
#117186272
#117185522
#117185382
#117185322
#117185182
#308013