Finite Group Databasebeta

← #13120702

Subgroups — #13120702

Order 1664.

500 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 131 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×#88201C211yesmaximal, characteristic
2C2×#88141C211yesmaximal, characteristic, complemented
3C13×#395C211yesmaximal, characteristic, complemented
4#11718890C211yesmaximal, characteristic, complemented
5#11718653C212yesmaximal, complemented
6#11718652C211yesmaximal, characteristic
7#11718651C212yesmaximal, complemented
8#11718334C212yesmaximal, complemented
9#11718333C211yesmaximal, characteristic
10#11718330C212yesmaximal, complemented
11#11718312C211yesmaximal, characteristic, complemented
12C22×Q8×C13C2211yescharacteristic
13C22×D52C2211yescharacteristic
14C22×Dic26C2211yescharacteristic
15C2×C13×C4⋊C4C2211yescharacteristic
16C2×C13×C4⋊C4C2212yes—
17#88201—24no—
18C2×#6103C2212yes—
19C2×#6103C2212yes—
20C2×#6103C2212yes—
21C2×#6103C2212yes—
22C2×#6103C2212yes—
23C2×#6103C2212yes—
24C2×#6101C2211yescharacteristic
25C2×#6101C2211yescharacteristic
26C2×#6101C2212yes—
27C2×C4×Dic13C2211yescharacteristic
28C2×C4×Dic13C2211yescharacteristic
29C2×C4×Dic13C2211yescharacteristic
30#88141—24no—
31C13×#95C2211yescharacteristic
32C13×#95C2212yes—
33#88071C2212yes—
34#88071C2212yes—
35#88071C2211yescharacteristic
36#88071C2212yes—
37C23×D13C2312yes—
38C2×Q8×C13—22no—
39C2×Q8×C13—42no—
40C22×C4×C13C2312yes—
41C22×C4×C13C2312yes—
42C22×C4×C13C2311yescharacteristic
43C22×C4×C13C2311yescharacteristic
44C22×C4×C13C2311yescharacteristic
45C22×Dic13C2312yes—
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2311yescharacteristic
48C22×Dic13C2311yescharacteristic
49C22×Dic13C2311yescharacteristic
50C2×D52—22no—
51C2×D52—42no—
52C2×Dic26—22no—
53C2×Dic26—22no—
54C2×Dic26—22no—
55C13×C4⋊C4—22no—
56C13×C4⋊C4—42no—
57#6103—24no—
58#6103—24no—
59#6103—24no—
60#6103—24no—
61#6103—42no—
62#6103—42no—
63#6101—22no—
64#6101—22no—
65#6101—42no—
66C4×Dic13—22no—
67C4×Dic13—22no—
68C4×Dic13D414yes—
69#2632—131nomaximal, Sylow
70C23×C13C2411yescharacteristic
71C22×D13—22no—
72C22×D13—22no—
73C22×D13—22no—
74C22×D13—22no—
75C22×D13—22no—
76C22×D13—22no—
77C22×D13—22no—
78Q8×C13—44no—
79C2×C4×C13—21no—
80C2×C4×C13—21no—
81C2×C4×C13—22no—
82C2×C4×C13—21no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13—22no—
87C2×C4×C13—22no—
88C2×C4×C13—22no—
89C2×C4×C13—22no—
90C2×C4×C13C2×D412yes—
91C2×C4×C13#4112yes—
92C2×Dic13—21no—
93C2×Dic13—21no—
94C2×Dic13—21no—
95C2×Dic13—21no—
96C2×Dic13—22no—
97C2×Dic13—21no—
98C2×Dic13—22no—
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13—22no—
102C2×Dic13C2×D412yes—
103C2×Dic13C2×D412yes—
104C2×Dic13#4112yes—
105D52—44no—
106Dic26—28no—
107C2×#124—131no—
108C2×C4×Q8—131no—
109#399—132no—
110#395—131no—
111#395D1311yescharacteristic, complemented
112#394—131no—
113#394—131no—
114#393—132no—
115#388—132no—
116#388—132no—
117#383—131no—
118C22×C13C22×D411yescharacteristic
119C22×C13C2×#4111yescharacteristic
120C22×C13C2×#4111yescharacteristic
121C22×C13H(5,𝔽2)12yes—
122C22×C132−1+412yes—
123C2×D13—42no—
124C2×D13—42no—
125C2×D13—42no—
126C2×D13—42no—
127C2×D13—42no—
128C2×D13—42no—
129C2×D13—42no—
130C4×C13—22no—
131C4×C13—22no—
132C4×C13—41no—
133C4×C13—42no—
134C4×C13—42no—
135Dic13—22no—
136Dic13—22no—
137Dic13—22no—
138Dic13—41no—
139Dic13—42no—
140C22×Q8—131no—
141C22×Q8C2×D1311yescharacteristic
142C22×D4—131no—
143#124—264no—
144C4×Q8—264no—
145C2×C4⋊C4—131no—
146C2×C4⋊C4—131no—
147C2×C4⋊C4—132no—
148C2×C4⋊C4C2×D1311yescharacteristic
149C2×C4⋊C4C2×D1312yes—
150C2×C22⋊C4—132no—
151C2×C22⋊C4—132no—
152C2×C22⋊C4—132no—
153C2×C22⋊C4—132no—
154C2×C22⋊C4—132no—
155C2×C22⋊C4—132no—
156C2×C42—131no—
157C2×C42—131no—
158C2×C42—131no—
159#95—132no—
160#95—132no—
161#95—131no—
162#95—132no—
163#95C2×D1311yescharacteristic
164#95C2×D1312yes—
165C2×C13#52511yescharacteristic
166C2×C13#53511yescharacteristic
167C2×C13#53611yescharacteristic
168C2×C13#54212yes—
169C2×C13#54212yes—
170D13—82no—
171C24—132no—
172C2×Q8—22no—
173C2×Q8—42no—
174C2×Q8—262no—
175C2×Q8—262no—
176C2×Q8—262no—
177C2×D4—262no—
178C2×D4—522no—
179C22×C4—131no—
180C22×C4—131no—
181C22×C4—131no—
182C22×C4—132no—
183C22×C4—131no—
184C22×C4C22×D1312yes—
185C22×C4C22×D1312yes—
186C22×C4C22×D1311yescharacteristic
187C22×C4C22×D1311yescharacteristic
188C22×C4C22×D1311yescharacteristic
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—262no—
192C4⋊C4—262no—
193C4⋊C4—522no—
194C22⋊C4—264no—
195C22⋊C4—264no—
196C22⋊C4—264no—
197C22⋊C4—264no—
198C22⋊C4—522no—
199C22⋊C4—522no—
200C42—134no—
201C42—262no—
202C42—262no—
203C13#263211yescharacteristic, complemented, Sylow
204C23—262no—
205C23—262no—
206C23—262no—
207C23—262no—
208C23—262no—
209C23—262no—
210C23—262no—
211C23C23×D1311yescharacteristic
212Q8—44no—
213Q8—268no—
214D4—524no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—132no—
227C2×C4—132no—
228C2×C4—132no—
229C2×C4—261no—
230C2×C4—261no—
231C2×C4—261no—
232C2×C4—262no—
233C2×C4—261no—
234C2×C4—261no—
235C2×C4—262no—
236C2×C4—262no—
237C2×C4—262no—
238C2×C4—262no—
239C2×C4D4×D1312yes—
240C2×C4#613112yes—
241C22—522no—
242C22—522no—
243C22—522no—
244C22—522no—
245C22—522no—
246C22—522no—
247C22—522no—
248C22C2×D4×D1311yescharacteristic
249C22#8825112yes—
250C22C2×#613111yescharacteristic
251C22#8825412yes—
252C22#41×D1311yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—262no—
259C4—262no—
260C4—262no—
261C4—522no—
262C4—521no—
263C2—1042no—
264C2#1171929711yescharacteristic
265C2#1171933211yescharacteristic
266C2#1171934512yes—
267C2#1171942911yescharacteristic
268C2#1171943812yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#395C2
prime radicalC22×D52C22
hypercenter#395D13
hypocenterC13#2632
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#88141C2semidirect
3. C13×#395C2semidirect
4. #11718890C2semidirect
5. #11718653C2semidirect
7. #11718651C2semidirect
8. #11718334C2semidirect
10. #11718330C2semidirect
11. #11718312C2semidirect
111. #395D13semidirect
203. C13#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#882012
C2×#881412
C13×#3952
#117188902
#117186532
#117186522
#117186512
#117183342
#117183332
#117183302
#117183122
#263213