Finite Group Databasebeta

← #13120591

Subgroups — #13120591

Order 1664.

728 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 149 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×#88194C211yesmaximal, characteristic, complemented
2C2×#88192C211yesmaximal, characteristic, complemented
3C2×#88149C211yesmaximal, characteristic, complemented
4C2×#88126C211yesmaximal, characteristic, complemented
5C13×#390C211yesmaximal, characteristic, complemented
6#11718638C211yesmaximal, characteristic, complemented
7#11718621C214yesmaximal, complemented
8#11718620C214yesmaximal, complemented
9#11718600C211yesmaximal, characteristic, complemented
10C22×D4×C13C2211yescharacteristic
11C22×C22⋊D13C2212yes—
12C22×C22⋊D13C2214yes—
13C22×D52C2211yescharacteristic
14C22×D52C2212yes—
15C22×C4×D13C2211yescharacteristic
16C2×C13×C4⋊C4C2211yescharacteristic
17C2×C13×C22⋊C4C2214yes—
18C2×C42×C13C2211yescharacteristic
19C2×C22⋊Dic13C2212yes—
20#88194—24no—
21#88192—24no—
22C2×#6103C2212yes—
23C2×#6103C2212yes—
24C2×#6103C2214yes—
25C2×C4⋊Dic13C2211yescharacteristic
26C2×#6101C2212yes—
27C2×C4×Dic13C2211yescharacteristic
28#88149—24no—
29#88126—24no—
30#88071C2214yes—
31C24×C13C2312yes—
32C23×D13C2312yes—
33C23×D13C2311yescharacteristic
34C2×D4×C13—22no—
35C2×D4×C13—22no—
36C2×D4×C13—22no—
37C22×C4×C13C2312yes—
38C22×C4×C13C2312yes—
39C22×C4×C13C2311yescharacteristic
40C2×C22⋊D13—24no—
41C2×C22⋊D13—28no—
42C2×C22⋊D13—44no—
43C2×C22⋊D13—44no—
44C2×C22⋊D13—44no—
45C22×Dic13C2312yes—
46C22×Dic13C2312yes—
47C22×Dic13C2311yescharacteristic
48C2×D52—24no—
49C2×D52—22no—
50C2×D52—22no—
51C2×D52—22no—
52C2×D52—44no—
53C2×C4×D13—22no—
54C2×C4×D13—22no—
55C2×C4×D13D414yescomplemented
56C13×C4⋊C4—22no—
57C13×C22⋊C4—44no—
58C42×C13—22no—
59C22⋊Dic13—24no—
60#6103—24no—
61#6103—24no—
62#6103—44no—
63C4⋊Dic13—22no—
64#6101—24no—
65C4×Dic13D414yescomplemented
66#2619—131nomaximal, Sylow
67C23×C13—22no—
68C23×C13—22no—
69C23×C13—22no—
70C23×C13—24no—
71C23×C13—24no—
72C23×C13C2411yescharacteristic
73C22×D13—21no—
74C22×D13—21no—
75C22×D13—22no—
76C22×D13—22no—
77C22×D13—22no—
78C22×D13—22no—
79C22×D13—22no—
80C22×D13—24no—
81C22×D13—24no—
82C22×D13C2×D412yes—
83D4×C13—44no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13—22no—
87C2×C4×C13—22no—
88C2×C4×C13—24no—
89C2×C4×C13C2×D412yes—
90C2×C4×C13C2×D412yes—
91C2×C4×C13#4112yes—
92C22⋊D13—416no—
93C22⋊D13—48no—
94C2×Dic13—21no—
95C2×Dic13—21no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—24no—
100C2×Dic13C2×D414yes—
101C2×Dic13C2×D412yes—
102D52—48no—
103D52—44no—
104C4×D13—28no—
105C2×#127—131no—
106C2×#121—131no—
107C2×#121—131no—
108C2×#117—131no—
109#394—134no—
110#392—134no—
111#390—131no—
112#390D1311yescharacteristic, complemented
113#382—131no—
114C22×C13—42no—
115C22×C13—44no—
116C22×C13—42no—
117C22×C13—42no—
118C22×C13—44no—
119C22×C13C22×D411yescharacteristic
120C22×C13C22×D411yescharacteristic
121C22×C13C2×#4111yescharacteristic
122C22×C13H(5,𝔽2)12yes—
123C22×C13H(5,𝔽2)12yes—
124C2×D13—22no—
125C2×D13—22no—
126C2×D13—22no—
127C2×D13—42no—
128C2×D13—42no—
129C2×D13—42no—
130C2×D13—42no—
131C2×D13—44no—
132C2×D13—42no—
133C2×D13—44no—
134C4×C13—42no—
135C4×C13—42no—
136C4×C13#12114yes—
137Dic13—22no—
138Dic13—24no—
139Dic13—42no—
140C22×D4—132no—
141C22×D4—134no—
142C22×D4—131no—
143C22×D4—132no—
144C22×D4C2×D1311yescharacteristic
145C23×C4—131no—
146#127—264no—
147#121—264no—
148#121—264no—
149#117—264no—
150C2×C4⋊C4—132no—
151C2×C4⋊C4—131no—
152C2×C4⋊C4C2×D1311yescharacteristic
153C2×C22⋊C4—132no—
154C2×C22⋊C4—132no—
155C2×C22⋊C4—132no—
156C2×C22⋊C4—134no—
157C2×C22⋊C4C2×D1314yes—
158C2×C42—131no—
159C2×C42C2×D1311yescharacteristic
160#95—134no—
161C2×C13—82no—
162C2×C13C2×#12111yescharacteristic
163C2×C13#53512yes—
164C2×C13#53511yescharacteristic
165C2×C13#53511yescharacteristic
166C2×C13#54012yes—
167D13—42no—
168D13—82no—
169C24—131no—
170C24—132no—
171C24C22×D1312yes—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—264no—
176C2×D4—268no—
177C2×D4—264no—
178C2×D4—262no—
179C2×D4—262no—
180C2×D4—262no—
181C2×D4—524no—
182C2×D4—524no—
183C2×D4—524no—
184C2×D4—524no—
185C22×C4—132no—
186C22×C4—131no—
187C22×C4—134no—
188C22×C4—132no—
189C22×C4—262no—
190C22×C4—262no—
191C22×C4C22×D1312yes—
192C22×C4C22×D1312yes—
193C22×C4C22×D1311yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—264no—
196C4⋊C4—262no—
197C22⋊C4—44no—
198C22⋊C4—264no—
199C22⋊C4—264no—
200C22⋊C4—264no—
201C22⋊C4—524no—
202C42—22no—
203C42—134no—
204C13#261911yescharacteristic, complemented, Sylow
205C23—22no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—24no—
210C23—132no—
211C23—261no—
212C23—261no—
213C23—262no—
214C23—262no—
215C23—262no—
216C23—262no—
217C23—262no—
218C23—264no—
219C23—264no—
220C23C23×D1311yescharacteristic
221D4—44no—
222D4—528no—
223D4—5216no—
224D4—528no—
225D4—524no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—134no—
232C2×C4—132no—
233C2×C4—261no—
234C2×C4—261no—
235C2×C4—262no—
236C2×C4—262no—
237C2×C4—268no—
238C2×C4—262no—
239C2×C4—264no—
240C2×C4#612712yes—
241C2×C4D4×D1312yes—
242C2×C4D4×D1312yes—
243C22—42no—
244C22—44no—
245C22—44no—
246C22—42no—
247C22—42no—
248C22—262no—
249C22—262no—
250C22—262no—
251C22—522no—
252C22—522no—
253C22—522no—
254C22—524no—
255C22—524no—
256C22—522no—
257C22—522no—
258C22C2×#612711yescharacteristic
259C22C2×D4×D1311yescharacteristic
260C22C2×D4×D1311yescharacteristic
261C22#8825112yes—
262C22#8825612yes—
263C4—42no—
264C4—42no—
265C4—262no—
266C4—264no—
267C4—522no—
268C4#8813814yes—
269C2—82no—
270C2—522no—
271C2—1042no—
272C2C2×#8813811yescharacteristic
273C2#1171926712yes—
274C2#1171931212yes—
275C2#1171937911yescharacteristic
276C2#1171942011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#390C2
hypercenter#390D13
hypocenterC13#2619
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#88194C2semidirect
2. C2×#88192C2semidirect
3. C2×#88149C2semidirect
4. C2×#88126C2semidirect
5. C13×#390C2semidirect
6. #11718638C2semidirect
7. #11718621C2semidirect
8. #11718620C2semidirect
9. #11718600C2semidirect
55. C2×C4×D13D4semidirect
65. C4×Dic13D4semidirect
112. #390D13semidirect
204. C13#2619semidirect

Maximal subgroups

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

subgroupindex
C2×#881942
C2×#881922
C2×#881492
C2×#881262
C13×#3902
#117186382
#117186212
#117186202
#117186002
#261913