Finite Group Databasebeta

← #13120403

Subgroups — #13120403

Order 1664.

604 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 135 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×#88147C211yesmaximal, characteristic, complemented
2C2×#88131C211yesmaximal, characteristic, complemented
3C13×#399C211yesmaximal, characteristic, complemented
4#11718651C212yesmaximal, complemented
5#11718650C212yesmaximal, complemented
6#11718643C211yesmaximal, characteristic, complemented
7#11718621C211yesmaximal, characteristic
8#11718619C212yesmaximal, complemented
9#11718617C212yesmaximal, complemented
10#11718333C211yesmaximal, characteristic, complemented
11#11718312C211yesmaximal, characteristic, complemented
12C22×C22⋊D13C2212yes—
13C23×Dic13C2211yescharacteristic
14C22×D52C2211yescharacteristic
15C22×C4×D13C2212yes—
16C2×C13×C4⋊C4C2211yescharacteristic
17C2×C13×C4⋊C4C2212yes—
18C2×C13×C22⋊C4C2211yescharacteristic
19C2×C13×C22⋊C4C2212yes—
20C2×C22⋊Dic13C2211yescharacteristic
21C2×#6103C2212yes—
22C2×#6103C2212yes—
23C2×#6103C2212yes—
24C2×#6103C2212yes—
25C2×#6101C2211yescharacteristic
26C2×#6101C2211yescharacteristic
27C2×C4×Dic13C2211yescharacteristic
28C2×C4×Dic13C2211yescharacteristic
29#88147—24no—
30#88131—24no—
31C13×#95C2211yescharacteristic
32#88071C2212yes—
33#88071C2212yes—
34#88071C2211yescharacteristic
35#88071C2212yes—
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C23×D13C2312yes—
39C22×C4×C13C2312yes—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C2×C22⋊D13—24no—
44C2×C22⋊D13—24no—
45C2×C22⋊D13—42no—
46C22×Dic13—22no—
47C22×Dic13—22no—
48C22×Dic13—22no—
49C22×Dic13C2312yes—
50C22×Dic13C2311yescharacteristic
51C22×Dic13C2311yescharacteristic
52C22×Dic13C2311yescharacteristic
53C22×Dic13C2311yescharacteristic
54C2×D52—24no—
55C2×D52—22no—
56C2×C4×D13—24no—
57C2×C4×D13—42no—
58C2×C4×D13—42no—
59C13×C4⋊C4—22no—
60C13×C4⋊C4—42no—
61C13×C22⋊C4—22no—
62C13×C22⋊C4—42no—
63C22⋊Dic13—22no—
64#6103—24no—
65#6103—24no—
66#6103—24no—
67#6103—42no—
68#6101—22no—
69#6101—22no—
70C4×Dic13—22no—
71C4×Dic13D414yescomplemented
72#2598—131nomaximal, Sylow
73C23×C13—21no—
74C23×C13—21no—
75C23×C13—22no—
76C23×C13—22no—
77C23×C13#4112yes—
78C23×C13C2411yescharacteristic
79C22×D13—22no—
80C22×D13—22no—
81C22×D13—22no—
82C22×D13—22no—
83C22×D13—22no—
84C22×D13—22no—
85C22×D13#4114yes—
86C2×C4×C13—22no—
87C2×C4×C13—21no—
88C2×C4×C13—22no—
89C2×C4×C13—22no—
90C2×C4×C13—22no—
91C2×C4×C13—22no—
92C2×C4×C13—22no—
93C2×C4×C13—22no—
94C2×C4×C13—22no—
95C2×C4×C13C2×D412yes—
96C22⋊D13—48no—
97C2×Dic13—28no—
98C2×Dic13—21no—
99C2×Dic13—21no—
100C2×Dic13—21no—
101C2×Dic13—21no—
102C2×Dic13—22no—
103C2×Dic13—21no—
104C2×Dic13—22no—
105C2×Dic13—22no—
106C2×Dic13—22no—
107C2×Dic13—22no—
108C2×Dic13C2×D412yes—
109C2×Dic13C2×D412yes—
110C2×Dic13#4112yes—
111D52—28no—
112C4×D13—48no—
113C2×C4×D4—131no—
114C2×#117—131no—
115#400—131no—
116#399D1311yescharacteristic, complemented
117#394—131no—
118#392—131no—
119#388—132no—
120#386—132no—
121#386—132no—
122#385—132no—
123#383—131no—
124C22×C13—22no—
125C22×C13—24no—
126C22×C13—41no—
127C22×C13—41no—
128C22×C13—42no—
129C22×C13C22×D411yescharacteristic
130C22×C13C2×#4111yescharacteristic
131C22×C13C2×#4112yes—
132C22×C13C2×#4111yescharacteristic
133C22×C13H(5,𝔽2)12yes—
134C2×D13—24no—
135C2×D13—24no—
136C2×D13—24no—
137C2×D13—42no—
138C2×D13—42no—
139C2×D13—42no—
140C2×D13—42no—
141C4×C13—22no—
142C4×C13—42no—
143C4×C13—41no—
144C4×C13—42no—
145Dic13—22no—
146Dic13—22no—
147Dic13—22no—
148Dic13—41no—
149Dic13—42no—
150C22×D4—132no—
151C22×D4—131no—
152C23×C4—131no—
153C23×C4—132no—
154C4×D4—264no—
155#117—264no—
156C2×C4⋊C4—131no—
157C2×C4⋊C4—131no—
158C2×C4⋊C4C2×D1311yescharacteristic
159C2×C4⋊C4C2×D1312yes—
160C2×C22⋊C4—131no—
161C2×C22⋊C4—132no—
162C2×C22⋊C4—132no—
163C2×C22⋊C4—132no—
164C2×C22⋊C4—132no—
165C2×C22⋊C4C2×D1311yescharacteristic
166C2×C22⋊C4C2×D1312yes—
167C2×C42—131no—
168C2×C42—131no—
169#95—132no—
170#95—132no—
171#95—131no—
172#95—132no—
173#95—132no—
174#95C2×D1311yescharacteristic
175C2×C13—42no—
176C2×C13#52511yescharacteristic
177C2×C13#52511yescharacteristic
178C2×C13#53511yescharacteristic
179C2×C13#55112yes—
180C2×C13#55312yes—
181D13—44no—
182C24—132no—
183C24C22×D1311yescharacteristic
184C2×D4—264no—
185C2×D4—264no—
186C2×D4—264no—
187C2×D4—262no—
188C2×D4—522no—
189C22×C4—131no—
190C22×C4—131no—
191C22×C4—131no—
192C22×C4—132no—
193C22×C4—131no—
194C22×C4—264no—
195C22×C4—262no—
196C22×C4—262no—
197C22×C4—262no—
198C22×C4—522no—
199C22×C4—522no—
200C22×C4C22×D1312yes—
201C22×C4C22×D1312yes—
202C22×C4C22×D1311yescharacteristic
203C22×C4C22×D1311yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4—262no—
207C4⋊C4—262no—
208C22⋊C4—22no—
209C22⋊C4—42no—
210C22⋊C4—264no—
211C22⋊C4—264no—
212C22⋊C4—264no—
213C22⋊C4—262no—
214C22⋊C4—522no—
215C42—134no—
216C42—262no—
217C13#259811yescharacteristic, complemented, Sylow
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—134no—
223C23—262no—
224C23—262no—
225C23—262no—
226C23—262no—
227C23—262no—
228C23—262no—
229C23#613112yes—
230C23C23×D1311yescharacteristic
231D4—268no—
232D4—528no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—132no—
243C2×C4—132no—
244C2×C4—132no—
245C2×C4—268no—
246C2×C4—261no—
247C2×C4—262no—
248C2×C4—261no—
249C2×C4—261no—
250C2×C4—261no—
251C2×C4—261no—
252C2×C4—262no—
253C2×C4—262no—
254C2×C4—262no—
255C2×C4—262no—
256C2×C4—528no—
257C2×C4D4×D1312yes—
258C22—22no—
259C22—24no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22—264no—
264C22—264no—
265C22—264no—
266C22—522no—
267C22—522no—
268C22—522no—
269C22—522no—
270C22C2×D4×D1311yescharacteristic
271C22#8825112yes—
272C22C2×#613111yescharacteristic
273C22#41×D1311yescharacteristic
274C22#41×D1312yes—
275C4—22no—
276C4—42no—
277C4—41no—
278C4—42no—
279C4—262no—
280C4—262no—
281C4—262no—
282C4—522no—
283C4—521no—
284C2—42no—
285C2—524no—
286C2#1171933211yescharacteristic
287C2#1171934212yes—
288C2#1171936111yescharacteristic
289C2#1171936612yes—
290C2#1171942011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#399C2
prime radical#11718621C2
hypercenter#399D13
hypocenterC13#2598
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#88147C2semidirect
2. C2×#88131C2semidirect
3. C13×#399C2semidirect
4. #11718651C2semidirect
5. #11718650C2semidirect
6. #11718643C2semidirect
8. #11718619C2semidirect
9. #11718617C2semidirect
10. #11718333C2semidirect
11. #11718312C2semidirect
71. C4×Dic13D4semidirect
116. #399D13semidirect
217. C13#2598semidirect

Maximal subgroups

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

subgroupindex
C2×#881472
C2×#881312
C13×#3992
#117186512
#117186502
#117186432
#117186212
#117186192
#117186172
#117183332
#117183122
#259813