Finite Group Databasebeta

← #13122892

Subgroups — #13122892

Order 1664.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88144C211yesmaximal, characteristic, complemented
2C2×#88139C211yesmaximal, characteristic
3C13×#397C211yesmaximal, characteristic, complemented
4#11718654C211yesmaximal, characteristic, complemented
5#11718622C212yesmaximal
6#11718610C211yesmaximal, characteristic, complemented
7#11718609C212yesmaximal, complemented
8#11718338C211yesmaximal, characteristic, complemented
9#11718337C212yesmaximal, complemented
10#11718317C212yesmaximal, complemented
11#11718311C211yesmaximal, characteristic, complemented
12C22×C22⋊D13C2211yescharacteristic
13C22×Dic26C2211yescharacteristic
14C2×C13×C4⋊C4C2211yescharacteristic
15C2×C13×C22⋊C4C2211yescharacteristic
16C2×C13×C22⋊C4C2212yes—
17C2×C22⋊Dic13C2211yescharacteristic
18C2×C22⋊Dic13C2212yes—
19C2×#6103C2211yescharacteristic
20C2×#6103C2211yescharacteristic
21C2×#6103C2212yes—
22C2×#6103C2212yes—
23C2×C4⋊Dic13C2212yes—
24C2×C4⋊Dic13C2211yescharacteristic
25C2×C4⋊Dic13C2212yes—
26C2×#6101C2212yes—
27C2×#6101C2212yes—
28C2×#6101C2212yes—
29C2×C4×Dic13C2211yescharacteristic
30#88144—24no—
31#88139—24no—
32C13×#95C2211yescharacteristic
33C13×#95C2212yes—
34#88071C2212yes—
35#88071C2211yescharacteristic
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C23×D13C2311yescharacteristic
39C22×C4×C13C2312yes—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C2×C22⋊D13—22no—
44C2×C22⋊D13—42no—
45C22×Dic13C2312yes—
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2312yes—
48C22×Dic13C2311yescharacteristic
49C22×Dic13C2311yescharacteristic
50C2×Dic26—22no—
51C2×Dic26—42no—
52C13×C4⋊C4—22no—
53C13×C22⋊C4—22no—
54C13×C22⋊C4—42no—
55C22⋊Dic13—22no—
56C22⋊Dic13—42no—
57#6103—22no—
58#6103—22no—
59#6103—42no—
60#6103—42no—
61C4⋊Dic13—22no—
62C4⋊Dic13—24no—
63C4⋊Dic13—42no—
64#6101—24no—
65#6101—42no—
66#6101—42no—
67C4×Dic13D414yes—
68#2820—131nomaximal, Sylow
69C23×C13—21no—
70C23×C13—22no—
71C23×C13—21no—
72C23×C13—22no—
73C23×C13—21no—
74C23×C13C2411yescharacteristic
75C22×D13—21no—
76C22×D13—21no—
77C22×D13—22no—
78C22×D13—22no—
79C22×D13—21no—
80C2×C4×C13—22no—
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×C13C2×D412yes—
90C22⋊D13—44no—
91C2×Dic13—21no—
92C2×Dic13—22no—
93C2×Dic13—21no—
94C2×Dic13—22no—
95C2×Dic13—21no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13—22no—
102C2×Dic13C2×D412yes—
103C2×Dic13C2×D412yes—
104Dic26—44no—
105C2×#125—131no—
106C2×#124—131no—
107#399—131no—
108#399—131no—
109#397—131no—
110#397D1311yescharacteristic, complemented
111#396—132no—
112#396—132no—
113#395—132no—
114#394—132no—
115#382—131no—
116C22×C13—41no—
117C22×C13—41no—
118C22×C13—41no—
119C22×C13—42no—
120C22×C13—42no—
121C22×C13C22×D411yescharacteristic
122C22×C13H(5,𝔽2)11yescharacteristic
123C22×C13H(5,𝔽2)12yes—
124C22×C132−1+412yes—
125C22×C132−1+411yescharacteristic
126C2×D13—41no—
127C2×D13—41no—
128C2×D13—41no—
129C2×D13—42no—
130C2×D13—42no—
131C4×C13—22no—
132C4×C13—41no—
133C4×C13—42no—
134C4×C13—42no—
135Dic13—22no—
136Dic13—22no—
137Dic13—41no—
138Dic13—42no—
139Dic13—42no—
140C22×Q8—131no—
141C22×D4—131no—
142#125—264no—
143#124—264no—
144C2×C4⋊C4—132no—
145C2×C4⋊C4—132no—
146C2×C4⋊C4—131no—
147C2×C4⋊C4—132no—
148C2×C4⋊C4—132no—
149C2×C4⋊C4—132no—
150C2×C4⋊C4C2×D1311yescharacteristic
151C2×C22⋊C4—131no—
152C2×C22⋊C4—131no—
153C2×C22⋊C4—131no—
154C2×C22⋊C4—132no—
155C2×C22⋊C4—132no—
156C2×C22⋊C4—132no—
157C2×C22⋊C4C2×D1311yescharacteristic
158C2×C22⋊C4C2×D1312yes—
159C2×C42—131no—
160#95—132no—
161#95—131no—
162#95—132no—
163#95C2×D1311yescharacteristic
164#95C2×D1312yes—
165C2×C13—81no—
166C2×C13#53511yescharacteristic
167C2×C13#53611yescharacteristic
168C2×C13#53711yescharacteristic
169C2×C13#56212yes—
170C2×C13#56312yes—
171D13—81no—
172C24—131no—
173C24C22×D1311yescharacteristic
174C2×Q8—262no—
175C2×Q8—522no—
176C2×D4—262no—
177C2×D4—522no—
178C22×C4—131no—
179C22×C4—131no—
180C22×C4—132no—
181C22×C4—131no—
182C22×C4—132no—
183C22×C4C22×D1312yes—
184C22×C4C22×D1312yes—
185C22×C4C22×D1311yescharacteristic
186C22×C4C22×D1311yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—264no—
189C4⋊C4—264no—
190C4⋊C4—262no—
191C4⋊C4—522no—
192C4⋊C4—522no—
193C4⋊C4—522no—
194C22⋊C4—22no—
195C22⋊C4—42no—
196C22⋊C4—262no—
197C22⋊C4—262no—
198C22⋊C4—262no—
199C22⋊C4—522no—
200C22⋊C4—522no—
201C22⋊C4—522no—
202C42—134no—
203C13#282011yescharacteristic, complemented, Sylow
204C23—21no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—22no—
209C23—261no—
210C23—261no—
211C23—262no—
212C23—261no—
213C23—262no—
214C23C23×D1311yescharacteristic
215Q8—524no—
216D4—524no—
217C2×C4—22no—
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—261no—
229C2×C4—262no—
230C2×C4—261no—
231C2×C4—261no—
232C2×C4—262no—
233C2×C4—262no—
234C2×C4—262no—
235C2×C4—262no—
236C2×C4—262no—
237C2×C4—262no—
238C2×C4—262no—
239C2×C4D4×D1312yes—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—41no—
244C22—42no—
245C22—521no—
246C22—521no—
247C22—522no—
248C22—521no—
249C22—522no—
250C22C2×D4×D1311yescharacteristic
251C22#8825111yescharacteristic
252C22#8825612yes—
253C22#8825711yescharacteristic
254C22#8825712yes—
255C4—22no—
256C4—42no—
257C4—42no—
258C4—41no—
259C4—262no—
260C4—262no—
261C4—522no—
262C4—522no—
263C4—521no—
264C2—81no—
265C2—1041no—
266C2#1171931311yescharacteristic
267C2#1171935311yescharacteristic
268C2#1171937212yes—
269C2#1171937911yescharacteristic
270C2#1171941712yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#397C2
prime radicalC22×C22⋊D13C22
hypercenter#397D13
hypocenterC13#2820
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#88144C2semidirect
3. C13×#397C2semidirect
4. #11718654C2semidirect
6. #11718610C2semidirect
7. #11718609C2semidirect
8. #11718338C2semidirect
9. #11718337C2semidirect
10. #11718317C2semidirect
11. #11718311C2semidirect
110. #397D13semidirect
203. C13#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#881442
C2×#881392
C13×#3972
#117186542
#117186222
#117186102
#117186092
#117183382
#117183372
#117183172
#117183112
#282013