Finite Group Databasebeta

← #13120771

Subgroups — #13120771

Order 1664.

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

normal subgroups only

This group has 500 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
1C2×#88142C211yesmaximal, characteristic, complemented
2C2×#88139C211yesmaximal, characteristic
3C13×#397C211yesmaximal, characteristic, complemented
4#11718654C211yesmaximal, characteristic, complemented
5#11718618C212yesmaximal
6#11718614C211yesmaximal, characteristic, complemented
7#11718608C212yesmaximal, complemented
8#11718335C211yesmaximal, characteristic, complemented
9#11718334C212yesmaximal, complemented
10#11718309C211yesmaximal, characteristic, complemented
11#11718307C212yesmaximal, complemented
12C22×C22⋊D13C2211yescharacteristic
13C22×Dic26C2211yescharacteristic
14C22×Dic26C2212yes—
15C2×C13×C4⋊C4C2211yescharacteristic
16C2×C13×C22⋊C4C2211yescharacteristic
17C2×C13×C22⋊C4C2212yes—
18C2×C22⋊Dic13C2211yescharacteristic
19C2×C22⋊Dic13C2212yes—
20C2×#6103C2211yescharacteristic
21C2×#6103C2211yescharacteristic
22C2×#6103C2212yes—
23C2×#6103C2212yes—
24C2×C4⋊Dic13C2211yescharacteristic
25C2×#6101C2212yes—
26C2×C4×Dic13C2212yes—
27C2×C4×Dic13C2211yescharacteristic
28#88142—24no—
29#88139—24no—
30C13×#95C2212yes—
31C13×#95C2211yescharacteristic
32#88071C2212yes—
33#88071C2212yes—
34#88071C2212yes—
35#88071C2212yes—
36#88071C2211yescharacteristic
37C24×C13C2311yescharacteristic
38C23×D13C2311yescharacteristic
39C22×C4×C13C2312yes—
40C22×C4×C13C2311yescharacteristic
41C22×C4×C13C2312yes—
42C22×C4×C13C2311yescharacteristic
43C2×C22⋊D13—22no—
44C2×C22⋊D13—24no—
45C22×Dic13C2312yes—
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2312yes—
48C22×Dic13C2311yescharacteristic
49C22×Dic13C2311yescharacteristic
50C2×Dic26—22no—
51C2×Dic26—24no—
52C2×Dic26—24no—
53C2×Dic26—42no—
54C2×Dic26—42no—
55C13×C4⋊C4—22no—
56C13×C22⋊C4—22no—
57C13×C22⋊C4—42no—
58C22⋊Dic13—22no—
59C22⋊Dic13—42no—
60#6103—22no—
61#6103—22no—
62#6103—42no—
63#6103—42no—
64C4⋊Dic13—22no—
65#6101—24no—
66C4×Dic13—24no—
67C4×Dic13D414yes—
68#2640—131nomaximal, Sylow
69C23×C13—21no—
70C23×C13—22no—
71C23×C13—21no—
72C23×C13—21no—
73C23×C13—22no—
74C23×C13C2411yescharacteristic
75C22×D13—21no—
76C22×D13—22no—
77C22×D13—21no—
78C22×D13—21no—
79C22×D13—22no—
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—21no—
93C2×Dic13—22no—
94C2×Dic13—21no—
95C2×Dic13—22no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13C2×D412yes—
102C2×Dic13C2×D412yes—
103C2×Dic13#4114yes—
104Dic26—48no—
105Dic26—44no—
106C2×C4⋊Q8—131no—
107C2×#124—131no—
108#397—131no—
109#397—131no—
110#397—131no—
111#397D1311yescharacteristic, complemented
112#393—132no—
113#393—132no—
114#391—132no—
115#390—132no—
116#376—131no—
117C22×C13—41no—
118C22×C13—41no—
119C22×C13—41no—
120C22×C13—42no—
121C22×C13—42no—
122C22×C13C22×D411yescharacteristic
123C22×C13C2×#4112yes—
124C22×C13H(5,𝔽2)11yescharacteristic
125C22×C13H(5,𝔽2)11yescharacteristic
126C22×C132−1+412yes—
127C2×D13—41no—
128C2×D13—41no—
129C2×D13—41no—
130C2×D13—42no—
131C2×D13—42no—
132C4×C13—22no—
133C4×C13—41no—
134C4×C13—42no—
135C4×C13—42no—
136Dic13—22no—
137Dic13—41no—
138Dic13—42no—
139Dic13—42no—
140Dic13#12414yes—
141C22×Q8—131no—
142C22×Q8—132no—
143C22×D4—131no—
144C4⋊Q8—264no—
145#124—264no—
146C2×C4⋊C4—132no—
147C2×C4⋊C4—131no—
148C2×C4⋊C4C2×D1311yescharacteristic
149C2×C22⋊C4—131no—
150C2×C22⋊C4—131no—
151C2×C22⋊C4—131no—
152C2×C22⋊C4—132no—
153C2×C22⋊C4—132no—
154C2×C22⋊C4—132no—
155C2×C22⋊C4C2×D1311yescharacteristic
156C2×C22⋊C4C2×D1312yes—
157C2×C42—131no—
158C2×C42—132no—
159#95—132no—
160#95—131no—
161#95—132no—
162#95—132no—
163#95—132no—
164#95C2×D1311yescharacteristic
165#95C2×D1312yes—
166C2×C13—81no—
167C2×C13C2×#12411yescharacteristic
168C2×C13#53511yescharacteristic
169C2×C13#53611yescharacteristic
170C2×C13#54212yes—
171C2×C13#54212yes—
172D13—81no—
173C24—131no—
174C24C22×D1311yescharacteristic
175C2×Q8—262no—
176C2×Q8—264no—
177C2×Q8—264no—
178C2×Q8—522no—
179C2×Q8—522no—
180C2×D4—262no—
181C2×D4—264no—
182C22×C4—131no—
183C22×C4—131no—
184C22×C4—132no—
185C22×C4—131no—
186C22×C4—132no—
187C22×C4C22×D1312yes—
188C22×C4C22×D1311yescharacteristic
189C22×C4C22×D1312yes—
190C22×C4C22×D1311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—264no—
193C4⋊C4—262no—
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—
203C42—264no—
204C13#264011yescharacteristic, complemented, Sylow
205C23—21no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—261no—
211C23—261no—
212C23—262no—
213C23—261no—
214C23—262no—
215C23C23×D1311yescharacteristic
216Q8—524no—
217Q8—528no—
218D4—524no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—132no—
229C2×C4—134no—
230C2×C4—132no—
231C2×C4—261no—
232C2×C4—261no—
233C2×C4—261no—
234C2×C4—262no—
235C2×C4—262no—
236C2×C4—262no—
237C2×C4—262no—
238C2×C4—262no—
239C2×C4—262no—
240C2×C4—262no—
241C2×C4D4×D1312yes—
242C22—41no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—42no—
247C22—521no—
248C22—521no—
249C22—522no—
250C22—521no—
251C22—522no—
252C22C2×D4×D1311yescharacteristic
253C22#8825111yescharacteristic
254C22#41×D1312yes—
255C22#8825611yescharacteristic
256C22#8825712yes—
257C4—22no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—134no—
262C4—262no—
263C4—522no—
264C4—521no—
265C4—522no—
266C2—81no—
267C2—1041no—
268C2#1171931211yescharacteristic
269C2#1171935311yescharacteristic
270C2#1171936712yes—
271C2D13×#12411yescharacteristic
272C2#1171941512yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#397C2
prime radicalC22×C22⋊D13C22
hypercenter#397D13
hypocenterC13#2640
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#88142C2semidirect
3. C13×#397C2semidirect
4. #11718654C2semidirect
6. #11718614C2semidirect
7. #11718608C2semidirect
8. #11718335C2semidirect
9. #11718334C2semidirect
10. #11718309C2semidirect
11. #11718307C2semidirect
111. #397D13semidirect
204. C13#2640semidirect

Maximal subgroups

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

subgroupindex
C2×#881422
C2×#881392
C13×#3972
#117186542
#117186182
#117186142
#117186082
#117183352
#117183342
#117183092
#117183072
#264013