Finite Group Databasebeta

← #13120774

Subgroups — #13120774

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×#88198C211yesmaximal, characteristic, complemented
2C2×#88139C211yesmaximal, characteristic
3C13×#393C211yesmaximal, characteristic, complemented
4#11718892C211yesmaximal, characteristic, complemented
5#11718618C212yesmaximal
6#11718613C212yesmaximal, complemented
7#11718608C211yesmaximal, characteristic, complemented
8#11718336C212yesmaximal, complemented
9#11718334C211yesmaximal, characteristic, complemented
10#11718309C211yesmaximal, characteristic, complemented
11#11718307C212yesmaximal, complemented
12C22×Q8×C13C2211yescharacteristic
13C22×C22⋊D13C2211yescharacteristic
14C22×Dic26C2211yescharacteristic
15C22×Dic26C2211yescharacteristic
16C2×C13×C22⋊C4C2211yescharacteristic
17C2×C13×C22⋊C4C2212yes—
18C2×C22⋊Dic13C2211yescharacteristic
19C2×C22⋊Dic13C2212yes—
20#88198—24no—
21C2×#6103C2211yescharacteristic
22C2×#6103C2211yescharacteristic
23C2×#6103C2212yes—
24C2×#6103C2212yes—
25C2×#6101C2212yes—
26C2×#6101C2212yes—
27C2×C4×Dic13C2212yes—
28C2×C4×Dic13C2211yescharacteristic
29#88139—24no—
30C13×#95C2212yes—
31C13×#95C2211yescharacteristic
32#88071C2212yes—
33#88071C2212yes—
34#88071C2212yes—
35#88071C2212yes—
36#88071C2211yescharacteristic
37C24×C13C2311yescharacteristic
38C23×D13C2311yescharacteristic
39C2×Q8×C13—22no—
40C2×Q8×C13—42no—
41C22×C4×C13C2312yes—
42C22×C4×C13C2311yescharacteristic
43C22×C4×C13C2312yes—
44C22×C4×C13C2311yescharacteristic
45C2×C22⋊D13—22no—
46C2×C22⋊D13—24no—
47C22×Dic13C2312yes—
48C22×Dic13C2311yescharacteristic
49C22×Dic13C2312yes—
50C22×Dic13C2311yescharacteristic
51C22×Dic13C2311yescharacteristic
52C2×Dic26—22no—
53C2×Dic26—22no—
54C2×Dic26—24no—
55C2×Dic26—42no—
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—
64#6101—24no—
65#6101—24no—
66C4×Dic13—24no—
67C4×Dic13D414yes—
68#2640—131nomaximal, Sylow
69C23×C13—21no—
70C23×C13—22no—
71C23×C13—22no—
72C23×C13—21no—
73C23×C13—21no—
74C23×C13C2411yescharacteristic
75C22×D13—21no—
76C22×D13—22no—
77C22×D13—22no—
78C22×D13—21no—
79C22×D13—21no—
80Q8×C13—44no—
81C2×C4×C13—22no—
82C2×C4×C13—22no—
83C2×C4×C13—21no—
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—
91C22⋊D13—44no—
92C2×Dic13—21no—
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—
104C2×Dic13#4114yes—
105Dic26—44no—
106Dic26—44no—
107C2×C4⋊Q8—131no—
108C2×#124—131no—
109#397—132no—
110#397—132no—
111#393—131no—
112#393—131no—
113#393—131no—
114#393D1311yescharacteristic, complemented
115#391—132no—
116#390—132no—
117#376—131no—
118C22×C13—41no—
119C22×C13—41no—
120C22×C13—41no—
121C22×C13—42no—
122C22×C13—42no—
123C22×C13C22×D411yescharacteristic
124C22×C13C2×#4112yes—
125C22×C13H(5,𝔽2)12yes—
126C22×C132−1+411yescharacteristic
127C22×C132−1+411yescharacteristic
128C2×D13—41no—
129C2×D13—41no—
130C2×D13—41no—
131C2×D13—42no—
132C2×D13—42no—
133C4×C13—22no—
134C4×C13—41no—
135C4×C13—42no—
136C4×C13—42no—
137Dic13—22no—
138Dic13—41no—
139Dic13—42no—
140Dic13—42no—
141Dic13#12414yes—
142C22×Q8—131no—
143C22×Q8—131no—
144C22×Q8C2×D1311yescharacteristic
145C22×D4—131no—
146C4⋊Q8—264no—
147#124—264no—
148C2×C4⋊C4—132no—
149C2×C4⋊C4—132no—
150C2×C22⋊C4—131no—
151C2×C22⋊C4—131no—
152C2×C22⋊C4—131no—
153C2×C22⋊C4—132no—
154C2×C22⋊C4—132no—
155C2×C22⋊C4—132no—
156C2×C22⋊C4C2×D1311yescharacteristic
157C2×C22⋊C4C2×D1312yes—
158C2×C42—131no—
159C2×C42—132no—
160#95—131no—
161#95—132no—
162#95—132no—
163#95—132no—
164#95—132no—
165#95C2×D1311yescharacteristic
166#95C2×D1312yes—
167C2×C13—81no—
168C2×C13C2×#12411yescharacteristic
169C2×C13#53511yescharacteristic
170C2×C13#53611yescharacteristic
171C2×C13#54212yes—
172C2×C13#54212yes—
173D13—81no—
174C24—131no—
175C24C22×D1311yescharacteristic
176C2×Q8—22no—
177C2×Q8—42no—
178C2×Q8—262no—
179C2×Q8—264no—
180C2×Q8—262no—
181C2×Q8—522no—
182C2×D4—262no—
183C2×D4—264no—
184C22×C4—131no—
185C22×C4—131no—
186C22×C4—132no—
187C22×C4—131no—
188C22×C4—132no—
189C22×C4C22×D1312yes—
190C22×C4C22×D1311yescharacteristic
191C22×C4C22×D1312yes—
192C22×C4C22×D1311yescharacteristic
193C4⋊C4—264no—
194C4⋊C4—264no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—262no—
198C22⋊C4—262no—
199C22⋊C4—262no—
200C22⋊C4—522no—
201C22⋊C4—522no—
202C22⋊C4—522no—
203C42—134no—
204C42—264no—
205C13#264011yescharacteristic, complemented, Sylow
206C23—21no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—261no—
212C23—262no—
213C23—261no—
214C23—262no—
215C23—261no—
216C23C23×D1311yescharacteristic
217Q8—44no—
218Q8—524no—
219Q8—524no—
220D4—524no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—132no—
231C2×C4—134no—
232C2×C4—132no—
233C2×C4—261no—
234C2×C4—261no—
235C2×C4—261no—
236C2×C4—262no—
237C2×C4—262no—
238C2×C4—262no—
239C2×C4—262no—
240C2×C4—262no—
241C2×C4—262no—
242C2×C4—262no—
243C2×C4D4×D1312yes—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—521no—
250C22—521no—
251C22—522no—
252C22—522no—
253C22—521no—
254C22C2×D4×D1311yescharacteristic
255C22#8825112yes—
256C22#8825411yescharacteristic
257C22#41×D1312yes—
258C22#8825711yescharacteristic
259C4—22no—
260C4—42no—
261C4—42no—
262C4—41no—
263C4—134no—
264C4—262no—
265C4—522no—
266C4—522no—
267C4—521no—
268C2—81no—
269C2—1041no—
270C2#1171929711yescharacteristic
271C2#1171933311yescharacteristic
272C2#1171934512yes—
273C2#1171937412yes—
274C2D13×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#393C2
prime radicalC22×C22⋊D13C22
hypercenter#393D13
hypocenterC13#2640
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#88198C2semidirect
3. C13×#393C2semidirect
4. #11718892C2semidirect
6. #11718613C2semidirect
7. #11718608C2semidirect
8. #11718336C2semidirect
9. #11718334C2semidirect
10. #11718309C2semidirect
11. #11718307C2semidirect
114. #393D13semidirect
205. C13#2640semidirect

Maximal subgroups

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

subgroupindex
C2×#881982
C2×#881392
C13×#3932
#117188922
#117186182
#117186132
#117186082
#117183362
#117183342
#117183092
#117183072
#264013