Finite Group Databasebeta

← #13120656

Subgroups — #13120656

Order 1664.

488 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 143 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×#88131C211yesmaximal, characteristic
3C13×#393C211yesmaximal, characteristic
4#11718890C211yesmaximal, characteristic, complemented
5#11718610C212yesmaximal
6#11718608C211yesmaximal, characteristic
7#11718603C212yesmaximal
8#11718319C212yesmaximal, complemented
9#11718317C211yesmaximal, characteristic, complemented
10#11718311C211yesmaximal, characteristic, complemented
11#11718308C212yesmaximal, complemented
12C22×Q8×C13C2211yescharacteristic
13C23×Dic13C2211yescharacteristic
14C22×Dic26C2211yescharacteristic
15C2×C13×C22⋊C4C2211yescharacteristic
16C2×C13×C22⋊C4C2212yes—
17C2×C22⋊Dic13C2211yescharacteristic
18C2×C22⋊Dic13C2212yes—
19#88198—24no—
20C2×C4⋊Dic13C2212yes—
21C2×#6101C2211yescharacteristic
22C2×#6101C2211yescharacteristic
23C2×#6101C2212yes—
24C2×#6101C2212yes—
25C2×#6101C2212yes—
26C2×#6101C2212yes—
27C2×C4×Dic13C2211yescharacteristic
28C2×C4×Dic13C2211yescharacteristic
29#88131—24no—
30C13×#95C2212yes—
31C13×#95C2211yescharacteristic
32#88071C2212yes—
33#88071C2212yes—
34#88071C2211yescharacteristic
35#88071C2212yes—
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C2×Q8×C13—22no—
39C2×Q8×C13—42no—
40C22×C4×C13C2311yescharacteristic
41C22×C4×C13C2312yes—
42C22×C4×C13C2312yes—
43C22×C4×C13C2311yescharacteristic
44C22×Dic13—22no—
45C22×Dic13—22no—
46C22×Dic13Q814yescomplemented
47C22×Dic13C2312yes—
48C22×Dic13C2312yes—
49C22×Dic13C2311yescharacteristic
50C22×Dic13C2311yescharacteristic
51C22×Dic13C2311yescharacteristic
52C22×Dic13C2311yescharacteristic
53C2×Dic26—22no—
54C2×Dic26—42no—
55C13×C22⋊C4—22no—
56C13×C22⋊C4—42no—
57C22⋊Dic13—22no—
58C22⋊Dic13—42no—
59C4⋊Dic13—42no—
60#6101—22no—
61#6101—22no—
62#6101—24no—
63#6101—24no—
64#6101—24no—
65#6101—42no—
66C4×Dic13—22no—
67C4×Dic13D414yes—
68#2628—131nomaximal, Sylow
69C23×C13—21no—
70C23×C13—21no—
71C23×C13—22no—
72C23×C13—22no—
73C23×C13C2×Q812yes—
74C23×C13C2411yescharacteristic
75Q8×C13—44no—
76C2×C4×C13—22no—
77C2×C4×C13—21no—
78C2×C4×C13—22no—
79C2×C4×C13—22no—
80C2×C4×C13—22no—
81C2×C4×C13—22no—
82C2×C4×C13—22no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13C2×D412yes—
86C2×Dic13—21no—
87C2×Dic13—21no—
88C2×Dic13—22no—
89C2×Dic13—28no—
90C2×Dic13—21no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13C2×D412yes—
99C2×Dic13C2×D412yes—
100C2×Dic13C2×Q812yes—
101C2×Dic13C2×Q812yes—
102C2×Dic13#4112yes—
103Dic26—44no—
104C2×C4⋊Q8—131no—
105C2×#117—131no—
106#400—132no—
107#399—132no—
108#395—131no—
109#395—131no—
110#393—131no—
111#393D1311yescharacteristic
112#389—132no—
113#382—131no—
114#380—132no—
115C22×C13—22no—
116C22×C13—24no—
117C22×C13—41no—
118C22×C13—41no—
119C22×C13—42no—
120C22×C13C22×D411yescharacteristic
121C22×C13C22×Q811yescharacteristic
122C22×C13C2×#4111yescharacteristic
123C22×C13H(5,𝔽2)12yes—
124C22×C132−1+412yes—
125C4×C13—22no—
126C4×C13—42no—
127C4×C13—41no—
128C4×C13—42no—
129Dic13—22no—
130Dic13—22no—
131Dic13—42no—
132Dic13—41no—
133Dic13—42no—
134Dic13C22⋊Q814yes—
135C22×Q8—131no—
136C22×Q8C2×D1311yescharacteristic
137C23×C4—131no—
138C4⋊Q8—264no—
139#117—264no—
140C2×C4⋊C4—131no—
141C2×C4⋊C4—131no—
142C2×C4⋊C4—132no—
143C2×C4⋊C4—132no—
144C2×C4⋊C4—132no—
145C2×C4⋊C4—132no—
146C2×C4⋊C4—132no—
147C2×C22⋊C4—131no—
148C2×C22⋊C4—132no—
149C2×C22⋊C4C2×D1311yescharacteristic
150C2×C22⋊C4C2×D1312yes—
151C2×C42—131no—
152C2×C42—131no—
153#95—132no—
154#95—132no—
155#95—131no—
156#95—132no—
157#95—132no—
158#95C2×D1311yescharacteristic
159#95C2×D1312yes—
160C2×C13—42no—
161C2×C13C2×C22⋊Q811yescharacteristic
162C2×C13#53511yescharacteristic
163C2×C13#53611yescharacteristic
164C2×C13#54212yes—
165C2×C13#54412yes—
166C24C22×D1311yescharacteristic
167C2×Q8—22no—
168C2×Q8—42no—
169C2×Q8—262no—
170C2×Q8—522no—
171C22×C4—131no—
172C22×C4—134no—
173C22×C4—131no—
174C22×C4—131no—
175C22×C4—132no—
176C22×C4—132no—
177C22×C4—131no—
178C22×C4—262no—
179C22×C4—262no—
180C22×C4C22×D1311yescharacteristic
181C22×C4C22×D1312yes—
182C22×C4C22×D1312yes—
183C22×C4C22×D1311yescharacteristic
184C4⋊C4—262no—
185C4⋊C4—264no—
186C4⋊C4—264no—
187C4⋊C4—264no—
188C4⋊C4—262no—
189C4⋊C4—522no—
190C4⋊C4—522no—
191C22⋊C4—22no—
192C22⋊C4—42no—
193C22⋊C4—262no—
194C22⋊C4—522no—
195C42—134no—
196C42—262no—
197C13#262811yescharacteristic, complemented, Sylow
198C23—21no—
199C23—22no—
200C23—21no—
201C23—22no—
202C23Q8×D1312yes—
203C23C23×D1311yescharacteristic
204Q8—44no—
205Q8—524no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—132no—
216C2×C4—132no—
217C2×C4—132no—
218C2×C4—132no—
219C2×C4—132no—
220C2×C4—268no—
221C2×C4—261no—
222C2×C4—261no—
223C2×C4—262no—
224C2×C4—261no—
225C2×C4—262no—
226C2×C4—262no—
227C2×C4—262no—
228C2×C4—262no—
229C2×C4—262no—
230C2×C4—262no—
231C2×C4—262no—
232C2×C4D4×D1312yes—
233C22—24no—
234C22—22no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22C2×D4×D1311yescharacteristic
239C22#8825112yes—
240C22C2×Q8×D1311yescharacteristic
241C22#41×D1311yescharacteristic
242C22#8825712yes—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—41no—
247C4—134no—
248C4—262no—
249C4—262no—
250C4—521no—
251C4—522no—
252C4—522no—
253C2—42no—
254C2#1171929711yescharacteristic
255C2#1171932812yes—
256C2D13×C22⋊Q811yescharacteristic
257C2#1171937412yes—
258C2#1171938011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#393C2
prime radicalC24×C13C23
hypercenter#393D13
hypocenterC13#2628
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#88198C2semidirect
4. #11718890C2semidirect
8. #11718319C2semidirect
9. #11718317C2semidirect
10. #11718311C2semidirect
11. #11718308C2semidirect
46. C22×Dic13Q8semidirect
197. C13#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#881982
C2×#881312
C13×#3932
#117188902
#117186102
#117186082
#117186032
#117183192
#117183172
#117183112
#117183082
#262813