Finite Group Databasebeta

← #13120215

Subgroups — #13120215

Order 1664.

544 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#6101C211yesmaximal, characteristic
2C2×#88132C211yesmaximal, characteristic
3C13×#385C211yesmaximal, characteristic
4#11718871C211yesmaximal, characteristic
5#11718640C212yesmaximal, complemented
6#11718634C212yesmaximal, complemented
7#11718611C211yesmaximal, characteristic
8#11718607C211yesmaximal, characteristic
9C22⋊C4×Dic13C211yesmaximal, characteristic
10#11718318C212yesmaximal, complemented
11#11718308C212yesmaximal, complemented
12C23×C4×C13C2211yescharacteristic
13C23×Dic13C2211yescharacteristic
14C23×Dic13C2211yescharacteristic
15C22×Dic26C2211yescharacteristic
16C2×C13×C4⋊C4C2212yes—
17C2×C13×C22⋊C4C2211yescharacteristic
18C2×C13×C22⋊C4C2211yescharacteristic
19C2×C22⋊Dic13C2211yescharacteristic
20C2×C22⋊Dic13C2211yescharacteristic
21C2×C4⋊Dic13C2211yescharacteristic
22C2×C4⋊Dic13C2212yes—
23C2×#6101—24no—
24C2×#6101C2211yescharacteristic
25C2×#6101C2211yescharacteristic
26C2×#6101C2212yes—
27C2×#6101C2212yes—
28C2×#6101C2212yes—
29C2×#6101C2212yes—
30C2×C4×Dic13C2212yes—
31#88132—24no—
32C13×#95C2212yes—
33#88071C2212yes—
34#88071C2212yes—
35#88071C2212yes—
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C22×C4×C13—22no—
39C22×C4×C13—42no—
40C22×C4×C13C2311yescharacteristic
41C22×C4×C13C2312yes—
42C22×C4×C13C2311yescharacteristic
43C22×C4×C13C2312yes—
44C22×Dic13—22no—
45C22×Dic13—24no—
46C22×Dic13—42no—
47C22×Dic13D414yes—
48C22×Dic13C2311yescharacteristic
49C22×Dic13C2311yescharacteristic
50C22×Dic13C2311yescharacteristic
51C22×Dic13C2311yescharacteristic
52C22×Dic13C2312yes—
53C22×Dic13C2312yes—
54C2×Dic26—22no—
55C2×Dic26—42no—
56C13×C4⋊C4—42no—
57C13×C22⋊C4—22no—
58C13×C22⋊C4Q814yescomplemented
59C22⋊Dic13—22no—
60C22⋊Dic13Q814yescomplemented
61C4⋊Dic13—22no—
62C4⋊Dic13—24no—
63#6101—22no—
64#6101—22no—
65#6101—24no—
66#6101—24no—
67#6101—24no—
68#6101—42no—
69C4×Dic13—42no—
70#2583—131nomaximal, Sylow
71C23×C13—22no—
72C23×C13—22no—
73C23×C13C2×D412yes—
74C23×C13C2×Q812yes—
75C23×C13C2×Q812yes—
76C23×C13C2411yescharacteristic
77C2×C4×C13—22no—
78C2×C4×C13—22no—
79C2×C4×C13—22no—
80C2×C4×C13—21no—
81C2×C4×C13—21no—
82C2×C4×C13—22no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13—44no—
87C2×C4×C13C2×Q814yes—
88C2×Dic13—28no—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—21no—
96C2×Dic13—21no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—44no—
100C2×Dic13C2×D412yes—
101C2×Dic13C2×D412yes—
102C2×Dic13C2×Q814yes—
103C2×Dic13#4114yes—
104Dic26—44no—
105C2×C22⋊Q8—131no—
106C22×C4⋊C4—131no—
107#398—132no—
108#395—132no—
109#389—132no—
110#389—132no—
111#385—131no—
112#385—131no—
113#385—131no—
114#385D1311yescharacteristic
115C4×C22⋊C4—131no—
116C22×C13—22no—
117C22×C13—22no—
118C22×C13—24no—
119C22×C13—24no—
120C22×C13C4⋊Q814yes—
121C22×C13C22×D411yescharacteristic
122C22×C13C22×Q811yescharacteristic
123C22×C13C22×Q811yescharacteristic
124C22×C13C2×#4112yes—
125C22×C13H(5,𝔽2)12yes—
126C4×C13—24no—
127C4×C13—41no—
128C4×C13—41no—
129C4×C13—42no—
130Dic13—24no—
131Dic13—22no—
132Dic13—22no—
133Dic13—24no—
134Dic13—41no—
135Dic13—41no—
136C22×Q8—131no—
137C23×C4—131no—
138C23×C4—131no—
139C23×C4C2×D1311yescharacteristic
140C22⋊Q8—264no—
141C2×C4⋊C4—132no—
142C2×C4⋊C4—132no—
143C2×C4⋊C4—131no—
144C2×C4⋊C4—131no—
145C2×C4⋊C4—131no—
146C2×C4⋊C4—132no—
147C2×C4⋊C4—132no—
148C2×C4⋊C4—132no—
149C2×C4⋊C4—264no—
150C2×C4⋊C4C2×D1312yes—
151C2×C22⋊C4—131no—
152C2×C22⋊C4—131no—
153C2×C22⋊C4C2×D1311yescharacteristic
154C2×C22⋊C4C2×D1311yescharacteristic
155C2×C42—132no—
156#95—132no—
157#95—132no—
158#95—132no—
159#95—132no—
160#95C2×D1312yes—
161C2×C13—24no—
162C2×C13#52511yescharacteristic
163C2×C13C2×C4⋊Q811yescharacteristic
164C2×C13#53411yescharacteristic
165C2×C13#55412yes—
166C2×C13#55412yes—
167C24C22×D1311yescharacteristic
168C2×Q8—262no—
169C2×Q8—522no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—132no—
173C22×C4—132no—
174C22×C4—131no—
175C22×C4—131no—
176C22×C4—134no—
177C22×C4—131no—
178C22×C4—131no—
179C22×C4—264no—
180C22×C4—262no—
181C22×C4—522no—
182C22×C4C22×D1311yescharacteristic
183C22×C4C22×D1312yes—
184C22×C4C22×D1311yescharacteristic
185C22×C4C22×D1312yes—
186C4⋊C4—42no—
187C4⋊C4—264no—
188C4⋊C4—264no—
189C4⋊C4—262no—
190C4⋊C4—262no—
191C4⋊C4—262no—
192C4⋊C4—264no—
193C4⋊C4—264no—
194C4⋊C4—522no—
195C22⋊C4—22no—
196C22⋊C4—134no—
197C22⋊C4—262no—
198C22⋊C4Dic2614yescomplemented
199C42—522no—
200C13#258311yescharacteristic, complemented, Sylow
201C23—22no—
202C23—22no—
203C23C2×Dic2612yes—
204C23D4×D1312yes—
205C23Q8×D1312yes—
206C23C23×D1311yescharacteristic
207Q8—524no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—44no—
218C2×C4—132no—
219C2×C4—132no—
220C2×C4—134no—
221C2×C4—134no—
222C2×C4—268no—
223C2×C4—262no—
224C2×C4—262no—
225C2×C4—262no—
226C2×C4—262no—
227C2×C4—262no—
228C2×C4—262no—
229C2×C4—261no—
230C2×C4—261no—
231C2×C4—262no—
232C2×C4—262no—
233C2×C4—524no—
234C2×C4C2×Dic2614yes—
235C22—24no—
236C22—22no—
237C22—24no—
238C22—22no—
239C22#8814214yes—
240C22C22×Dic2611yescharacteristic
241C22C2×D4×D1311yescharacteristic
242C22#8825112yes—
243C22C2×Q8×D1311yescharacteristic
244C22#41×D1312yes—
245C4—24no—
246C4—42no—
247C4—41no—
248C4—41no—
249C4—262no—
250C4—262no—
251C4—264no—
252C4—264no—
253C4—521no—
254C4—521no—
255C2—24no—
256C2C2×#8814211yescharacteristic
257C2#1171924412yes—
258C2#1171929411yescharacteristic
259C2#1171934112yes—
260C2#1171936111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#385C2
prime radicalC24×C13C23
hypercenter#385D13
hypocenterC13#2583
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
5. #11718640C2semidirect
6. #11718634C2semidirect
10. #11718318C2semidirect
11. #11718308C2semidirect
58. C13×C22⋊C4Q8semidirect
60. C22⋊Dic13Q8semidirect
198. C22⋊C4Dic26semidirect
200. C13#2583semidirect

Maximal subgroups

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

subgroupindex
C22×#61012
C2×#881322
C13×#3852
#117188712
#117186402
#117186342
#117186112
#117186072
C22⋊C4×Dic132
#117183182
#117183082
#258313