Finite Group Databasebeta

← #13123210

Subgroups — #13123210

Order 1664.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#88201C211yesmaximal, characteristic
2C2×#88144C211yesmaximal, characteristic, complemented
3C13×#391C211yesmaximal, characteristic, complemented
4#11718891C211yesmaximal, characteristic, complemented
5#11718649C211yesmaximal, characteristic
6#11718600C211yesmaximal, characteristic
7C42×Dic13C211yesmaximal, characteristic, complemented
8#11718331C214yesmaximal, complemented
9#11718330C214yesmaximal, complemented
10C22×Q8×C13C2211yescharacteristic
11C22×D52C2211yescharacteristic
12C2×C13×C4⋊C4C2211yescharacteristic
13C2×C42×C13C2211yescharacteristic
14#88201—24no—
15C2×#6103C2212yes—
16C2×#6103C2212yes—
17C2×#6103C2214yes—
18C2×#6103C2214yes—
19C2×C4⋊Dic13C2212yes—
20C2×C4⋊Dic13C2211yescharacteristic
21C2×#6101C2212yes—
22C2×C4×Dic13C2212yes—
23C2×C4×Dic13C2212yes—
24C2×C4×Dic13C2211yescharacteristic
25C2×C4×Dic13C2211yescharacteristic
26#88144—24no—
27C13×#95C2214yes—
28#88071C2214yes—
29C23×D13C2312yes—
30C2×Q8×C13—22no—
31C2×Q8×C13—22no—
32C2×Q8×C13—22no—
33C22×C4×C13C2312yes—
34C22×C4×C13C2312yes—
35C22×C4×C13C2312yes—
36C22×C4×C13C2311yescharacteristic
37C22×Dic13C2312yes—
38C22×Dic13C2311yescharacteristic
39C22×Dic13C2311yescharacteristic
40C22×Dic13C2312yes—
41C2×D52—22no—
42C2×D52—22no—
43C2×D52—22no—
44C13×C4⋊C4—22no—
45C42×C13—22no—
46#6103—28no—
47#6103—24no—
48#6103—24no—
49#6103—28no—
50C4⋊Dic13—22no—
51C4⋊Dic13—24no—
52#6101—24no—
53C4×Dic13—22no—
54C4×Dic13—24no—
55C4×Dic13—24no—
56C4×Dic13D414yes—
57#2861—131nomaximal, Sylow
58C23×C13C2411yescharacteristic
59C22×D13—22no—
60C22×D13—24no—
61C22×D13—24no—
62C22×D13—22no—
63C22×D13—22no—
64Q8×C13—44no—
65C2×C4×C13—22no—
66C2×C4×C13—22no—
67C2×C4×C13—22no—
68C2×C4×C13—22no—
69C2×C4×C13—24no—
70C2×C4×C13—22no—
71C2×C4×C13—22no—
72C2×C4×C13C2×D412yes—
73C2×C4×C13#4114yes—
74C2×C4×C13#4112yes—
75C2×C4×C13#4112yes—
76C2×Dic13—22no—
77C2×Dic13—22no—
78C2×Dic13—22no—
79C2×Dic13—22no—
80C2×Dic13—22no—
81C2×Dic13—24no—
82C2×Dic13C2×D412yes—
83C2×Dic13C2×D412yes—
84C2×Dic13#4114yes—
85D52—44no—
86C2×#125—131no—
87C2×#124—131no—
88#391—131no—
89#391D1311yescharacteristic, complemented
90#390—131no—
91#390—131no—
92#388—134no—
93#388—134no—
94C43—131no—
95C22×C13C22×D411yescharacteristic
96C22×C13C2×#4111yescharacteristic
97C22×C13C2×#4111yescharacteristic
98C22×C13C2×#4112yes—
99C22×C13C2×#4112yes—
100C2×D13—42no—
101C2×D13—44no—
102C2×D13—44no—
103C2×D13—42no—
104C2×D13—42no—
105C4×C13—24no—
106C4×C13—42no—
107C4×C13—42no—
108C4×C13#12414yes—
109Dic13—24no—
110Dic13—22no—
111Dic13—22no—
112Dic13—42no—
113C22×Q8C2×D1311yescharacteristic
114C22×D4—131no—
115#125—264no—
116#124—264no—
117C2×C4⋊C4—132no—
118C2×C4⋊C4—132no—
119C2×C4⋊C4—131no—
120C2×C4⋊C4C2×D1311yescharacteristic
121C2×C22⋊C4—132no—
122C2×C22⋊C4—132no—
123C2×C22⋊C4—134no—
124C2×C22⋊C4—134no—
125C2×C42—131no—
126C2×C42—132no—
127C2×C42—132no—
128C2×C42—131no—
129C2×C42C2×D1311yescharacteristic
130#95—134no—
131#95C2×D1314yes—
132C2×C13C2×#12411yescharacteristic
133C2×C13#52912yes—
134C2×C13#52912yes—
135C2×C13#53211yescharacteristic
136C2×C13#53211yescharacteristic
137D13—82no—
138C24—132no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×D4—262no—
143C2×D4—262no—
144C2×D4—262no—
145C22×C4—132no—
146C22×C4—131no—
147C22×C4—131no—
148C22×C4—132no—
149C22×C4C22×D1312yes—
150C22×C4C22×D1312yes—
151C22×C4C22×D1312yes—
152C22×C4C22×D1311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—264no—
155C4⋊C4—264no—
156C4⋊C4—262no—
157C22⋊C4—264no—
158C22⋊C4—264no—
159C22⋊C4—268no—
160C22⋊C4—268no—
161C42—22no—
162C42—134no—
163C42—264no—
164C42—262no—
165C42—264no—
166C13#286111yescharacteristic, complemented, Sylow
167C23—262no—
168C23—262no—
169C23—262no—
170C23—264no—
171C23—264no—
172C23C23×D1311yescharacteristic
173Q8—44no—
174D4—524no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—132no—
183C2×C4—132no—
184C2×C4—134no—
185C2×C4—262no—
186C2×C4—262no—
187C2×C4—262no—
188C2×C4—262no—
189C2×C4—262no—
190C2×C4—264no—
191C2×C4#612712yes—
192C2×C4D4×D1312yes—
193C2×C4#612912yes—
194C2×C4#613114yes—
195C22—522no—
196C22—522no—
197C22—524no—
198C22—524no—
199C22—522no—
200C22C2×#612711yescharacteristic
201C22C2×D4×D1311yescharacteristic
202C22C2×#612911yescharacteristic
203C22C2×#613112yes—
204C22#41×D1312yes—
205C4—24no—
206C4—42no—
207C4—42no—
208C4—262no—
209C4—264no—
210C4—262no—
211C4—522no—
212C4#8813914yes—
213C2—1042no—
214C2C2×#8813911yescharacteristic
215C2#1171928512yes—
216C2#1171932912yes—
217C2#1171937311yescharacteristic
218C2#1171943011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#391C2
prime radicalC22×D52C22
hypercenter#391D13
hypocenterC13#2861
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#88144C2semidirect
3. C13×#391C2semidirect
4. #11718891C2semidirect
7. C42×Dic13C2semidirect
8. #11718331C2semidirect
9. #11718330C2semidirect
89. #391D13semidirect
166. C13#2861semidirect

Maximal subgroups

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

subgroupindex
C2×#882012
C2×#881442
C13×#3912
#117188912
#117186492
#117186002
C42×Dic132
#117183312
#117183302
#286113