Finite Group Databasebeta

← #13122807

Subgroups — #13122807

Order 1664.

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

normal subgroups only

This group has 408 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
1C13×#399C211yesmaximal, characteristic
2#11718646C212yesmaximal, complemented
3#11718644C212yesmaximal, complemented
4#11718643C212yesmaximal, complemented
5#11718611C212yesmaximal
6#11718610C211yesmaximal, characteristic
7#11718607C211yesmaximal, characteristic
8#11718606C212yesmaximal
9#11718308C212yesmaximal, complemented
10C23×Dic13C2211yescharacteristic
11C2×C13×C4⋊C4C2212yes—
12C2×C13×C4⋊C4C2211yescharacteristic
13C2×C13×C22⋊C4C2212yes—
14C2×C13×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic13C2212yes—
16C2×C22⋊Dic13C2211yescharacteristic
17C2×C4⋊Dic13C2212yes—
18C2×C4⋊Dic13C2211yescharacteristic
19C2×C4⋊Dic13C2212yes—
20C2×#6101C2212yes—
21C2×#6101C2212yes—
22C2×#6101C2212yes—
23C2×C4×Dic13C2212yes—
24C13×#95C2211yescharacteristic
25#88071C2212yes—
26#88071C2212yes—
27#88071C2212yes—
28#88071C2212yes—
29#88071C2211yescharacteristic
30#88071C2212yes—
31C24×C13C2311yescharacteristic
32C22×C4×C13C2312yes—
33C22×C4×C13C2311yescharacteristic
34C22×C4×C13C2311yescharacteristic
35C22×C4×C13C2312yes—
36C22×Dic13—42no—
37C22×Dic13—41no—
38C22×Dic13C2312yes—
39C22×Dic13C2312yes—
40C22×Dic13C2312yes—
41C22×Dic13C2312yes—
42C13×C4⋊C4—24no—
43C13×C4⋊C4—41no—
44C13×C22⋊C4—24no—
45C13×C22⋊C4—41no—
46C22⋊Dic13—42no—
47C22⋊Dic13Q814yes—
48C4⋊Dic13—42no—
49C4⋊Dic13—41no—
50C4⋊Dic13—42no—
51#6101—24no—
52#6101—42no—
53#6101—42no—
54C4×Dic13—42no—
55#2807—131nomaximal, Sylow
56C23×C13—22no—
57C23×C13—22no—
58C23×C13—21no—
59C23×C13—21no—
60C23×C13C2×Q812yes—
61C23×C13C2411yescharacteristic
62C2×C4×C13—22no—
63C2×C4×C13—22no—
64C2×C4×C13—22no—
65C2×C4×C13—22no—
66C2×C4×C13—22no—
67C2×C4×C13—22no—
68C2×C4×C13—21no—
69C2×C4×C13—22no—
70C2×C4×C13—21no—
71C2×C4×C13#4114yes—
72C2×Dic13—22no—
73C2×Dic13—22no—
74C2×Dic13—22no—
75C2×Dic13—22no—
76C2×Dic13—22no—
77C2×Dic13—22no—
78C2×Dic13—22no—
79C2×Dic13—22no—
80C2×Dic13—22no—
81C2×Dic13—22no—
82C2×Dic13—22no—
83C2×Dic13—44no—
84C2×Dic13C2×Q814yes—
85#400—132no—
86#399—131no—
87#399D1311yescharacteristic
88#398—132no—
89#389—132no—
90#389—132no—
91#388—132no—
92#385—131no—
93#385—132no—
94C22×C13—22no—
95C22×C13—24no—
96C22×C13—42no—
97C22×C13—41no—
98C22×C13—41no—
99C22×C13C22×Q811yescharacteristic
100C22×C13C2×#4112yes—
101C22×C13H(5,𝔽2)11yescharacteristic
102C22×C13H(5,𝔽2)12yes—
103C22×C132−1+411yescharacteristic
104C4×C13—24no—
105C4×C13—42no—
106C4×C13—41no—
107C4×C13—41no—
108Dic13—24no—
109Dic13—42no—
110Dic13—42no—
111Dic13—42no—
112C23×C4—131no—
113C2×C4⋊C4—132no—
114C2×C4⋊C4—132no—
115C2×C4⋊C4—132no—
116C2×C4⋊C4—132no—
117C2×C4⋊C4—132no—
118C2×C4⋊C4—131no—
119C2×C4⋊C4C2×D1312yes—
120C2×C4⋊C4C2×D1311yescharacteristic
121C2×C22⋊C4—131no—
122C2×C22⋊C4—132no—
123C2×C22⋊C4C2×D1312yes—
124C2×C22⋊C4C2×D1311yescharacteristic
125C2×C42—132no—
126#95—132no—
127#95—132no—
128#95—132no—
129#95—132no—
130#95—132no—
131#95—131no—
132#95C2×D1311yescharacteristic
133C2×C13—42no—
134C2×C13#53912yes—
135C2×C13#54411yescharacteristic
136C2×C13#55412yes—
137C2×C13#55911yescharacteristic
138C2×C13#56011yescharacteristic
139C24C22×D1311yescharacteristic
140C22×C4—132no—
141C22×C4—132no—
142C22×C4—132no—
143C22×C4—132no—
144C22×C4—521no—
145C22×C4—522no—
146C22×C4C22×D1312yes—
147C22×C4C22×D1311yescharacteristic
148C22×C4C22×D1311yescharacteristic
149C22×C4C22×D1312yes—
150C4⋊C4—24no—
151C4⋊C4—41no—
152C4⋊C4—264no—
153C4⋊C4—522no—
154C4⋊C4—522no—
155C4⋊C4—522no—
156C4⋊C4—521no—
157C4⋊C4—522no—
158C22⋊C4—24no—
159C22⋊C4—41no—
160C22⋊C4—134no—
161C22⋊C4—522no—
162C42—522no—
163C13#280711yescharacteristic, complemented, Sylow
164C23—22no—
165C23—21no—
166C23—22no—
167C23—21no—
168C23Q8×D1312yes—
169C23C23×D1311yescharacteristic
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—134no—
180C2×C4—262no—
181C2×C4—262no—
182C2×C4—262no—
183C2×C4—262no—
184C2×C4—262no—
185C2×C4—262no—
186C2×C4—262no—
187C2×C4—262no—
188C2×C4—262no—
189C2×C4—262no—
190C2×C4—262no—
191C2×C4—524no—
192C2×C4#612914yes—
193C22—22no—
194C22—24no—
195C22—42no—
196C22—41no—
197C22—41no—
198C22C2×#612912yes—
199C22#8825111yescharacteristic
200C22C2×Q8×D1311yescharacteristic
201C22#8825612yes—
202C22#8825711yescharacteristic
203C4—24no—
204C4—41no—
205C4—42no—
206C4—41no—
207C4—264no—
208C4—522no—
209C4—522no—
210C4—522no—
211C2—42no—
212C2#1171932811yescharacteristic
213C2#1171934012yes—
214C2#1171937012yes—
215C2#1171937111yescharacteristic
216C2#1171941811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#399C2
prime radicalC24×C13C23
hypercenter#399D13
hypocenterC13#2807
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #11718646C2semidirect
3. #11718644C2semidirect
4. #11718643C2semidirect
9. #11718308C2semidirect
163. C13#2807semidirect

Maximal subgroups

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

subgroupindex
C13×#3992
#117186462
#117186442
#117186432
#117186112
#117186102
#117186072
#117186062
#117183082
#280713