Finite Group Databasebeta

← #13122049

Subgroups — #13122049

Order 1664.

464 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 137 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×#390C211yesmaximal, characteristic
2#11718881C211yesmaximal, characteristic
3#11718881C211yesmaximal, characteristic
4#11718645C212yesmaximal, complemented
5#11718610C212yesmaximal, complemented
6#11718609C212yesmaximal, complemented
7#11718607C212yesmaximal, complemented
8#11718603C212yesmaximal, complemented
9#11718594C212yesmaximal, complemented
10C22×D4×C13C2211yescharacteristic
11C23×Dic13C2211yescharacteristic
12C23×Dic13C2211yescharacteristic
13C2×C13×C4⋊C4C2211yescharacteristic
14C2×C13×C22⋊C4C2212yes—
15C2×C13×C22⋊C4C2212yes—
16C2×C42×C13C2211yescharacteristic
17C2×C22⋊Dic13C2212yes—
18C2×C22⋊Dic13C2212yes—
19C2×C22⋊Dic13C2211yescharacteristic
20C2×C22⋊Dic13C2211yescharacteristic
21C2×C4⋊Dic13C2212yes—
22C2×#6101C2212yes—
23C2×#6101C2212yes—
24C2×#6101C2212yes—
25C2×#6101C2212yes—
26#88071C2212yes—
27#88071C2212yes—
28#88071C2212yes—
29#88071C2212yes—
30#88071C2212yes—
31C24×C13C2311yescharacteristic
32C24×C13C2311yescharacteristic
33C2×D4×C13—41no—
34C2×D4×C13—41no—
35C2×D4×C13Q814yes—
36C22×C4×C13C2312yes—
37C22×C4×C13C2312yes—
38C22×C4×C13C2311yescharacteristic
39C22×Dic13—24no—
40C22×Dic13—41no—
41C22×Dic13—41no—
42C22×Dic13—42no—
43C22×Dic13C2312yes—
44C22×Dic13C2312yes—
45C22×Dic13C2312yes—
46C22×Dic13C2312yes—
47C13×C4⋊C4—41no—
48C13×C22⋊C4—42no—
49C13×C22⋊C4—42no—
50C42×C13—41no—
51C22⋊Dic13—24no—
52C22⋊Dic13—41no—
53C22⋊Dic13—41no—
54C22⋊Dic13—42no—
55C4⋊Dic13—42no—
56#6101—24no—
57#6101—42no—
58#6101—42no—
59#6101—42no—
60#2748—131nomaximal, Sylow
61C23×C13—22no—
62C23×C13—22no—
63C23×C13—22no—
64C23×C13—21no—
65C23×C13—21no—
66C23×C13—22no—
67C23×C13—21no—
68C23×C13—21no—
69C23×C13C2×Q812yes—
70C23×C13C2×Q812yes—
71C23×C13C2411yescharacteristic
72D4×C13—44no—
73C2×C4×C13—21no—
74C2×C4×C13—21no—
75C2×C4×C13—22no—
76C2×C4×C13—22no—
77C2×C4×C13—22no—
78C2×C4×C13—22no—
79C2×C4×C13—22no—
80C2×C4×C13—22no—
81C2×C4×C13C2×Q812yes—
82C2×Dic13—22no—
83C2×Dic13—22no—
84C2×Dic13—22no—
85C2×Dic13—22no—
86C2×Dic13—22no—
87C2×Dic13—22no—
88C2×Dic13—22no—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—44no—
94C2×Dic13—44no—
95C2×Dic13#4114yes—
96#399—132no—
97#398—132no—
98#396—132no—
99#390D1311yescharacteristic
100#387—132no—
101#386—131no—
102#386—131no—
103#385—132no—
104#380—132no—
105C22×C13—22no—
106C22×C13—22no—
107C22×C13—24no—
108C22×C13—24no—
109C22×C13—42no—
110C22×C13—42no—
111C22×C13—41no—
112C22×C13—41no—
113C22×C13—41no—
114C22×C13—41no—
115C22×C13C22×Q811yescharacteristic
116C22×C13C2×#4112yes—
117C22×C13H(5,𝔽2)11yescharacteristic
118C22×C13H(5,𝔽2)12yes—
119C22×C13H(5,𝔽2)11yescharacteristic
120C4×C13—22no—
121C4×C13—42no—
122C4×C13—42no—
123Dic13—24no—
124Dic13—42no—
125Dic13—42no—
126Dic13—42no—
127C22×D4C2×D1311yescharacteristic
128C23×C4—131no—
129C23×C4—131no—
130C2×C4⋊C4—132no—
131C2×C4⋊C4—132no—
132C2×C4⋊C4—132no—
133C2×C4⋊C4—132no—
134C2×C4⋊C4—132no—
135C2×C4⋊C4C2×D1311yescharacteristic
136C2×C22⋊C4—132no—
137C2×C22⋊C4—132no—
138C2×C22⋊C4—131no—
139C2×C22⋊C4—131no—
140C2×C22⋊C4C2×D1312yes—
141C2×C22⋊C4C2×D1312yes—
142C2×C42C2×D1311yescharacteristic
143#95—132no—
144#95—132no—
145#95—132no—
146#95—132no—
147#95—132no—
148C2×C13—42no—
149C2×C13—42no—
150C2×C13#54012yes—
151C2×C13#54311yescharacteristic
152C2×C13#55111yescharacteristic
153C2×C13#55412yes—
154C2×C13#56011yescharacteristic
155C24C22×D1311yescharacteristic
156C24C22×D1311yescharacteristic
157C2×D4—41no—
158C2×D4—41no—
159C2×D4Dic2614yes—
160C22×C4—132no—
161C22×C4—132no—
162C22×C4—132no—
163C22×C4—132no—
164C22×C4—264no—
165C22×C4—521no—
166C22×C4—522no—
167C22×C4—521no—
168C22×C4C22×D1312yes—
169C22×C4C22×D1312yes—
170C22×C4C22×D1311yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—264no—
173C4⋊C4—522no—
174C4⋊C4—522no—
175C4⋊C4—522no—
176C4⋊C4—522no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—264no—
180C22⋊C4—521no—
181C22⋊C4—521no—
182C22⋊C4—522no—
183C42—41no—
184C13#274811yescharacteristic, complemented, Sylow
185C23—22no—
186C23—22no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23C2×Dic2612yes—
194C23C2×Dic2612yes—
195C23C23×D1311yescharacteristic
196D4—44no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—134no—
206C2×C4—262no—
207C2×C4—262no—
208C2×C4—262no—
209C2×C4—262no—
210C2×C4—262no—
211C2×C4—262no—
212C2×C4—262no—
213C2×C4—262no—
214C2×C4—262no—
215C2×C4—262no—
216C2×C4—262no—
217C2×C4—524no—
218C2×C4—524no—
219C2×C4C2×Dic2612yes—
220C22—22no—
221C22—22no—
222C22—24no—
223C22—24no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22C22×Dic2611yescharacteristic
231C22#8825111yescharacteristic
232C22#8825112yes—
233C22#8825111yescharacteristic
234C22#41×D1312yes—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—264no—
239C4—522no—
240C4—522no—
241C4—522no—
242C2—42no—
243C2—42no—
244C2#1171918811yescharacteristic
245C2#1171924412yes—
246C2#1171932012yes—
247C2#1171938411yescharacteristic
248C2#1171942511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#390C2
prime radicalC22×D4×C13C22
hypercenter#390D13
hypocenterC13#2748
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
4. #11718645C2semidirect
5. #11718610C2semidirect
6. #11718609C2semidirect
7. #11718607C2semidirect
8. #11718603C2semidirect
9. #11718594C2semidirect
184. C13#2748semidirect

Maximal subgroups

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

subgroupindex
C13×#3902
#117188812
#117186452
#117186102
#117186092
#117186072
#117186032
#117185942
#274813