Finite Group Databasebeta

← #13120019

Subgroups — #13120019

Order 1664.

584 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 147 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×#88151C211yesmaximal, characteristic, complemented
2C2×#88147C211yesmaximal, characteristic
3C13×#400C211yesmaximal, characteristic, complemented
4#11718653C211yesmaximal, characteristic, complemented
5#11718640C211yesmaximal, characteristic, complemented
6#11718334C211yesmaximal, characteristic, complemented
7#11718329C212yesmaximal, complemented
8#11718328C212yesmaximal, complemented
9#11718326C212yesmaximal
10D13×#95C211yesmaximal, characteristic, complemented
11#11718316C212yesmaximal, complemented
12C22×D52C2211yescharacteristic
13C22×C4×D13C2212yes—
14C22×C4×D13C2211yescharacteristic
15C22×C4×D13C2211yescharacteristic
16C22×Dic26C2211yescharacteristic
17C2×C13×C4⋊C4C2211yescharacteristic
18C2×C13×C4⋊C4C2211yescharacteristic
19C2×#6103C2211yescharacteristic
20C2×#6103C2211yescharacteristic
21C2×#6103C2212yes—
22C2×#6103C2212yes—
23C2×#6103C2212yes—
24C2×C4⋊Dic13C2212yes—
25C2×#6101C2212yes—
26C2×#6101C2212yes—
27C2×C4×Dic13C2211yescharacteristic
28#88151—24no—
29#88147—24no—
30C13×#95C2212yes—
31C13×#95C2211yescharacteristic
32C13×#95C2212yes—
33#88071C2212yes—
34#88071C2212yes—
35#88071C2212yes—
36#88071C2211yescharacteristic
37C23×D13C2311yescharacteristic
38C23×D13C2311yescharacteristic
39C22×C4×C13C2312yes—
40C22×C4×C13C2311yescharacteristic
41C22×C4×C13C2312yes—
42C22×C4×C13C2311yescharacteristic
43C22×C4×C13C2311yescharacteristic
44C22×Dic13C2311yescharacteristic
45C22×Dic13C2312yes—
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2312yes—
48C2×D52—22no—
49C2×D52—22no—
50C2×D52Q814yescomplemented
51C2×C4×D13—24no—
52C2×C4×D13—22no—
53C2×C4×D13—24no—
54C2×C4×D13—42no—
55C2×C4×D13—42no—
56C2×C4×D13—42no—
57C2×C4×D13D414yes—
58C2×Dic26—22no—
59C2×Dic26—42no—
60C13×C4⋊C4—22no—
61C13×C4⋊C4—22no—
62#6103—22no—
63#6103—24no—
64#6103—22no—
65#6103—24no—
66#6103—42no—
67C4⋊Dic13—24no—
68#6101—24no—
69#6101—42no—
70C4×Dic13—22no—
71#2564—131nomaximal, Sylow
72C23×C13C2411yescharacteristic
73C22×D13—22no—
74C22×D13—22no—
75C22×D13—21no—
76C22×D13—22no—
77C22×D13—21no—
78C22×D13—22no—
79C22×D13C2×D412yes—
80C22×D13C2×Q812yes—
81C22×D13C2×Q812yes—
82C22×D13#4112yes—
83C2×C4×C13—21no—
84C2×C4×C13—21no—
85C2×C4×C13—22no—
86C2×C4×C13—22no—
87C2×C4×C13—21no—
88C2×C4×C13—22no—
89C2×C4×C13—22no—
90C2×C4×C13—22no—
91C2×C4×C13—22no—
92C2×C4×C13—22no—
93C2×C4×C13—22no—
94C2×C4×C13C2×D412yes—
95C2×C4×C13C2×Q812yes—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—21no—
100C2×Dic13—22no—
101C2×Dic13—22no—
102C2×Dic13—22no—
103C2×Dic13—22no—
104C2×Dic13C2×D412yes—
105C2×Dic13#4114yes—
106D52—28no—
107C4×D13—44no—
108C4×D13—48no—
109C4×D13—44no—
110Dic26—44no—
111C2×C22⋊Q8—131no—
112C2×C4×D4—131no—
113#400D1311yescharacteristic, complemented
114#399—131no—
115#395—131no—
116#393—131no—
117#387—132no—
118#386—132no—
119#385—132no—
120#380—132no—
121C2×#95—131no—
122C22×C13C22×D411yescharacteristic
123C22×C13C22×Q811yescharacteristic
124C22×C13C2×#4111yescharacteristic
125C22×C13C2×#4112yes—
126C22×C13H(5,𝔽2)12yes—
127C2×D13—22no—
128C2×D13—22no—
129C2×D13—24no—
130C2×D13—24no—
131C2×D13—22no—
132C2×D13—24no—
133C2×D13—41no—
134C2×D13—41no—
135C2×D13—42no—
136C2×D13C22⋊Q814yes—
137C4×C13—22no—
138C4×C13—22no—
139C4×C13—42no—
140C4×C13—41no—
141C4×C13—42no—
142Dic13—22no—
143Dic13—24no—
144Dic13—41no—
145Dic13—42no—
146C22×Q8—131no—
147C22×D4—131no—
148C23×C4—131no—
149C23×C4—132no—
150C23×C4—131no—
151C22⋊Q8—264no—
152C4×D4—264no—
153C2×C4⋊C4—132no—
154C2×C4⋊C4—132no—
155C2×C4⋊C4—132no—
156C2×C4⋊C4C2×D1311yescharacteristic
157C2×C4⋊C4C2×D1311yescharacteristic
158C2×C22⋊C4—131no—
159C2×C22⋊C4—131no—
160C2×C22⋊C4—132no—
161C2×C22⋊C4—132no—
162C2×C22⋊C4—132no—
163C2×C42—131no—
164#95—132no—
165#95—131no—
166#95—132no—
167#95—132no—
168#95C2×D1311yescharacteristic
169#95C2×D1312yes—
170#95C2×D1312yes—
171C2×C13C2×C22⋊Q811yescharacteristic
172C2×C13#53211yescharacteristic
173C2×C13#53411yescharacteristic
174C2×C13#55112yes—
175C2×C13#55412yes—
176D13—24no—
177D13—42no—
178C24—131no—
179C24—131no—
180C2×Q8—262no—
181C2×Q8—522no—
182C2×D4—134no—
183C2×D4—262no—
184C2×D4—262no—
185C22×C4—132no—
186C22×C4—131no—
187C22×C4—134no—
188C22×C4—131no—
189C22×C4—132no—
190C22×C4—262no—
191C22×C4—264no—
192C22×C4—264no—
193C22×C4—522no—
194C22×C4—522no—
195C22×C4—522no—
196C22×C4C22×D1312yes—
197C22×C4C22×D1312yes—
198C22×C4C22×D1311yescharacteristic
199C22×C4C22×D1311yescharacteristic
200C22×C4C22×D1311yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—264no—
204C4⋊C4—264no—
205C4⋊C4—522no—
206C22⋊C4—264no—
207C22⋊C4—262no—
208C22⋊C4—264no—
209C22⋊C4—262no—
210C22⋊C4—522no—
211C42—262no—
212C13#256411yescharacteristic, complemented, Sylow
213C23—132no—
214C23—132no—
215C23—132no—
216C23—132no—
217C23—262no—
218C23—262no—
219C23—261no—
220C23—262no—
221C23—261no—
222C23—262no—
223C23C23×D1311yescharacteristic
224Q8—524no—
225D4—268no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—132no—
238C2×C4—134no—
239C2×C4—262no—
240C2×C4—262no—
241C2×C4—262no—
242C2×C4—261no—
243C2×C4—262no—
244C2×C4—262no—
245C2×C4—262no—
246C2×C4—262no—
247C2×C4—524no—
248C2×C4—528no—
249C2×C4—524no—
250C2×C4D4×D1312yes—
251C2×C4Q8×D1312yes—
252C22—134no—
253C22—262no—
254C22—262no—
255C22—264no—
256C22—264no—
257C22—262no—
258C22—264no—
259C22—521no—
260C22—521no—
261C22—522no—
262C22C2×D4×D1311yescharacteristic
263C22C2×Q8×D1311yescharacteristic
264C22#41×D1311yescharacteristic
265C22#41×D1312yes—
266C22#8825612yes—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—42no—
271C4—42no—
272C4—262no—
273C4—264no—
274C4—522no—
275C4—521no—
276C2—264no—
277C2—522no—
278C2D13×C22⋊Q811yescharacteristic
279C2#1171935811yescharacteristic
280C2#1171937311yescharacteristic
281C2#1171939512yes—
282C2#1171941012yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#400C2
prime radicalC22×D52C22
hypercenter#400D13
hypocenterC13#2564
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#88151C2semidirect
3. C13×#400C2semidirect
4. #11718653C2semidirect
5. #11718640C2semidirect
6. #11718334C2semidirect
7. #11718329C2semidirect
8. #11718328C2semidirect
10. D13×#95C2semidirect
11. #11718316C2semidirect
50. C2×D52Q8semidirect
113. #400D13semidirect
212. C13#2564semidirect

Maximal subgroups

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

subgroupindex
C2×#881512
C2×#881472
C13×#4002
#117186532
#117186402
#117183342
#117183292
#117183282
#117183262
D13×#952
#117183162
#256413