Finite Group Databasebeta

← #13120456

Subgroups — #13120456

Order 1664.

582 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 141 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×D4×Dic13C211yesmaximal, characteristic, complemented
2C2×#88133C211yesmaximal, characteristic, complemented
3C13×#392C211yesmaximal, characteristic
4#11718883C211yesmaximal, characteristic, complemented
5#11718882C212yesmaximal, complemented
6#11718881C212yesmaximal, complemented
7#11718614C211yesmaximal, characteristic, complemented
8#11718611C212yesmaximal, complemented
9#11718605C212yesmaximal, complemented
10#11718322C211yesmaximal, characteristic, complemented
11#11718309C211yesmaximal, characteristic, complemented
12C22×D4×C13C2211yescharacteristic
13C22×D4×C13C2212yes—
14C23×Dic13C2212yes—
15C2×C13×C22⋊C4C2211yescharacteristic
16C2×C13×C22⋊C4C2212yes—
17C2×C22⋊Dic13C2211yescharacteristic
18C2×C22⋊Dic13C2211yescharacteristic
19C2×C22⋊Dic13C2212yes—
20C2×C22⋊Dic13C2212yes—
21C2×C22⋊Dic13C2212yes—
22C2×C22⋊Dic13C2212yes—
23D4×Dic13—24no—
24C2×C4⋊Dic13C2211yescharacteristic
25C2×C4⋊Dic13C2211yescharacteristic
26C2×C4⋊Dic13C2212yes—
27C2×#6101C2211yescharacteristic
28C2×#6101C2211yescharacteristic
29C2×C4×Dic13C2212yes—
30C2×C4×Dic13C2211yescharacteristic
31#88133—24no—
32C13×#95C2211yescharacteristic
33#88071C2212yes—
34#88071C2211yescharacteristic
35#88071C2212yes—
36#88071C2212yes—
37C24×C13C2311yescharacteristic
38C24×C13C2312yes—
39C2×D4×C13—24no—
40C2×D4×C13—24no—
41C2×D4×C13—22no—
42C2×D4×C13—24no—
43C2×D4×C13—42no—
44C22×C4×C13C2311yescharacteristic
45C22×C4×C13C2311yescharacteristic
46C22×C4×C13C2312yes—
47C22×Dic13—24no—
48C22×Dic13—42no—
49C22×Dic13—42no—
50C22×Dic13C2312yes—
51C22×Dic13C2311yescharacteristic
52C22×Dic13C2312yes—
53C22×Dic13C2311yescharacteristic
54C22×Dic13C2311yescharacteristic
55C22×Dic13C2311yescharacteristic
56C13×C22⋊C4—22no—
57C13×C22⋊C4—24no—
58C22⋊Dic13—24no—
59C22⋊Dic13—22no—
60C22⋊Dic13—22no—
61C22⋊Dic13—24no—
62C22⋊Dic13—42no—
63C22⋊Dic13—42no—
64C4⋊Dic13—22no—
65C4⋊Dic13—22no—
66C4⋊Dic13—42no—
67#6101—22no—
68#6101—22no—
69C4×Dic13—24no—
70C4×Dic13D414yescomplemented
71#2603—131nomaximal, Sylow
72C23×C13—22no—
73C23×C13—22no—
74C23×C13—22no—
75C23×C13—22no—
76C23×C13—22no—
77C23×C13—22no—
78C23×C13—21no—
79C23×C13—22no—
80C23×C13—22no—
81C23×C13—21no—
82C23×C13—21no—
83C23×C13#4114yes—
84C23×C13C2411yescharacteristic
85D4×C13—28no—
86D4×C13—48no—
87C2×C4×C13—22no—
88C2×C4×C13—22no—
89C2×C4×C13—22no—
90C2×C4×C13—21no—
91C2×C4×C13—22no—
92C2×C4×C13C2×D412yes—
93C2×C4×C13#4114yes—
94C2×Dic13—22no—
95C2×Dic13—21no—
96C2×Dic13—21no—
97C2×Dic13—21no—
98C2×Dic13—21no—
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13—22no—
102C2×Dic13—22no—
103C2×Dic13—21no—
104C2×Dic13—22no—
105C2×Dic13—21no—
106C2×Dic13—22no—
107C2×Dic13—22no—
108C2×Dic13—48no—
109C2×Dic13C2×D412yes—
110C2×Dic13C2×D412yes—
111C2×#126—131no—
112C2×C4×D4—131no—
113#398—131no—
114#397—131no—
115#394—131no—
116#392D1311yescharacteristic
117#390—132no—
118#388—132no—
119#386—132no—
120#385—132no—
121#376—131no—
122C22×C13—24no—
123C22×C13—24no—
124C22×C13—24no—
125C22×C13—42no—
126C22×C13—42no—
127C22×C13—42no—
128C22×C13—42no—
129C22×C13—41no—
130C22×C13—41no—
131C22×C13—41no—
132C22×C13—42no—
133C22×C13—42no—
134C22×C13C22×D411yescharacteristic
135C22×C13C2×#4112yes—
136C22×C13C2×#4112yes—
137C22×C13H(5,𝔽2)11yescharacteristic
138C22×C13H(5,𝔽2)11yescharacteristic
139C4×C13—22no—
140C4×C13—24no—
141C4×C13—41no—
142Dic13—22no—
143Dic13—22no—
144Dic13—41no—
145Dic13—41no—
146Dic13—42no—
147Dic13—42no—
148C22×D4C2×D1311yescharacteristic
149C22×D4C2×D1312yes—
150C23×C4—132no—
151#126—264no—
152C4×D4—264no—
153C2×C4⋊C4—131no—
154C2×C4⋊C4—131no—
155C2×C4⋊C4—131no—
156C2×C4⋊C4—131no—
157C2×C4⋊C4—132no—
158C2×C22⋊C4—132no—
159C2×C22⋊C4—131no—
160C2×C22⋊C4—131no—
161C2×C22⋊C4—132no—
162C2×C22⋊C4—132no—
163C2×C22⋊C4—132no—
164C2×C22⋊C4C2×D1311yescharacteristic
165C2×C22⋊C4C2×D1312yes—
166C2×C42—131no—
167C2×C42—132no—
168#95—132no—
169#95—131no—
170#95—132no—
171#95—132no—
172#95C2×D1311yescharacteristic
173C2×C13—44no—
174C2×C13—81no—
175C2×C13#52511yescharacteristic
176C2×C13#53211yescharacteristic
177C2×C13#53511yescharacteristic
178C2×C13#55112yes—
179C2×C13#55312yes—
180C24C22×D1311yescharacteristic
181C24C22×D1312yes—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—42no—
187C22×C4—131no—
188C22×C4—131no—
189C22×C4—132no—
190C22×C4—131no—
191C22×C4—132no—
192C22×C4—131no—
193C22×C4—264no—
194C22×C4—522no—
195C22×C4—522no—
196C22×C4C22×D1311yescharacteristic
197C22×C4C22×D1311yescharacteristic
198C22×C4C22×D1312yes—
199C4⋊C4—262no—
200C4⋊C4—262no—
201C4⋊C4—262no—
202C4⋊C4—262no—
203C4⋊C4—522no—
204C22⋊C4—22no—
205C22⋊C4—24no—
206C22⋊C4—264no—
207C22⋊C4—262no—
208C22⋊C4—262no—
209C22⋊C4—264no—
210C22⋊C4—522no—
211C22⋊C4—522no—
212C42—134no—
213C42—264no—
214C13#260311yescharacteristic, complemented, Sylow
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23#612914yes—
227C23C23×D1311yescharacteristic
228D4—28no—
229D4—48no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—132no—
236C2×C4—132no—
237C2×C4—262no—
238C2×C4—261no—
239C2×C4—261no—
240C2×C4—261no—
241C2×C4—261no—
242C2×C4—262no—
243C2×C4—262no—
244C2×C4—262no—
245C2×C4—262no—
246C2×C4—261no—
247C2×C4—262no—
248C2×C4—261no—
249C2×C4—262no—
250C2×C4—262no—
251C2×C4—528no—
252C2×C4D4×D1312yes—
253C2×C4#612914yes—
254C22—24no—
255C22—24no—
256C22—24no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—41no—
262C22—42no—
263C22—41no—
264C22—42no—
265C22—41no—
266C22C2×D4×D1311yescharacteristic
267C22C2×#612912yes—
268C22C2×#612912yes—
269C22#8825111yescharacteristic
270C22#8825611yescharacteristic
271C4—22no—
272C4—24no—
273C4—41no—
274C4—262no—
275C4—262no—
276C4—521no—
277C4—521no—
278C4—522no—
279C4—522no—
280C2—44no—
281C2—81no—
282C2#1171929111yescharacteristic
283C2#1171929212yes—
284C2#1171931211yescharacteristic
285C2#1171932312yes—
286C2#1171942111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
Fitting, prime radicalC13×#392C2
hypercenter#392D13
hypocenterC13#2603
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic13C2semidirect
2. C2×#88133C2semidirect
4. #11718883C2semidirect
5. #11718882C2semidirect
6. #11718881C2semidirect
7. #11718614C2semidirect
8. #11718611C2semidirect
9. #11718605C2semidirect
10. #11718322C2semidirect
11. #11718309C2semidirect
70. C4×Dic13D4semidirect
214. C13#2603semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic132
C2×#881332
C13×#3922
#117188832
#117188822
#117188812
#117186142
#117186112
#117186052
#117183222
#117183092
#260313