Finite Group Databasebeta

← #13120564

Subgroups — #13120564

Order 1664.

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

normal subgroups only

This group has 816 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
1C22×#6129C211yesmaximal, characteristic
2C2×#88150C211yesmaximal, characteristic, complemented
3C2×#88132C212yesmaximal, complemented
4C13×#394C211yesmaximal, characteristic, complemented
5#11718883C211yesmaximal, characteristic, complemented
6#11718640C211yesmaximal, characteristic, complemented
7#11718622C212yesmaximal, complemented
8#11718620C212yesmaximal, complemented
9#11718603C212yesmaximal, complemented
10#11718328C211yesmaximal, characteristic, complemented
11#11718317C211yesmaximal, characteristic, complemented
12C22×D4×C13C2211yescharacteristic
13C22×C22⋊D13C2212yes—
14C23×Dic13C2212yes—
15C2×#6129—28no—
16C2×#6129—24no—
17C22×C4×D13C2211yescharacteristic
18C22×Dic26C2211yescharacteristic
19C22×Dic26C2211yescharacteristic
20C2×C13×C4⋊C4C2211yescharacteristic
21C2×C13×C22⋊C4C2212yes—
22C2×C13×C22⋊C4C2212yes—
23C2×C22⋊Dic13C2212yes—
24C2×C22⋊Dic13C2212yes—
25C2×#6103C2211yescharacteristic
26C2×#6103C2211yescharacteristic
27C2×#6103C2212yes—
28C2×C4⋊Dic13C2212yes—
29C2×#6101C2212yes—
30C2×#6101C2212yes—
31C2×#6101C2211yescharacteristic
32C2×#6101C2211yescharacteristic
33#88150—24no—
34#88132—28no—
35C13×#95C2211yescharacteristic
36#88071C2212yes—
37#88071C2211yescharacteristic
38#88071C2212yes—
39C24×C13C2312yes—
40C23×D13C2311yescharacteristic
41C2×D4×C13—24no—
42C2×D4×C13—22no—
43C22×C4×C13C2311yescharacteristic
44C22×C4×C13C2312yes—
45C22×C4×C13C2311yescharacteristic
46C22×C4×C13C2311yescharacteristic
47C2×C22⋊D13—24no—
48C2×C22⋊D13—24no—
49C2×C22⋊D13—24no—
50C22×Dic13—24no—
51C22×Dic13—24no—
52C22×Dic13D418yes—
53C22×Dic13C2311yescharacteristic
54C22×Dic13C2312yes—
55C22×Dic13C2311yescharacteristic
56C22×Dic13C2312yes—
57C22×Dic13C2311yescharacteristic
58#6129—232no—
59C2×C4×D13—24no—
60C2×C4×D13D414yes—
61C2×Dic26—22no—
62C2×Dic26—24no—
63C2×Dic26—24no—
64C2×Dic26—22no—
65C13×C4⋊C4—22no—
66C13×C22⋊C4—24no—
67C13×C22⋊C4—42no—
68C22⋊Dic13—24no—
69C22⋊Dic13—42no—
70#6103—22no—
71#6103—22no—
72#6103—42no—
73C4⋊Dic13—24no—
74#6101—22no—
75#6101—24no—
76#6101—24no—
77#6101—22no—
78#2616—131nomaximal, Sylow
79C23×C13—22no—
80C23×C13—22no—
81C23×C13—22no—
82C23×C13—22no—
83C23×C13—22no—
84C23×C13—22no—
85C23×C13C2×D414yes—
86C23×C13C2411yescharacteristic
87C22×D13—22no—
88C22×D13—21no—
89C22×D13—22no—
90C22×D13—21no—
91C22×D13C2×D412yes—
92D4×C13—28no—
93C2×C4×C13—22no—
94C2×C4×C13—22no—
95C2×C4×C13—21no—
96C2×C4×C13—22no—
97C2×C4×C13—21no—
98C2×C4×C13—22no—
99C2×C4×C13—21no—
100C2×C4×C13—21no—
101C2×C4×C13—22no—
102C2×C4×C13C2×D412yes—
103C22⋊D13—216no—
104C2×Dic13—22no—
105C2×Dic13—22no—
106C2×Dic13—216no—
107C2×Dic13—21no—
108C2×Dic13—22no—
109C2×Dic13—21no—
110C2×Dic13—22no—
111C2×Dic13—22no—
112C2×Dic13—21no—
113C2×Dic13—22no—
114C2×Dic13—21no—
115C2×Dic13C2×D414yes—
116C2×Dic13C2×D414yes—
117C2×Dic13C2×D412yes—
118C4×D13—28no—
119Dic26—28no—
120Dic26—44no—
121C22×#41—131no—
122C2×C22⋊Q8—131no—
123C2×C22⋊Q8—132no—
124#395—131no—
125#395—131no—
126#394—131no—
127#394—132no—
128#394—132no—
129#394D1311yescharacteristic, complemented
130#380—132no—
131#380—131no—
132C22×C13—24no—
133C22×C13—24no—
134C22×C13—24no—
135C22×C13—42no—
136C22×C13—42no—
137C22×C13—42no—
138C22×C13—42no—
139C22×C13C22×D411yescharacteristic
140C22×C13C22×D412yes—
141C22×C13H(5,𝔽2)11yescharacteristic
142C22×C132−1+412yes—
143C22×C132−1+411yescharacteristic
144C2×D13—22no—
145C2×D13—22no—
146C2×D13—22no—
147C2×D13—42no—
148C2×D13—42no—
149C4×C13—22no—
150C4×C13—41no—
151C4×C13—42no—
152C4×C13—41no—
153Dic13—24no—
154Dic13—41no—
155Dic13—42no—
156Dic13—41no—
157Dic13#12014yes—
158C2×#41—268no—
159C2×#41—264no—
160C22×Q8—131no—
161C22×Q8—131no—
162C22×D4—132no—
163C22×D4C2×D1311yescharacteristic
164C23×C4—132no—
165C23×C4—131no—
166C22⋊Q8—268no—
167C22⋊Q8—264no—
168C2×C4⋊C4—131no—
169C2×C4⋊C4—132no—
170C2×C4⋊C4—132no—
171C2×C4⋊C4—131no—
172C2×C4⋊C4—132no—
173C2×C4⋊C4C2×D1311yescharacteristic
174C2×C22⋊C4—132no—
175C2×C22⋊C4—131no—
176C2×C22⋊C4—131no—
177C2×C22⋊C4—132no—
178C2×C22⋊C4—132no—
179C2×C22⋊C4C2×D1312yes—
180C2×C22⋊C4C2×D1312yes—
181#95—132no—
182#95—132no—
183#95—131no—
184#95C2×D1311yescharacteristic
185C2×C13—44no—
186C2×C13C2×#12011yescharacteristic
187C2×C13#53612yes—
188C2×C13#53611yescharacteristic
189C2×C13#53712yes—
190C2×C13#53711yescharacteristic
191D13—42no—
192C24—131no—
193C24C22×D1312yes—
194#41—2632no—
195C2×Q8—264no—
196C2×Q8—264no—
197C2×Q8—262no—
198C2×Q8—262no—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—264no—
202C2×D4—264no—
203C2×D4—264no—
204C22×C4—131no—
205C22×C4—138no—
206C22×C4—132no—
207C22×C4—134no—
208C22×C4—132no—
209C22×C4—131no—
210C22×C4—131no—
211C22×C4—264no—
212C22×C4—264no—
213C22×C4—264no—
214C22×C4C22×D1311yescharacteristic
215C22×C4C22×D1312yes—
216C22×C4C22×D1311yescharacteristic
217C22×C4C22×D1311yescharacteristic
218C4⋊C4—22no—
219C4⋊C4—264no—
220C4⋊C4—264no—
221C4⋊C4—262no—
222C4⋊C4—264no—
223C4⋊C4—262no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4—262no—
227C22⋊C4—262no—
228C22⋊C4—264no—
229C22⋊C4—522no—
230C22⋊C4—522no—
231C13#261611yescharacteristic, complemented, Sylow
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—132no—
239C23—262no—
240C23—261no—
241C23—261no—
242C23—262no—
243C23D4×D1314yes—
244C23C23×D1311yescharacteristic
245Q8—268no—
246Q8—524no—
247D4—28no—
248D4—2616no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—134no—
259C2×C4—132no—
260C2×C4—134no—
261C2×C4—268no—
262C2×C4—2616no—
263C2×C4—262no—
264C2×C4—262no—
265C2×C4—262no—
266C2×C4—261no—
267C2×C4—262no—
268C2×C4—261no—
269C2×C4—262no—
270C2×C4—261no—
271C2×C4—262no—
272C2×C4—261no—
273C2×C4D4×D1312yes—
274C22—24no—
275C22—24no—
276C22—24no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—262no—
282C22—262no—
283C22—262no—
284C22—522no—
285C22—522no—
286C22C2×D4×D1312yes—
287C22C2×D4×D1311yescharacteristic
288C22#8825111yescharacteristic
289C22#8825712yes—
290C22#8825711yescharacteristic
291C4—22no—
292C4—41no—
293C4—42no—
294C4—41no—
295C4—134no—
296C4—264no—
297C4—521no—
298C4—522no—
299C4—521no—
300C2—44no—
301C2—522no—
302C2D13×#12011yescharacteristic
303C2#1171930212yes—
304C2#1171931311yescharacteristic
305C2#1171935312yes—
306C2#1171938011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#394C2
prime radicalC22×#6129C2
hypercenter#394D13
hypocenterC13#2616
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#88150C2semidirect
3. C2×#88132C2semidirect
4. C13×#394C2semidirect
5. #11718883C2semidirect
6. #11718640C2semidirect
7. #11718622C2semidirect
8. #11718620C2semidirect
9. #11718603C2semidirect
10. #11718328C2semidirect
11. #11718317C2semidirect
129. #394D13semidirect
231. C13#2616semidirect

Maximal subgroups

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

subgroupindex
C22×#61292
C2×#881502
C2×#881322
C13×#3942
#117188832
#117186402
#117186222
#117186202
#117186032
#117183282
#117183172
#261613