Finite Group Databasebeta

← #15830788

Subgroups — #15830788

Order 3328.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13128646C211yesmaximal, characteristic
2#13123921C211yesmaximal, characteristic
3#13113399C212yesmaximal
4#13113386C214yesmaximal
5#13113347C214yesmaximal
6#13113334C212yesmaximal
7C13×#2356C211yesmaximal, characteristic
8C2×C4×Q8×C13C2211yescharacteristic
9#11719517—21no—
10#11719275C414yescomplemented
11#11719274—24no—
12Q8×Dic26C414yescomplemented
13#11719205—21no—
14#11719205—22no—
15#11719175C2214yes—
16#11719175C2212yes—
17C13×#391—21no—
18C13×#387—22no—
19C13×#383C2212yes—
20C13×#382C2211yescharacteristic
21C4×C13×C4⋊C4—21no—
22C4×C13×C4⋊C4C2211yescharacteristic
23C13×#376C2212yes—
24#11718685—24no—
25#11718587C2214yes—
26#11718586C2212yes—
27#11718044C2214yes—
28#11718044C2218yes—
29#11718044C2214yes—
30C22×Q8×C13D411yescharacteristic
31C4×Q8×C13C2×C414yes—
32C4×Q8×C13D414yes—
33C2×C13×C4⋊C4—21no—
34C2×C13×C4⋊C4—21no—
35C2×C13×C4⋊C4D411yescharacteristic
36C2×C13×C4⋊C4D412yes—
37C2×C42×C13—21no—
38C2×C42×C13—22no—
39C2×C42×C13C2311yescharacteristic
40C2×C42×C13C2312yes—
41Q8×Dic13—24no—
42Q8×Dic13—24no—
43#88180—24no—
44#88180—22no—
45#88178—24no—
46#88178—22no—
47#88178C2314yes—
48#88144—28no—
49#88144—24no—
50#88142—28no—
51#88142—24no—
52#88124—22no—
53#88124—24no—
54#88123C2×C414yes—
55#88123C2×C412yes—
56#88123C2×C414yes—
57#88123C2×C412yes—
58C4×Dic26—22no—
59C4×Dic26—24no—
60C4×Dic26C2×C418yes—
61C4×Dic26C2×C414yes—
62C13×#95—22no—
63C13×#95—21no—
64C13×#95—22no—
65C13×#95—22no—
66C13×#95C2311yescharacteristic
67C13×#95C2311yescharacteristic
68C13×#95C2312yes—
69C22⋊#1055C2314yes—
70C4⋊#1055—22no—
71C4⋊#1055—24no—
72#88043—24no—
73C4×#1055—22no—
74#17632—131nomaximal, Sylow
75C2×Q8×C13—22no—
76C2×Q8×C13—22no—
77C2×Q8×C13C22⋊C414yes—
78C22×C4×C13—21no—
79C22×C4×C13—22no—
80C22×C4×C13—21no—
81C22×C4×C13C2×D412yes—
82C22×C4×C13C2×D412yes—
83C22×C4×C13C2×D411yescharacteristic
84C22×C4×C13C2×D411yescharacteristic
85C22×C4×C13C2411yescharacteristic
86C2×Dic26—22no—
87C2×Dic26—24no—
88C2×Dic26C22×C414yes—
89C13×C4⋊C4—42no—
90C13×C4⋊C4—42no—
91C13×C4⋊C4C22⋊C418yes—
92C13×C4⋊C4C22⋊C414yes—
93C42×C13—44no—
94C42×C13—42no—
95C42×C13C22×C414yes—
96C42×C13C22×C412yes—
97C42×C13C2×D412yes—
98C42×C13C2×D414yes—
99C22⋊Dic13—24no—
100C4⋊Dic13—22no—
101C4⋊Dic13—24no—
102C4⋊Dic13—24no—
103C4⋊Dic13—22no—
104C4⋊Dic13—24no—
105C4⋊Dic13—22no—
106C4⋊Dic13C22×C414yes—
107#6101—24no—
108#6101—28no—
109#6101—24no—
110C4×Dic13—28no—
111C4×Dic13—24no—
112C2×#1055—24no—
113#3502—131no—
114#2983—131no—
115#2356D1311yescharacteristic
116#1518—132no—
117#1515—134no—
118#1506—134no—
119#1503—132no—
120C23×C13C22×D411yescharacteristic
121Q8×C13—28no—
122C2×C4×C13—22no—
123C2×C4×C13—22no—
124C2×C4×C13—22no—
125C2×C4×C13—42no—
126C2×C4×C13—41no—
127C2×C4×C13—41no—
128C2×C4×C13—41no—
129C2×C4×C13—41no—
130C2×C4×C13—42no—
131C2×C4×C13—41no—
132C2×C4×C13—41no—
133C2×C4×C13—42no—
134C2×C4×C13—41no—
135C2×C4×C13—42no—
136C2×C4×C13C2×C22⋊C414yes—
137C2×C4×C13C2×C22⋊C412yes—
138C2×C4×C13C2×C22⋊C414yes—
139C2×C4×C13C2×C22⋊C412yes—
140C2×C4×C13#13712yes—
141C2×C4×C13#13712yes—
142C2×C4×C13C23×C411yescharacteristic
143C2×C4×C13C22×D411yescharacteristic
144C2×Dic13—24no—
145C2×Dic13—24no—
146Dic26—44no—
147Dic26—84no—
148Dic26—82no—
149#1055—44no—
150Q82—134no—
151#557—134no—
152#557—264no—
153#544—261no—
154#544—262no—
155#533—132no—
156#533—134no—
157#519—261no—
158C2×C4×Q8C2×D1311yescharacteristic
159#446—264no—
160#432—134no—
161#431—132no—
162#391—21no—
163#387—22no—
164#383C2×D1312yes—
165#382C2×D1311yescharacteristic
166C4×C4⋊C4—21no—
167C4×C4⋊C4C2×D1311yescharacteristic
168#376C2×D1312yes—
169#328—134no—
170#328—134no—
171#328—138no—
172C22×C13—21no—
173C22×C13—21no—
174C22×C13C22×C22⋊C411yescharacteristic
175C22×C13C2×#13711yescharacteristic
176C22×C13C2×#13711yescharacteristic
177C4×C13—21no—
178C4×C13—21no—
179C4×C13—24no—
180C4×C13—24no—
181C4×C13—22no—
182C4×C13—42no—
183C4×C13—81no—
184C4×C13—82no—
185C4×C13#41914yes—
186Dic13—44no—
187Dic13—44no—
188C22×Q8C22⋊D1311yescharacteristic
189C4⋊Q8—132no—
190C4⋊Q8—134no—
191C4⋊Q8—132no—
192C4⋊Q8—134no—
193C4⋊Q8—268no—
194C4⋊Q8—264no—
195#125—268no—
196#125—264no—
197#125—262no—
198#125—264no—
199C22⋊Q8—134no—
200C22⋊Q8—264no—
201C22⋊Q8—262no—
202C4×Q8—134no—
203C4×Q8—138no—
204C4×Q8—264no—
205C4×Q8—264no—
206C4×Q8—262no—
207C4×Q8—264no—
208C4×Q8C4×D1314yes—
209C4×Q8D5214yes—
210#117—264no—
211#117—262no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C22⋊D1311yescharacteristic
215C2×C4⋊C4C22⋊D1312yes—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42C22×D1312yes—
219C2×C42C22×D1311yescharacteristic
220C4⋊C8—262no—
221C4⋊C8—264no—
222C22⋊C8—134no—
223#97—264no—
224C4×C8—262no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—22no—
229#95C22×D1311yescharacteristic
230#95C22×D1312yes—
231#95C22×D1311yescharacteristic
232C2×C13—21no—
233C2×C13—21no—
234C2×C13#288411yescharacteristic
235C2×C13C2×#41911yescharacteristic
236C2×C13#290111yescharacteristic
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—134no—
240C2×Q8—262no—
241C2×Q8—264no—
242C2×Q8#610314yes—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4C2×C22⋊D1312yes—
247C22×C4C2×C22⋊D1312yes—
248C22×C4C2×C22⋊D1311yescharacteristic
249C22×C4C2×C22⋊D1311yescharacteristic
250C22×C4C23×D1311yescharacteristic
251C2×C8—264no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—134no—
255C4⋊C4—264no—
256C4⋊C4—262no—
257C4⋊C4—264no—
258C4⋊C4—262no—
259C4⋊C4—264no—
260C4⋊C4—268no—
261C4⋊C4—262no—
262C4⋊C4—264no—
263C4⋊C4—264no—
264C4⋊C4#610318yes—
265C4⋊C4#610314yes—
266C22⋊C4—264no—
267C42—44no—
268C42—42no—
269C42—268no—
270C42—264no—
271C42C2×C4×D1314yes—
272C42C2×C4×D1312yes—
273C42C2×D5212yes—
274C42C2×D5214yes—
275C13#1763211yescharacteristic, complemented, Sylow
276C23C22×C22⋊D1311yescharacteristic
277Q8—28no—
278Q8—524no—
279Q8—1044no—
280Q8—1042no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—264no—
296C2×C4—264no—
297C2×C4C2×#610314yes—
298C2×C4C2×#610312yes—
299C2×C4C2×#610314yes—
300C2×C4C2×#610312yes—
301C2×C4#8820512yes—
302C2×C4#8820512yes—
303C2×C4C22×C4×D1311yescharacteristic
304C2×C4C22×D5211yescharacteristic
305C8—524no—
306C22—21no—
307C22—21no—
308C22C22×#610311yescharacteristic
309C22C2×#8820511yescharacteristic
310C22C2×#8820511yescharacteristic
311C4—24no—
312C4—24no—
313C4—22no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—82no—
319C4—524no—
320C4—524no—
321C4#1171866514yes—
322C2—21no—
323C2—21no—
324C2#1312329911yescharacteristic
325C2C2×#1171866511yescharacteristic
326C2#1312338411yescharacteristic
subgroupG/H
centerC22C22×#6103
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2356C2
socleC22×C13C22×C22⋊C4
prime radicalC23×C13C22×D4
cosocleC22×C4×C13C24
hypercenter#2356D13
hypocenterC13#17632
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2356

Complements

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

NH ≅ G/Ntype
10. #11719275C4semidirect
12. Q8×Dic26C4semidirect
275. C13#17632semidirect

Maximal subgroups

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

subgroupindex
#131286462
#131239212
#131133992
#131133862
#131133472
#131133342
C13×#23562
#1763213