Finite Group Databasebeta

← #15781201

Subgroups — #15781201

Order 3328.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#13119144C211yesmaximal, characteristic
2C2×#11718064C214yesmaximal
3C4×#88073C211yesmaximal, characteristic
4C13×#1799C211yesmaximal, characteristic
5C2×C4×D4×C13C2211yescharacteristic
6C2×#88189C2212yes—
7C22×C4⋊Dic13C411yescharacteristic, complemented
8C2×C13×#99C2212yes—
9C13×#380—21no—
10C2×#88073C2212yes—
11#11718873—21no—
12#11718873C411yescharacteristic, complemented
13#11718644—21no—
14#11718614—21no—
15#11718604C412yescomplemented
16#11718589—21no—
17C4×#6099—21no—
18C13×#342—21no—
19#11718122—21no—
20#11718064—28no—
21C22×D4×C13C2311yescharacteristic
22C23×C4×C13C2×C411yescharacteristic
23C23×C4×C13D411yescharacteristic
24C23×Dic13C2×C412yes—
25C4×D4×C13—41no—
26C4×D4×C13—41no—
27C2×C13×C4⋊C4—41no—
28C2×C13×C4⋊C4D411yescharacteristic
29C2×C13×C22⋊C4—22no—
30C2×C13×C22⋊C4C2×C411yescharacteristic
31C2×C13×C22⋊C4D411yescharacteristic
32C2×C42×C13D411yescharacteristic
33C2×C22⋊Dic13—22no—
34C2×C22⋊Dic13C2×C412yes—
35#88189—24no—
36#88189—24no—
37C2×C4⋊Dic13—21no—
38C2×C4⋊Dic13—21no—
39C2×C4⋊Dic13—21no—
40C2×C4⋊Dic13—41no—
41C2×C4⋊Dic13—41no—
42C2×#6101—22no—
43C2×#6099—22no—
44C13×#99—24no—
45C13×#95—41no—
46#88073—22no—
47#88073D414yes—
48#88071—22no—
49#88071—22no—
50#88071—21no—
51#88043—41no—
52C4×#1055—41no—
53#13217—131nomaximal, Sylow
54C24×C13C22×C411yescharacteristic
55C24×C13C2×D411yescharacteristic
56C2×D4×C13—41no—
57C2×D4×C13—41no—
58C2×D4×C13C2×D412yes—
59C2×D4×C13#4112yes—
60C22×C4×C13—21no—
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13—21no—
64C22×C4×C13—42no—
65C22×C4×C13—42no—
66C22×C4×C13—42no—
67C22×C4×C13C22⋊C411yescharacteristic
68C22×C4×C13C22⋊C411yescharacteristic
69C22×C4×C13C22⋊C411yescharacteristic
70C22×C4×C13C22⋊C411yescharacteristic
71C22×C4×C13C2×D411yescharacteristic
72C22×Dic13—22no—
73C22×Dic13—22no—
74C22×Dic13—24no—
75C22×Dic13—42no—
76C22×Dic13—42no—
77C13×C4⋊C4—41no—
78C13×C4⋊C4—82no—
79C13×C22⋊C4—41no—
80C13×C22⋊C4—41no—
81C13×C22⋊C4—42no—
82C13×C22⋊C4—42no—
83C42×C13—21no—
84C42×C13—21no—
85C22⋊Dic13—24no—
86C22⋊Dic13—42no—
87C22⋊Dic13—42no—
88C4⋊Dic13—42no—
89C4⋊Dic13—42no—
90C4⋊Dic13—41no—
91C4⋊Dic13—41no—
92#6101—44no—
93#6099—24no—
94#6099—42no—
95C2×#1055—42no—
96#2486—131no—
97C2×#352—134no—
98#1799D1311yescharacteristic
99C4×#100—131no—
100C23×C13—21no—
101C23×C13—21no—
102C23×C13—21no—
103C23×C13—21no—
104C23×C13—41no—
105C23×C13—41no—
106C23×C13—41no—
107C23×C13—41no—
108C23×C13C2×C22⋊C411yescharacteristic
109C23×C13C4×D412yes—
110C23×C13#12411yescharacteristic
111C23×C13#12711yescharacteristic
112D4×C13—42no—
113D4×C13—42no—
114C2×C4×C13—21no—
115C2×C4×C13—21no—
116C2×C4×C13—22no—
117C2×C4×C13—21no—
118C2×C4×C13—21no—
119C2×C4×C13—21no—
120C2×C4×C13—42no—
121C2×C4×C13—42no—
122C2×C4×C13—42no—
123C2×C4×C13—41no—
124C2×C4×C13—41no—
125C2×C4×C13—41no—
126C2×C4×C13—81no—
127C2×C4×C13—82no—
128C2×C4×C13—82no—
129C2×C4×C13—82no—
130C2×C4×C13#9911yescharacteristic
131C2×C4×C13#9911yescharacteristic
132C2×C4×C13#12112yes—
133C2×Dic13—42no—
134C2×Dic13—42no—
135C2×Dic13—42no—
136C2×Dic13—42no—
137C2×Dic13—42no—
138C2×Dic13—42no—
139C2×Dic13—84no—
140#1055—44no—
141C2×#123—132no—
142C2×C4×D4C2×D1311yescharacteristic
143C22×C4⋊C4—131no—
144C2×#100—132no—
145C2×#99C2×D1312yes—
146C4×M16—261no—
147#398—261no—
148#397—261no—
149#389—261no—
150#381—132no—
151#380—21no—
152#380—131no—
153#380—261no—
154#352—268no—
155#343—261no—
156#342—21no—
157C22×C13—21no—
158C22×C13—21no—
159C22×C13—22no—
160C22×C13—21no—
161C22×C13—21no—
162C22×C13—42no—
163C22×C13—41no—
164C22×C13—41no—
165C22×C13—42no—
166C22×C13—41no—
167C22×C13—41no—
168C22×C13—41no—
169C22×C13—41no—
170C22×C13—81no—
171C22×C13#39011yescharacteristic
172C22×C13C2×#9911yescharacteristic
173C22×C13#41011yescharacteristic
174C4×C13—21no—
175C4×C13—22no—
176C4×C13—41no—
177C4×C13—41no—
178C4×C13—41no—
179C4×C13—162no—
180C4×C13#35212yescomplemented
181Dic13—82no—
182Dic13—82no—
183C22×D4C22×D1311yescharacteristic
184C23×C4—132no—
185C23×C4C4×D1311yescharacteristic
186C23×C4C22⋊D1311yescharacteristic
187C2×M16—262no—
188#123—264no—
189#123—264no—
190C4×D4—41no—
191C4×D4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—261no—
194C2×C4⋊C4—261no—
195C2×C4⋊C4—261no—
196C2×C4⋊C4—262no—
197C2×C4⋊C4—521no—
198C2×C4⋊C4—521no—
199C2×C4⋊C4D5211yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—132no—
202C2×C22⋊C4—262no—
203C2×C22⋊C4C4×D1311yescharacteristic
204C2×C22⋊C4C22⋊D1311yescharacteristic
205C2×C42D5211yescharacteristic
206#100—134no—
207#100—262no—
208#99—24no—
209#97—521no—
210C4×C8—521no—
211#95—41no—
212#95—261no—
213#95—262no—
214#95—262no—
215C2×C13—21no—
216C2×C13—21no—
217C2×C13—41no—
218C2×C13—42no—
219C2×C13—41no—
220C2×C13#196411yescharacteristic
221C2×C13C2×#35211yescharacteristic
222C2×C13#212311yescharacteristic
223C24C2×C4×D1311yescharacteristic
224C24C2×C22⋊D1311yescharacteristic
225C2×D4—41no—
226C2×D4—41no—
227C2×D4D4×D1312yes—
228C2×D4#613112yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—262no—
237C22×C4—262no—
238C22×C4—264no—
239C22×C4—522no—
240C22×C4—522no—
241C22×C4#610311yescharacteristic
242C22×C4#610311yescharacteristic
243C22×C4#610311yescharacteristic
244C22×C4#610311yescharacteristic
245C22×C4C2×D5211yescharacteristic
246M16—264no—
247M16—522no—
248C2×C8—522no—
249C4⋊C4—41no—
250C4⋊C4—82no—
251C4⋊C4—522no—
252C4⋊C4—522no—
253C4⋊C4—524no—
254C4⋊C4—521no—
255C4⋊C4—521no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—264no—
261C22⋊C4—522no—
262C22⋊C4—522no—
263C42—21no—
264C42—21no—
265C13#1321711yescharacteristic, complemented, Sylow
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23#8814712yes—
275C23C2×#610311yescharacteristic
276C23#8819411yescharacteristic
277C23#8820111yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—522no—
297C2×C4—522no—
298C2×C4—522no—
299C2×C4—522no—
300C2×C4—522no—
301C2×C4—522no—
302C2×C4—1044no—
303C2×C4#8804611yescharacteristic
304C2×C4#8804611yescharacteristic
305C2×C4#8814912yes—
306C8—524no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22C2×#8804611yescharacteristic
322C22#1171864911yescharacteristic
323C22#1171866711yescharacteristic
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—162no—
330C4—1042no—
331C4—1042no—
332C4#1171806412yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#1311567111yescharacteristic
339C2C2×#1171806411yescharacteristic
340C2#1311648911yescharacteristic
subgroupG/H
centerC22C2×#88046
commutatorC24×C13C22×C4
FrattiniC22×D4C22×D13
FittingC13×#1799C2
socleC22×C13C2×#99
prime radical, cosocleC22×D4×C13C23
hypercenter#1799D13
hypocenterC13#13217
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
7. C22×C4⋊Dic13C4semidirect
12. #11718873C4semidirect
15. #11718604C4semidirect
180. C4×C13#352semidirect
265. C13#13217semidirect
332. C4#11718064semidirect

Maximal subgroups

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

subgroupindex
#131191442
C2×#117180642
C4×#880732
C13×#17992
#1321713