Finite Group Databasebeta

← #15758972

Subgroups — #15758972

Order 3328.

612 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 115 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#13120641C211yesmaximal, characteristic
2#13113407C212yesmaximal, complemented
3C2×#11718117C211yesmaximal, characteristic
4#13112703C212yesmaximal, complemented
5C13×#2312C211yesmaximal, characteristic
6C22×C13×C22⋊C4C2211yescharacteristic
7C2×#88178C2211yescharacteristic
8C13×#388—24no—
9C13×#380C2212yes—
10C13×#379—21no—
11C4×C13×C22⋊C4—21no—
12#11718908—21no—
13#11718873C2211yescharacteristic
14#11718871—21no—
15C2×C22⋊#1055C2212yes—
16#11718610C412yescomplemented
17#11718609—22no—
18#11718608C412yescomplemented
19#11718117—22no—
20#11718044—24no—
21C25×C13D411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×C4×C13C2311yescharacteristic
24C22×Dic26C2×C411yescharacteristic
25C2×C13×C4⋊C4—22no—
26C2×C13×C22⋊C4—22no—
27C2×C13×C22⋊C4—22no—
28C2×C13×C22⋊C4—22no—
29C2×C13×C22⋊C4—22no—
30C2×C13×C22⋊C4—22no—
31C2×C13×C22⋊C4C2×C412yes—
32C2×C13×C22⋊C4D412yes—
33C2×C42×C13—22no—
34C2×C22⋊Dic13—21no—
35C2×C22⋊Dic13—21no—
36C2×C22⋊Dic13—22no—
37C2×C22⋊Dic13—22no—
38C2×C4⋊Dic13—21no—
39C2×C4⋊Dic13C2×C411yescharacteristic
40#88178—22no—
41#88178—22no—
42C2×#6101—21no—
43C2×#6101—22no—
44C22×#1055—22no—
45C13×#95—22no—
46C13×#95D414yes—
47#88071—22no—
48#88071C2×C412yes—
49C22⋊#1055—22no—
50C22⋊#1055—22no—
51#9201—131nomaximal, Sylow
52C24×C13—21no—
53C24×C13—21no—
54C24×C13—22no—
55C24×C13—22no—
56C24×C13—81no—
57C24×C13—81no—
58C24×C13C22⋊C412yes—
59C24×C13C2×D411yescharacteristic
60C22×C4×C13—21no—
61C22×C4×C13—21no—
62C22×C4×C13—24no—
63C22×C4×C13—81no—
64C22×C4×C13—81no—
65C22×C4×C13C22⋊C412yes—
66C22×C4×C13C22×C411yescharacteristic
67C22×C4×C13C2×D411yescharacteristic
68C22×C4×C13C2×D412yes—
69C22×C4×C13#4112yes—
70C22×Dic13—21no—
71C22×Dic13—22no—
72C22×Dic13—21no—
73C2×Dic26—22no—
74C2×Dic26—81no—
75C13×C4⋊C4—82no—
76C13×C22⋊C4—24no—
77C13×C22⋊C4—24no—
78C13×C22⋊C4—42no—
79C13×C22⋊C4—42no—
80C13×C22⋊C4—82no—
81C13×C22⋊C4—82no—
82C42×C13—82no—
83C22⋊Dic13—41no—
84C22⋊Dic13—41no—
85C22⋊Dic13—42no—
86C22⋊Dic13—82no—
87C22⋊Dic13—82no—
88C4⋊Dic13—22no—
89C4⋊Dic13—81no—
90#6101—42no—
91#6101—82no—
92C2×#1055—42no—
93C2×#1055—82no—
94#2626—131no—
95#2312D1311yescharacteristic
96#1520—132no—
97C2×#323—131no—
98#1289—132no—
99C23×C13—22no—
100C23×C13—22no—
101C23×C13—22no—
102C23×C13—41no—
103C23×C13—41no—
104C23×C13—42no—
105C23×C13—42no—
106C23×C13—42no—
107C23×C13—42no—
108C23×C13—42no—
109C23×C13—42no—
110C23×C13—81no—
111C23×C13—81no—
112C23×C13—84no—
113C23×C13—84no—
114C23×C13C2×C22⋊C411yescharacteristic
115C23×C13#12011yescharacteristic
116C23×C13#12311yescharacteristic
117C23×C13#13512yes—
118C23×C13#13712yes—
119C2×C4×C13—22no—
120C2×C4×C13—42no—
121C2×C4×C13—41no—
122C2×C4×C13—42no—
123C2×C4×C13—41no—
124C2×C4×C13—42no—
125C2×C4×C13—42no—
126C2×C4×C13—44no—
127C2×C4×C13—44no—
128C2×C4×C13—44no—
129C2×C4×C13—81no—
130C2×C4×C13C4×D412yes—
131C2×C4×C13#12112yes—
132C2×Dic13—42no—
133C2×Dic13—42no—
134C2×Dic13—42no—
135C2×Dic13—42no—
136C2×Dic13—41no—
137C2×Dic13—41no—
138C2×Dic13—42no—
139Dic26—82no—
140#1055—82no—
141C2×C22⋊Q8—131no—
142C22×C22⋊C4C2×D1311yescharacteristic
143C2×C22⋊C8—132no—
144#399—132no—
145#396—262no—
146#393—132no—
147#388—24no—
148#385—261no—
149#380—131no—
150#380C2×D1312yes—
151#379—21no—
152#379—261no—
153C4×C22⋊C4—21no—
154#328—264no—
155#323—262no—
156C22×C13—21no—
157C22×C13—21no—
158C22×C13—21no—
159C22×C13—21no—
160C22×C13—21no—
161C22×C13—21no—
162C22×C13—42no—
163C22×C13—42no—
164C22×C13—44no—
165C22×C13—42no—
166C22×C13—42no—
167C22×C13—42no—
168C22×C13—82no—
169C22×C13—84no—
170C22×C13—84no—
171C22×C13#38611yescharacteristic
172C22×C13#41611yescharacteristic
173C22×C13#41911yescharacteristic
174C22×C13#45212yes—
175C22×C13#48512yes—
176C4×C13—22no—
177C4×C13—22no—
178C4×C13—42no—
179C4×C13—84no—
180Dic13—82no—
181Dic13—81no—
182Dic13—81no—
183C25C22⋊D1311yescharacteristic
184C22×Q8—131no—
185C23×C4C22⋊D1311yescharacteristic
186C23×C4C22×D1311yescharacteristic
187C22×C8—262no—
188C22⋊Q8—262no—
189C22⋊Q8—262no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—131no—
192C2×C4⋊C4—261no—
193C2×C4⋊C4—262no—
194C2×C4⋊C4—261no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—261no—
201C2×C22⋊C4—261no—
202C2×C22⋊C4—262no—
203C2×C22⋊C4—262no—
204C2×C22⋊C4C4×D1312yes—
205C2×C22⋊C4D5212yes—
206C2×C42—22no—
207C22⋊C8—262no—
208C22⋊C8—262no—
209#95—22no—
210#95—132no—
211#95—262no—
212#95C22⋊D1314yes—
213C2×C13—21no—
214C2×C13—22no—
215C2×C13—21no—
216C2×C13—22no—
217C2×C13—82no—
218C2×C13#187311yescharacteristic
219C2×C13#191111yescharacteristic
220C2×C13#191311yescharacteristic
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—81no—
226C24—81no—
227C24#610312yes—
228C24C2×C22⋊D1311yescharacteristic
229C2×Q8—262no—
230C2×Q8—1041no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—262no—
237C22×C4—261no—
238C22×C4—261no—
239C22×C4#610312yes—
240C22×C4C2×C4×D1311yescharacteristic
241C22×C4C2×D5211yescharacteristic
242C22×C4#612712yes—
243C22×C4C2×C22⋊D1312yes—
244C2×C8—522no—
245C2×C8—1042no—
246C4⋊C4—82no—
247C4⋊C4—262no—
248C4⋊C4—522no—
249C4⋊C4—1042no—
250C4⋊C4—1041no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C22⋊C4—521no—
258C22⋊C4—522no—
259C22⋊C4—521no—
260C22⋊C4—1042no—
261C22⋊C4—1042no—
262C42—82no—
263C13#920111yescharacteristic, complemented, Sylow
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×#610311yescharacteristic
280C23#8818311yescharacteristic
281C23#8820412yes—
282C23#8820512yes—
283C23#8820711yescharacteristic
284Q8—1042no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—81no—
296C2×C4—522no—
297C2×C4—522no—
298C2×C4—522no—
299C2×C4—522no—
300C2×C4—522no—
301C2×C4—521no—
302C2×C4—521no—
303C2×C4C4×C22⋊D1312yes—
304C2×C4#8818412yes—
305C8—1042no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#1171865712yes—
322C22#1171866311yescharacteristic
323C22#1171866511yescharacteristic
324C22#1171887411yescharacteristic
325C22#1171890012yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—84no—
330C4—1042no—
331C4—1041no—
332C4—1041no—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#1311514811yescharacteristic
339C2#1311536811yescharacteristic
340C2#1311537811yescharacteristic
subgroupG/H
centerC22#11718874
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2312C2
socleC22×C13#386
prime radicalC25×C13D4
cosocleC23×C4×C13C23
hypercenter#2312D13
hypocenterC13#9201
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #13113407C2semidirect
4. #13112703C2semidirect
16. #11718610C4semidirect
18. #11718608C4semidirect
263. C13#9201semidirect

Maximal subgroups

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

subgroupindex
#131206412
#131134072
C2×#117181172
#131127032
C13×#23122
#920113