Finite Group Databasebeta

← #15773527

Subgroups — #15773527

Order 3328.

590 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 87 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#13118201C211yesmaximal, characteristic
2#13113177C211yesmaximal, characteristic
3#13112856C214yesmaximal, complemented
4C4×C13×#99C211yesmaximal, characteristic
5C23×Dic26C411yescharacteristic, complemented
6C2×C4×D4×C13C2211yescharacteristic
7C2×#88178—21no—
8C2×C13×#99—21no—
9C4×C13×C22⋊C4—21no—
10#11718881—21no—
11#11718873C411yescharacteristic, complemented
12#11718640C412yescomplemented
13#11718608C2212yes—
14#11718607—21no—
15#11718590—21no—
16#11718587—21no—
17C13×#342C2212yes—
18#11718117—21no—
19#11718117C2212yes—
20C22×D4×C13D411yescharacteristic
21C23×C4×C13C2×C411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×Dic13—21no—
24C22×Dic26—21no—
25C22×Dic26—44no—
26C22×Dic26—41no—
27C22×Dic26—41no—
28C22×Dic26C2×C412yes—
29C4×D4×C13—41no—
30C4×D4×C13—41no—
31C2×C13×C4⋊C4C2×C411yescharacteristic
32C2×C13×C22⋊C4—22no—
33C2×C13×C22⋊C4D411yescharacteristic
34C2×C13×C22⋊C4C2311yescharacteristic
35C2×C42×C13—41no—
36C2×C42×C13D411yescharacteristic
37C2×C22⋊Dic13—21no—
38C2×C22⋊Dic13—22no—
39C2×C4⋊Dic13—21no—
40#88178—44no—
41C2×#6101—22no—
42C13×#99—24no—
43C13×#95—41no—
44#88071—22no—
45#88071—22no—
46#88071C2×C412yes—
47C22⋊#1055—22no—
48C4⋊#1055—41no—
49#88043—41no—
50#11785—131nomaximal, Sylow
51C24×C13C22⋊C411yescharacteristic
52C24×C13C2×D411yescharacteristic
53C2×D4×C13—22no—
54C2×D4×C13—22no—
55C2×D4×C13—41no—
56C22×C4×C13—21no—
57C22×C4×C13—21no—
58C22×C4×C13—21no—
59C22×C4×C13—21no—
60C22×C4×C13—42no—
61C22×C4×C13—42no—
62C22×C4×C13—42no—
63C22×C4×C13C22⋊C411yescharacteristic
64C22×C4×C13C22⋊C411yescharacteristic
65C22×C4×C13C22⋊C411yescharacteristic
66C22×C4×C13C22×C411yescharacteristic
67C22×C4×C13C2×D411yescharacteristic
68C22×Dic13—22no—
69C22×Dic13—22no—
70C22×Dic13—44no—
71C22×Dic13—42no—
72C2×Dic26—24no—
73C2×Dic26—42no—
74C2×Dic26—48no—
75C2×Dic26—44no—
76C2×Dic26—41no—
77C2×Dic26—41no—
78C2×Dic26—42no—
79C2×Dic26—44no—
80C2×Dic26—44no—
81C13×C4⋊C4—41no—
82C13×C22⋊C4—24no—
83C13×C22⋊C4—41no—
84C13×C22⋊C4—41no—
85C13×C22⋊C4—42no—
86C42×C13—21no—
87C42×C13—21no—
88C42×C13—82no—
89C22⋊Dic13—44no—
90C22⋊Dic13—42no—
91C4⋊Dic13—42no—
92#6101—44no—
93C2×#1055—42no—
94#2409—131no—
95C4×#99D1311yescharacteristic
96#1466—131no—
97#1351—134no—
98C23×C13—21no—
99C23×C13—21no—
100C23×C13—21no—
101C23×C13—21no—
102C23×C13—41no—
103C23×C13—41no—
104C23×C13—41no—
105C23×C13—41no—
106C23×C13#10412yes—
107C23×C13C2×C22⋊C411yescharacteristic
108C23×C13#13712yes—
109D4×C13—44no—
110C2×C4×C13—22no—
111C2×C4×C13—21no—
112C2×C4×C13—21no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—21no—
116C2×C4×C13—21no—
117C2×C4×C13—42no—
118C2×C4×C13—42no—
119C2×C4×C13—42no—
120C2×C4×C13—41no—
121C2×C4×C13—41no—
122C2×C4×C13—41no—
123C2×C4×C13—44no—
124C2×C4×C13—81no—
125C2×C4×C13—82no—
126C2×C4×C13—82no—
127C2×C4×C13#9911yescharacteristic
128C2×C4×C13#9911yescharacteristic
129C2×Dic13—24no—
130C2×Dic13—42no—
131C2×Dic13—48no—
132C2×Dic13—42no—
133C2×Dic13—42no—
134C2×Dic13—42no—
135C2×Dic13—42no—
136Dic26—44no—
137Dic26—44no—
138Dic26—44no—
139Dic26—44no—
140#1055—82no—
141C23×Q8—131no—
142C2×C22⋊Q8—261no—
143C2×C4×D4C2×D1311yescharacteristic
144#432—261no—
145C2×#99—21no—
146#395—132no—
147#393—132no—
148#391—261no—
149#386—261no—
150#385—261no—
151#380—131no—
152C4×C22⋊C4—21no—
153#342C2×D1312yes—
154#323—132no—
155#323—261no—
156C22×C13—21no—
157C22×C13—21no—
158C22×C13—22no—
159C22×C13—22no—
160C22×C13—42no—
161C22×C13—42no—
162C22×C13—42no—
163C22×C13—41no—
164C22×C13—41no—
165C22×C13—41no—
166C22×C13—41no—
167C22×C13—81no—
168C22×C13C2×#9911yescharacteristic
169C22×C13#41911yescharacteristic
170C22×C13C2×#10411yescharacteristic
171C4×C13—21no—
172C4×C13—41no—
173C4×C13—41no—
174C4×C13—41no—
175C4×C13—41no—
176C4×C13—84no—
177C4×C13AΣL(1,𝔽16)12yes—
178Dic13—44no—
179Dic13—82no—
180C22×Q8—132no—
181C22×Q8—261no—
182C22×Q8—521no—
183C22×Q8—521no—
184C22×Q8—524no—
185C22×D4C22⋊D1311yescharacteristic
186C23×C4—261no—
187C23×C4C4×D1311yescharacteristic
188C23×C4C22⋊D1311yescharacteristic
189C22⋊Q8—524no—
190C4×D4—41no—
191C4×D4—41no—
192C2×C4⋊C4—262no—
193C2×C4⋊C4—261no—
194C2×C4⋊C4C4×D1311yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—261no—
197C2×C22⋊C4—262no—
198C2×C22⋊C4D5211yescharacteristic
199C2×C22⋊C4C22×D1311yescharacteristic
200C2×C42—41no—
201C2×C42D5211yescharacteristic
202C4⋊C8—521no—
203#99—24no—
204C22⋊C8—262no—
205#97—521no—
206#95—41no—
207#95—132no—
208#95—262no—
209#95—262no—
210C2×C13—21no—
211C2×C13—21no—
212C2×C13—42no—
213C2×C13—42no—
214C2×C13#150811yescharacteristic
215C2×C13C2×AΣL(1,𝔽16)11yescharacteristic
216C2×C13#212911yescharacteristic
217C24#610311yescharacteristic
218C24C2×C22⋊D1311yescharacteristic
219C2×Q8—264no—
220C2×Q8—522no—
221C2×Q8—522no—
222C2×Q8—524no—
223C2×Q8—524no—
224C2×Q8—528no—
225C2×Q8—524no—
226C2×Q8—521no—
227C2×Q8—521no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—262no—
239C22×C4—262no—
240C22×C4—524no—
241C22×C4—522no—
242C22×C4#610311yescharacteristic
243C22×C4#610311yescharacteristic
244C22×C4#610311yescharacteristic
245C22×C4C2×C4×D1311yescharacteristic
246C22×C4C2×D5211yescharacteristic
247C2×C8—522no—
248C4⋊C4—41no—
249C4⋊C4—522no—
250C4⋊C4—524no—
251C22⋊C4—24no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—524no—
256C22⋊C4—522no—
257C42—21no—
258C42—21no—
259C42—82no—
260C13#1178511yescharacteristic, complemented, Sylow
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#8804512yes—
270C23C2×#610311yescharacteristic
271C23#8820512yes—
272Q8—524no—
273Q8—524no—
274Q8—524no—
275Q8—524no—
276D4—44no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—264no—
295C2×C4—522no—
296C2×C4—528no—
297C2×C4—522no—
298C2×C4—522no—
299C2×C4—522no—
300C2×C4—522no—
301C2×C4#8804611yescharacteristic
302C2×C4#8804611yescharacteristic
303C8—1042no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×#8804511yescharacteristic
317C22C2×#8804611yescharacteristic
318C22#1171866511yescharacteristic
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—84no—
325C4—524no—
326C4—1042no—
327C4#1171806312yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2#1311335511yescharacteristic
333C2C2×#1171806311yescharacteristic
334C2#1311651011yescharacteristic
subgroupG/H
centerC22C2×#88046
commutatorC22×C4×C13C22×C4
FrattiniC2×C22⋊C4C22×D13
FittingC4×C13×#99C2
socleC22×C13C2×#99
prime radicalC22×D4×C13D4
cosocleC2×C13×C22⋊C4C23
hypercenterC4×#99D13
hypocenterC13#11785
LCSG ▷ C22×C4×C13 ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #13112856C2semidirect
5. C23×Dic26C4semidirect
11. #11718873C4semidirect
12. #11718640C4semidirect
260. C13#11785semidirect

Maximal subgroups

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

subgroupindex
#131182012
#131131772
#131128562
C4×C13×#992
#1178513