Finite Group Databasebeta

← #15767338

Subgroups — #15767338

Order 3328.

604 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 121 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#13120950C211yesmaximal, characteristic
2#13115031C212yesmaximal
3#13113100C211yesmaximal, characteristic
4#13112716C212yesmaximal
5C13×#1280C211yesmaximal, characteristic
6C22×C4⋊Dic13C2211yescharacteristic
7C2×#88178—21no—
8C22×#6101—21no—
9C2×C13×C22⋊C8C2212yes—
10C2×C13×#95C2211yescharacteristic
11#11718873—21no—
12#11718873C2212yes—
13C2×#88071—21no—
14C2×#88071C2211yescharacteristic
15C2×#88057—22no—
16#11718322—22no—
17#11718317—22no—
18#11718143—22no—
19C23×C4×C13D411yescharacteristic
20C23×C4×C13D411yescharacteristic
21C23×C4×C13C2311yescharacteristic
22C23×Dic13—21no—
23C23×Dic13—21no—
24C22×Dic26—21no—
25C22×C8×C13—22no—
26C2×C22⋊Dic13—22no—
27C2×C4⋊Dic13—22no—
28C2×C4⋊Dic13—21no—
29C2×C4⋊Dic13—21no—
30C2×C4⋊Dic13—21no—
31C2×C4⋊Dic13—42no—
32C2×C4⋊Dic13D412yes—
33C2×C4⋊Dic13D412yes—
34#88178—42no—
35#88178—42no—
36C2×#6101—22no—
37C2×#6101—42no—
38C2×#6101—42no—
39C2×#6101—42no—
40C13×C22⋊C8—22no—
41C13×C22⋊C8Q814yescomplemented
42C13×#95—22no—
43#88071—22no—
44#88071—22no—
45#88071—21no—
46#88071—42no—
47#88071Q812yescomplemented
48#88057—44no—
49#10570—131nomaximal, Sylow
50C24×C13C2×D411yescharacteristic
51C22×C4×C13—21no—
52C22×C4×C13—21no—
53C22×C4×C13—21no—
54C22×C4×C13—21no—
55C22×C4×C13—41no—
56C22×C4×C13—41no—
57C22×C4×C13—41no—
58C22×C4×C13—41no—
59C22×C4×C13—41no—
60C22×C4×C13—41no—
61C22×C4×C13SD1612yes—
62C22×C4×C13SD1612yes—
63C22×C4×C13C2×D412yes—
64C22×C4×C13C2×Q812yes—
65C22×C4×C13C2×Q811yescharacteristic
66C22×C4×C13#4111yescharacteristic
67C22×Dic13—22no—
68C22×Dic13—22no—
69C22×Dic13—22no—
70C22×Dic13—22no—
71C22×Dic13—22no—
72C22×Dic13—22no—
73C22×Dic13—41no—
74C22×Dic13—41no—
75C22×Dic13—41no—
76C22×Dic13—41no—
77C22×Dic13—41no—
78C22×Dic13—41no—
79C22×Dic13—41no—
80C22×Dic13—41no—
81C22×Dic13—41no—
82C22×Dic13—41no—
83C2×Dic26—42no—
84C2×Dic26—81no—
85C2×Dic26—81no—
86C2×C8×C13—24no—
87C2×C8×C13—42no—
88C2×C8×C13—42no—
89C22⋊Dic13—42no—
90C22⋊Dic13—42no—
91C4⋊Dic13—24no—
92C4⋊Dic13—42no—
93C4⋊Dic13—42no—
94C4⋊Dic13—42no—
95C4⋊Dic13—42no—
96C4⋊Dic13—81no—
97C4⋊Dic13—81no—
98#6101—44no—
99#6101—48no—
100#2659—131no—
101#1853—132no—
102#1444—131no—
103#1294—132no—
104#1280D1311yescharacteristic
105C23×C13—21no—
106C23×C13—21no—
107C23×C13—21no—
108C23×C13—21no—
109C23×C13#12011yescharacteristic
110C23×C13C22⋊Q811yescharacteristic
111C23×C13C4⋊Q811yescharacteristic
112C23×C13C2×SD1611yescharacteristic
113C23×C13C2×SD1611yescharacteristic
114C23×C13AGL(1,ℤ/8)11yescharacteristic
115C23×C13#13711yescharacteristic
116C2×C4×C13—22no—
117C2×C4×C13—22no—
118C2×C4×C13—22no—
119C2×C4×C13—22no—
120C2×C4×C13—42no—
121C2×C4×C13—41no—
122C2×C4×C13—41no—
123C2×C4×C13—41no—
124C2×C4×C13—41no—
125C2×C4×C13—41no—
126C2×C4×C13—41no—
127C2×C4×C13—42no—
128C2×C4×C13—42no—
129C2×C4×C13—81no—
130C2×C4×C13—81no—
131C2×C4×C13C22⋊Q812yes—
132C2×C4×C13C4⋊Q812yes—
133C2×Dic13—24no—
134C2×Dic13—42no—
135C2×Dic13—42no—
136C2×Dic13—42no—
137C2×Dic13—42no—
138C2×Dic13—42no—
139C2×Dic13—42no—
140C2×Dic13—42no—
141C2×Dic13—42no—
142C2×Dic13—42no—
143C2×Dic13—42no—
144C2×Dic13—42no—
145C2×Dic13—42no—
146C2×Dic13—42no—
147C2×Dic13—42no—
148C2×Dic13—42no—
149C2×Dic13—81no—
150C2×Dic13—81no—
151C2×Dic13—81no—
152C2×Dic13—81no—
153Dic26—84no—
154C8×C13—44no—
155C2×C22⋊Q8—261no—
156C22×C4⋊C4—131no—
157C22×C4⋊C4—261no—
158C2×#106—262no—
159C2×C22⋊C8C2×D1312yes—
160#398—262no—
161#395—262no—
162#380—132no—
163#380—261no—
164C2×#95—131no—
165C2×#95—261no—
166C2×#95C2×D1311yescharacteristic
167#340—262no—
168C22×C13—21no—
169C22×C13—21no—
170C22×C13—21no—
171C22×C13—21no—
172C22×C13—21no—
173C22×C13—21no—
174C22×C13—21no—
175C22×C13—21no—
176C22×C13—21no—
177C22×C13—21no—
178C22×C13—21no—
179C22×C13—21no—
180C22×C13#39511yescharacteristic
181C22×C13#44811yescharacteristic
182C22×C13#45011yescharacteristic
183C22×C13#45311yescharacteristic
184C22×C13#45611yescharacteristic
185C22×C13B(4,𝔽2)11yescharacteristic
186C22×C13#45811yescharacteristic
187C22×C13#47511yescharacteristic
188C22×C13#47611yescharacteristic
189C22×C13#49811yescharacteristic
190C22×C13#50111yescharacteristic
191C4×C13—22no—
192C4×C13—22no—
193C4×C13—42no—
194C4×C13—42no—
195Dic13—44no—
196Dic13—82no—
197Dic13—82no—
198Dic13—82no—
199C22×Q8—261no—
200C23×C4—261no—
201C23×C4—261no—
202C23×C4D5211yescharacteristic
203C23×C4D5211yescharacteristic
204C23×C4C22×D1311yescharacteristic
205C22×C8—22no—
206C22⋊Q8—522no—
207C22⋊Q8—522no—
208C2×C4⋊C4—132no—
209C2×C4⋊C4—132no—
210C2×C4⋊C4—262no—
211C2×C4⋊C4—262no—
212C2×C4⋊C4—261no—
213C2×C4⋊C4—261no—
214C2×C4⋊C4—261no—
215C2×C4⋊C4—522no—
216C2×C4⋊C4—522no—
217C2×C4⋊C4—522no—
218C2×C4⋊C4—522no—
219C2×C22⋊C4—262no—
220#106—524no—
221C22⋊C8—22no—
222C22⋊C8Dic2614yescomplemented
223#95—22no—
224#95—132no—
225#95—262no—
226#95—262no—
227#95—261no—
228#95—522no—
229C2×C13—21no—
230C2×C13—21no—
231C2×C13—21no—
232C2×C13—21no—
233C2×C13#159611yescharacteristic
234C2×C13#160411yescharacteristic
235C2×C13#161111yescharacteristic
236C2×C13#161311yescharacteristic
237C2×C13#205811yescharacteristic
238C2×C13#206011yescharacteristic
239C2×C13#206211yescharacteristic
240C24C2×D5211yescharacteristic
241C2×Q8—522no—
242C2×Q8—1041no—
243C2×Q8—1041no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—262no—
255C22×C4—262no—
256C22×C4—262no—
257C22×C4—262no—
258C22×C4—262no—
259C22×C4—262no—
260C22×C4—521no—
261C22×C4—521no—
262C22×C4—521no—
263C22×C4—521no—
264C22×C4—521no—
265C22×C4—521no—
266C22×C4—521no—
267C22×C4—521no—
268C22×C4—521no—
269C22×C4—521no—
270C22×C4#609512yes—
271C22×C4#609512yes—
272C22×C4C2×Dic2612yes—
273C22×C4D4×D1312yes—
274C22×C4#612911yescharacteristic
275C22×C4Q8×D1311yescharacteristic
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C4⋊C4—264no—
280C4⋊C4—522no—
281C4⋊C4—522no—
282C4⋊C4—522no—
283C4⋊C4—524no—
284C4⋊C4—522no—
285C4⋊C4—528no—
286C4⋊C4—1041no—
287C4⋊C4—1041no—
288C22⋊C4—522no—
289C22⋊C4—522no—
290C13#1057011yescharacteristic, complemented, Sylow
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#8812311yescharacteristic
296C23#8813611yescharacteristic
297C23#8815111yescharacteristic
298C23C2×#609511yescharacteristic
299C23C2×#609511yescharacteristic
300C23#8816211yescharacteristic
301C23#8816311yescharacteristic
302Q8—1044no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—264no—
319C2×C4—522no—
320C2×C4—522no—
321C2×C4—522no—
322C2×C4—522no—
323C2×C4—522no—
324C2×C4—522no—
325C2×C4—522no—
326C2×C4—522no—
327C2×C4—522no—
328C2×C4—522no—
329C2×C4—522no—
330C2×C4—522no—
331C2×C4—522no—
332C2×C4—522no—
333C2×C4—522no—
334C2×C4—1041no—
335C2×C4—1041no—
336C2×C4—1041no—
337C2×C4—1041no—
338C2×C4#8813212yes—
339C2×C4#8814212yes—
340C8—44no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1171831711yescharacteristic
354C22#1171834111yescharacteristic
355C22#1171836311yescharacteristic
356C22#1171839211yescharacteristic
357C22#1171839911yescharacteristic
358C22#1171840211yescharacteristic
359C22#1171840311yescharacteristic
360C22#1171840811yescharacteristic
361C22#1171841311yescharacteristic
362C22#1171850311yescharacteristic
363C22#1171851211yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—524no—
369C4—1042no—
370C4—1042no—
371C4—1042no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#1311376111yescharacteristic
377C2#1311380711yescharacteristic
378C2#1311384611yescharacteristic
379C2#1311385811yescharacteristic
380C2#1311617011yescharacteristic
381C2#1311618211yescharacteristic
382C2#1311619011yescharacteristic
subgroupG/H
centerC23#88136
commutator, cosocleC23×C4×C13C23
FrattiniC23×C4C22×D13
FittingC13×#1280C2
socleC23×C13#120
prime radicalC24×C13C2×D4
hypercenter#1280D13
hypocenterC13#10570
LCSG ▷ C23×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
41. C13×C22⋊C8Q8semidirect
47. #88071Q8semidirect
222. C22⋊C8Dic26semidirect
290. C13#10570semidirect

Maximal subgroups

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

subgroupindex
#131209502
#131150312
#131131002
#131127162
C13×#12802
#1057013