Finite Group Databasebeta

← #353×Dic53 #27299256

Subgroups — #353×Dic53 #27299256

Order 13568.

720 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 127 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#127×Dic53C211yesmaximal, characteristic
2C2×#16810416C211yesmaximal, characteristic
3#99×Dic53C212yesmaximal, complemented
4#20272839C212yesmaximal, complemented
5C2×C53×#353C211yesmaximal, characteristic
6C2×D4×Dic53—22no—
7C2×D4×Dic53C411yescharacteristic, complemented
8C2×D4×Dic53C2211yescharacteristic
9C2×#13133132C2212yes—
10#16810526—22no—
11#16810526C411yescharacteristic, complemented
12#16810526C2211yescharacteristic
13#16810493C411yescharacteristic, complemented
14C22⋊C4×Dic53—22no—
15C42×Dic53C411yescharacteristic, complemented
16#16810416C414yescomplemented
17#16810389—22no—
18C2×#127×C53C2211yescharacteristic
19C2×#99×C53C2212yes—
20C53×#353C414yescomplemented
21C23×Dic53—21no—
22C23×Dic53D412yes—
23D4×Dic53—41no—
24D4×Dic53—41no—
25D4×Dic53—48no—
26D4×Dic53—41no—
27D4×Dic53—41no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53D412yes—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—42no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53C2×C411yescharacteristic
41#13133132—22no—
42#13133132C2×C414yescomplemented
43#13133124—42no—
44C22×D4×C53—22no—
45C22×D4×C53C2×C411yescharacteristic
46C22×D4×C53C2311yescharacteristic
47#127×C53—22no—
48#127×C53C2×C412yes—
49#127×C53C2×C412yes—
50C2×C22⋊C4×C53—22no—
51C2×C42×C53C2×C411yescharacteristic
52#99×C53—22no—
53#99×C53C2×C414yescomplemented
54C24×C53—21no—
55C24×C53C2×D412yes—
56C2×D4×C53—24no—
57C2×D4×C53—24no—
58C2×D4×C53—21no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53—44no—
62C2×D4×C53—44no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53C4212yescomplemented
66C2×D4×C53C22×C411yescharacteristic
67C2×D4×C53C22×C411yescharacteristic
68C2×D4×C53#4112yes—
69C22×C4×C53—21no—
70C22×C4×C53—42no—
71C22×C4×C53C22×C411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—21no—
76C22×Dic53—44no—
77C22×Dic53—44no—
78C22×Dic53—42no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C22⋊C412yes—
84C22⋊C4×C53—24no—
85C22⋊C4×C53—42no—
86C42×C53—21no—
87C42×C53C4212yescomplemented
88C22⋊Dic53—24no—
89C22⋊Dic53—42no—
90C22⋊Dic53—44no—
91C22⋊Dic53—44no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C4⋊Dic53—41no—
95C4⋊Dic53—44no—
96C4⋊Dic53—41no—
97C4×Dic53—24no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—42no—
101C4×Dic53—41no—
102C4×Dic53—84no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53—42no—
109C23×C53—42no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53C2×C22⋊C411yescharacteristic
113C23×C53C4×D412yes—
114C23×C53C4×D412yes—
115D4×C53—41no—
116D4×C53—41no—
117D4×C53—42no—
118D4×C53—42no—
119D4×C53—48no—
120D4×C53—48no—
121D4×C53—42no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—44no—
128C2×C4×C53—82no—
129C2×C4×C53—82no—
130C2×C4×C53C2×C4211yescharacteristic
131C2×C4×C53#11711yescharacteristic
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—21no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—44no—
139C2×Dic53—42no—
140C2×Dic53—41no—
141C2×Dic53—42no—
142C2×Dic53—44no—
143C2×Dic53—41no—
144C2×Dic53—82no—
145C2×Dic53—82no—
146C2×Dic53—82no—
147C2×Dic53—82no—
148C2×Dic53—82no—
149C2×Dic53#9911yescharacteristic
150C2×Dic53#9911yescharacteristic
151C4×#353—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—42no—
157C22×C53—42no—
158C22×C53—44no—
159C22×C53—42no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53—82no—
163C22×C53—82no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53C4×C22⋊C411yescharacteristic
167C22×C53C2×#9911yescharacteristic
168C22×C53#41011yescharacteristic
169C4×C53—24no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—84no—
173Dic53—21no—
174Dic53—42no—
175Dic53—42no—
176Dic53—41no—
177Dic53—84no—
178Dic53—81no—
179Dic53#35312yesdirect product factor
180C4×#127—531no—
181C2×#353—531no—
182C2×#353D5311yescharacteristic
183C4×#99—532no—
184#1393—532no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—42no—
188C2×C53—42no—
189C2×C53—82no—
190C2×C53C4×#9911yescharacteristic
191C2×C53C2×#35311yescharacteristic
192C2×C53#212611yescharacteristic
193C2×#127C2×D5311yescharacteristic
194C2×C4×D4—531no—
195C2×C4×D4—531no—
196C2×C4×D4—1062no—
197C2×#99—532no—
198C2×#99C2×D5312yes—
199#390—531no—
200#390—531no—
201#390—1062no—
202#384—531no—
203C4×C22⋊C4—1062no—
204C43—531no—
205#353—534no—
206#353Dic5314yesdirect product factor
207#342—1062no—
208C53C4×#35311yescharacteristic, complemented, Sylow
209C22×D4—22no—
210C22×D4C4×D5311yescharacteristic
211C22×D4C22×D5311yescharacteristic
212C23×C4—532no—
213C23×C4—1061no—
214#127—22no—
215#127C4×D5312yes—
216#127C2×Dic5312yes—
217C4×D4—2121no—
218C4×D4—2121no—
219C4×D4—2128no—
220C4×D4—2121no—
221C4×D4—2121no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1062no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—532no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4—1061no—
229C2×C22⋊C4—1062no—
230C2×C22⋊C4—1062no—
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—1061no—
234C2×C42—1061no—
235C2×C42—2122no—
236C2×C42C4×D5311yescharacteristic
237#99—22no—
238#99—534no—
239#99—1062no—
240#99C2×Dic5314yescomplemented
241#95—2122no—
242C24—21no—
243C24D4×D5312yes—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4C4×Dic5312yescomplemented
254C2×D4C2×C4×D5311yescharacteristic
255C2×D4#1171994712yes—
256C2×D4C22×Dic5311yescharacteristic
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—532no—
262C22×C4—1062no—
263C22×C4—1062no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—2121no—
267C22×C4—2121no—
268C22×C4—2124no—
269C22×C4—2124no—
270C22×C4—2122no—
271C22×C4C2×C4×D5311yescharacteristic
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C4⋊C4—2124no—
275C22⋊C4—24no—
276C22⋊C4—42no—
277C22⋊C4—1064no—
278C22⋊C4—2122no—
279C22⋊C4—2124no—
280C22⋊C4—2124no—
281C22⋊C4—2122no—
282C22⋊C4—2122no—
283C42—21no—
284C42—1061no—
285C42—1061no—
286C42—1064no—
287C42—4244no—
288C42—2121no—
289C42—2122no—
290C42C4×Dic5312yescomplemented
291C23—22no—
292C23—22no—
293C23—22no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—41no—
299C23—41no—
300C23C22⋊C4×D5311yescharacteristic
301C23#1313317712yes—
302C23D4×Dic5312yes—
303D4—41no—
304D4—42no—
305D4—41no—
306D4—42no—
307D4—48no—
308D4—48no—
309D4—42no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—44no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—531no—
319C2×C4—531no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1061no—
323C2×C4—1062no—
324C2×C4—4242no—
325C2×C4—4242no—
326C2×C4—4242no—
327C2×C4—4242no—
328C2×C4—4242no—
329C2×C4—2124no—
330C2×C4—2124no—
331C2×C4—2122no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2122no—
337C2×C4C2×C4×Dic5311yescharacteristic
338C2×C4#1313317311yescharacteristic
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—42no—
344C22—42no—
345C22—44no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—82no—
350C22—82no—
351C22—81no—
352C22—81no—
353C22C22⋊C4×Dic5311yescharacteristic
354C22#99×D5311yescharacteristic
355C22#1681059511yescharacteristic
356C4—24no—
357C4—21no—
358C4—21no—
359C4—84no—
360C4—532no—
361C4—1061no—
362C4—4244no—
363C4—4241no—
364C4—2122no—
365C4—2122no—
366C4—2121no—
367C2—21no—
368C2—21no—
369C2—42no—
370C2—42no—
371C2—82no—
372C2#99×Dic5311yescharacteristic
373C2#353×D5311yescharacteristic
374C2#2027637711yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC2×C53×#353C2
socleC22×C53C2×#99
prime radicalC2×#127×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#353D53
hypocenterC53C4×#353
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#353

Complements

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

NH ≅ G/Ntype
3. #99×Dic53C2semidirect
4. #20272839C2semidirect
7. C2×D4×Dic53C4semidirect
11. #16810526C4semidirect
13. #16810493C4semidirect
15. C42×Dic53C4semidirect
16. #16810416C4semidirect
20. C53×#353C4semidirect
42. #13133132C2×C4semidirect
53. #99×C53C2×C4semidirect
65. C2×D4×C53C42semidirect
87. C42×C53C42semidirect
179. Dic53#353direct (N × H)
206. #353Dic53direct (N × H)
208. C53C4×#353semidirect
240. #99C2×Dic53semidirect
253. C2×D4C4×Dic53semidirect
290. C42C4×Dic53semidirect

Maximal subgroups

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

subgroupindex
#127×Dic532
C2×#168104162
#99×Dic532
#202728392
C2×C53×#3532
C4×#35353