Finite Group Databasebeta

← #27302777

Subgroups — #27302777

Order 13568.

420 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 93 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#20279925C211yesmaximal, characteristic
2#20274709C211yesmaximal, characteristic
3#101×Dic53C211yesmaximal, characteristic
4#20272842C211yesmaximal, characteristic
5#20272841C211yesmaximal, characteristic
6#20272841C211yesmaximal, characteristic
7C53×#1394C211yesmaximal, characteristic
8C2×#13133216C411yescharacteristic
9C2×Q8×Dic53C2211yescharacteristic
10#16810646—21no—
11C2×#13133136C2211yescharacteristic
12C2×#13133136C2211yescharacteristic
13M16×Dic53—21no—
14#16810544—21no—
15#16810544—21no—
16#16810530C2211yescharacteristic
17#16810520—21no—
18#16810520—21no—
19#16810493C411yescharacteristic
20#16810492C411yescharacteristic
21C4×#11719919C411yescharacteristic
22#16810391—21no—
23#16810391—21no—
24C2×#101×C53C2211yescharacteristic
25C2×#101×C53C2211yescharacteristic
26C53×#391C2211yescharacteristic
27C22×Dic106—21no—
28C2×#11719917—21no—
29C2×#11719917—21no—
30#13133216—41no—
31#13133216—41no—
32Q8×Dic53—41no—
33Q8×Dic53—41no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D411yescharacteristic
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48#13133146—41no—
49#13133146—41no—
50#13133136—21no—
51#13133136—21no—
52#13133136—21no—
53#13133136Q812yescomplemented
54#13133127—41no—
55C8×Dic53—41no—
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59C22×Q8×C53C2311yescharacteristic
60C2×M16×C53—21no—
61C2×M16×C53—21no—
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C42×C53C2×C411yescharacteristic
64#101×C53—21no—
65#101×C53—21no—
66#101×C53—21no—
67#101×C53Q812yescomplemented
68#95×C53—21no—
69#95×C53—21no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53C2×Q811yescharacteristic
73C2×Q8×C53C2×Q811yescharacteristic
74C2×Q8×C53#4111yescharacteristic
75C2×Q8×C53#4111yescharacteristic
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53C22⋊C411yescharacteristic
84C22×Dic53C22⋊C411yescharacteristic
85C22×Dic53C22⋊C411yescharacteristic
86C22×Dic53C22⋊C411yescharacteristic
87C2×Dic106—42no—
88C2×Dic106—81no—
89C2×Dic106—81no—
90M16×C53—22no—
91M16×C53—41no—
92M16×C53—41no—
93M16×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C4⋊C4×C53—41no—
97C42×C53—41no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102C4⋊Dic53—81no—
103C4⋊Dic53—81no—
104#11719919—24no—
105#11719919—41no—
106#11719919—41no—
107#11719919—81no—
108#11719919—81no—
109C4×Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53—21no—
114C4×Dic53—21no—
115C4×Dic53—41no—
116#11719917—22no—
117#11719917—41no—
118#11719917—41no—
119#11719917—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C23×C53C2×C22⋊C411yescharacteristic
123Q8×C53—41no—
124Q8×C53—41no—
125Q8×C53—41no—
126Q8×C53—41no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—22no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53C4×Q811yescharacteristic
140C2×C4×C53C4×Q811yescharacteristic
141C2×C4×C53C22⋊Q811yescharacteristic
142C2×C4×C53C22⋊Q811yescharacteristic
143C2×C4×C53#12411yescharacteristic
144C2×C4×C53C4⋊Q811yescharacteristic
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53#9911yescharacteristic
159C2×Dic53#9911yescharacteristic
160Dic106—84no—
161C8×C53—42no—
162C8×C53—81no—
163#88322—42no—
164#88322—81no—
165#13302—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53#39111yescharacteristic
169C22×C53C2×#9911yescharacteristic
170C22×C53#41011yescharacteristic
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—42no—
178C4×C53—81no—
179Dic53—21no—
180Dic53—42no—
181Dic53—41no—
182Dic53—41no—
183Dic53—41no—
184Dic53—81no—
185Dic53#35612yes—
186#2567—531no—
187#1801—531no—
188C4×#101—531no—
189#1394—531no—
190#1394—531no—
191#1394—531no—
192#1394D5311yescharacteristic
193C2×C53—21no—
194C2×C53—21no—
195C2×C53#199611yescharacteristic
196C2×C53C2×#35611yescharacteristic
197C2×C53#213411yescharacteristic
198C2×C4⋊Q8—531no—
199C2×C4×Q8—531no—
200#423—1061no—
201C2×#101—531no—
202C2×#101—531no—
203C2×#101C2×D5311yescharacteristic
204C2×#101C2×D5311yescharacteristic
205C4×M16—1061no—
206#395—1061no—
207#395—1061no—
208#391—531no—
209#391C2×D5311yescharacteristic
210#389—1061no—
211#389—1061no—
212#384—531no—
213#384—531no—
214C4×C4⋊C4—531no—
215#343—1061no—
216#343—1061no—
217C53#1330211yescharacteristic, complemented, Sylow
218C22×Q8—1061no—
219C22×Q8C22×D5311yescharacteristic
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—1061no—
223C2×M16—1061no—
224C4⋊Q8—2121no—
225C4⋊Q8—2121no—
226C4×Q8—2121no—
227C4×Q8—2121no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—531no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4C4×D5311yescharacteristic
239C2×C42—531no—
240C2×C42—531no—
241C2×C42—531no—
242C2×C42—531no—
243C2×C42C4×D5311yescharacteristic
244C4⋊C8—2121no—
245C4⋊C8—2121no—
246#101—21no—
247#101—21no—
248#101—21no—
249#101—532no—
250#101—1061no—
251#101—1061no—
252#101—1061no—
253#101Dic10612yescomplemented
254#97—2121no—
255C4×C8—2121no—
256#95—21no—
257#95—21no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—4241no—
264C2×Q8—4241no—
265C2×Q8—2122no—
266C2×Q8C2×Dic10611yescharacteristic
267C2×Q8#1171994511yescharacteristic
268C2×Q8#1171994711yescharacteristic
269C2×Q8Q8×D5311yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4C2×C4×D5311yescharacteristic
279C22×C4D4×D5311yescharacteristic
280C22×C4D4×D5311yescharacteristic
281M16—22no—
282M16—41no—
283M16—41no—
284M16—41no—
285M16—1062no—
286M16—2121no—
287M16—2121no—
288M16—2121no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—2121no—
292C2×C8—2121no—
293C4⋊C4—41no—
294C4⋊C4—1064no—
295C4⋊C4—4241no—
296C4⋊C4—4241no—
297C4⋊C4—4241no—
298C4⋊C4—4241no—
299C4⋊C4—2121no—
300C4⋊C4—2121no—
301C4⋊C4—2121no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C42—41no—
306C42—1061no—
307C42—1061no—
308C42—1061no—
309C42—1061no—
310C42—1061no—
311C42—1061no—
312C42—2121no—
313C23C22⋊C4×D5311yescharacteristic
314Q8—41no—
315Q8—41no—
316Q8—41no—
317Q8—41no—
318Q8—4244no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—1062no—
334C2×C4—1062no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4#1313318211yescharacteristic
347C2×C4C4×Dic10611yescharacteristic
348C2×C4#1313319411yescharacteristic
349C2×C4#1313319511yescharacteristic
350C2×C4#1313320311yescharacteristic
351C2×C4#1313321611yescharacteristic
352C8—42no—
353C8—81no—
354C8—4241no—
355C8—2122no—
356C22—21no—
357C22—21no—
358C22#1681052711yescharacteristic
359C22#99×D5311yescharacteristic
360C22#1681059511yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—42no—
368C4—81no—
369C4—532no—
370C4—1061no—
371C4—4241no—
372C4—2122no—
373C4—2121no—
374C4—2121no—
375C4—2121no—
376C2—21no—
377C2—21no—
378C2#2027569811yescharacteristic
379C2#356×D5311yescharacteristic
380C2#2027640511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#1394C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#1394D53
hypocenterC53#13302
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
53. #13133136Q8semidirect
67. #101×C53Q8semidirect
217. C53#13302semidirect
253. #101Dic106semidirect

Maximal subgroups

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

subgroupindex
#202799252
#202747092
#101×Dic532
#202728422
#202728412
C53×#13942
#1330253