Finite Group Databasebeta

← #27282134

Subgroups — #27282134

Order 13568.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#20283763C211yesmaximal, characteristic
2#20274801C211yesmaximal, characteristic, complemented
3#20274795C211yesmaximal, characteristic, complemented
4#20273258C211yesmaximal, characteristic
5#20272622C211yesmaximal, characteristic, complemented
6#20272591C211yesmaximal, characteristic, complemented
7C53×#3092C211yesmaximal, characteristic
8#16810701—21no—
9#16810695—21no—
10#16810695—21no—
11D4×#88322—21no—
12#16810684—21no—
13#16810590—21no—
14#16810587C2211yescharacteristic
15#16810587C2211yescharacteristic
16C2×C22⋊#88322C2211yescharacteristic
17#16810483C2211yescharacteristic
18#16810392—21no—
19#16810392—21no—
20#16810385—21no—
21#16810385—21no—
22#16810360—21no—
23#16810357—21no—
24C53×#525C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#484C412yescomplemented
27C53×#483—21no—
28C53×#481C412yescomplemented
29C53×#480—21no—
30C53×#428—21no—
31C53×#417—21no—
32C53×#407C2211yescharacteristic
33C2×#11719917—21no—
34C2×#11719917—21no—
35C2×#11719917D411yescharacteristic
36C2×#11719917D411yescharacteristic
37C22×#88322—21no—
38C22×#88322D411yescharacteristic
39C22×#88322D411yescharacteristic
40C2×C4⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52#13133128—21no—
53C4×#88322—21no—
54#13133124—21no—
55C2×#41×C53D411yescharacteristic
56C23×C4×C53—21no—
57C23×C4×C53C2311yescharacteristic
58C2×M16×C53—21no—
59#123×C53—21no—
60C22⋊Q8×C53C2×C411yescharacteristic
61C22⋊Q8×C53D411yescharacteristic
62#121×C53C2×C411yescharacteristic
63#121×C53D411yescharacteristic
64#120×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67#117×C53D411yescharacteristic
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53C2×C412yes—
74#107×C53—22no—
75#106×C53—22no—
76#103×C53—22no—
77#102×C53—22no—
78C22⋊C8×C53—21no—
79C22⋊C8×C53C2×C412yes—
80C24×C53#4111yescharacteristic
81#41×C53—41no—
82#41×C53—41no—
83C2×Q8×C53C22⋊C411yescharacteristic
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53C22⋊C411yescharacteristic
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—81no—
95C22×C4×C53—81no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C22×Dic53—41no—
103M16×C53—41no—
104M16×C53—41no—
105C2×C8×C53—22no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53C22⋊C411yescharacteristic
113C4⋊C4×C53C22⋊C411yescharacteristic
114C22⋊C4×C53—21no—
115C22⋊C4×C53—21no—
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C42×C53—21no—
120C22⋊Dic53—81no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123#11719917—41no—
124#11719917—41no—
125#11719917—41no—
126#11719917—41no—
127#11719917—81no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322—21no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—81no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53C4×D411yescharacteristic
142C23×C53#12111yescharacteristic
143C23×C53#12311yescharacteristic
144Q8×C53—41no—
145D4×C53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—81no—
149D4×C53—81no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—82no—
166C2×C4×C53—81no—
167C2×C4×C53—81no—
168C2×C4×C53—82no—
169C2×C4×C53C2×C22⋊C411yescharacteristic
170C2×C4×C53C4×D411yescharacteristic
171C2×C4×C53#12011yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×Dic53—81no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176C8×C53—42no—
177#88322—41no—
178#88322—41no—
179#88322—41no—
180#88322—41no—
181#88322—81no—
182#9522—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—42no—
192C22×C53—42no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53#38611yescharacteristic
197C22×C53#47111yescharacteristic
198C22×C53#47211yescharacteristic
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—41no—
203C4×C53—41no—
204C4×C53—41no—
205C4×C53—82no—
206C4×C53#45811yescharacteristic
207C4×C53#45811yescharacteristic
208Dic53—161no—
209#3092D5311yescharacteristic
210#2976—531no—
211#1823—531no—
212#1821—531no—
213#1513—531no—
214#1305—531no—
215#1293—531no—
216C2×C53—21no—
217C2×C53—21no—
218C2×C53—41no—
219C2×C53—41no—
220C2×C53—81no—
221C2×C53#189011yescharacteristic
222C2×C53#190211yescharacteristic
223C2×C53#190711yescharacteristic
224#525C2×D5311yescharacteristic
225C22×C4⋊C4C2×D5311yescharacteristic
226#484Dic5312yescomplemented
227#483—21no—
228#481Dic5312yescomplemented
229#480—21no—
230#436—1061no—
231#435—1061no—
232#435—1061no—
233C8×D4—1061no—
234#433—1061no—
235#428—21no—
236#417—21no—
237#408—1061no—
238#407—531no—
239#407—531no—
240#407C2×D5311yescharacteristic
241C2×C22⋊C8—531no—
242#380—531no—
243#343—1061no—
244#343—1061no—
245#340—1061no—
246#340—1061no—
247#328—1061no—
248#327—1061no—
249C53#952211yescharacteristic, complemented, Sylow
250C2×#41C22⋊D5311yescharacteristic
251C23×C4—21no—
252C23×C4C22×D5311yescharacteristic
253C2×M16—21no—
254C2×M16—531no—
255C2×M16—531no—
256C2×M16—1061no—
257C2×M16—1061no—
258C22×C8—531no—
259C22×C8—531no—
260C22×C8—1061no—
261#123—21no—
262C22⋊Q8C2×Dic5311yescharacteristic
263C22⋊Q8C22⋊D5311yescharacteristic
264#121C2×Dic5311yescharacteristic
265#121C22⋊D5311yescharacteristic
266#120—21no—
267C4×D4—21no—
268C4×D4—21no—
269#117C22⋊D5311yescharacteristic
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4C2×Dic5312yes—
277C2×C22⋊C4—1061no—
278#107—22no—
279#106—22no—
280C4⋊C8—1061no—
281C4⋊C8—1061no—
282#103—22no—
283#102—22no—
284C22⋊C8—21no—
285C22⋊C8—1061no—
286C22⋊C8—1061no—
287C22⋊C8—1061no—
288C22⋊C8—1061no—
289C22⋊C8—1061no—
290C22⋊C8—1061no—
291C22⋊C8—1061no—
292C22⋊C8—1061no—
293C22⋊C8C2×Dic5312yes—
294#97—1061no—
295C4×C8—1061no—
296#95—1061no—
297C24#1171994711yescharacteristic
298#41—41no—
299#41—41no—
300C2×Q8C22⋊Dic5311yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C22⋊Dic5311yescharacteristic
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—2121no—
314C22×C4D4×D5311yescharacteristic
315C22×C4D4×D5311yescharacteristic
316C22×C4#1171994711yescharacteristic
317C22×C4C22×Dic5311yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319C22×C4C2×C22⋊D5311yescharacteristic
320M16—41no—
321M16—41no—
322M16—4241no—
323M16—2121no—
324M16—2121no—
325M16—2121no—
326M16—2121no—
327C2×C8—22no—
328C2×C8—1061no—
329C2×C8—1061no—
330C2×C8—1061no—
331C2×C8—1061no—
332C2×C8—4241no—
333C2×C8—2121no—
334C2×C8—2121no—
335C2×C8—2121no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—2121no—
343C4⋊C4—2121no—
344C4⋊C4C22⋊Dic5311yescharacteristic
345C4⋊C4C22⋊Dic5311yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—4241no—
352C42—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23D4×Dic5311yescharacteristic
359C23#1313318911yescharacteristic
360C23#1313319811yescharacteristic
361Q8—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365D4—81no—
366D4—81no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—4241no—
387C2×C4—4241no—
388C2×C4—4241no—
389C2×C4C2×C22⋊Dic5311yescharacteristic
390C2×C4D4×Dic5311yescharacteristic
391C2×C4#1313318511yescharacteristic
392C2×C4#1313318911yescharacteristic
393C8—42no—
394C8—4241no—
395C8—2121no—
396C8—2121no—
397C8—2121no—
398C8—2121no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22#1681050311yescharacteristic
413C22#1681088411yescharacteristic
414C22#1681089111yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—82no—
422C4—8481no—
423C4#1681081211yescharacteristic
424C4#1681081211yescharacteristic
425C2—21no—
426C2—21no—
427C2—41no—
428C2—41no—
429C2—81no—
430C2#2027512911yescharacteristic
431C2#2027519011yescharacteristic
432C2#2027522511yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3092C2
socleC22×C53#386
prime radicalC53×#525C22
cosocleC23×C4×C53C23
hypercenter#3092D53
hypocenterC53#9522
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3092

Complements

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

NH ≅ G/Ntype
2. #20274801C2semidirect
3. #20274795C2semidirect
5. #20272622C2semidirect
6. #20272591C2semidirect
26. C53×#484C4semidirect
28. C53×#481C4semidirect
226. #484Dic53semidirect
228. #481Dic53semidirect
249. C53#9522semidirect

Maximal subgroups

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

subgroupindex
#202837632
#202748012
#202747952
#202732582
#202726222
#202725912
C53×#30922
#952253