Finite Group Databasebeta

← #28299504

Subgroups — #28299504

Order 13568.

662 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 122 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#20290572C211yesmaximal, characteristic, complemented
2#20290376C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289531C212yesmaximal, complemented
5C4×#13133188C211yesmaximal, characteristic
6C22×D53⋊C8—21no—
7C2×#13133284—21no—
8C2×#13133274—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133188C411yescharacteristic
13C2×C4×C22⋊D53—21no—
14C2×C42×D53C2211yescharacteristic
15#16810526C2211yescharacteristic
16#16810519C2211yescharacteristic
17#16810511—21no—
18#16810499—21no—
19#16810480C411yescharacteristic
20C4×C4⋊Dic53C411yescharacteristic
21C4×C22⋊Dic53—21no—
22C2×C4×D4×C53C411yescharacteristic
23C2×D53⋊C8—24no—
24C2×D53⋊C8—42no—
25C2×D53⋊C8—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28#13133284—42no—
29#13133274—42no—
30C22×C22⋊D53—21no—
31C22×C4×D53C2×C411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D411yescharacteristic
34#13133188—22no—
35#13133188C814yescomplemented
36C4×C22⋊D53—42no—
37C4×C22⋊D53—42no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44C2×#11719921—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53C2311yescharacteristic
49C42×D53—42no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C22×D4×C53C2×C411yescharacteristic
54C23×C4×C53—21no—
55C4×D4×C53—42no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53—21no—
58C2×C42×C53C2×C411yescharacteristic
59C24×C53—21no—
60C23×D53C22⋊C411yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53C2×C812yescomplemented
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C2×C22⋊D53—24no—
73C2×C22⋊D53—42no—
74C2×C22⋊D53—42no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C2×D411yescharacteristic
79C22×Dic53C2×D411yescharacteristic
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53C2×C812yes—
89C2×C4×D53M1612yes—
90C2×C4×D53D812yes—
91C2×C4×D53SD1612yes—
92C2×#88324—24no—
93C2×#88324—42no—
94C2×#88324—42no—
95C2×#88324—42no—
96C2×#88324—42no—
97C2×#88324—42no—
98D53⋊C8—48no—
99C4⋊C4×C53—41no—
100C22⋊C4×C53—42no—
101C42×C53—22no—
102C22⋊Dic53—24no—
103C22⋊Dic53—41no—
104C22⋊Dic53—41no—
105C22⋊Dic53—41no—
106C22⋊Dic53—41no—
107#11719921—42no—
108C4⋊Dic53—21no—
109C4⋊Dic53—41no—
110C4⋊Dic53C2×C812yescomplemented
111#11719919—41no—
112#11719919—41no—
113C4×Dic53—22no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116C4×Dic53—42no—
117C23×C53—22no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53C2×C22⋊C411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53C22⋊C812yes—
130C22×D53#10212yes—
131C22×D53#10412yes—
132D4×C53—42no—
133D4×C53—42no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—84no—
144C2×C4×C53#9912yes—
145C2×C4×C53#10212yes—
146C2×C4×C53C22×C811yescharacteristic
147C2×C4×C53C2×M1611yescharacteristic
148C22⋊D53—48no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—22no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53C22⋊C812yes—
162C2×Dic53C2×D811yescharacteristic
163C2×Dic53C2×SD1611yescharacteristic
164C2×Dic53AGL(1,ℤ/8)11yescharacteristic
165C2×Dic53#13711yescharacteristic
166C4×D53—24no—
167C4×D53—24no—
168C4×D53—42no—
169C4×D53—42no—
170C4×D53—42no—
171C4×D53—42no—
172#88324—44no—
173#88324—82no—
174#7586—531nomaximal, Sylow
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53C2×C22⋊C811yescharacteristic
183C22×C53C2×#9911yescharacteristic
184C22×C53#41311yescharacteristic
185C22×C53C2×#10211yescharacteristic
186C22×C53#41711yescharacteristic
187C22×C53C2×#10411yescharacteristic
188C22×C53#42111yescharacteristic
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—22no—
192C2×D53—22no—
193C2×D53—22no—
194C2×D53—22no—
195C2×D53D4⋊C814yes—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—22no—
199C4×C53—22no—
200C4×C53—81no—
201Dic53—22no—
202Dic53—21no—
203Dic53—21no—
204Dic53—21no—
205Dic53—21no—
206Dic53—42no—
207Dic53—81no—
208C4×#121—531no—
209#1797—531no—
210C2×#336—531no—
211#1271—532no—
212#1270—532no—
213C2×C53—82no—
214C2×C53#145911yescharacteristic
215C2×C53C2×D4⋊C811yescharacteristic
216C2×C53#147711yescharacteristic
217C2×C53#150411yescharacteristic
218C2×C53#151911yescharacteristic
219C2×C53#153211yescharacteristic
220C2×C53#154711yescharacteristic
221D53—24no—
222C23×C8—1061no—
223C2×#121—531no—
224C2×C4×D4—1061no—
225C2×C4×D4#615711yescharacteristic
226C22×C42—531no—
227C2×C4⋊C8—1061no—
228C2×C22⋊C8—1061no—
229#390—531no—
230#389—531no—
231#388—1061no—
232#386—1061no—
233#380—531no—
234C4×C4⋊C4—531no—
235C4×C22⋊C4—1061no—
236#336—532no—
237#336—532no—
238#336—1061no—
239C53#758611yescharacteristic, complemented, Sylow
240C22×D4—1061no—
241C22×D4C2×#615711yescharacteristic
242C23×C4—21no—
243C23×C4—531no—
244C23×C4—531no—
245C23×C4—531no—
246C22×C8—1062no—
247C22×C8—1064no—
248C22×C8—1062no—
249C22×C8—2122no—
250C22×C8—2122no—
251#121—534no—
252#121—1062no—
253C4×D4—42no—
254C4×D4—2122no—
255C4×D4—2122no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—531no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4C2×#615711yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—1061no—
262C2×C22⋊C4—1061no—
263C2×C22⋊C4—1061no—
264C2×C22⋊C4—1061no—
265C2×C42—531no—
266C2×C42—531no—
267C2×C42—531no—
268C2×C42—1061no—
269C2×C42—2122no—
270C2×C42C2×#615711yescharacteristic
271C4⋊C8—2122no—
272C22⋊C8—2122no—
273#95—1061no—
274#95—1061no—
275#95—1061no—
276C24—21no—
277C24—531no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—1064no—
282C2×D4—2122no—
283C2×D4—2122no—
284C2×D4D53⋊C812yescomplemented
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—532no—
293C22×C4—532no—
294C22×C4—532no—
295C22×C4—532no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4—1062no—
299C22×C4—1062no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4—2121no—
303C22×C4—2121no—
304C22×C4—2121no—
305C22×C4—2121no—
306C22×C4C22⋊#615711yescharacteristic
307C22×C4C22⋊#615711yescharacteristic
308C22×C4C22×#615711yescharacteristic
309C2×C8—1064no—
310C2×C8—2122no—
311C2×C8—2122no—
312C2×C8—2122no—
313C2×C8—2122no—
314C2×C8—2122no—
315C2×C8—2128no—
316C4⋊C4—41no—
317C4⋊C4—532no—
318C4⋊C4—1061no—
319C4⋊C4—2121no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C22⋊C4—42no—
323C22⋊C4—1064no—
324C22⋊C4—2122no—
325C22⋊C4—2121no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C22⋊C4—2121no—
329C42—22no—
330C42—1062no—
331C42—2121no—
332C42—2121no—
333C42—2122no—
334C23—22no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—532no—
342C23—532no—
343C23—532no—
344C23—1061no—
345C23—1061no—
346C23—1061no—
347C23—1061no—
348C23C2×C22⋊#615711yescharacteristic
349D4—42no—
350D4—42no—
351D4—2128no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—84no—
362C2×C4—532no—
363C2×C4—531no—
364C2×C4—531no—
365C2×C4—531no—
366C2×C4—531no—
367C2×C4—1064no—
368C2×C4—1064no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—1061no—
373C2×C4—1061no—
374C2×C4—1061no—
375C2×C4—1062no—
376C2×C4—2122no—
377C2×C4—2122no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4#1313327512yes—
386C2×C4#1313327912yes—
387C2×C4C2×D53⋊C811yescharacteristic
388C2×C4D53⋊M1611yescharacteristic
389C8—4242no—
390C8—2124no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—534no—
399C22—1062no—
400C22—1062no—
401C22—1062no—
402C22—1062no—
403C22—1062no—
404C22—1062no—
405C22D53⋊(C22⋊C8)11yescharacteristic
406C22C2×#1313327511yescharacteristic
407C22#1681160911yescharacteristic
408C22C2×#1313327911yescharacteristic
409C22#1681161411yescharacteristic
410C22D53⋊#10411yescharacteristic
411C22#1681162111yescharacteristic
412C4—22no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—81no—
417C4—1061no—
418C4—1061no—
419C4—1061no—
420C4—1061no—
421C4—1062no—
422C4—4241no—
423C4—2122no—
424C2—82no—
425C2—1064no—
426C2#2028995011yescharacteristic
427C2D53⋊(D4⋊C8)11yescharacteristic
428C2#2029001111yescharacteristic
429C2#2029010111yescharacteristic
430C2#2029015111yescharacteristic
431C2#2029019011yescharacteristic
432C2#2029022611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133188C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×D4#6157
hypocenterC53#7586
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20290572C2semidirect
2. #20290376C2semidirect
3. #20289534C2semidirect
4. #20289531C2semidirect
35. #13133188C8semidirect
64. C2×D4×C53C2×C8semidirect
110. C4⋊Dic53C2×C8semidirect
239. C53#7586semidirect
284. C2×D4D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202903762
#202895342
#202895312
C4×#131331882
#758653