Finite Group Databasebeta

← #27272316

Subgroups — #27272316

Order 13568.

594 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 109 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#389×D53C211yesmaximal, characteristic
2#20274476C211yesmaximal, characteristic
3#20274476C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic, complemented
5#20272538C211yesmaximal, characteristic, complemented
6#20272537C211yesmaximal, characteristic, complemented
7C53×#1271C211yesmaximal, characteristic, complemented
8C2×C4⋊C4×D53—21no—
9C2×C4⋊C4×D53C411yescharacteristic
10C2×C4⋊C4×D53C411yescharacteristic
11C2×C42×D53C2211yescharacteristic
12C2×#13133129—21no—
13C2×#13133129—21no—
14C2×#13133127—21no—
15C2×#13133127—21no—
16#16810522C411yescharacteristic
17#16810521C2211yescharacteristic
18#16810520—21no—
19#16810519C411yescharacteristic
20#16810518—21no—
21#95×D53—21no—
22#16810378C2211yescharacteristic
23#16810378C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×C4×D53—21no—
28C22×C4×D53—21no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32C22×#88322—21no—
33C22×#88322—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C4⋊C4×D53—22no—
42C4⋊C4×D53—22no—
43C4⋊C4×D53—42no—
44C4⋊C4×D53—42no—
45C4⋊C4×D53—42no—
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C42×D53—42no—
50#13133129—24no—
51#13133129—24no—
52#13133127—42no—
53#13133127—42no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C22×C8×C53—21no—
58C22×C8×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53C2×C411yescharacteristic
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C42×C53C2311yescharacteristic
63#95×C53—21no—
64C23×D53C22⋊C411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C2×C4×D53—22no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—22no—
79C2×C4×D53—24no—
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—42no—
89C2×C4×D53—42no—
90C2×C4×D53—42no—
91C2×C4×D53—42no—
92C2×C4×D53—42no—
93C2×C8×C53—22no—
94C2×C8×C53—22no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53M1612yes—
104C42×C53—41no—
105C4⋊Dic53—21no—
106C4⋊Dic53—42no—
107C4⋊Dic53—41no—
108C4⋊Dic53M1612yes—
109#11719919—24no—
110#11719919—42no—
111#11719919—41no—
112#11719919—41no—
113#11719919—42no—
114C4×Dic53—22no—
115C4×Dic53—22no—
116C4×Dic53—41no—
117C2×#88322—22no—
118C2×#88322—22no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—41no—
123C23×C53C2×C22⋊C411yescharacteristic
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53#10112yes—
129C22×D53#10412yes—
130C22×D53#10412yes—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53C2×M1611yescharacteristic
144C2×C4×C53C2×M1611yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53#13711yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—22no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53#9912yes—
164C4×D53—24no—
165C4×D53—24no—
166C4×D53—42no—
167C4×D53—42no—
168C4×D53—42no—
169C4×D53—42no—
170C4×D53—48no—
171C4×D53—84no—
172C8×C53—42no—
173C8×C53—42no—
174#88322—42no—
175#88322—42no—
176#7627—531nomaximal, Sylow
177C22×C53#40711yescharacteristic
178C22×C53C2×#9911yescharacteristic
179C22×C53C2×#10111yescharacteristic
180C22×C53#41811yescharacteristic
181C22×C53#41911yescharacteristic
182C22×C53C2×#10411yescharacteristic
183C22×C53C2×#10411yescharacteristic
184C2×D53—22no—
185C2×D53—22no—
186C2×D53—22no—
187C2×D53—22no—
188C2×D53—22no—
189C2×D53—22no—
190C2×D53#33014yes—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—42no—
198C4×C53—81no—
199Dic53—22no—
200Dic53—42no—
201Dic53—41no—
202Dic53—41no—
203Dic53—81no—
204C2×#389—531no—
205#1749—531no—
206#1749—531no—
207#1271—531no—
208#1271—531no—
209#1271—531no—
210#1271D5311yescharacteristic, complemented
211C2×C53#146311yescharacteristic
212C2×C53#149011yescharacteristic
213C2×C53#149111yescharacteristic
214C2×C53#150711yescharacteristic
215C2×C53#150811yescharacteristic
216C2×C53C2×#33011yescharacteristic
217C2×C53#155611yescharacteristic
218D53—24no—
219C22×C4⋊C4—531no—
220C22×C4⋊C4—531no—
221C22×C4⋊C4—1061no—
222C22×C42—531no—
223C2×C22⋊C8—1061no—
224C2×C22⋊C8—1061no—
225C2×#97—1061no—
226C2×#97—1061no—
227#389—531no—
228#389—531no—
229#389—531no—
230#389—1061no—
231#389—1061no—
232#389C2×D5311yescharacteristic
233C2×#95—1061no—
234#336—531no—
235#336—531no—
236#336C2×D5311yescharacteristic
237#336C2×D5311yescharacteristic
238C53#762711yescharacteristic, complemented, Sylow
239C23×C4—531no—
240C23×C4—531no—
241C23×C4—531no—
242C23×C4—1061no—
243C23×C4—1061no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—1061no—
247C22×C8—1061no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—1062no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4—1062no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—2122no—
259C2×C4⋊C4—2122no—
260C2×C4⋊C4—2122no—
261C2×C4⋊C4C4×D5311yescharacteristic
262C2×C4⋊C4C4×D5311yescharacteristic
263C2×C42—531no—
264C2×C42—531no—
265C2×C42—531no—
266C2×C42—2122no—
267C2×C42C22×D5311yescharacteristic
268C22⋊C8—1064no—
269C22⋊C8—1064no—
270#97—2122no—
271#97—2122no—
272#95—21no—
273#95—1061no—
274#95—1061no—
275#95—1061no—
276C24—531no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—1062no—
286C22×C4—1064no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4—2122no—
298C22×C4—2122no—
299C22×C4—2122no—
300C22×C4—2122no—
301C22×C4—2122no—
302C22×C4C2×C4×D5311yescharacteristic
303C22×C4D4×D5311yescharacteristic
304C22×C4D4×D5311yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—1062no—
312C2×C8—1062no—
313C2×C8—2121no—
314C2×C8—2121no—
315C2×C8—2121no—
316C2×C8—2121no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—532no—
322C4⋊C4—1064no—
323C4⋊C4—1061no—
324C4⋊C4—2121no—
325C4⋊C4—2122no—
326C4⋊C4—2122no—
327C4⋊C4—2122no—
328C4⋊C4—2121no—
329C4⋊C4—2121no—
330C4⋊C4#1171991212yes—
331C42—41no—
332C42—1062no—
333C42—1062no—
334C42—2121no—
335C23—532no—
336C23—532no—
337C23—532no—
338C23—1061no—
339C23—1061no—
340C23—1061no—
341C23—1061no—
342C23C22⋊C4×D5311yescharacteristic
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—532no—
356C2×C4—1064no—
357C2×C4—1064no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1062no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—4244no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4—2122no—
372C2×C4—2128no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4M16×D5311yescharacteristic
379C2×C4C2×#1171991211yescharacteristic
380C2×C4#1313324111yescharacteristic
381C2×C4#1313324411yescharacteristic
382C2×C4#1313324611yescharacteristic
383C2×C4#1313324911yescharacteristic
384C8—42no—
385C8—42no—
386C8—2122no—
387C8—2122no—
388C22—534no—
389C22—1062no—
390C22—1062no—
391C22—1062no—
392C22—1062no—
393C22—1062no—
394C22—1062no—
395C22#1681058511yescharacteristic
396C22#99×D5311yescharacteristic
397C22#101×D5311yescharacteristic
398C22#1681062611yescharacteristic
399C22#1681063011yescharacteristic
400C22#104×D5311yescharacteristic
401C22#104×D5311yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—42no—
409C4—81no—
410C4—1062no—
411C4—4241no—
412C4—2122no—
413C4—2121no—
414C4—2121no—
415C2—1064no—
416C2#2027304311yescharacteristic
417C2#2027315611yescharacteristic
418C2#2027316011yescharacteristic
419C2#2027322911yescharacteristic
420C2#2027323311yescharacteristic
421C2#330×D5311yescharacteristic
422C2#2027343011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7627
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #20272539C2semidirect
5. #20272538C2semidirect
6. #20272537C2semidirect
7. C53×#1271C2semidirect
210. #1271D53semidirect
238. C53#7627semidirect

Maximal subgroups

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

subgroupindex
#389×D532
#202744762
#202725392
#202725382
#202725372
C53×#12712
#762753