Finite Group Databasebeta

← #27294913

Subgroups — #27294913

Order 13568.

450 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 87 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#20274707C211yesmaximal, characteristic, complemented
2#20273053C211yesmaximal, characteristic, complemented
3#20272742C212yesmaximal, complemented
4#20272718C212yesmaximal, complemented
5C53×#2635C211yesmaximal, characteristic
6#16810647—21no—
7C2×#13133134—21no—
8#16810587—21no—
9#16810392C2212yes—
10#16810349C2212yes—
11#16810347—21no—
12C2×#121×C53C411yescharacteristic
13C2×C4×D4×C53C2211yescharacteristic
14C53×#400C2211yescharacteristic
15C53×#399C411yescharacteristic
16C53×#394—21no—
17C53×#394C2211yescharacteristic
18C53×#392—21no—
19C53×#392C411yescharacteristic
20C53×#388—21no—
21C53×#386—21no—
22C53×#381C411yescharacteristic
23C2×#11719917—22no—
24C2×#11719917—41no—
25C4⋊#88322—41no—
26C4⋊#88322—41no—
27#13133134—42no—
28C22⋊#88322—22no—
29C22⋊#88322—41no—
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53C2×C411yescharacteristic
33C22×D4×C53D411yescharacteristic
34C23×C4×C53C2×C411yescharacteristic
35C23×C4×C53D411yescharacteristic
36#121×C53—41no—
37#121×C53—41no—
38C4×D4×C53—41no—
39C4×D4×C53—41no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53—21no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C22⋊C411yescharacteristic
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—42no—
66C2×D4×C53—42no—
67C2×D4×C53—81no—
68C2×D4×C53—81no—
69C2×D4×C53—81no—
70C2×D4×C53—81no—
71C2×D4×C53—81no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—81no—
80C22×C4×C53—81no—
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—81no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—81no—
96C22⋊C4×C53—81no—
97C22⋊C4×C53—81no—
98C22⋊C4×C53—81no—
99C42×C53—21no—
100C42×C53—21no—
101#11719917—42no—
102#11719917—42no—
103#11719917—82no—
104C2×#88322—42no—
105C2×#88322—42no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53C2×C22⋊C411yescharacteristic
124D4×C53—42no—
125D4×C53—42no—
126D4×C53—82no—
127D4×C53—84no—
128D4×C53—84no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53#9911yescharacteristic
152C2×C4×C53#9911yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53#13711yescharacteristic
155#88322—82no—
156#88322—82no—
157#11832—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—42no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53C2×#9911yescharacteristic
178C22×C53#41711yescharacteristic
179C22×C53#42111yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—81no—
188C4×C53—81no—
189C4×C53—81no—
190#2635D5311yescharacteristic
191#1800—531no—
192#1465—531no—
193#1353—532no—
194#1345—532no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—81no—
198C2×C53—81no—
199C2×C53—161no—
200C2×C53#151311yescharacteristic
201C2×C53#212111yescharacteristic
202C2×C53#213411yescharacteristic
203C2×#121Dic5311yescharacteristic
204C2×C4×D4C2×D5311yescharacteristic
205#423—1061no—
206C2×#100—1061no—
207#407—1061no—
208#400C2×D5311yescharacteristic
209#399Dic5311yescharacteristic
210#394—21no—
211#394C2×D5311yescharacteristic
212#392—21no—
213#392Dic5311yescharacteristic
214#388—21no—
215#386—21no—
216#381Dic5311yescharacteristic
217#343—532no—
218#324—532no—
219#323—1061no—
220C53#1183211yescharacteristic, complemented, Sylow
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2×Dic5311yescharacteristic
224C22×D4C22⋊D5311yescharacteristic
225C23×C4C2×Dic5311yescharacteristic
226C23×C4C22⋊D5311yescharacteristic
227C2×M16—1062no—
228C2×M16—2121no—
229#121—41no—
230#121—41no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C2×Dic5311yescharacteristic
235C2×C4⋊C4C22×D5311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4C2×Dic5311yescharacteristic
243C2×C22⋊C4C22⋊D5311yescharacteristic
244C2×C42C22⋊D5311yescharacteristic
245C4⋊C8—2121no—
246C4⋊C8—2121no—
247#100—2122no—
248C22⋊C8—1062no—
249C22⋊C8—2121no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95C2×Dic5311yescharacteristic
254#95C2×Dic5311yescharacteristic
255C24—21no—
256C24C22⋊Dic5311yescharacteristic
257C24C22⋊Dic5311yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—81no—
266C2×D4—81no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—81no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4C22⋊Dic5311yescharacteristic
280C22×C4C22⋊Dic5311yescharacteristic
281C22×C4C22×Dic5311yescharacteristic
282C22×C4C2×C22⋊D5311yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284M16—4242no—
285M16—2122no—
286M16—2122no—
287C2×C8—2122no—
288C2×C8—2122no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—81no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C42—21no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C2×C22⋊Dic5311yescharacteristic
322D4—42no—
323D4—42no—
324D4—82no—
325D4—84no—
326D4—84no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4#1313313211yescharacteristic
350C2×C4#1313313211yescharacteristic
351C2×C4#1313324211yescharacteristic
352C2×C4#1313324711yescharacteristic
353C8—4242no—
354C8—4242no—
355C22—21no—
356C22—21no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22C2×#1313313211yescharacteristic
375C22#1681062411yescharacteristic
376C22#1681064011yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—81no—
385C4—81no—
386C4—81no—
387C2—21no—
388C2—21no—
389C2—81no—
390C2—81no—
391C2—161no—
392C2#2027325811yescharacteristic
393C2#2027636111yescharacteristic
394C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
Fitting, prime radicalC53×#2635C2
socleC22×C53C2×#99
cosocleC2×C4⋊C4×C53C23
hypercenter#2635D53
hypocenterC53#11832
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2635

Complements

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

NH ≅ G/Ntype
1. #20274707C2semidirect
2. #20273053C2semidirect
3. #20272742C2semidirect
4. #20272718C2semidirect
220. C53#11832semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202730532
#202727422
#202727182
C53×#26352
#1183253