Finite Group Databasebeta

← #27294476

Subgroups — #27294476

Order 13568.

470 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#13133134C211yesmaximal, characteristic, complemented
2#20273033C211yesmaximal, characteristic, complemented
3#20272742C212yesmaximal, complemented
4#20272718C212yesmaximal, complemented
5C53×#2635C211yesmaximal, characteristic
6C2×#13133134—21no—
7C2×C22⋊#88322—21no—
8C4×#11719917—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—
24C22×#88322—41no—
25#13133134—24no—
26C22⋊#88322—22no—
27C22⋊#88322—41no—
28#13133128—41no—
29C4×#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—22no—
61C2×D4×C53—22no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—42no—
65C2×D4×C53—42no—
66C2×D4×C53—81no—
67C2×D4×C53—81no—
68C2×D4×C53—81no—
69C2×D4×C53—81no—
70C2×D4×C53—81no—
71C22×C4×C53—21no—
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—81no—
79C22×C4×C53—81no—
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—81no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—81no—
94C22⋊C4×C53—81no—
95C22⋊C4×C53—81no—
96C22⋊C4×C53—81no—
97C42×C53—21no—
98C42×C53—21no—
99#11719917—24no—
100#11719917—42no—
101C2×#88322—42no—
102C2×#88322—42no—
103C2×#88322—82no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
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#10412yes—
121C23×C53C2×C22⋊C411yescharacteristic
122D4×C53—44no—
123D4×C53—82no—
124D4×C53—84no—
125D4×C53—84no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
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—81no—
146C2×C4×C53—81no—
147C2×C4×C53—81no—
148C2×C4×C53#9911yescharacteristic
149C2×C4×C53#9911yescharacteristic
150C2×C4×C53#13512yes—
151#88322—44no—
152#88322—82no—
153#11737—531nomaximal, Sylow
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—22no—
157C22×C53—42no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53C2×#9911yescharacteristic
174C22×C53#41611yescharacteristic
175C22×C53C2×#10411yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—22no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—81no—
183C4×C53—81no—
184C4×C53—81no—
185#2635D5311yescharacteristic
186C4×#100—531no—
187#1459—531no—
188#1353—532no—
189#1345—532no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—42no—
193C2×C53—81no—
194C2×C53—161no—
195C2×C53#150111yescharacteristic
196C2×C53#212111yescharacteristic
197C2×C53#213411yescharacteristic
198C2×#121Dic5311yescharacteristic
199C2×C4×D4C2×D5311yescharacteristic
200C2×#100—1061no—
201C2×C22⋊C8—1061no—
202C4×M16—1061no—
203#400C2×D5311yescharacteristic
204#399Dic5311yescharacteristic
205#394—21no—
206#394C2×D5311yescharacteristic
207#392—21no—
208#392Dic5311yescharacteristic
209#388—21no—
210#386—21no—
211#381Dic5311yescharacteristic
212#343—532no—
213#324—532no—
214#323—1061no—
215C53#1173711yescharacteristic, complemented, Sylow
216C22×D4—21no—
217C22×D4—21no—
218C22×D4C2×Dic5311yescharacteristic
219C22×D4C22⋊D5311yescharacteristic
220C23×C4C2×Dic5311yescharacteristic
221C23×C4C22⋊D5311yescharacteristic
222C2×M16—1062no—
223C22×C8—2121no—
224#121—41no—
225#121—41no—
226C4×D4—41no—
227C4×D4—41no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C2×Dic5311yescharacteristic
230C2×C4⋊C4C22×D5311yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4C2×Dic5311yescharacteristic
238C2×C22⋊C4C22⋊D5311yescharacteristic
239C2×C42C22⋊D5311yescharacteristic
240#100—1064no—
241C22⋊C8—1062no—
242C22⋊C8—2121no—
243#97—2121no—
244C4×C8—2121no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95C2×Dic5311yescharacteristic
249#95C2×Dic5311yescharacteristic
250C24—21no—
251C24C22⋊Dic5311yescharacteristic
252C24C22⋊Dic5311yescharacteristic
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—81no—
260C2×D4—81no—
261C2×D4—81no—
262C2×D4—81no—
263C2×D4—81no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4C22⋊Dic5311yescharacteristic
274C22×C4C22⋊Dic5311yescharacteristic
275C22×C4C22×Dic5311yescharacteristic
276C22×C4C2×C22⋊D5311yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278M16—1064no—
279M16—2122no—
280C2×C8—4242no—
281C2×C8—2122no—
282C2×C8—2122no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—81no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C42—21no—
296C42—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23#1313314412yes—
314C23C2×C22⋊Dic5311yescharacteristic
315D4—44no—
316D4—82no—
317D4—84no—
318D4—84no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4#1313313211yescharacteristic
342C2×C4#1313313211yescharacteristic
343C2×C4#1313323712yes—
344C8—4242no—
345C8—2124no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C2×#1313313211yescharacteristic
366C22#1681061911yescharacteristic
367C22C2×#1313314411yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—81no—
375C4—81no—
376C4—81no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—81no—
381C2—161no—
382C2#2027320611yescharacteristic
383C2#2027636111yescharacteristic
384C2#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#11737
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. C4×#13133134C2semidirect
2. #20273033C2semidirect
3. #20272742C2semidirect
4. #20272718C2semidirect
215. C53#11737semidirect

Maximal subgroups

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

subgroupindex
C4×#131331342
#202730332
#202727422
#202727182
C53×#26352
#1173753