Finite Group Databasebeta

← #27294786

Subgroups — #27294786

Order 13568.

448 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280457C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic, complemented
3#20272720C212yesmaximal, complemented
4#20272717C212yesmaximal, complemented
5C53×#1821C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133191C411yescharacteristic
8#16810587—21no—
9#16810553C411yescharacteristic
10#16810553C2211yescharacteristic
11#16810545—21no—
12#16810545—21no—
13#16810540C411yescharacteristic
14#16810540C2211yescharacteristic
15#16810502—21no—
16#16810483C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#343—21no—
22C53×#343C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C2×#11719917—41no—
26#13133200—44no—
27#13133191—41no—
28#13133191—41no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C22⋊#88322—22no—
40C22⋊#88322—41no—
41#13133124—21no—
42#13133124—21no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C2×M16×C53—22no—
49C2×C4⋊C4×C53—41no—
50C2×C4⋊C4×C53—41no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53C2311yescharacteristic
55C22⋊C8×C53—41no—
56C22⋊C8×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C23×D53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—41no—
67C22×C4×C53—81no—
68C22×C4×C53—81no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C2×C22⋊D53—42no—
76C2×C22⋊D53—42no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53—41no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C2×D212—41no—
83C2×D212—81no—
84M16×C53—42no—
85M16×C53—42no—
86C2×C8×C53—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C22⋊Dic53—42no—
92#11719921—42no—
93#11719921—42no—
94#11719921—42no—
95#11719921—81no—
96#11719921—81no—
97C4⋊Dic53—41no—
98C4⋊Dic53—81no—
99#11719919—42no—
100#11719919—42no—
101#11719917—82no—
102C2×#88322—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#9911yescharacteristic
108C23×C53#9911yescharacteristic
109C23×C53C2×C22⋊C411yescharacteristic
110C22×D53—41no—
111C22×D53—41no—
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53AGL(1,ℤ/8)12yes—
137C22⋊D53—82no—
138C22⋊D53—82no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148D212—82no—
149C8×C53—82no—
150#88322—82no—
151#11805—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53C2×#9911yescharacteristic
163C22×C53#41811yescharacteristic
164C22×C53#42111yescharacteristic
165C2×D53—81no—
166C2×D53—81no—
167C2×D53—81no—
168C2×D53—81no—
169C2×D53—81no—
170C2×D53—81no—
171C2×D53—81no—
172C4×C53—22no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178Dic53—81no—
179Dic53—81no—
180Dic53—81no—
181#2618—531no—
182#1821D5311yescharacteristic, complemented
183#1463—531no—
184#1346—532no—
185#1345—532no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—41no—
189C2×C53—41no—
190C2×C53#151011yescharacteristic
191C2×C53#212111yescharacteristic
192C2×C53#213311yescharacteristic
193D53—161no—
194C2×#123—1061no—
195C2×#121—531no—
196C22×C4⋊C4C2×D5311yescharacteristic
197#407—21no—
198#407—1061no—
199#398—531no—
200#398—531no—
201#396—1061no—
202#396—1061no—
203#394—531no—
204#394—531no—
205#386—1061no—
206#380—531no—
207#343—21no—
208#343C2×D5312yes—
209#324—532no—
210#324—1061no—
211C53#1180511yescharacteristic, complemented, Sylow
212C22×D4—531no—
213C22×D4—1061no—
214C23×C4C4×D5311yescharacteristic
215C23×C4D21211yescharacteristic
216C23×C4C22⋊D5311yescharacteristic
217C2×M16—22no—
218C2×M16—2121no—
219#123—2124no—
220#121—2121no—
221#121—2121no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4C4×D5311yescharacteristic
229C2×C4⋊C4D21211yescharacteristic
230C2×C4⋊C4C22⋊D5311yescharacteristic
231C2×C4⋊C4C22×D5311yescharacteristic
232C2×C22⋊C4—1061no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8—1062no—
241C22⋊C8—2121no—
242#95—531no—
243#95—531no—
244#95—1061no—
245#95—1061no—
246C24—1061no—
247C24#1171992111yescharacteristic
248C2×D4—4241no—
249C2×D4—2121no—
250C2×D4—2122no—
251C2×D4—2122no—
252C2×D4—2121no—
253C2×D4—2121no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4#1171992111yescharacteristic
268C22×C4#1171992111yescharacteristic
269C22×C4#1171992111yescharacteristic
270C22×C4C2×C4×D5311yescharacteristic
271C22×C4C2×D21211yescharacteristic
272C22×C4C2×C22⋊D5311yescharacteristic
273M16—42no—
274M16—42no—
275M16—4242no—
276C2×C8—42no—
277C2×C8—2122no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—4241no—
283C4⋊C4—2121no—
284C4⋊C4—2122no—
285C4⋊C4—2122no—
286C22⋊C4—4241no—
287C22⋊C4—4241no—
288C22⋊C4—2122no—
289C22⋊C4—2122no—
290C22⋊C4—2122no—
291C22⋊C4—2122no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—2121no—
297C23—2121no—
298C23—2121no—
299C23—2121no—
300C23—2121no—
301C23—2121no—
302C23—2121no—
303C23#1313313111yescharacteristic
304C23#1313313111yescharacteristic
305C23C2×#1171992111yescharacteristic
306D4—4242no—
307D4—4242no—
308D4—4242no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4#1313324212yes—
338C8—82no—
339C8—4242no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—4241no—
351C22—4241no—
352C22—4241no—
353C22—4241no—
354C22—4241no—
355C22—4241no—
356C22—4241no—
357C22C2×#1313313111yescharacteristic
358C22#1681062711yescharacteristic
359C22#1681064111yescharacteristic
360C4—22no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—4241no—
367C4—4241no—
368C4—4241no—
369C2—21no—
370C2—21no—
371C2—41no—
372C2—41no—
373C2—8481no—
374C2#2027324311yescharacteristic
375C2#2027636211yescharacteristic
376C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1821C2
socleC22×C53C2×#99
prime radicalC2×#13133191C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1821D53
hypocenterC53#11805
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #20273045C2semidirect
3. #20272720C2semidirect
4. #20272717C2semidirect
5. C53×#1821C2semidirect
182. #1821D53semidirect
211. C53#11805semidirect

Maximal subgroups

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

subgroupindex
#202804572
#202730452
#202727202
#202727172
C53×#18212
#1180553