Finite Group Databasebeta

← #27279196

Subgroups — #27279196

Order 13568.

566 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277951C211yesmaximal, characteristic
2#20273139C212yesmaximal, complemented
3#20273029C211yesmaximal, characteristic, complemented
4#20272616C212yesmaximal, complemented
5C53×#2921C211yesmaximal, characteristic, complemented
6C2×C4×D212C2211yescharacteristic
7C2×C4×C22⋊D53—21no—
8#16810676C2212yes—
9#16810553C411yescharacteristic
10#16810548C411yescharacteristic
11#16810545—21no—
12#16810540C411yescharacteristic
13#16810539—21no—
14#16810532C411yescharacteristic
15C4×C4⋊Dic53C2211yescharacteristic
16C4×#11719921—21no—
17#16810350—24no—
18#16810349—21no—
19#16810347—21no—
20C2×#117×C53C2211yescharacteristic
21C53×#433—24no—
22C53×#423C2212yes—
23C53×#407—21no—
24C4×M16×C53—21no—
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C22×C4×D53—21no—
28C4×D212—22no—
29C4×D212—22no—
30C4×C22⋊D53—44no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53—21no—
40C4⋊#88322—22no—
41C22⋊#88322—22no—
42C4×#88322—22no—
43#13133124—21no—
44#13133124—21no—
45#13133124C2×C411yescharacteristic
46#13133124C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C2×M16×C53—22no—
49#117×C53—22no—
50#117×C53—22no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—22no—
58C4⋊C8×C53D414yescomplemented
59C22⋊C8×C53—24no—
60#97×C53—22no—
61C4×C8×C53—22no—
62C24×C53C22⋊C411yescharacteristic
63C23×D53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C2×C22⋊D53—42no—
75C2×C22⋊D53—42no—
76C2×C22⋊D53—42no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×D212—22no—
81C2×D212—42no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—42no—
85C2×C4×D53—42no—
86M16×C53—42no—
87M16×C53—42no—
88C2×C8×C53—24no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—22no—
91C22⋊C4×C53—22no—
92C22⋊C4×C53—22no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53C2×D412yes—
96C42×C53#4112yes—
97C22⋊Dic53—42no—
98#11719921—42no—
99#11719921—42no—
100#11719921—42no—
101#11719921—42no—
102C4⋊Dic53—22no—
103C4⋊Dic53—42no—
104#11719919—42no—
105C4×Dic53—42no—
106C2×#88322—42no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53C2×C22⋊C411yescharacteristic
113C22×D53—22no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53C4×D412yes—
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12112yes—
134C2×C4×C53#12311yescharacteristic
135C2×C4×C53#13512yes—
136C2×C4×C53AGL(1,ℤ/8)12yes—
137C22⋊D53—48no—
138C2×Dic53—22no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147D212—24no—
148D212—42no—
149C4×D53—82no—
150C4×D53—82no—
151C8×C53—44no—
152#88322—82no—
153#9036—531nomaximal, Sylow
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—42no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53#38611yescharacteristic
162C22×C53#41611yescharacteristic
163C22×C53#41811yescharacteristic
164C2×D53—42no—
165C2×D53—42no—
166C2×D53—42no—
167C2×D53—81no—
168C2×D53—81no—
169C2×D53—81no—
170C2×D53—81no—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53#44912yes—
177C4×C53#48212yes—
178Dic53—42no—
179Dic53—81no—
180Dic53—81no—
181#2921D5311yescharacteristic, complemented
182#2392—531no—
183#1486—532no—
184#1458—531no—
185#1303—532no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—81no—
189C2×C53#187511yescharacteristic
190C2×C53#188411yescharacteristic
191C2×C53#189111yescharacteristic
192D53—82no—
193C2×C4×D4—531no—
194C2×C4×D4—1061no—
195C2×#117C2×D5311yescharacteristic
196#433—24no—
197#431—532no—
198#423C2×D5312yes—
199#407—21no—
200C4×M16—21no—
201#398—531no—
202#397—531no—
203#396—1061no—
204#394—531no—
205#394—1061no—
206#392—531no—
207C4×C4⋊C4—531no—
208C4×C22⋊C4—1061no—
209D4⋊C8—1064no—
210#324—1061no—
211#323—1061no—
212C53#903611yescharacteristic, complemented, Sylow
213C22×D4—531no—
214C22×D4—1061no—
215C23×C4—1061no—
216C23×C4C22⋊D5311yescharacteristic
217C2×M16—22no—
218C4×D4—1062no—
219C4×D4—1062no—
220C4×D4—2124no—
221#117—22no—
222#117—22no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4C4×D5311yescharacteristic
227C2×C4⋊C4D21211yescharacteristic
228C2×C22⋊C4—1061no—
229C2×C22⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C22⋊C4—1061no—
232C2×C22⋊C4—1061no—
233C2×C22⋊C4C4×D5311yescharacteristic
234C2×C22⋊C4D21211yescharacteristic
235C2×C42—1061no—
236C2×C42C22⋊D5311yescharacteristic
237C2×C42C22×D5311yescharacteristic
238C4⋊C8—22no—
239C4⋊C8—1062no—
240C4⋊C8C22⋊D5314yescomplemented
241C22⋊C8—24no—
242C22⋊C8—1062no—
243#97—22no—
244C4×C8—22no—
245C4×C8—1062no—
246#95—531no—
247#95—531no—
248#95—1061no—
249#95—1061no—
250C24—1061no—
251C24#1171992111yescharacteristic
252C2×D4—1062no—
253C2×D4—2122no—
254C2×D4—2122no—
255C2×D4—2122no—
256C2×D4—2122no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—2121no—
265C22×C4—2121no—
266C22×C4—2122no—
267C22×C4—2122no—
268C22×C4#1171992111yescharacteristic
269C22×C4#1171992111yescharacteristic
270C22×C4#1171992111yescharacteristic
271C22×C4C2×C4×D5311yescharacteristic
272C22×C4C2×D21211yescharacteristic
273C22×C4C2×C22⋊D5311yescharacteristic
274M16—42no—
275M16—42no—
276C2×C8—24no—
277C2×C8—2122no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—1062no—
281C4⋊C4—2122no—
282C4⋊C4—2122no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—2122no—
286C22⋊C4—2122no—
287C22⋊C4—2122no—
288C22⋊C4—2122no—
289C22⋊C4—2122no—
290C42—21no—
291C42—21no—
292C42—2122no—
293C42#1171994512yes—
294C42C2×C22⋊D5312yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23—1062no—
301C23—2121no—
302C23—2121no—
303C23—2121no—
304C23—2121no—
305C23—2121no—
306C23—2121no—
307C23C2×#1171992111yescharacteristic
308D4—1064no—
309D4—2122no—
310D4—2128no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—1062no—
323C2×C4—4242no—
324C2×C4—4242no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4C4×C22⋊D5312yes—
334C2×C4#1313318611yescharacteristic
335C2×C4#1313319112yes—
336C2×C4#1313320011yescharacteristic
337C2×C4#1313323712yes—
338C2×C4#1313324212yes—
339C8—44no—
340C8—4242no—
341C22—21no—
342C22—21no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—4241no—
349C22—4241no—
350C22—4241no—
351C22—4241no—
352C22—2122no—
353C22—2122no—
354C22—2122no—
355C22#1681050211yescharacteristic
356C22#1681061711yescharacteristic
357C22#1681062711yescharacteristic
358C4—22no—
359C4—22no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—4241no—
364C4—4241no—
365C4—2122no—
366C4#1681076812yes—
367C4#1681094712yes—
368C2—21no—
369C2—21no—
370C2—81no—
371C2—4242no—
372C2#2027503711yescharacteristic
373C2#2027509811yescharacteristic
374C2#2027513311yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2921C2
socleC22×C53#386
prime radical#16810532C4
cosocleC2×C42×C53C23
hypercenter#2921D53
hypocenterC53#9036
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #20273139C2semidirect
3. #20273029C2semidirect
4. #20272616C2semidirect
5. C53×#2921C2semidirect
58. C4⋊C8×C53D4semidirect
181. #2921D53semidirect
212. C53#9036semidirect
240. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202779512
#202731392
#202730292
#202726162
C53×#29212
#903653