Finite Group Databasebeta

← #27295135

Subgroups — #27295135

Order 13568.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#20279620C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic, complemented
3#20272722C212yesmaximal, complemented
4#20272719C212yesmaximal
5C53×#1461C211yesmaximal, characteristic, complemented
6C2×#13133193—21no—
7C2×C4×D212C411yescharacteristic
8#16810647—21no—
9C2×#13133136—21no—
10#16810553C411yescharacteristic
11#16810553C2211yescharacteristic
12#16810540C411yescharacteristic
13#16810540C2211yescharacteristic
14#16810514—21no—
15#16810514—21no—
16#16810489C411yescharacteristic
17#16810480—21no—
18#16810392C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#431—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C4×D53—21no—
25C2×#11719917—22no—
26#13133193—42no—
27#13133193—42no—
28C4×D212—41no—
29C4×D212—41no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C4⋊#88322—41no—
40C4⋊#88322—41no—
41#13133136—42no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124C2×C411yescharacteristic
46#13133124C2×C411yescharacteristic
47C22×Q8×C53D411yescharacteristic
48C4×Q8×C53—41no—
49C4×Q8×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C4⋊C4×C53C2311yescharacteristic
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C4⋊C8×C53—41no—
57C22⋊C8×C53—22no—
58C4×C8×C53—41no—
59C23×D53—21no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C2×Q8×C53D812yes—
63C2×Q8×C53SD1612yes—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C2×D212—41no—
75C2×D212—42no—
76C2×C4×D53—42no—
77C2×C4×D53—42no—
78C2×C4×D53—42no—
79C2×C8×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89#11719921—42no—
90#11719921—42no—
91#11719921—41no—
92#11719921—41no—
93#11719921—42no—
94#11719921—41no—
95#11719921—41no—
96C4⋊Dic53—42no—
97C4⋊Dic53—41no—
98#11719919—42no—
99#11719919—41no—
100#11719919—41no—
101#11719917—42no—
102#11719917—42no—
103C2×#88322—42no—
104C23×C53C2×C22⋊C411yescharacteristic
105C22×D53—22no—
106C22×D53—41no—
107C22×D53—41no—
108C22×D53—41no—
109C22×D53—41no—
110C22×D53—41no—
111C22×D53—41no—
112Q8×C53—42no—
113Q8×C53—42no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
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—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53#9911yescharacteristic
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#10212yes—
130C2×C4×C53#10212yes—
131C2×C4×C53C2×D811yescharacteristic
132C2×C4×C53C2×SD1611yescharacteristic
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142D212—42no—
143D212—42no—
144C4×D53—84no—
145C8×C53—82no—
146#88322—82no—
147#11879—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53C2×#9911yescharacteristic
151C22×C53C2×#10211yescharacteristic
152C22×C53#42111yescharacteristic
153C2×D53—42no—
154C2×D53—42no—
155C2×D53—42no—
156C2×D53—81no—
157C2×D53—81no—
158C2×D53—81no—
159C2×D53—81no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—22no—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169Dic53—81no—
170Dic53—81no—
171Dic53—81no—
172#2536—531no—
173#1801—531no—
174#1461D5311yescharacteristic, complemented
175#1346—532no—
176#1345—532no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53#151911yescharacteristic
180C2×C53#212011yescharacteristic
181C2×C53#213311yescharacteristic
182D53—82no—
183C2×C22⋊Q8—1061no—
184C2×C4×Q8C2×D5311yescharacteristic
185C2×C4×D4—531no—
186#431—21no—
187#423—1061no—
188C2×#101—1061no—
189#398—531no—
190#398—531no—
191#394—531no—
192#394—531no—
193#388—1061no—
194#388—1061no—
195#382—531no—
196#380—1061no—
197#343—532no—
198#324—21no—
199#324C2×D5312yes—
200C53#1187911yescharacteristic, complemented, Sylow
201C22×Q8D21211yescharacteristic
202C22×D4—531no—
203C23×C4—1061no—
204C2×M16—1062no—
205C22⋊Q8—2122no—
206C22⋊Q8—2122no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—2121no—
210C4×D4—2121no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4C4×D5311yescharacteristic
216C2×C4⋊C4C22⋊D5311yescharacteristic
217C2×C4⋊C4C22×D5311yescharacteristic
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1061no—
221C2×C22⋊C4—1061no—
222C2×C22⋊C4—1061no—
223C2×C42C4×D5311yescharacteristic
224C2×C42D21211yescharacteristic
225C2×C42C22⋊D5311yescharacteristic
226C4⋊C8—41no—
227C4⋊C8—2121no—
228C4⋊C8—2121no—
229#101—2122no—
230C22⋊C8—22no—
231C4×C8—41no—
232#95—531no—
233#95—531no—
234#95—1061no—
235#95—1061no—
236#95—1061no—
237C24—1061no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8#1171991312yes—
241C2×Q8D42412yes—
242C2×D4—2121no—
243C2×D4—2122no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—2122no—
248C22×C4—2122no—
249C22×C4—2122no—
250C22×C4#1171992111yescharacteristic
251C22×C4#1171992111yescharacteristic
252C22×C4#1171992111yescharacteristic
253C22×C4#1171992111yescharacteristic
254C22×C4C2×C4×D5311yescharacteristic
255C22×C4C2×D21211yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257M16—2122no—
258M16—2122no—
259C2×C8—42no—
260C2×C8—2122no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—2121no—
265C4⋊C4—2122no—
266C4⋊C4—2122no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C22⋊C4—2122no—
270C22⋊C4—2122no—
271C22⋊C4—2121no—
272C22⋊C4—2121no—
273C22⋊C4—2122no—
274C22⋊C4—2121no—
275C22⋊C4—2121no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—1062no—
283C23—2121no—
284C23—2121no—
285C23—2121no—
286C23—2121no—
287C23—2121no—
288C23—2121no—
289C23C2×#1171992111yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—2122no—
293D4—2122no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—4244no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4#1313313111yescharacteristic
318C2×C4#1313313111yescharacteristic
319C2×C4#1313313912yes—
320C2×C4#1313313912yes—
321C2×C4C2×D42411yescharacteristic
322C2×C4C2×#1171991311yescharacteristic
323C8—82no—
324C8—4242no—
325C22—21no—
326C22—21no—
327C22—4241no—
328C22—4241no—
329C22—4241no—
330C22—4241no—
331C22—2122no—
332C22—2122no—
333C22—2122no—
334C22C2×#1313313111yescharacteristic
335C22C2×#1313313911yescharacteristic
336C22#1681063911yescharacteristic
337C4—21no—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—4241no—
347C4—4241no—
348C4—4241no—
349C2—21no—
350C2—21no—
351C2—4242no—
352C2#2027328411yescharacteristic
353C2#2027635411yescharacteristic
354C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1461C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1461D53
hypocenterC53#11879
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #20274711C2semidirect
3. #20272722C2semidirect
5. C53×#1461C2semidirect
174. #1461D53semidirect
200. C53#11879semidirect

Maximal subgroups

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

subgroupindex
#202796202
#202747112
#202727222
#202727192
C53×#14612
#1187953