Finite Group Databasebeta

← #27294471

Subgroups — #27294471

Order 13568.

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

normal subgroups only

This group has 402 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
1C4×#13133136C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272745C212yesmaximal
4#20272721C212yesmaximal
5C53×#2399C211yesmaximal, characteristic
6#16810676—21no—
7C2×#13133136—21no—
8C4×#11719917—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×C4×Q8×C53C411yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#398—21no—
15C53×#398C411yescharacteristic
16C53×#398C2211yescharacteristic
17C53×#395—21no—
18C53×#395C411yescharacteristic
19C53×#395C2211yescharacteristic
20C4×C4⋊C4×C53—21no—
21C4×C4⋊C4×C53—21no—
22C53×#376C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25#13133136—24no—
26C22⋊#88322—22no—
27#13133128—41no—
28C4×#88322—41no—
29C4×#88322—41no—
30C22×Q8×C53C2×C411yescharacteristic
31C22×Q8×C53D411yescharacteristic
32C4×Q8×C53—41no—
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35C4×Q8×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53C2311yescharacteristic
45C2×C42×C53—21no—
46C2×C42×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53—41no—
60C2×Q8×C53—42no—
61C2×Q8×C53—41no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C42×C53—24no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—41no—
90C42×C53—41no—
91#11719917—24no—
92#11719917—42no—
93C2×#88322—42no—
94C2×#88322—42no—
95C23×C53C2×C22⋊C411yescharacteristic
96Q8×C53—44no—
97Q8×C53—42no—
98Q8×C53—42no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53#9911yescharacteristic
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#10412yes—
128C2×C4×C53#13512yes—
129#88322—44no—
130#88322—82no—
131#11736—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53C2×#9911yescharacteristic
135C22×C53#41611yescharacteristic
136C22×C53C2×#10411yescharacteristic
137C4×C53—21no—
138C4×C53—22no—
139C4×C53—22no—
140C4×C53—42no—
141C4×C53—42no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53—42no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—81no—
148C4×C53#35612yes—
149#2399D5311yescharacteristic
150C4×#101—531no—
151#1461—531no—
152#1354—532no—
153#1346—532no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53#150111yescharacteristic
157C2×C53#212011yescharacteristic
158C2×C53C2×#35611yescharacteristic
159C2×C4×Q8Dic5311yescharacteristic
160C2×C4×Q8C2×D5311yescharacteristic
161#431—1061no—
162C2×#101—1061no—
163C4×M16—1061no—
164#398—21no—
165#398Dic5311yescharacteristic
166#398C2×D5311yescharacteristic
167#395—21no—
168#395Dic5311yescharacteristic
169#395C2×D5311yescharacteristic
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172#376Dic5311yescharacteristic
173#343—532no—
174#324—532no—
175#324—1061no—
176C53#1173611yescharacteristic, complemented, Sylow
177C22×Q8C2×Dic5311yescharacteristic
178C22×Q8C22⋊D5311yescharacteristic
179C2×M16—1062no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C2×Dic5311yescharacteristic
190C2×C4⋊C4C2×Dic5311yescharacteristic
191C2×C4⋊C4C22⋊D5311yescharacteristic
192C2×C4⋊C4C22×D5311yescharacteristic
193C2×C42—21no—
194C2×C42—21no—
195C2×C42C2×Dic5311yescharacteristic
196C2×C42C22⋊D5311yescharacteristic
197C2×C42C22⋊D5311yescharacteristic
198C4⋊C8—2121no—
199#101—1064no—
200C22⋊C8—1062no—
201#97—2121no—
202C4×C8—2121no—
203C4×C8—2121no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95C2×Dic5311yescharacteristic
210#95C2×Dic5311yescharacteristic
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—41no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4C22⋊Dic5311yescharacteristic
221C22×C4C22⋊Dic5311yescharacteristic
222C22×C4C22⋊Dic5311yescharacteristic
223C22×C4C22⋊Dic5311yescharacteristic
224C22×C4C22×Dic5311yescharacteristic
225C22×C4C2×C22⋊D5311yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227M16—1064no—
228M16—2122no—
229C2×C8—2122no—
230C2×C8—2122no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C42—24no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—41no—
248C42—41no—
249C23C2×C22⋊Dic5311yescharacteristic
250Q8—44no—
251Q8—42no—
252Q8—42no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4#1313313211yescharacteristic
280C2×C4#1313313211yescharacteristic
281C2×C4#1313314412yes—
282C2×C4#1313323712yes—
283C8—4242no—
284C8—2124no—
285C22—21no—
286C22—21no—
287C22C2×#1313313211yescharacteristic
288C22#1681061911yescharacteristic
289C22C2×#1313314411yescharacteristic
290C4—22no—
291C4—22no—
292C4—21no—
293C4—42no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4#1681042212yes—
302C2—21no—
303C2—21no—
304C2#2027320611yescharacteristic
305C2#2027635611yescharacteristic
306C2C2×#1681042211yescharacteristic
subgroupG/H
centerC2×C4#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2399C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2399D53
hypocenterC53#11736
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2399

Complements

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

NH ≅ G/Ntype
176. C53#11736semidirect

Maximal subgroups

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

subgroupindex
C4×#131331362
#202730392
#202727452
#202727212
C53×#23992
#1173653