Finite Group Databasebeta

← #27294469

Subgroups — #27294469

Order 13568.

402 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 87 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#20278006C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272744C212yesmaximal
4#20272720C212yesmaximal
5C4×#101×C53C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic
7#16810676—21no—
8#16810554—21no—
9#16810553C411yescharacteristic
10#16810553C2211yescharacteristic
11#16810543C411yescharacteristic
12#16810543C2211yescharacteristic
13#16810542—21no—
14C4×#11719919—21no—
15C4×#11719919—21no—
16#16810469C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C2×#101×C53—21no—
21C4×M16×C53—21no—
22C53×#343C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C4×Dic106—41no—
25C4×Dic106—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C4⋊#88322—41no—
35C22⋊#88322—22no—
36C4×#88322—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×Q8×C53D411yescharacteristic
45C2×M16×C53—22no—
46C4×Q8×C53—41no—
47C4×Q8×C53—41no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54#101×C53—24no—
55#97×C53—41no—
56C4×C8×C53—41no—
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53—41no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C2×Dic106—41no—
72C2×Dic106—42no—
73M16×C53—24no—
74M16×C53—42no—
75C2×C8×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C4⋊Dic53—41no—
86#11719919—42no—
87#11719919—42no—
88#11719919—42no—
89#11719919—42no—
90#11719919—42no—
91C4×Dic53—24no—
92C4×Dic53—41no—
93C4×Dic53—41no—
94C2×#88322—42no—
95C23×C53C2×C22⋊C411yescharacteristic
96Q8×C53—44no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53#9911yescharacteristic
113C2×C4×C53#10412yes—
114C2×C4×C53#13512yes—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127Dic106—42no—
128Dic106—42no—
129C8×C53—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—41no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53#35612yes—
145Dic53—42no—
146Dic53—42no—
147Dic53—42no—
148Dic53—81no—
149#2399—531no—
150C4×#101D5311yescharacteristic
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×Q8—531no—
160C2×C4×Q8C2×D5311yescharacteristic
161#431—1061no—
162C2×#101—21no—
163C4×M16—21no—
164#398—531no—
165#398—531no—
166#398—1061no—
167#395—531no—
168#395—531no—
169#395—1061no—
170C4×C4⋊C4—1061no—
171C4×C4⋊C4—1061no—
172#376—531no—
173#343C2×D5312yes—
174#324—532no—
175#324—1061no—
176C53#1173611yescharacteristic, complemented, Sylow
177C22×Q8—531no—
178C22×Q8C22⋊D5311yescharacteristic
179C2×M16—22no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—2121no—
183C4×Q8—2121no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4C4×D5311yescharacteristic
191C2×C4⋊C4D21211yescharacteristic
192C2×C4⋊C4C22×D5311yescharacteristic
193C2×C42—1061no—
194C2×C42—1061no—
195C2×C42C4×D5311yescharacteristic
196C2×C42D21211yescharacteristic
197C2×C42C22⋊D5311yescharacteristic
198C4⋊C8—2121no—
199#101—24no—
200C22⋊C8—1062no—
201#97—41no—
202C4×C8—41no—
203C4×C8—2121no—
204#95—531no—
205#95—531no—
206#95—1061no—
207#95—1061no—
208#95—1061no—
209#95—1061no—
210#95—1061no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—2121no—
215C2×Q8—2122no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4#1171992111yescharacteristic
221C22×C4#1171992111yescharacteristic
222C22×C4#1171992111yescharacteristic
223C22×C4#1171992111yescharacteristic
224C22×C4C2×C4×D5311yescharacteristic
225C22×C4C2×D21211yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227M16—24no—
228M16—42no—
229C2×C8—42no—
230C2×C8—2122no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—2121no—
235C4⋊C4—2122no—
236C4⋊C4—2122no—
237C4⋊C4—2122no—
238C4⋊C4—2122no—
239C4⋊C4—2122no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—1064no—
247C42—2121no—
248C42—2121no—
249C23C2×#1171992111yescharacteristic
250Q8—44no—
251Q8—2122no—
252Q8—2122no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4#1313313111yescharacteristic
280C2×C4#1313313111yescharacteristic
281C2×C4#1313314512yes—
282C2×C4#1313323712yes—
283C8—44no—
284C8—4242no—
285C22—21no—
286C22—21no—
287C22C2×#1313313111yescharacteristic
288C22#1681061711yescharacteristic
289C22C2×#1313314511yescharacteristic
290C4—22no—
291C4—22no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—4241no—
298C4—2122no—
299C4—2122no—
300C4—2122no—
301C4#1681042112yes—
302C2—21no—
303C2—21no—
304C2#2027320411yescharacteristic
305C2#2027635711yescharacteristic
306C2C2×#1681042111yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC4×#101×C53C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC4×#101D53
hypocenterC53#11736
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

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
#202780062
#202730392
#202727442
#202727202
C4×#101×C532
#1173653