Finite Group Databasebeta

← #27284653

Subgroups — #27284653

Order 13568.

804 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#16810779C212yesmaximal
2#20280829C211yesmaximal, characteristic
3C2×#16810359C212yesmaximal
4#20272558C211yesmaximal, characteristic
5C53×#1446C211yesmaximal, characteristic
6C22×#11719925—22no—
7C2×#13133196C2212yes—
8#16810779—24no—
9#16810779—24no—
10C2×#13133141—22no—
11C2×C22⋊#88322C2212yes—
12#16810555—22no—
13#16810544—22no—
14#16810483—22no—
15#16810359—24no—
16C23×Q8×C53C2211yescharacteristic
17C2×#95×C53—22no—
18C2×#95×C53C2211yescharacteristic
19C2×#95×C53C2211yescharacteristic
20C22×Dic106D412yes—
21C2×#11719925—28no—
22C2×#11719925—48no—
23C22×#88322—22no—
24#13133196—22no—
25#13133196—22no—
26#13133196—22no—
27#13133196—22no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53D412yes—
31C2×#11719919—22no—
32C2×C22⋊Dic53—22no—
33#13133141—28no—
34C22⋊#88322—22no—
35C22⋊#88322D414yes—
36#13133124—24no—
37C22×Q8×C53—21no—
38C22×Q8×C53—22no—
39C22×Q8×C53—22no—
40C22×Q8×C53—44no—
41C22×Q8×C53D412yes—
42C23×C4×C53—21no—
43C23×C4×C53—21no—
44C23×C4×C53D412yes—
45C23×C4×C53C2311yescharacteristic
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—44no—
49C24×C53C2×D411yescharacteristic
50C2×Q8×C53—28no—
51C2×Q8×C53—24no—
52C2×Q8×C53—44no—
53C2×Q8×C53—44no—
54C2×Q8×C53—44no—
55C2×Q8×C53—48no—
56C2×Q8×C53—42no—
57C22×C4×C53—24no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—42no—
72C22×C4×C53Q1614yescomplemented
73C22×C4×C53C2×D412yes—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D412yes—
76C22×C4×C53#4111yescharacteristic
77C22×Dic53—22no—
78C22×Dic53—22no—
79C2×Dic106—22no—
80C2×Dic106—22no—
81C2×Dic106—44no—
82C22⋊Dic53—42no—
83C22⋊Dic53—42no—
84#11719925—416no—
85C4⋊Dic53—22no—
86C4⋊Dic53—22no—
87C4⋊Dic53—82no—
88C4⋊Dic53—82no—
89#11719919—42no—
90#11719919—42no—
91C2×#88322—24no—
92C2×#88322—44no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—22no—
96C23×C53#12011yescharacteristic
97C23×C53#12111yescharacteristic
98C23×C53#12711yescharacteristic
99C23×C53C2×Q1612yes—
100C23×C53#13712yes—
101Q8×C53—28no—
102Q8×C53—44no—
103Q8×C53—48no—
104C2×C4×C53—24no—
105C2×C4×C53—24no—
106C2×C4×C53—24no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—44no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—44no—
115C2×C4×C53—42no—
116C2×C4×C53—44no—
117C2×C4×C53—44no—
118C2×C4×C53—42no—
119C2×C4×C53—82no—
120C2×C4×C53—82no—
121C2×C4×C53—81no—
122C2×C4×C53—81no—
123C2×C4×C53#12012yes—
124C2×C4×C53#12112yes—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—44no—
130Dic106—44no—
131Dic106—82no—
132#88322—44no—
133#9967—531nomaximal, Sylow
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53#39211yescharacteristic
143C22×C53#45112yes—
144C22×C53#45511yescharacteristic
145C22×C53#45611yescharacteristic
146C22×C53B(4,𝔽2)12yes—
147C22×C53#46212yes—
148C22×C53#49411yescharacteristic
149C22×C53#49711yescharacteristic
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—44no—
153C4×C53—44no—
154C4×C53—82no—
155Dic53—82no—
156Dic53—82no—
157C2×#451—532no—
158#2659—531no—
159C2×#328—532no—
160#1446D5311yescharacteristic
161#1280—531no—
162C2×C53—21no—
163C2×C53—22no—
164C2×C53—21no—
165C2×C53#160212yes—
166C2×C53#160512yes—
167C2×C53#200711yescharacteristic
168C2×C53#201611yescharacteristic
169C2×C53#202111yescharacteristic
170C23×Q8C2×D5311yescharacteristic
171C22×Q16—1062no—
172C2×C22⋊Q8—532no—
173#451—1064no—
174#451—1064no—
175C2×#103—1062no—
176C2×C22⋊C8—532no—
177#398—1062no—
178#395—1062no—
179#380—1062no—
180C2×#95—22no—
181C2×#95C2×D5311yescharacteristic
182C2×#95C2×D5311yescharacteristic
183#328—1064no—
184C53#996711yescharacteristic, complemented, Sylow
185C22×Q8—21no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—44no—
189C22×Q8—532no—
190C22×Q8D21212yes—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4C22⋊D5312yes—
194C23×C4C22×D5311yescharacteristic
195C2×Q16—1068no—
196C2×Q16—2128no—
197C22×C8—1062no—
198C22⋊Q8—1062no—
199C22⋊Q8—1062no—
200C22⋊Q8—1062no—
201C22⋊Q8—1062no—
202C2×C4⋊C4—532no—
203C2×C4⋊C4—1062no—
204C2×C4⋊C4—1062no—
205C2×C4⋊C4—1062no—
206C2×C22⋊C4—1062no—
207#103—1068no—
208C22⋊C8—534no—
209C22⋊C8—1062no—
210#95—22no—
211#95—22no—
212#95—44no—
213#95—1064no—
214C24C2×C22⋊D5311yescharacteristic
215C2×Q8—28no—
216C2×Q8—24no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×Q8—48no—
221C2×Q8—42no—
222C2×Q8—1062no—
223C2×Q8—1062no—
224C2×Q8—2124no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—1062no—
241C22×C4—1062no—
242C22×C4#1171992514yescomplemented
243C22×C4C2×D21211yescharacteristic
244C22×C4#1171994511yescharacteristic
245C22×C4D4×D5312yes—
246C22×C4D4×D5312yes—
247Q16—21216no—
248C2×C8—1064no—
249C2×C8—2124no—
250C4⋊C4—1062no—
251C4⋊C4—1062no—
252C4⋊C4—4242no—
253C4⋊C4—4242no—
254C4⋊C4—2122no—
255C4⋊C4—2122no—
256C22⋊C4—2122no—
257C22⋊C4—2122no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23#1313318511yescharacteristic
262C23#1313319111yescharacteristic
263C23#1313321311yescharacteristic
264C23C2×#1171992512yes—
265C23#1313324712yes—
266Q8—28no—
267Q8—44no—
268Q8—48no—
269Q8—4242no—
270Q8—2124no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—2122no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4—2124no—
295C2×C4#1313318712yes—
296C2×C4#1313319012yes—
297C8—2124no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22#1681053211yescharacteristic
307C22#1681077812yes—
308C22#1681080011yescharacteristic
309C22#1681080511yescharacteristic
310C22#1681080912yes—
311C22#1681083312yes—
312C22#1681100211yescharacteristic
313C22#1681101811yescharacteristic
314C4—22no—
315C4—22no—
316C4—44no—
317C4—44no—
318C4—82no—
319C4—4242no—
320C4—4242no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2#2027367112yes—
325C2#2027368812yes—
326C2#2027576111yescharacteristic
327C2#2027581411yescharacteristic
328C2#2027584111yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1446C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#1446D53
hypocenterC53#9967
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1446

Complements

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

NH ≅ G/Ntype
72. C22×C4×C53Q16semidirect
184. C53#9967semidirect
242. C22×C4#11719925semidirect

Maximal subgroups

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

subgroupindex
C2×#168107792
#202808292
C2×#168103592
#202725582
C53×#14462
#996753