Finite Group Databasebeta

← #27284685

Subgroups — #27284685

Order 13568.

632 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 121 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×#16810951C212yesmaximal
2C2×#16810360C212yesmaximal
3#20272982C211yesmaximal, characteristic
4#20272558C211yesmaximal, characteristic
5C53×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7#16810951—24no—
8#16810951—24no—
9C2×#13133148—22no—
10C2×#13133141—22no—
11C2×C22⋊#88322C2212yes—
12C2×#13133124—22no—
13C2×#13133124C2211yescharacteristic
14#16810360—24no—
15C2×C22⋊Q8×C53C2212yes—
16C53×#398—22no—
17C53×#395—22no—
18C53×#380—22no—
19C2×#95×C53C2211yescharacteristic
20C23×Dic53—21no—
21C23×Dic53—21no—
22C22×#88322—22no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53D412yes—
27#13133148—28no—
28#13133141—28no—
29C22⋊#88322—22no—
30C22⋊#88322D414yes—
31#13133124—22no—
32#13133124—44no—
33C22×Q8×C53D412yes—
34C23×C4×C53D412yes—
35C23×C4×C53C2311yescharacteristic
36C22⋊Q8×C53—22no—
37C22⋊Q8×C53—22no—
38C22⋊Q8×C53—22no—
39C22⋊Q8×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53D412yes—
44C2×C22⋊C4×C53—22no—
45#95×C53—24no—
46#95×C53—22no—
47C24×C53C2×D411yescharacteristic
48C2×Q8×C53—22no—
49C2×Q8×C53—22no—
50C2×Q8×C53—44no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53SD1614yes—
59C22×C4×C53C2×D412yes—
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D412yes—
62C22×C4×C53#4111yescharacteristic
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—41no—
66C22×Dic53—42no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53—41no—
70C22×Dic53—42no—
71C22×Dic53—42no—
72C22×Dic53—42no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—82no—
78C4⋊C4×C53—82no—
79C22⋊C4×C53—42no—
80C22⋊C4×C53—42no—
81C4⋊Dic53—24no—
82C4⋊Dic53—42no—
83C2×#88322—24no—
84C2×#88322—44no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—22no—
88C23×C53#12011yescharacteristic
89C23×C53#12111yescharacteristic
90C23×C53#12711yescharacteristic
91C23×C53C2×SD1612yes—
92C23×C53#13712yes—
93Q8×C53—44no—
94Q8×C53—82no—
95C2×C4×C53—24no—
96C2×C4×C53—24no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—44no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—44no—
108C2×C4×C53—82no—
109C2×C4×C53#12012yes—
110C2×C4×C53#12112yes—
111C2×Dic53—44no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—44no—
115C2×Dic53—82no—
116C2×Dic53—81no—
117C2×Dic53—82no—
118C2×Dic53—81no—
119C2×Dic53—81no—
120C2×Dic53—81no—
121#88322—44no—
122#9973—531nomaximal, Sylow
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53#39211yescharacteristic
132C22×C53#44812yes—
133C22×C53#45511yescharacteristic
134C22×C53#45611yescharacteristic
135C22×C53#45812yes—
136C22×C53#46112yes—
137C22×C53#49211yescharacteristic
138C22×C53#49711yescharacteristic
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—44no—
142C4×C53—82no—
143C4×C53—82no—
144Dic53—82no—
145Dic53—82no—
146C2×#483—532no—
147#2659D5311yescharacteristic
148C2×#328—532no—
149#1445—531no—
150#1280—531no—
151C2×C53—21no—
152C2×C53—22no—
153C2×C53—21no—
154C2×C53#160812yes—
155C2×C53#161112yes—
156C2×C53#200711yescharacteristic
157C2×C53#201311yescharacteristic
158C2×C53#202211yescharacteristic
159C2×C22⋊Q8C2×D5312yes—
160C22×C4⋊C4—531no—
161#483—1064no—
162#483—1064no—
163C2×#106—1062no—
164C2×#103—1062no—
165C2×C22⋊C8—532no—
166#398—22no—
167#395—22no—
168#380—22no—
169C2×#95—531no—
170C2×#95—1062no—
171C2×#95C2×D5311yescharacteristic
172#328—1064no—
173C53#997311yescharacteristic, complemented, Sylow
174C22×Q8C22⋊D5312yes—
175C23×C4—1061no—
176C23×C4—1061no—
177C23×C4C22⋊D5312yes—
178C23×C4C22×D5311yescharacteristic
179C22×C8—1062no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—532no—
188C2×C4⋊C4—1062no—
189C2×C4⋊C4—1062no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4C22⋊D5312yes—
192C2×C22⋊C4—22no—
193#106—1068no—
194#103—1068no—
195C22⋊C8—534no—
196C22⋊C8—1062no—
197#95—24no—
198#95—22no—
199#95—1062no—
200#95—2124no—
201C24C2×C22⋊D5311yescharacteristic
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—44no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—1062no—
213C22×C4—1062no—
214C22×C4—2121no—
215C22×C4—2122no—
216C22×C4—2121no—
217C22×C4—2121no—
218C22×C4—2121no—
219C22×C4—2122no—
220C22×C4—2122no—
221C22×C4—2122no—
222C22×C4#1171992414yes—
223C22×C4D4×D5311yescharacteristic
224C22×C4D4×D5312yes—
225C22×C4#1171994711yescharacteristic
226C22×C4C2×C22⋊D5312yes—
227C2×C8—1064no—
228C2×C8—2124no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—1064no—
236C4⋊C4—2122no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1313318611yescharacteristic
243C23#1313318811yescharacteristic
244C23#1313321311yescharacteristic
245C23C2×#1171992412yes—
246C23#1313324812yes—
247Q8—44no—
248Q8—82no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—82no—
263C2×C4—4242no—
264C2×C4—4241no—
265C2×C4—4241no—
266C2×C4—4242no—
267C2×C4—4241no—
268C2×C4—4241no—
269C2×C4—2122no—
270C2×C4—2124no—
271C2×C4—2122no—
272C2×C4—2124no—
273C2×C4#1313318512yes—
274C2×C4#1313318912yes—
275C8—2124no—
276C22—22no—
277C22—22no—
278C22—21no—
279C22—21no—
280C22—22no—
281C22—22no—
282C22—21no—
283C22—21no—
284C22#1681053111yescharacteristic
285C22#1681076212yes—
286C22#1681080111yescharacteristic
287C22#1681080611yescharacteristic
288C22#1681081312yes—
289C22#1681082512yes—
290C22#1681099711yescharacteristic
291C22#1681101511yescharacteristic
292C4—22no—
293C4—22no—
294C4—44no—
295C4—82no—
296C4—82no—
297C4—4242no—
298C4—4242no—
299C2—21no—
300C2—21no—
301C2—22no—
302C2#2027370612yes—
303C2#2027372312yes—
304C2#2027576211yescharacteristic
305C2#2027579811yescharacteristic
306C2#2027584411yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2659C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#2659D53
hypocenterC53#9973
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
173. C53#9973semidirect

Maximal subgroups

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

subgroupindex
C2×#168109512
C2×#168103602
#202729822
#202725582
C53×#26592
#997353