Finite Group Databasebeta

← #27279157

Subgroups — #27279157

Order 13568.

696 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 135 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#20277313C211yesmaximal, characteristic
2#20274801C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272594C212yesmaximal, complemented
5C2×C53×#397C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7#16810516—24no—
8#16810483C2212yes—
9#16810479—21no—
10C4×C22⋊Dic53—21no—
11#16810385—24no—
12#16810347—22no—
13C22×C4⋊C4×C53C2211yescharacteristic
14C22×C22⋊C4×C53—21no—
15C22×C22⋊C4×C53C2211yescharacteristic
16C53×#397—22no—
17C53×#397C414yescomplemented
18C2×#95×C53—21no—
19C2×#95×C53C2211yescharacteristic
20C22×#88322—22no—
21C2×C4⋊Dic53D414yes—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C4×Dic53—22no—
26C22⋊#88322—22no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—22no—
30C25×C53D411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53—21no—
33C23×C4×C53D411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C22⋊C4×C53—24no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—44no—
43C2×C22⋊C4×C53C2×C412yes—
44C2×C22⋊C4×C53D412yes—
45#95×C53—24no—
46#95×C53—21no—
47#95×C53C2×C412yes—
48C24×C53—21no—
49C24×C53—22no—
50C24×C53—21no—
51C24×C53—22no—
52C24×C53—42no—
53C24×C53—42no—
54C24×C53C22⋊C412yes—
55C24×C53C2×D411yescharacteristic
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
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×C53C22⋊C412yes—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D412yes—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4112yes—
74C22×Dic53—24no—
75C4⋊C4×C53—24no—
76C4⋊C4×C53—42no—
77C22⋊C4×C53—24no—
78C22⋊C4×C53—24no—
79C22⋊C4×C53—48no—
80C22⋊Dic53—42no—
81C22⋊Dic53—42no—
82C22⋊Dic53—82no—
83C22⋊Dic53—82no—
84C4⋊Dic53—44no—
85C4×Dic53—82no—
86C2×#88322—44no—
87C2×#88322—42no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—48no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—81no—
101C23×C53—81no—
102C23×C53—84no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53#12011yescharacteristic
105C23×C53#12311yescharacteristic
106C23×C53#13512yes—
107C23×C53AGL(1,ℤ/8)12yes—
108C2×C4×C53—24no—
109C2×C4×C53—22no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—81no—
122C2×C4×C53—84no—
123C2×C4×C53—82no—
124C2×C4×C53—82no—
125C2×C4×C53C4×D412yes—
126C2×C4×C53#12112yes—
127C2×Dic53—44no—
128C2×Dic53—44no—
129C2×Dic53—44no—
130#88322—82no—
131#9028—531nomaximal, Sylow
132C22×C53—24no—
133C22×C53—24no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—44no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—82no—
143C22×C53—84no—
144C22×C53—84no—
145C22×C53#38611yescharacteristic
146C22×C53#41611yescharacteristic
147C22×C53#41811yescharacteristic
148C22×C53#44912yes—
149C22×C53#45512yes—
150C22×C53#45812yes—
151C22×C53#48212yes—
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—42no—
155C4×C53—82no—
156C4×C53—82no—
157Dic53—84no—
158C2×#397D5311yescharacteristic
159#2312—531no—
160#1823—532no—
161C2×#323—531no—
162#1294—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—22no—
166C2×C53—82no—
167C2×C53#161214yes—
168C2×C53#187511yescharacteristic
169C2×C53#188511yescharacteristic
170C2×C53#189011yescharacteristic
171C22×C4⋊C4C2×D5311yescharacteristic
172C22×C22⋊C4—21no—
173C22×C22⋊C4C2×D5311yescharacteristic
174C2×C22⋊C8—532no—
175#397—22no—
176#397Dic5314yescomplemented
177#388—1064no—
178#380—532no—
179#379—1061no—
180C4×C22⋊C4—1061no—
181C2×#95—21no—
182C2×#95C2×D5311yescharacteristic
183#340—1064no—
184#323—1062no—
185C53#902811yescharacteristic, complemented, Sylow
186C25C22⋊D5311yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C22⋊D5311yescharacteristic
190C23×C4C22×D5311yescharacteristic
191C22×C8—1062no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—534no—
196C2×C4⋊C4C2×Dic5312yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—1062no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4—1062no—
204C2×C22⋊C4C2×Dic5312yes—
205C2×C22⋊C4C22⋊D5312yes—
206C2×C42—1062no—
207C22⋊C8—1062no—
208C22⋊C8—1062no—
209#95—21no—
210#95—24no—
211#95—1062no—
212#95—1062no—
213#95C2×Dic5312yes—
214C24—21no—
215C24—22no—
216C24—21no—
217C24—22no—
218C24—42no—
219C24—42no—
220C24C22⋊Dic5312yes—
221C24C2×C22⋊D5311yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1064no—
236C22×C4C22⋊Dic5312yes—
237C22×C4D4×D5312yes—
238C22×C4#1171994712yes—
239C22×C4C22×Dic5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241C2×C8—2124no—
242C2×C8—2122no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—2124no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—48no—
249C22⋊C4—4242no—
250C22⋊C4—4242no—
251C22⋊C4—2122no—
252C22⋊C4—2122no—
253C42—4242no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—48no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—81no—
267C23—81no—
268C23—84no—
269C23C2×C22⋊Dic5311yescharacteristic
270C23#1313318511yescharacteristic
271C23#1313319811yescharacteristic
272C23#1313323712yes—
273C23#1313324312yes—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—84no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—2124no—
292C2×C4—2124no—
293C2×C4—2124no—
294C2×C4D4×Dic5312yes—
295C2×C4#1313318912yes—
296C8—4242no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—84no—
309C22—84no—
310C22#1681050311yescharacteristic
311C22#1681061911yescharacteristic
312C22#1681062811yescharacteristic
313C22#1681076612yes—
314C22#1681080012yes—
315C22#1681081212yes—
316C22#1681094412yes—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—82no—
322C4—4244no—
323C2—21no—
324C2—21no—
325C2—22no—
326C2—82no—
327C2#2027372714yes—
328C2#2027503811yescharacteristic
329C2#2027510411yescharacteristic
330C2#2027512911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#397C2
socleC23×C53#120
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenterC2×#397D53
hypocenterC53#9028
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. #20274801C2semidirect
4. #20272594C2semidirect
17. C53×#397C4semidirect
176. #397Dic53semidirect
185. C53#9028semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202748012
C2×#168103472
#202725942
C2×C53×#3972
#902853