Finite Group Databasebeta

← #27302417

Subgroups — #27302417

Order 13568.

472 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 103 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#20279700C211yesmaximal, characteristic
2C2×#16810421C214yesmaximal
3C4×#13133136C211yesmaximal, characteristic
4C53×#1801C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6C2×#13133215C2212yes—
7C2×#13133136C2212yes—
8C4×#11719917—21no—
9#16810553—21no—
10#16810553—21no—
11#16810528C412yes—
12#16810521—21no—
13#16810521—21no—
14C4×C4⋊Dic53C411yescharacteristic
15#16810421—28no—
16#16810392—21no—
17C2×C4×Q8×C53C2211yescharacteristic
18C53×#423—21no—
19C2×#101×C53C2212yes—
20C53×#343—21no—
21C22×Dic106C2×C412yes—
22C2×#11719917—22no—
23#13133217—41no—
24#13133217—41no—
25#13133215—24no—
26#13133215—24no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×#11719919—22no—
31C2×#11719919—22no—
32C2×C4×Dic53C2×C412yes—
33#13133136—22no—
34#13133136D414yes—
35#13133128—41no—
36C4×#88322—41no—
37#13133124—22no—
38#13133124—22no—
39#13133124—21no—
40C22×Q8×C53C2311yescharacteristic
41C2×M16×C53—22no—
42C4×Q8×C53—41no—
43C4×Q8×C53—41no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C4⋊C8×C53—41no—
51C4⋊C8×C53—41no—
52#101×C53—24no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×Q8×C53C2×D412yes—
56C2×Q8×C53#4112yes—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C2×Dic106—24no—
67C2×Dic106—42no—
68C2×Dic106—42no—
69M16×C53—42no—
70M16×C53—42no—
71C2×C8×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C4⋊Dic53—24no—
82C4⋊Dic53—42no—
83C4⋊Dic53—41no—
84C4⋊Dic53—41no—
85#11719919—44no—
86#11719919—42no—
87#11719919—42no—
88C4×Dic53—24no—
89#11719917—24no—
90#11719917—42no—
91C2×#88322—42no—
92C23×C53C2×C22⋊C411yescharacteristic
93Q8×C53—42no—
94Q8×C53—42no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
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—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53C4×D412yes—
111C2×C4×C53#12112yes—
112C2×C4×C53#12411yescharacteristic
113C2×C4×C53#12711yescharacteristic
114C2×Dic53—24no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120Dic106—84no—
121C8×C53—82no—
122#88322—44no—
123#13227—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53#39011yescharacteristic
127C22×C53C2×#9911yescharacteristic
128C22×C53#41011yescharacteristic
129C4×C53—21no—
130C4×C53—22no—
131C4×C53—21no—
132C4×C53—22no—
133C4×C53—21no—
134C4×C53—41no—
135C4×C53—41no—
136C4×C53—41no—
137C4×C53#35612yescomplemented
138Dic53—44no—
139Dic53—82no—
140#2546—531no—
141C2×#356—534no—
142#1801D5311yescharacteristic
143C4×#101—531no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53#196411yescharacteristic
147C2×C53C2×#35611yescharacteristic
148C2×C53#213511yescharacteristic
149C2×C4⋊Q8—532no—
150C2×C4⋊Q8—531no—
151C2×C4×Q8C2×D5311yescharacteristic
152#423—21no—
153C2×#101—532no—
154C2×#101C2×D5312yes—
155C4×M16—1061no—
156#398—1061no—
157#398—1061no—
158#391—532no—
159#389—1061no—
160#389—1061no—
161C4×C4⋊C4—531no—
162#356—1068no—
163#343—21no—
164#343—1061no—
165C53#1322711yescharacteristic, complemented, Sylow
166C22×Q8—532no—
167C22×Q8C22×D5311yescharacteristic
168C2×M16—22no—
169C2×M16—1062no—
170C4⋊Q8—1064no—
171C4⋊Q8—1064no—
172C4⋊Q8—2121no—
173C4⋊Q8—2121no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—1061no—
177C2×C4⋊C4—1062no—
178C2×C4⋊C4—1062no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4C4×D5311yescharacteristic
182C2×C4⋊C4D21211yescharacteristic
183C2×C4⋊C4C22⋊D5311yescharacteristic
184C2×C42—532no—
185C2×C42C4×D5311yescharacteristic
186C2×C42D21211yescharacteristic
187C2×C42C22⋊D5311yescharacteristic
188C4⋊C8—41no—
189C4⋊C8—41no—
190#101—24no—
191#101—534no—
192#101—1062no—
193#97—2121no—
194C4×C8—2121no—
195#95—1061no—
196#95—1062no—
197#95—1062no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—1064no—
201C2×Q8—2122no—
202C2×Q8—2122no—
203C2×Q8D4×D5312yes—
204C2×Q8#1171994912yes—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4#1171992111yescharacteristic
208C22×C4#1171992111yescharacteristic
209C22×C4#1171992111yescharacteristic
210C22×C4#1171992111yescharacteristic
211C22×C4C2×C4×D5311yescharacteristic
212C22×C4C2×D21211yescharacteristic
213C22×C4C2×C22⋊D5311yescharacteristic
214M16—42no—
215M16—42no—
216M16—1064no—
217M16—2122no—
218C2×C8—42no—
219C2×C8—2122no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—1064no—
224C4⋊C4—2122no—
225C4⋊C4—2121no—
226C4⋊C4—2121no—
227C4⋊C4—2124no—
228C4⋊C4—2122no—
229C4⋊C4—2122no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—1064no—
237C23C2×#1171992111yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—4244no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1064no—
255C2×C4—2122no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2122no—
259C2×C4—2122no—
260C2×C4#1313313111yescharacteristic
261C2×C4#1313313111yescharacteristic
262C2×C4#1313317912yes—
263C2×C4#1313319212yes—
264C2×C4#1313320611yescharacteristic
265C2×C4#1313321311yescharacteristic
266C8—82no—
267C8—2124no—
268C22—21no—
269C22—21no—
270C22#1681052411yescharacteristic
271C22C2×#1313313111yescharacteristic
272C22#1681059611yescharacteristic
273C4—21no—
274C4—22no—
275C4—22no—
276C4—21no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—4242no—
282C4—2124no—
283C4#1681042112yescomplemented
284C2—21no—
285C2—21no—
286C2#2027555011yescharacteristic
287C2C2×#1681042111yescharacteristic
288C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#1801C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#1801D53
hypocenterC53#13227
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
137. C4×C53#356semidirect
165. C53#13227semidirect
283. C4#16810421semidirect

Maximal subgroups

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

subgroupindex
#202797002
C2×#168104212
C4×#131331362
C53×#18012
#1322753