Finite Group Databasebeta

← #27302419

Subgroups — #27302419

Order 13568.

472 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16810422C214yesmaximal
2#20274711C211yesmaximal, characteristic
3C4×#13133136C211yesmaximal, characteristic
4C53×#2546C211yesmaximal, characteristic
5#16810647—21no—
6C2×#13133136C2212yes—
7C2×#13133136C2212yes—
8C4×#11719917—21no—
9#16810422—28no—
10#16810392—21no—
11#16810392—21no—
12C2×C4⋊Q8×C53C411yescharacteristic
13C2×C4⋊Q8×C53C2212yes—
14C2×C4×Q8×C53C2211yescharacteristic
15C53×#398—21no—
16C53×#398—21no—
17C53×#391C412yes—
18C53×#389—21no—
19C53×#389—21no—
20C4×C4⋊C4×C53C411yescharacteristic
21C2×#11719917—22no—
22C2×#11719917—22no—
23C4⋊#88322—41no—
24C4⋊#88322—41no—
25#13133136—24no—
26#13133136—22no—
27#13133136D414yes—
28#13133128—41no—
29C4×#88322—41no—
30C22×Q8×C53C2×C412yes—
31C22×Q8×C53C2311yescharacteristic
32C4⋊Q8×C53—24no—
33C4⋊Q8×C53—24no—
34C4⋊Q8×C53—41no—
35C4⋊Q8×C53—41no—
36C4×Q8×C53—41no—
37C4×Q8×C53—41no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C42×C53C2×C412yes—
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—21no—
51#95×C53—22no—
52#95×C53—22no—
53C2×Q8×C53—24no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×Q8×C53—42no—
57C2×Q8×C53—42no—
58C2×Q8×C53C2×D412yes—
59C2×Q8×C53#4112yes—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C4⋊C4×C53—24no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—44no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—24no—
86#11719917—24no—
87#11719917—42no—
88#11719917—42no—
89#11719917—42no—
90C2×#88322—42no—
91C2×#88322—42no—
92C23×C53C2×C22⋊C411yescharacteristic
93Q8×C53—42no—
94Q8×C53—42no—
95Q8×C53—84no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—24no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53#9911yescharacteristic
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53C4×D412yes—
118C2×C4×C53#12112yes—
119C2×C4×C53#12411yescharacteristic
120C2×C4×C53#12711yescharacteristic
121#88322—44no—
122#88322—82no—
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—44no—
135C4×C53—41no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53—82no—
139C4×C53#35612yescomplemented
140#2546D5311yescharacteristic
141C2×#356—534no—
142#1801—531no—
143C4×#101—531no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53#196411yescharacteristic
147C2×C53C2×#35611yescharacteristic
148C2×C53#213511yescharacteristic
149C2×C4⋊Q8Dic5311yescharacteristic
150C2×C4⋊Q8C2×D5312yes—
151C2×C4×Q8C2×D5311yescharacteristic
152#423—1061no—
153C2×#101—532no—
154C2×#101—532no—
155C4×M16—1061no—
156#398—21no—
157#398—21no—
158#391Dic5312yes—
159#389—21no—
160#389—21no—
161C4×C4⋊C4Dic5311yescharacteristic
162#356—1068no—
163#343—1061no—
164#343—1061no—
165C53#1322711yescharacteristic, complemented, Sylow
166C22×Q8C2×Dic5312yes—
167C22×Q8C22×D5311yescharacteristic
168C2×M16—1062no—
169C2×M16—1062no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C2×Dic5311yescharacteristic
182C2×C4⋊C4C22⋊D5311yescharacteristic
183C2×C4⋊C4C22⋊D5311yescharacteristic
184C2×C42C2×Dic5312yes—
185C2×C42C2×Dic5311yescharacteristic
186C2×C42C22⋊D5311yescharacteristic
187C2×C42C22⋊D5311yescharacteristic
188C4⋊C8—2121no—
189C4⋊C8—2121no—
190#101—534no—
191#101—1062no—
192#101—1064no—
193#97—2121no—
194C4×C8—2121no—
195#95—21no—
196#95—22no—
197#95—22no—
198C2×Q8—24no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8D4×D5312yes—
204C2×Q8#1171994712yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4C22⋊Dic5311yescharacteristic
208C22×C4C22⋊Dic5311yescharacteristic
209C22×C4C22⋊Dic5311yescharacteristic
210C22×C4C22⋊Dic5311yescharacteristic
211C22×C4C22×Dic5311yescharacteristic
212C22×C4C2×C22⋊D5311yescharacteristic
213C22×C4C2×C22⋊D5311yescharacteristic
214M16—1064no—
215M16—2122no—
216M16—2122no—
217M16—2122no—
218C2×C8—2122no—
219C2×C8—2122no—
220C4⋊C4—24no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—24no—
237C23C2×C22⋊Dic5311yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—84no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4#1313313211yescharacteristic
261C2×C4#1313313211yescharacteristic
262C2×C4D4×Dic5312yes—
263C2×C4#1313318812yes—
264C2×C4#1313320411yescharacteristic
265C2×C4#1313321311yescharacteristic
266C8—4242no—
267C8—2124no—
268C22—21no—
269C22—21no—
270C22#1681052611yescharacteristic
271C22C2×#1313313211yescharacteristic
272C22#1681059711yescharacteristic
273C4—21no—
274C4—22no—
275C4—22no—
276C4—21no—
277C4—21no—
278C4—44no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—82no—
283C4#1681042212yescomplemented
284C2—21no—
285C2—21no—
286C2#2027555211yescharacteristic
287C2C2×#1681042211yescharacteristic
288C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2546C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#2546D53
hypocenterC53#13227
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
139. C4×C53#356semidirect
165. C53#13227semidirect
283. C4#16810422semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
#202747112
C4×#131331362
C53×#25462
#1322753