Finite Group Databasebeta

← #27284933

Subgroups — #27284933

Order 13568.

734 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 105 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#20284715C212yesmaximal, complemented
2#20275547C211yesmaximal, characteristic, complemented
3#20273140C212yesmaximal, complemented
4#20272544C211yesmaximal, characteristic, complemented
5C53×#2397C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810936—22no—
8#16810903—22no—
9#16810818—22no—
10#16810757—22no—
11C4×#11719922—24no—
12#16810676C2212yes—
13#16810647C2211yescharacteristic
14C2×#13133145—22no—
15D4⋊#88322—24no—
16C2×C4×D4×C53—22no—
17C2×C4×D4×C53C2212yes—
18C53×#399—22no—
19C53×#392—22no—
20C53×#376C2211yescharacteristic
21C2×#11719945—21no—
22C2×#11719922—24no—
23C2×#11719917—21no—
24#13133217D411yescharacteristic, complemented
25#13133214D411yescharacteristic, complemented
26#13133205—21no—
27#13133191—22no—
28C4×D212—22no—
29#13133150—24no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32#13133145—44no—
33#13133138—24no—
34#13133137—24no—
35C22⋊#88322—22no—
36C4×#88322D414yescomplemented
37C22×D4×C53—22no—
38C22×D4×C53D412yes—
39C23×C4×C53—22no—
40C4×D4×C53—22no—
41C4×D4×C53—22no—
42C4×D4×C53—22no—
43C4×D4×C53—22no—
44C4×D4×C53—44no—
45C4×D4×C53—44no—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53D412yes—
48C2×C22⋊C4×C53—24no—
49C2×C22⋊C4×C53—22no—
50C2×C42×C53D412yes—
51C2×C42×C53C2311yescharacteristic
52#95×C53—22no—
53C24×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53—22no—
56C2×D4×C53—44no—
57C2×D4×C53—44no—
58C2×D4×C53—48no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—44no—
64C22×C4×C53—44no—
65C22×C4×C53C2×D412yes—
66C22×C4×C53C2×D411yescharacteristic
67C2×C22⋊D53—41no—
68#11719945—42no—
69#11719945—42no—
70C2×D212—22no—
71C2×D212—21no—
72C2×C4×D53—41no—
73C2×Dic106—21no—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—44no—
77C22⋊C4×C53—48no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—42no—
80C42×C53—22no—
81C42×C53D814yescomplemented
82C42×C53C2×D411yescharacteristic
83C42×C53C2×D412yes—
84C42×C53#4111yescharacteristic
85#11719922—48no—
86#11719921—42no—
87C4⋊Dic53—22no—
88#11719917—42no—
89C2×#88322—24no—
90C2×#88322—41no—
91C23×C53—24no—
92C23×C53—44no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—44no—
96C23×C53#12011yescharacteristic
97C22×D53—41no—
98D4×C53—24no—
99D4×C53—24no—
100D4×C53—42no—
101D4×C53—42no—
102D4×C53—416no—
103C2×C4×C53—24no—
104C2×C4×C53—42no—
105C2×C4×C53—44no—
106C2×C4×C53—42no—
107C2×C4×C53—44no—
108C2×C4×C53—84no—
109C2×C4×C53—82no—
110C2×C4×C53—82no—
111C2×C4×C53#12012yes—
112C2×C4×C53#12111yescharacteristic
113C2×C4×C53#12112yes—
114C2×C4×C53#12711yescharacteristic
115C2×C4×C53C2×D812yes—
116C2×C4×C53AGL(1,ℤ/8)12yes—
117C22⋊D53—82no—
118C2×Dic53—41no—
119D212—44no—
120D212—42no—
121C4×D53—81no—
122C4×D53—81no—
123Dic106—42no—
124#88322—44no—
125#88322—81no—
126#10012—531nomaximal, Sylow
127C22×C53—22no—
128C22×C53—48no—
129C22×C53—44no—
130C22×C53—84no—
131C22×C53—82no—
132C22×C53—82no—
133C22×C53#39211yescharacteristic
134C22×C53#44712yes—
135C2×D53—82no—
136C2×D53—81no—
137C4×C53—22no—
138C4×C53—22no—
139C4×C53—44no—
140C4×C53—82no—
141C4×C53#45912yes—
142C4×C53#49311yescharacteristic
143C4×C53#49611yescharacteristic
144Dic53—81no—
145#3100—532no—
146#2397D5311yescharacteristic, complemented
147#1963—531no—
148#1486—532no—
149#1274—531no—
150C2×C53—22no—
151C2×C53—84no—
152C2×C53#201012yes—
153C2×C53#201111yescharacteristic
154D53—161no—
155#532—531no—
156C2×C4×D4—22no—
157C2×C4×D4C2×D5312yes—
158#480—1062no—
159#474—1062no—
160#459—1062no—
161#447—1062no—
162C4×D8—1064no—
163#431—532no—
164#423—531no—
165C2×#104—1062no—
166#399—22no—
167#392—22no—
168#376C2×D5311yescharacteristic
169D4⋊C8—1064no—
170C53#1001211yescharacteristic, complemented, Sylow
171C2×#41—1061no—
172C22×D4—22no—
173C22×D4C22⋊D5312yes—
174C23×C4—22no—
175C2×D8—1064no—
176C2×M16—1061no—
177C4⋊Q8—531no—
178#127—531no—
179#124—1061no—
180#121—1062no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—1062no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊D5312yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C42C22⋊D5312yes—
193C2×C42C22×D5311yescharacteristic
194#107—1064no—
195C4⋊C8—1062no—
196C4⋊C8—1062no—
197#104—2124no—
198#102—1064no—
199#102—1064no—
200C22⋊C8—1062no—
201C4×C8—534no—
202#95—22no—
203C24—22no—
204#41—2122no—
205#41—2122no—
206C2×Q8—1061no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—48no—
212C2×D4—1062no—
213C2×D4—1061no—
214C2×D4—2121no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—2121no—
222C22×C4C2×C22⋊D5312yes—
223C22×C4C2×C22⋊D5311yescharacteristic
224D8—2128no—
225M16—2122no—
226C2×C8—1064no—
227C2×C8—2121no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—44no—
231C4⋊C4—1062no—
232C22⋊C4—48no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—2122no—
236C42—22no—
237C42#1171992214yescomplemented
238C42D4×D5311yescharacteristic
239C42D4×D5312yes—
240C42#1171994711yescharacteristic
241C23—24no—
242C23—44no—
243C23—42no—
244C23—44no—
245C23—42no—
246C23—2121no—
247C23#1313318611yescharacteristic
248Q8—2122no—
249D4—24no—
250D4—24no—
251D4—42no—
252D4—42no—
253D4—416no—
254D4—4242no—
255D4—2124no—
256D4—2122no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—84no—
264C2×C4—82no—
265C2×C4—4241no—
266C2×C4—4241no—
267C2×C4—2121no—
268C2×C4#1313318512yes—
269C2×C4#1313318811yescharacteristic
270C2×C4#1313318912yes—
271C2×C4#1313321311yescharacteristic
272C2×C4C2×#1171992212yes—
273C2×C4#1313324312yes—
274C8—4241no—
275C8—2124no—
276C22—22no—
277C22—48no—
278C22—44no—
279C22—84no—
280C22—82no—
281C22—82no—
282C22—4242no—
283C22—4241no—
284C22#1681053111yescharacteristic
285C22#1681075712yes—
286C4—22no—
287C4—22no—
288C4—44no—
289C4—82no—
290C4—4241no—
291C4#1681081512yes—
292C4#1681099911yescharacteristic
293C4#1681101011yescharacteristic
294C2—22no—
295C2—84no—
296C2—8481no—
297C2#2027578012yes—
298C2#2027578611yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2397C2
socleC22×C53#392
hypercenter#2397D53
hypocenterC53#10012
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #20284715C2semidirect
2. #20275547C2semidirect
3. #20273140C2semidirect
4. #20272544C2semidirect
5. C53×#2397C2semidirect
24. #13133217D4semidirect
25. #13133214D4semidirect
36. C4×#88322D4semidirect
81. C42×C53D8semidirect
146. #2397D53semidirect
170. C53#10012semidirect
237. C42#11719922semidirect

Maximal subgroups

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

subgroupindex
#202847152
#202755472
#202731402
#202725442
C53×#23972
#1001253