Finite Group Databasebeta

← #27284932

Subgroups — #27284932

Order 13568.

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

normal subgroups only

This group has 734 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#20284711C212yesmaximal, complemented
2#20277996C211yesmaximal, characteristic, complemented
3#20273139C212yesmaximal, complemented
4#20272544C211yesmaximal, characteristic, complemented
5C53×#1963C211yesmaximal, characteristic, complemented
6C2×C4×D212—22no—
7C2×C4×D212C2212yes—
8#16810934—22no—
9#16810900—22no—
10#16810815—22no—
11#16810756—22no—
12C4×#11719922—24no—
13#16810676C2212yes—
14#16810560—22no—
15#16810533—22no—
16#16810350—24no—
17C53×#532C2211yescharacteristic
18C53×#423C2211yescharacteristic
19C2×#104×C53—22no—
20C53×#376C2211yescharacteristic
21C22×D212—22no—
22C22×D212D412yes—
23C22×C4×D53—22no—
24C2×#11719922—24no—
25C4×D212—22no—
26C4×D212—22no—
27C4×D212—22no—
28C4×D212—22no—
29C4×D212—44no—
30C4×D212—44no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53D412yes—
33C2×#11719921—24no—
34C2×#11719921—22no—
35#13133150—24no—
36C4⋊#88322—22no—
37#13133138—24no—
38#13133137—24no—
39C22⋊#88322—22no—
40C4×#88322D414yescomplemented
41C2×#41×C53—21no—
42C2×M16×C53—21no—
43C4⋊Q8×C53D411yescharacteristic, complemented
44#127×C53D411yescharacteristic, complemented
45#124×C53—21no—
46#121×C53—22no—
47C4×D4×C53—22no—
48C2×C42×C53D412yes—
49C2×C42×C53C2311yescharacteristic
50C4⋊C8×C53—22no—
51#104×C53—44no—
52#95×C53—22no—
53C23×D53—22no—
54#41×C53—42no—
55#41×C53—42no—
56C2×Q8×C53—21no—
57C2×D4×C53—22no—
58C2×D4×C53—21no—
59C2×D4×C53—41no—
60C22×C4×C53—22no—
61C22×C4×C53—41no—
62C22×C4×C53C2×D412yes—
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—22no—
65C2×D212—22no—
66C2×D212—22no—
67C2×D212—44no—
68C2×D212—44no—
69C2×D212—48no—
70C2×C4×D53—42no—
71C2×C4×D53—42no—
72C2×C4×D53—44no—
73C2×C4×D53—44no—
74M16×C53—42no—
75C2×C8×C53—41no—
76C4⋊C4×C53—22no—
77C22⋊C4×C53—42no—
78C42×C53—22no—
79C42×C53D814yescomplemented
80C42×C53C2×D411yescharacteristic
81C42×C53C2×D412yes—
82C42×C53#4111yescharacteristic
83#11719922—48no—
84#11719921—48no—
85#11719921—42no—
86#11719921—42no—
87C4⋊Dic53—22no—
88C4⋊Dic53—22no—
89C4⋊Dic53—44no—
90C2×#88322—24no—
91C23×C53—41no—
92C23×C53#12011yescharacteristic
93C22×D53—24no—
94C22×D53—44no—
95C22×D53—42no—
96C22×D53—42no—
97C22×D53—44no—
98Q8×C53—42no—
99D4×C53—44no—
100D4×C53—42no—
101D4×C53—82no—
102C2×C4×C53—24no—
103C2×C4×C53—42no—
104C2×C4×C53—44no—
105C2×C4×C53—41no—
106C2×C4×C53—81no—
107C2×C4×C53—81no—
108C2×C4×C53#12012yes—
109C2×C4×C53#12112yes—
110C2×C4×C53#12111yescharacteristic
111C2×C4×C53#12711yescharacteristic
112C2×C4×C53C2×D812yes—
113C2×C4×C53AGL(1,ℤ/8)12yes—
114C2×Dic53—42no—
115C2×Dic53—44no—
116D212—24no—
117D212—24no—
118D212—42no—
119D212—42no—
120D212—416no—
121C4×D53—84no—
122C4×D53—82no—
123C4×D53—82no—
124C8×C53—81no—
125#88322—44no—
126#10012—531nomaximal, Sylow
127C22×C53—22no—
128C22×C53—82no—
129C22×C53—81no—
130C22×C53#39211yescharacteristic
131C22×C53#44712yes—
132C2×D53—48no—
133C2×D53—44no—
134C2×D53—84no—
135C2×D53—82no—
136C2×D53—82no—
137C4×C53—22no—
138C4×C53—22no—
139C4×C53—44no—
140C4×C53—81no—
141C4×C53#45912yes—
142C4×C53#49311yescharacteristic
143C4×C53#49611yescharacteristic
144Dic53—82no—
145#3100—532no—
146#2397—531no—
147#1963D5311yescharacteristic, complemented
148#1486—532no—
149#1274—531no—
150C2×C53—22no—
151C2×C53—161no—
152C2×C53#201012yes—
153C2×C53#201111yescharacteristic
154D53—84no—
155#532C2×D5311yescharacteristic
156C2×C4×D4—532no—
157C2×C4×D4—1062no—
158#480—1062no—
159#474—1062no—
160#459—1062no—
161#447—1062no—
162C4×D8—1064no—
163#431—532no—
164#423C2×D5311yescharacteristic
165C2×#104—22no—
166#399—1062no—
167#392—1062no—
168#376C2×D5311yescharacteristic
169D4⋊C8—1064no—
170C53#1001211yescharacteristic, complemented, Sylow
171C2×#41—21no—
172C22×D4—532no—
173C22×D4—1062no—
174C23×C4—1062no—
175C2×D8—1064no—
176C2×M16—21no—
177C4⋊Q8D21211yescharacteristic, complemented
178#127D21211yescharacteristic, complemented
179#124—21no—
180#121—22no—
181C4×D4—22no—
182C4×D4—1062no—
183C4×D4—1062no—
184C4×D4—1062no—
185C4×D4—1062no—
186C4×D4—2124no—
187C4×D4—2124no—
188C2×C4⋊C4—532no—
189C2×C4⋊C4—1062no—
190C2×C22⋊C4—1064no—
191C2×C22⋊C4—1062no—
192C2×C42C22⋊D5312yes—
193C2×C42C22×D5311yescharacteristic
194#107—1064no—
195C4⋊C8—22no—
196C4⋊C8—1062no—
197#104—44no—
198#102—1064no—
199#102—1064no—
200C22⋊C8—1062no—
201C4×C8—534no—
202#95—22no—
203C24—1062no—
204#41—42no—
205#41—42no—
206C2×Q8—21no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—1062no—
211C2×D4—1062no—
212C2×D4—2124no—
213C2×D4—2124no—
214C2×D4—2128no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—1062no—
218C22×C4—2122no—
219C22×C4—2122no—
220C22×C4—2124no—
221C22×C4—2124no—
222C22×C4D4×D5312yes—
223C22×C4C2×C22⋊D5311yescharacteristic
224D8—2128no—
225M16—42no—
226C2×C8—41no—
227C2×C8—1064no—
228C4⋊C4—22no—
229C4⋊C4—1062no—
230C4⋊C4—1062no—
231C4⋊C4—2124no—
232C22⋊C4—42no—
233C22⋊C4—2128no—
234C22⋊C4—2122no—
235C22⋊C4—2122no—
236C42—22no—
237C42#1171992214yescomplemented
238C42C2×D21211yescharacteristic
239C42#1171994511yescharacteristic
240C42D4×D5312yes—
241C23—41no—
242C23—1064no—
243C23—2122no—
244C23—2124no—
245C23—2124no—
246C23—2122no—
247C23#1313318511yescharacteristic
248Q8—42no—
249D4—44no—
250D4—42no—
251D4—82no—
252D4—1064no—
253D4—1064no—
254D4—2122no—
255D4—2122no—
256D4—21216no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—4242no—
264C2×C4—4244no—
265C2×C4—4242no—
266C2×C4—2124no—
267C2×C4—2122no—
268C2×C4#1313318712yes—
269C2×C4#1313319012yes—
270C2×C4#1313319111yescharacteristic
271C2×C4#1313321311yescharacteristic
272C2×C4C2×#1171992212yes—
273C2×C4#1313324212yes—
274C8—81no—
275C8—2124no—
276C22—22no—
277C22—82no—
278C22—81no—
279C22—4244no—
280C22—4242no—
281C22—4242no—
282C22—2128no—
283C22—2124no—
284C22#1681053211yescharacteristic
285C22#1681075612yes—
286C4—22no—
287C4—22no—
288C4—44no—
289C4—81no—
290C4—4242no—
291C4#1681081812yes—
292C4#1681099911yescharacteristic
293C4#1681101311yescharacteristic
294C2—22no—
295C2—161no—
296C2—4244no—
297C2#2027578112yes—
298C2#2027578511yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1963C2
socleC22×C53#392
hypercenter#1963D53
hypocenterC53#10012
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
1. #20284711C2semidirect
2. #20277996C2semidirect
3. #20273139C2semidirect
4. #20272544C2semidirect
5. C53×#1963C2semidirect
40. C4×#88322D4semidirect
43. C4⋊Q8×C53D4semidirect
44. #127×C53D4semidirect
79. C42×C53D8semidirect
147. #1963D53semidirect
170. C53#10012semidirect
177. C4⋊Q8D212semidirect
178. #127D212semidirect
237. C42#11719922semidirect

Maximal subgroups

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

subgroupindex
#202847112
#202779962
#202731392
#202725442
C53×#19632
#1001253