Finite Group Databasebeta

← #27285034

Subgroups — #27285034

Order 13568.

512 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 85 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#20284833C212yesmaximal, complemented
2#20275654C211yesmaximal, characteristic, complemented
3#20273326C212yesmaximal, complemented
4#20272585C211yesmaximal, characteristic, complemented
5C53×#3245C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7#16810934—22no—
8#16810915—22no—
9#16810820—22no—
10#16810761—22no—
11#16810723—24no—
12#16810680C2212yes—
13C2×#13133128C2211yescharacteristic
14#16810377—22no—
15#16810363—24no—
16C53×#532C2212yes—
17C53×#474—22no—
18C53×#466—22no—
19C53×#459—22no—
20C2×C4⋊C8×C53C2211yescharacteristic
21C22×Dic106D411yescharacteristic
22C2×#11719923—24no—
23C22×#88322—21no—
24C4×Dic106—22no—
25C4×Dic106—22no—
26C2×C4⋊Dic53D411yescharacteristic
27C2×#11719919—21no—
28C2×C4×Dic53—21no—
29#13133150—24no—
30C4⋊#88322—22no—
31#13133140—24no—
32#13133138—24no—
33C22⋊#88322—22no—
34#13133128—22no—
35#13133128D414yes—
36C2×#41×C53—22no—
37C2×D8×C53—22no—
38C22×C8×C53D412yes—
39C4⋊Q8×C53D412yes—
40#127×C53D412yes—
41#124×C53—22no—
42#121×C53—22no—
43#121×C53—22no—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46C2×C42×C53C2311yescharacteristic
47#107×C53—22no—
48C4⋊C8×C53—22no—
49#102×C53—24no—
50#41×C53—84no—
51C2×Q8×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53#4111yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C2×Dic106—22no—
63C2×Dic106—42no—
64D8×C53—82no—
65C2×C8×C53—22no—
66C2×C8×C53—42no—
67C4⋊C4×C53—22no—
68C4⋊C4×C53—22no—
69C22⋊C4×C53—42no—
70C22⋊C4×C53—42no—
71C42×C53C2×D412yes—
72C42×C53C2×D412yes—
73#11719923—48no—
74C4⋊Dic53—22no—
75#11719919—42no—
76C4×Dic53—41no—
77C4×Dic53—41no—
78C2×#88322—22no—
79C2×#88322—24no—
80C2×#88322—41no—
81C2×#88322—41no—
82C23×C53—42no—
83C23×C53#12111yescharacteristic
84Q8×C53—82no—
85D4×C53—44no—
86D4×C53—42no—
87D4×C53—42no—
88D4×C53—82no—
89D4×C53—84no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—84no—
96C2×C4×C53#12012yes—
97C2×C4×C53#12011yescharacteristic
98C2×C4×C53#12112yes—
99C2×C4×C53#12711yescharacteristic
100C2×Dic53—22no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—42no—
105Dic106—24no—
106Dic106—42no—
107C8×C53—42no—
108#88322—42no—
109#88322—44no—
110#10028—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—21no—
113C22×C53—82no—
114C22×C53—82no—
115C22×C53—82no—
116C22×C53#39211yescharacteristic
117C22×C53#47212yes—
118C4×C53—21no—
119C4×C53—22no—
120C4×C53—22no—
121C4×C53—82no—
122C4×C53#45511yescharacteristic
123C4×C53#45611yescharacteristic
124Dic53—42no—
125Dic53—81no—
126#3245D5311yescharacteristic
127#3112—532no—
128#1985—531no—
129#1530—532no—
130#1290—531no—
131C2×C53—22no—
132C2×C53—162no—
133C2×C53#200711yescharacteristic
134C2×C53#201011yescharacteristic
135C2×C53#201811yescharacteristic
136#532C2×D5312yes—
137C2×C4×Q8—531no—
138#480—1062no—
139#477—1062no—
140#474—22no—
141#466—22no—
142#460—1062no—
143#459—22no—
144#448—1062no—
145#440—1064no—
146#432—532no—
147C2×C4⋊C8C2×D5311yescharacteristic
148C2×#97—531no—
149#336—1062no—
150#329—1064no—
151C53#1002811yescharacteristic, complemented, Sylow
152C2×#41—22no—
153C22×Q8—531no—
154C2×SD16—1064no—
155C2×D8—22no—
156C22×C8—1061no—
157C22×C8C22⋊D5312yes—
158C4⋊Q8C22⋊D5312yes—
159#127C22⋊D5312yes—
160#124—22no—
161#121—22no—
162#121—22no—
163C4×Q8—1062no—
164C4×Q8—1062no—
165C4×D4—22no—
166C4×D4—22no—
167C2×C4⋊C4—531no—
168C2×C4⋊C4—1061no—
169C2×C42—1061no—
170C2×C42C22×D5311yescharacteristic
171#107—22no—
172#107—1064no—
173C4⋊C8—22no—
174C4⋊C8—1062no—
175#103—1064no—
176#102—24no—
177#102—1064no—
178C22⋊C8—1062no—
179#97—534no—
180#97—1062no—
181#41—84no—
182C2×Q8—22no—
183C2×Q8—1062no—
184C2×Q8—2122no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—42no—
189C22×C4—42no—
190C22×C4—1061no—
191C22×C4—1061no—
192C22×C4D4×D5311yescharacteristic
193C22×C4#1171994711yescharacteristic
194C22×C4C2×C22⋊D5311yescharacteristic
195SD16—2128no—
196D8—82no—
197C2×C8—22no—
198C2×C8—42no—
199C2×C8—1062no—
200C2×C8—1064no—
201C2×C8—2121no—
202C2×C8—2121no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—1062no—
206C4⋊C4—2122no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—2121no—
210C42—2121no—
211C42D4×D5312yes—
212C42C2×C22⋊D5312yes—
213C23—42no—
214C23#1313318811yescharacteristic
215Q8—82no—
216Q8—1064no—
217Q8—2122no—
218D4—44no—
219D4—42no—
220D4—42no—
221D4—82no—
222D4—84no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—1062no—
230C2×C4—2121no—
231C2×C4—2122no—
232C2×C4—2121no—
233C2×C4—2121no—
234C2×C4#1313318512yes—
235C2×C4#1313318611yescharacteristic
236C2×C4#1313318912yes—
237C2×C4#1313321311yescharacteristic
238C8—42no—
239C8—2122no—
240C8—2124no—
241C22—21no—
242C22—21no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22#1681053111yescharacteristic
247C22#1681089012yes—
248C4—21no—
249C4—22no—
250C4—22no—
251C4—82no—
252C4—4241no—
253C4—2122no—
254C4#1681080111yescharacteristic
255C4#1681080611yescharacteristic
256C2—22no—
257C2—162no—
258C2#2027576211yescharacteristic
259C2#2027577911yescharacteristic
260C2#2027582211yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3245C2
socleC22×C53#392
hypercenter#3245D53
hypocenterC53#10028
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #20284833C2semidirect
2. #20275654C2semidirect
3. #20273326C2semidirect
4. #20272585C2semidirect
151. C53#10028semidirect

Maximal subgroups

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

subgroupindex
#202848332
#202756542
#202733262
#202725852
C53×#32452
#1002853