Finite Group Databasebeta

← #27286339

Subgroups — #27286339

Order 13568.

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

normal subgroups only

This group has 542 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#20285653C212yesmaximal
2#20275539C211yesmaximal, characteristic
3#20273279C212yesmaximal, complemented
4#20272560C211yesmaximal, characteristic, complemented
5C53×#3162C211yesmaximal, characteristic, complemented
6C2×#13133214C2211yescharacteristic
7#16810997—22no—
8#16810971—22no—
9#16810823—24no—
10#16810773—24no—
11#16810676C2212yes—
12C2×#13133139—22no—
13#16810489C2211yescharacteristic
14#16810360—24no—
15C53×#533C2212yes—
16C53×#501—22no—
17C53×#501—22no—
18C53×#427—22no—
19C2×#97×C53C2211yescharacteristic
20C22×D212—21no—
21C22×D212—22no—
22C2×#11719924—28no—
23#13133214—22no—
24#13133214—21no—
25#13133214D412yes—
26C2×C4⋊Dic53—21no—
27C4⋊#88322—22no—
28#13133139—44no—
29#13133137—28no—
30C22⋊#88322D414yes—
31C4×#88322—22no—
32#13133124—22no—
33C22×C8×C53D412yes—
34C4⋊Q8×C53D412yes—
35C4⋊Q8×C53D412yes—
36#125×C53—22no—
37C22⋊Q8×C53—22no—
38C22⋊Q8×C53—22no—
39C4×Q8×C53—22no—
40C4×Q8×C53—22no—
41#117×C53—22no—
42C2×C42×C53C2311yescharacteristic
43#107×C53—24no—
44#106×C53—22no—
45#106×C53—22no—
46#97×C53—22no—
47C23×D53—41no—
48C2×Q8×C53—22no—
49C2×Q8×C53—22no—
50C22×C4×C53C2×D412yes—
51C22×C4×C53C2×D411yescharacteristic
52C22×Dic53—41no—
53C2×D212—24no—
54C2×D212—22no—
55C2×D212—42no—
56C2×D212—42no—
57C2×D212—41no—
58C2×D212—42no—
59C2×D212—84no—
60C2×C8×C53—22no—
61C2×C8×C53—22no—
62C2×C8×C53SD1614yescomplemented
63C4⋊C4×C53—22no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—22no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C22⋊C4×C53—42no—
70C42×C53—42no—
71C42×C53C2×D411yescharacteristic
72C42×C53C2×D412yes—
73C42×C53#4111yescharacteristic
74#11719924—88no—
75C4⋊Dic53—42no—
76C2×#88322—24no—
77C23×C53#12711yescharacteristic
78C22×D53—42no—
79C22×D53—84no—
80C22×D53—81no—
81C22×D53—81no—
82Q8×C53—44no—
83Q8×C53—82no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53#12012yes—
88C2×C4×C53#12011yescharacteristic
89C2×C4×C53#12111yescharacteristic
90C2×C4×C53#12112yes—
91C2×C4×C53C2×SD1612yes—
92C2×C4×C53#13712yes—
93C2×Dic53—82no—
94C2×Dic53—81no—
95D212—48no—
96D212—44no—
97D212—84no—
98D212—82no—
99C8×C53—24no—
100#88322—44no—
101#10220—531nomaximal, Sylow
102C22×C53—22no—
103C22×C53#39211yescharacteristic
104C22×C53#49211yescharacteristic
105C22×C53#49711yescharacteristic
106C2×D53—84no—
107C2×D53—82no—
108C2×D53—162no—
109C2×D53—162no—
110C4×C53—22no—
111C4×C53—22no—
112C4×C53—82no—
113C4×C53—82no—
114C4×C53#44812yes—
115C4×C53#46112yes—
116Dic53—161no—
117#3208—532no—
118#3162D5311yescharacteristic, complemented
119#1961—531no—
120#1518—532no—
121#1281—531no—
122C2×C53—21no—
123C2×C53—21no—
124C2×C53#200812yes—
125C2×C53#201311yescharacteristic
126D53—162no—
127#533C2×D5312yes—
128C2×#127—531no—
129#501—22no—
130#501—22no—
131#492—1062no—
132#486—1062no—
133#460—1064no—
134#450—1064no—
135#431—532no—
136#427—22no—
137C2×#102—1062no—
138C2×#97C2×D5311yescharacteristic
139#382—531no—
140#328—1064no—
141C53#1022011yescharacteristic, complemented, Sylow
142C22×D4—1062no—
143C22×D4—1061no—
144C2×SD16—1068no—
145C22×C8C22⋊D5312yes—
146C4⋊Q8C22⋊D5312yes—
147C4⋊Q8C22⋊D5312yes—
148#127—532no—
149#127—1061no—
150#127—1062no—
151#125—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156#117—22no—
157C2×C4⋊C4—1061no—
158C2×C42C22×D5311yescharacteristic
159#107—24no—
160#106—22no—
161#106—22no—
162C4⋊C8—1062no—
163#102—1068no—
164#102—2124no—
165C22⋊C8—534no—
166#97—22no—
167C4×C8—1062no—
168#95—1062no—
169C24—2121no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×D4—1064no—
173C2×D4—1062no—
174C2×D4—4244no—
175C2×D4—2121no—
176C2×D4—2122no—
177C2×D4—2122no—
178C2×D4—2122no—
179C22×C4—2121no—
180C22×C4D4×D5312yes—
181C22×C4C2×C22⋊D5311yescharacteristic
182SD16—4248no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—1064no—
186C2×C8#1171992414yescomplemented
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—2122no—
194C22⋊C4—42no—
195C42—42no—
196C42D4×D5311yescharacteristic
197C42#1171994711yescharacteristic
198C42C2×C22⋊D5312yes—
199C23—4244no—
200C23—4241no—
201C23—4241no—
202C23—2122no—
203C23#1313321311yescharacteristic
204Q8—44no—
205Q8—82no—
206D4—4244no—
207D4—4242no—
208D4—2128no—
209D4—2124no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—4242no—
214C2×C4—4241no—
215C2×C4#1313318512yes—
216C2×C4#1313318611yescharacteristic
217C2×C4#1313318811yescharacteristic
218C2×C4#1313318912yes—
219C2×C4C2×#1171992412yes—
220C2×C4#1313324812yes—
221C8—24no—
222C8—2124no—
223C22—22no—
224C22—8482no—
225C22—8482no—
226C22—4244no—
227C22—4242no—
228C22#1681053111yescharacteristic
229C22#1681099711yescharacteristic
230C22#1681101511yescharacteristic
231C4—22no—
232C4—22no—
233C4—82no—
234C4—82no—
235C4—8481no—
236C4#1681076212yes—
237C4#1681082512yes—
238C2—21no—
239C2—21no—
240C2—8482no—
241C2#2027576512yes—
242C2#2027579811yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3162C2
socleC22×C53#392
prime radicalC2×#13133214C22
hypercenter#3162D53
hypocenterC53#10220
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
3. #20273279C2semidirect
4. #20272560C2semidirect
5. C53×#3162C2semidirect
62. C2×C8×C53SD16semidirect
118. #3162D53semidirect
141. C53#10220semidirect
186. C2×C8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202755392
#202732792
#202725602
C53×#31622
#1022053