Finite Group Databasebeta

← #27285082

Subgroups — #27285082

Order 13568.

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

normal subgroups only

This group has 528 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#20285239C211yesmaximal, characteristic, complemented
2#20284723C212yesmaximal
3#20273117C212yesmaximal, complemented
4#20272546C211yesmaximal, characteristic, complemented
5C53×#1955C211yesmaximal, characteristic
6#16811180C2212yes—
7#16811032—22no—
8#16811032—22no—
9#16810953—22no—
10#16810913—22no—
11#16810828—22no—
12#16810774—22no—
13C4×#11719923—24no—
14#16810676C2212yes—
15#16810661—22no—
16#16810353—24no—
17C2×C4×D4×C53C2211yescharacteristic
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×C22⋊C8×C53—22no—
20C2×#97×C53C2211yescharacteristic
21C2×#11719923—24no—
22#13133217D412yes—
23#13133217D412yes—
24#13133209—22no—
25#13133196—22no—
26#13133196—22no—
27C4×Dic106—22no—
28C4×Dic106—22no—
29#13133176—22no—
30#13133151—22no—
31#13133151—22no—
32#13133149—24no—
33#13133148—24no—
34C4⋊#88322—22no—
35#13133140—24no—
36#13133138—24no—
37C22⋊#88322—22no—
38C4×#88322D414yes—
39C22×D4×C53D411yescharacteristic
40C23×C4×C53—21no—
41C22×C8×C53—21no—
42C22×C8×C53D412yes—
43C4×D4×C53—22no—
44C4×D4×C53—22no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—22no—
49C22⋊C8×C53—44no—
50#97×C53—22no—
51C24×C53—21no—
52C2×D4×C53—22no—
53C2×D4×C53—42no—
54C22×C4×C53—21no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53#4111yescharacteristic
62C2×Dic106—22no—
63C2×Dic106—22no—
64C2×C8×C53—22no—
65C2×C8×C53—22no—
66C2×C8×C53—41no—
67C2×C8×C53—42no—
68C2×C8×C53SD1614yescomplemented
69C4⋊C4×C53—22no—
70C22⋊C4×C53—42no—
71C42×C53C2×D412yes—
72C42×C53C2×D412yes—
73C22⋊Dic53—42no—
74#11719923—48no—
75C4⋊Dic53—22no—
76C4⋊Dic53—22no—
77C4⋊Dic53—22no—
78C4⋊Dic53—22no—
79#11719919—42no—
80#11719919—42no—
81C4×Dic53—42no—
82C2×#88322—24no—
83C23×C53—22no—
84C23×C53—41no—
85C23×C53—42no—
86C23×C53—41no—
87C23×C53—42no—
88C23×C53#12111yescharacteristic
89D4×C53—24no—
90D4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—82no—
96C2×C4×C53—81no—
97C2×C4×C53—81no—
98C2×C4×C53#12011yescharacteristic
99C2×C4×C53#12012yes—
100C2×C4×C53#12112yes—
101C2×C4×C53#12711yescharacteristic
102C2×C4×C53C2×SD1612yes—
103C2×C4×C53#13712yes—
104C2×Dic53—42no—
105C2×Dic53—42no—
106Dic106—44no—
107Dic106—44no—
108C8×C53—24no—
109C8×C53—81no—
110#88322—44no—
111#10036—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53—44no—
115C22×C53—42no—
116C22×C53—82no—
117C22×C53—81no—
118C22×C53—81no—
119C22×C53#39211yescharacteristic
120C22×C53#46112yes—
121C4×C53—22no—
122C4×C53—22no—
123C4×C53—81no—
124C4×C53#44812yes—
125C4×C53#49211yescharacteristic
126C4×C53#49711yescharacteristic
127Dic53—82no—
128Dic53—82no—
129#3162—531no—
130#3101—532no—
131#1955D5311yescharacteristic
132#1481—532no—
133#1275—531no—
134C2×C53—22no—
135C2×C53—82no—
136C2×C53#201011yescharacteristic
137C2×C53#201311yescharacteristic
138C2×C53#201711yescharacteristic
139#533—532no—
140C2×C4×D4C2×D5311yescharacteristic
141#501—1062no—
142#501—1062no—
143#483—1062no—
144#476—1062no—
145#461—1062no—
146#450—1062no—
147C4×SD16—1064no—
148#431—532no—
149#427—1062no—
150C2×C4⋊C8C2×D5311yescharacteristic
151C2×C22⋊C8—22no—
152C2×#97C2×D5311yescharacteristic
153Q8⋊C8—1064no—
154C53#1003611yescharacteristic, complemented, Sylow
155C22×D4D21211yescharacteristic
156C23×C4—21no—
157C2×SD16—1064no—
158C22×C8—21no—
159C22×C8C22⋊D5312yes—
160C4⋊Q8—532no—
161C4⋊Q8—532no—
162#125—1062no—
163C22⋊Q8—1062no—
164C22⋊Q8—1062no—
165C4×Q8—1062no—
166C4×Q8—1062no—
167C4×D4—22no—
168C4×D4—22no—
169#117—1062no—
170C2×C4⋊C4D21211yescharacteristic
171C2×C22⋊C4—21no—
172C2×C42C22×D5311yescharacteristic
173#107—1062no—
174#107—1062no—
175#106—1064no—
176#106—1064no—
177C4⋊C8—22no—
178C4⋊C8—1062no—
179#103—1064no—
180#102—1064no—
181C22⋊C8—44no—
182C22⋊C8—1062no—
183#97—22no—
184C4×C8—534no—
185C24—21no—
186C2×Q8—1062no—
187C2×Q8—1062no—
188C2×D4—22no—
189C2×D4—42no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C2×D21211yescharacteristic
196C22×C4#1171994511yescharacteristic
197C22×C4C2×C22⋊D5311yescharacteristic
198SD16—2128no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—1064no—
204C2×C8#1171992314yescomplemented
205C4⋊C4—22no—
206C4⋊C4—1062no—
207C4⋊C4—1062no—
208C4⋊C4—1062no—
209C4⋊C4—1062no—
210C4⋊C4—2122no—
211C4⋊C4—2122no—
212C22⋊C4—42no—
213C22⋊C4—2122no—
214C42—2122no—
215C42D4×D5312yes—
216C42D4×D5312yes—
217C23—22no—
218C23—41no—
219C23—42no—
220C23—41no—
221C23—42no—
222C23#1313319111yescharacteristic
223Q8—2124no—
224Q8—2124no—
225D4—24no—
226D4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4#1313318511yescharacteristic
237C2×C4#1313318712yes—
238C2×C4#1313319012yes—
239C2×C4#1313321311yescharacteristic
240C2×C4C2×#1171992312yes—
241C2×C4#1313324712yes—
242C8—24no—
243C8—81no—
244C8—2124no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1681053211yescharacteristic
253C22#1681082812yes—
254C4—22no—
255C4—22no—
256C4—81no—
257C4—4242no—
258C4—4242no—
259C4#1681076112yes—
260C4#1681099511yescharacteristic
261C4#1681101811yescharacteristic
262C2—22no—
263C2—82no—
264C2#2027578211yescharacteristic
265C2#2027579711yescharacteristic
266C2#2027581911yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1955C2
socleC22×C53#392
prime radicalC22×D4×C53D4
hypercenter#1955D53
hypocenterC53#10036
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. #20285239C2semidirect
3. #20273117C2semidirect
4. #20272546C2semidirect
68. C2×C8×C53SD16semidirect
154. C53#10036semidirect
204. C2×C8#11719923semidirect

Maximal subgroups

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

subgroupindex
#202852392
#202847232
#202731172
#202725462
C53×#19552
#1003653