Finite Group Databasebeta

← #27281689

Subgroups — #27281689

Order 13568.

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

normal subgroups only

This group has 504 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#20285175C211yesmaximal, characteristic
2#20283782C211yesmaximal, characteristic
3#20273401C212yesmaximal, complemented
4#20272604C212yesmaximal, complemented
5C53×#1560C211yesmaximal, characteristic
6C2×#13133247—22no—
7C2×#13133217C2211yescharacteristic
8#16811018C414yes—
9#16811002—21no—
10#16810995—21no—
11#16810701—22no—
12D4×#88322—22no—
13#16810676C2212yes—
14#16810590—22no—
15C4×#11719917C2211yescharacteristic
16#16810368—22no—
17#16810368—22no—
18C53×#532C2211yescharacteristic
19C53×#423C2212yes—
20C53×#333—21no—
21C53×#331—21no—
22C22×Dic106—21no—
23#13133247—44no—
24#13133247—44no—
25C2×#11719925—24no—
26C2×#11719923—24no—
27C2×#11719917—22no—
28C2×#11719917—21no—
29C22×#88322—22no—
30#13133217—21no—
31#13133217—22no—
32#13133217C2×C412yes—
33C2×C4⋊Dic53—22no—
34C4⋊#88322D412yes—
35C4⋊#88322D412yes—
36C22⋊#88322—24no—
37C22⋊#88322—22no—
38#13133128C2×C412yes—
39C4×#88322—21no—
40C4×#88322—22no—
41C2×#41×C53D412yes—
42C2×M16×C53—22no—
43C4⋊Q8×C53D411yescharacteristic
44#127×C53D411yescharacteristic
45#124×C53C2×C412yes—
46#121×C53—22no—
47C4×D4×C53—22no—
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—22no—
50C4⋊C8×C53—22no—
51#41×C53—42no—
52#41×C53—42no—
53C2×Q8×C53C22⋊C412yes—
54C2×D4×C53—22no—
55C2×D4×C53—22no—
56C2×D4×C53C22⋊C412yes—
57C22×C4×C53—22no—
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53#4112yes—
60C22×Dic53—41no—
61C2×Dic106—22no—
62C2×Dic106—42no—
63C2×Dic106—42no—
64C2×Dic106—41no—
65M16×C53—82no—
66C2×C8×C53—42no—
67C4⋊C4×C53—22no—
68C22⋊C4×C53—24no—
69C42×C53C22×C411yescharacteristic
70C42×C53C2×D411yescharacteristic
71C42×C53C2×D412yes—
72#11719925—48no—
73#11719923—48no—
74C4⋊Dic53—24no—
75C4⋊Dic53—42no—
76#11719917—22no—
77#11719917—22no—
78#11719917—82no—
79C2×#88322—22no—
80C2×#88322—22no—
81C2×#88322—22no—
82C2×#88322—82no—
83C23×C53—22no—
84C23×C53#12311yescharacteristic
85Q8×C53—42no—
86D4×C53—42no—
87D4×C53—42no—
88D4×C53—42no—
89D4×C53—82no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53C2×C22⋊C411yescharacteristic
98C2×C4×C53C4×D412yes—
99C2×C4×C53#12011yescharacteristic
100C2×C4×C53#12112yes—
101C2×Dic53—42no—
102C2×Dic53—82no—
103Dic106—44no—
104Dic106—81no—
105Dic106—81no—
106C8×C53—82no—
107#88322—24no—
108#88322—42no—
109#88322—42no—
110#9435—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—22no—
113C22×C53—42no—
114C22×C53—44no—
115C22×C53#38611yescharacteristic
116C4×C53—21no—
117C4×C53—22no—
118C4×C53—42no—
119C4×C53#45311yescharacteristic
120C4×C53#45311yescharacteristic
121C4×C53#47112yes—
122C4×C53#47312yes—
123Dic53—82no—
124#3152—531no—
125#2979—531no—
126#1560D5311yescharacteristic
127#1549—532no—
128#1299—532no—
129C2×C53—41no—
130C2×C53—82no—
131C2×C53#188711yescharacteristic
132C2×C53#190812yes—
133C2×#137—1062no—
134#532C2×D5311yescharacteristic
135C2×C4⋊Q8—531no—
136#497—534no—
137#494—1061no—
138#492—1061no—
139#436—1062no—
140C8×D4—1062no—
141#431—532no—
142#423C2×D5312yes—
143#408—1062no—
144C4×M16—531no—
145#333—21no—
146#332—1062no—
147#332—1062no—
148#331—21no—
149C53#943511yescharacteristic, complemented, Sylow
150C2×#41D21212yes—
151C22×Q8—1061no—
152#137—2124no—
153#137—2124no—
154C2×Q16—1064no—
155C2×SD16—1064no—
156C2×M16—22no—
157C2×M16—1061no—
158C2×M16—1062no—
159C22×C8—1062no—
160C4⋊Q8—532no—
161C4⋊Q8—1062no—
162C4⋊Q8—1061no—
163C4⋊Q8C22⋊D5311yescharacteristic
164#127C22⋊D5311yescharacteristic
165#124C4×D5312yes—
166#121—22no—
167C4×D4—22no—
168C2×C4⋊C4—1062no—
169C2×C42C22×D5311yescharacteristic
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—532no—
173C4⋊C8—532no—
174C22⋊C8—1064no—
175C22⋊C8—1062no—
176#97—532no—
177C4×C8—1062no—
178C4×C8—1061no—
179#41—42no—
180#41—42no—
181C2×Q8—1062no—
182C2×Q8—2122no—
183C2×Q8—2122no—
184C2×Q8—2121no—
185C2×Q8#1171992112yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4#1171992112yes—
189C22×C4—22no—
190C22×C4—2121no—
191C22×C4C2×D21211yescharacteristic
192C22×C4#1171994712yes—
193Q16—2128no—
194SD16—2128no—
195M16—82no—
196M16—1062no—
197M16—1062no—
198M16—4242no—
199C2×C8—42no—
200C2×C8—1062no—
201C2×C8—1062no—
202C2×C8—1062no—
203C2×C8—4242no—
204C4⋊C4—22no—
205C4⋊C4—1064no—
206C4⋊C4—2122no—
207C22⋊C4—24no—
208C42C2×C4×D5311yescharacteristic
209C42D4×D5312yes—
210C42C2×C22⋊D5311yescharacteristic
211C23—22no—
212C23#1313320111yescharacteristic
213Q8—42no—
214Q8—4241no—
215Q8—4241no—
216Q8—2124no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—82no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—4242no—
229C2×C4—2122no—
230C2×C4C2×#1171992111yescharacteristic
231C2×C4#1313317712yes—
232C2×C4#1313318711yescharacteristic
233C2×C4#1313318912yes—
234C8—82no—
235C8—1064no—
236C8—2122no—
237C8—2122no—
238C22—22no—
239C22—21no—
240C22—44no—
241C22—42no—
242C22#1681050011yescharacteristic
243C4—22no—
244C4—21no—
245C4—42no—
246C4—4242no—
247C4#1681079311yescharacteristic
248C4#1681079311yescharacteristic
249C4#1681088412yes—
250C4#1681089612yes—
251C2—41no—
252C2—82no—
253C2#2027511111yescharacteristic
254C2#2027522812yes—
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1560C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#1560D53
hypocenterC53#9435
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1560

Complements

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

NH ≅ G/Ntype
3. #20273401C2semidirect
4. #20272604C2semidirect
149. C53#9435semidirect

Maximal subgroups

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

subgroupindex
#202851752
#202837822
#202734012
#202726042
C53×#15602
#943553