Finite Group Databasebeta

← #27279401

Subgroups — #27279401

Order 13568.

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

normal subgroups only

This group has 444 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#20285037C211yesmaximal, characteristic
2#20283782C211yesmaximal, characteristic
3#20273137C212yesmaximal, complemented
4#20272553C212yesmaximal, complemented
5C53×#1525C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810977—22no—
8#16810973C412yes—
9#16810969C412yes—
10#16810701—22no—
11D4×#88322—22no—
12#16810676C2212yes—
13#16810619—22no—
14#16810590—22no—
15C2×C4×#88322C2211yescharacteristic
16#16810355—22no—
17#16810355—22no—
18C53×#532C2211yescharacteristic
19C4×M16×C53C2212yes—
20C53×#329—22no—
21C2×#11719917—22no—
22C22×#88322—22no—
23C22×#88322—21no—
24#13133217C2×C412yes—
25#13133209—21no—
26#13133196—22no—
27C4×Dic106—22no—
28#13133176—21no—
29C4⋊#88322D412yes—
30C4⋊#88322D412yes—
31#13133141—24no—
32#13133138—24no—
33C22⋊#88322—24no—
34C22⋊#88322—22no—
35C4×#88322—21no—
36C4×#88322—22no—
37C4×#88322C2×C412yes—
38C2×#41×C53D412yes—
39C2×M16×C53—22no—
40C4⋊Q8×C53C2×C411yescharacteristic
41#127×C53C2×C411yescharacteristic
42#124×C53D412yes—
43#121×C53—22no—
44C4×D4×C53—22no—
45C2×C42×C53C2311yescharacteristic
46#97×C53—22no—
47C4×C8×C53—22no—
48#41×C53—42no—
49#41×C53—42no—
50C2×Q8×C53C22⋊C412yes—
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53C22⋊C412yes—
54C22×C4×C53—22no—
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53#4112yes—
57C2×Dic106—22no—
58M16×C53—24no—
59M16×C53—42no—
60C2×C8×C53—42no—
61C4⋊C4×C53—22no—
62C22⋊C4×C53—24no—
63C42×C53C22×C411yescharacteristic
64C42×C53C2×D411yescharacteristic
65C42×C53C2×D412yes—
66C22⋊Dic53—41no—
67C4⋊Dic53—22no—
68C4⋊Dic53—22no—
69#11719919—42no—
70C4×Dic53—41no—
71#11719917—82no—
72C2×#88322—22no—
73C2×#88322—22no—
74C2×#88322—22no—
75C2×#88322—41no—
76C2×#88322—41no—
77C2×#88322—82no—
78C23×C53—22no—
79C23×C53#12311yescharacteristic
80Q8×C53—42no—
81D4×C53—42no—
82D4×C53—42no—
83D4×C53—42no—
84D4×C53—82no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53C2×C22⋊C411yescharacteristic
93C2×C4×C53C4×D412yes—
94C2×C4×C53#12011yescharacteristic
95C2×C4×C53#12112yes—
96C2×Dic53—42no—
97Dic106—42no—
98Dic106—42no—
99C8×C53—44no—
100#88322—42no—
101#88322—42no—
102#88322—42no—
103#9067—531nomaximal, Sylow
104C22×C53—22no—
105C22×C53—21no—
106C22×C53—44no—
107C22×C53—42no—
108C22×C53#38611yescharacteristic
109C4×C53—22no—
110C4×C53—21no—
111C4×C53—42no—
112C4×C53#44312yes—
113C4×C53#44412yes—
114C4×C53#45511yescharacteristic
115C4×C53#45511yescharacteristic
116Dic53—82no—
117#3131—531no—
118#2979—531no—
119#1525D5311yescharacteristic
120#1485—532no—
121#1278—532no—
122C2×C53—41no—
123C2×C53—82no—
124C2×C53#187612yes—
125C2×C53#188511yescharacteristic
126#533—531no—
127#532C2×D5311yescharacteristic
128#488—1062no—
129#487—532no—
130#486—532no—
131#436—1062no—
132C8×D4—1062no—
133#431—532no—
134#416—1062no—
135#408—1062no—
136C4×M16C2×D5312yes—
137C2×C4×C8—531no—
138#329—22no—
139Q8⋊C8—1062no—
140Q8⋊C8—1062no—
141C53#906711yescharacteristic, complemented, Sylow
142C2×#41D21212yes—
143C2×M16—22no—
144C2×M16—1062no—
145C22×C8—1062no—
146C22×C8—1061no—
147C4⋊Q8—532no—
148C4⋊Q8C4×D5311yescharacteristic
149#127C4×D5311yescharacteristic
150#125—1061no—
151#124C22⋊D5312yes—
152C22⋊Q8—1062no—
153#121—22no—
154C4×Q8—1062no—
155C4×D4—22no—
156#117—1061no—
157C2×C42C22×D5311yescharacteristic
158C4⋊C8—532no—
159C4⋊C8—532no—
160#103—1064no—
161#102—1064no—
162C22⋊C8—1064no—
163C22⋊C8—1062no—
164#97—22no—
165C4×C8—22no—
166C4×C8—532no—
167C4×C8—1062no—
168C4×C8—1061no—
169#41—42no—
170#41—42no—
171C2×Q8—1062no—
172C2×Q8#1171992112yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4#1171992112yes—
176C22×C4—22no—
177C22×C4C2×D21211yescharacteristic
178C22×C4#1171994712yes—
179M16—24no—
180M16—42no—
181M16—4242no—
182C2×C8—42no—
183C2×C8—1062no—
184C2×C8—1062no—
185C2×C8—1062no—
186C2×C8—4242no—
187C2×C8—2121no—
188C2×C8—2121no—
189C4⋊C4—22no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—2122no—
193C22⋊C4—24no—
194C22⋊C4—2121no—
195C42—2121no—
196C42C2×C4×D5311yescharacteristic
197C42D4×D5312yes—
198C42C2×C22⋊D5311yescharacteristic
199C23—22no—
200C23#1313320111yescharacteristic
201Q8—42no—
202Q8—2122no—
203Q8—2122no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—2122no—
216C2×C4C2×#1171992111yescharacteristic
217C2×C4#1313317712yes—
218C2×C4#1313318711yescharacteristic
219C2×C4#1313318912yes—
220C8—44no—
221C8—2122no—
222C8—2122no—
223C8—2122no—
224C22—22no—
225C22—21no—
226C22—44no—
227C22—42no—
228C22#1681050011yescharacteristic
229C4—22no—
230C4—21no—
231C4—42no—
232C4—4242no—
233C4#1681073712yes—
234C4#1681074212yes—
235C4#1681080311yescharacteristic
236C4#1681080311yescharacteristic
237C2—41no—
238C2—82no—
239C2#2027504012yes—
240C2#2027510111yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1525C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#1525D53
hypocenterC53#9067
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1525

Complements

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

NH ≅ G/Ntype
3. #20273137C2semidirect
4. #20272553C2semidirect
141. C53#9067semidirect

Maximal subgroups

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

subgroupindex
#202850372
#202837822
#202731372
#202725532
C53×#15252
#906753