Finite Group Databasebeta

← #27277693

Subgroups — #27277693

Order 13568.

426 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 115 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#20274707C211yesmaximal, characteristic
2#20273036C211yesmaximal, characteristic
3#20272618C212yesmaximal, complemented
4#20272573C212yesmaximal, complemented
5C53×#2357C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133134—21no—
9#16810392—21no—
10#16810347—22no—
11C2×C4×D4×C53C2211yescharacteristic
12C53×#390—21no—
13C53×#388—22no—
14C53×#382C2211yescharacteristic
15C53×#381C412yes—
16C4×C4⋊C4×C53—21no—
17C4×C4⋊C4×C53C2211yescharacteristic
18C4×C22⋊C4×C53C412yes—
19C2×#11719917—22no—
20C4⋊#88322—22no—
21C4⋊#88322—22no—
22C4⋊#88322—22no—
23#13133134—42no—
24C22⋊#88322—22no—
25C4×#88322—22no—
26C22×D4×C53D411yescharacteristic
27C23×C4×C53C2×C412yes—
28C4×D4×C53—22no—
29C4×D4×C53—22no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53D411yescharacteristic
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53D412yes—
36C2×C42×C53—22no—
37C2×C42×C53—21no—
38C2×C42×C53C2311yescharacteristic
39#95×C53—22no—
40#95×C53—21no—
41#95×C53C2×C411yescharacteristic
42#95×C53C2×C412yes—
43#95×C53C2×C411yescharacteristic
44C24×C53C22⋊C412yes—
45C2×D4×C53—22no—
46C2×D4×C53—22no—
47C2×D4×C53—21no—
48C2×D4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—21no—
54C22×C4×C53—82no—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C4⋊C4×C53—22no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C22⋊C4×C53—24no—
64C22⋊C4×C53—44no—
65C22⋊C4×C53—82no—
66C42×C53—42no—
67C42×C53—82no—
68C42×C53#4112yes—
69C42×C53#4112yes—
70#11719917—42no—
71#11719917—42no—
72C2×#88322—42no—
73C2×#88322—42no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53C2×C22⋊C411yescharacteristic
80D4×C53—42no—
81D4×C53—42no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×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—82no—
97C2×C4×C53#11711yescharacteristic
98C2×C4×C53#11711yescharacteristic
99C2×C4×C53#12312yes—
100C2×C4×C53#12312yes—
101C2×C4×C53#13512yes—
102C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C53#13711yescharacteristic
104#88322—82no—
105#88322—82no—
106#8771—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—42no—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53—82no—
113C22×C53—82no—
114C22×C53#38111yescharacteristic
115C22×C53#41611yescharacteristic
116C22×C53#41711yescharacteristic
117C4×C53—22no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53—42no—
122C4×C53—81no—
123C4×C53—82no—
124C4×C53#48212yes—
125C4×C53#48512yes—
126#2357D5311yescharacteristic
127#1800—531no—
128#1460—531no—
129#1303—532no—
130#1286—532no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—82no—
134C2×C53#182511yescharacteristic
135C2×C53#182711yescharacteristic
136C2×C53#182911yescharacteristic
137C2×C4×D4C2×D5311yescharacteristic
138#431—532no—
139#423—532no—
140C2×#100—1061no—
141#390—21no—
142#388—22no—
143#382C2×D5311yescharacteristic
144#381Dic5312yes—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2×D5311yescharacteristic
147C4×C22⋊C4Dic5312yes—
148#343—1061no—
149#323—1062no—
150C53#877111yescharacteristic, complemented, Sylow
151C22×D4C22⋊D5311yescharacteristic
152C23×C4C2×Dic5312yes—
153C2×M16—1062no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4C22⋊D5311yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4C22⋊D5312yes—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42C22×D5311yescharacteristic
165C4⋊C8—1062no—
166C4⋊C8—1062no—
167C4⋊C8—1062no—
168#100—2122no—
169C22⋊C8—1062no—
170C4×C8—1062no—
171#95—21no—
172#95—22no—
173#95C2×Dic5311yescharacteristic
174#95C2×Dic5312yes—
175#95C2×Dic5311yescharacteristic
176C24C22⋊Dic5312yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—82no—
187C22×C4C22⋊Dic5312yes—
188C22×C4C22×Dic5311yescharacteristic
189C22×C4C2×C22⋊D5311yescharacteristic
190C22×C4C2×C22⋊D5311yescharacteristic
191M16—2122no—
192M16—2122no—
193C2×C8—2122no—
194C2×C8—2122no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C22⋊C4—24no—
200C22⋊C4—44no—
201C22⋊C4—82no—
202C42—42no—
203C42—82no—
204C42#1171994512yes—
205C42#1171994512yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×C22⋊Dic5311yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4#1313317611yescharacteristic
230C2×C4#1313317611yescharacteristic
231C2×C4#1313320012yes—
232C2×C4#1313320012yes—
233C2×C4#1313323712yes—
234C2×C4#1313324211yescharacteristic
235C2×C4#1313324711yescharacteristic
236C8—4242no—
237C8—4242no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22#1681048611yescharacteristic
246C22#1681061911yescharacteristic
247C22#1681062411yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—82no—
255C4#1681094712yes—
256C4#1681096412yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2027481011yescharacteristic
261C2#2027481911yescharacteristic
262C2#2027482611yescharacteristic
subgroupG/H
centerC22#16810486
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2357C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2357D53
hypocenterC53#8771
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2357

Complements

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

NH ≅ G/Ntype
3. #20272618C2semidirect
4. #20272573C2semidirect
150. C53#8771semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202730362
#202726182
#202725732
C53×#23572
#877153