Finite Group Databasebeta

← #27277685

Subgroups — #27277685

Order 13568.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277688C211yesmaximal, characteristic
2#20273036C211yesmaximal, characteristic
3#20272616C212yesmaximal, complemented
4#20272575C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810526—21no—
8#16810516—22no—
9#16810489C2211yescharacteristic
10#16810486C412yes—
11C4⋊C4×Dic53—21no—
12C4×C4⋊Dic53C2211yescharacteristic
13C4×C22⋊Dic53C412yes—
14#16810347—22no—
15C2×C4×D4×C53C2211yescharacteristic
16C53×#423C2212yes—
17C2×#100×C53—21no—
18C53×#343—21no—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—21no—
21C2×C22⋊Dic53—22no—
22C2×C22⋊Dic53—22no—
23C2×C4×Dic53—22no—
24C2×C4×Dic53—21no—
25C4⋊#88322—22no—
26C22⋊#88322—22no—
27C4×#88322—22no—
28#13133124—22no—
29#13133124—21no—
30#13133124C2×C411yescharacteristic
31#13133124C2×C412yes—
32#13133124C2×C411yescharacteristic
33C22×D4×C53D411yescharacteristic
34C23×C4×C53C2×C412yes—
35C2×M16×C53—22no—
36C4×D4×C53—22no—
37C4×D4×C53—22no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C22⋊C4×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—22no—
42C4⋊C8×C53—22no—
43#100×C53—42no—
44C24×C53C22⋊C412yes—
45C2×D4×C53—22no—
46C2×D4×C53—22no—
47C2×D4×C53—21no—
48C2×D4×C53—21no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—82no—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22×C411yescharacteristic
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53—22no—
58C22×Dic53—21no—
59M16×C53—42no—
60M16×C53—42no—
61C2×C8×C53—42no—
62C4⋊C4×C53—22no—
63C22⋊C4×C53—24no—
64C42×C53#4112yes—
65C42×C53#4112yes—
66C22⋊Dic53—44no—
67C22⋊Dic53—82no—
68C4⋊Dic53—42no—
69C4⋊Dic53—42no—
70C4×Dic53—41no—
71C4×Dic53—41no—
72C4×Dic53—82no—
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—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—82no—
87C2×C4×C53—82no—
88C2×C4×C53#11711yescharacteristic
89C2×C4×C53#11711yescharacteristic
90C2×C4×C53#12312yes—
91C2×C4×C53#12312yes—
92C2×C4×C53#13512yes—
93C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
94C2×C4×C53#13711yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—42no—
99C2×Dic53—41no—
100C2×Dic53—42no—
101C2×Dic53—41no—
102C2×Dic53—42no—
103C2×Dic53—41no—
104C8×C53—82no—
105#88322—82no—
106#8770—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#48212yes—
122C4×C53#48512yes—
123Dic53—42no—
124Dic53—81no—
125Dic53—82no—
126#2357—531no—
127#1800D5311yescharacteristic
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#423C2×D5312yes—
140C2×#100—21no—
141#390—1061no—
142#388—1062no—
143#382—531no—
144#381—532no—
145C4×C4⋊C4—531no—
146C4×C4⋊C4—1061no—
147C4×C22⋊C4—532no—
148#343—21no—
149#323—1062no—
150C53#877011yescharacteristic, complemented, Sylow
151C22×D4C22⋊D5311yescharacteristic
152C23×C4C4×D5312yes—
153C2×M16—22no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—1061no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4C22⋊D5311yescharacteristic
159C2×C22⋊C4—1062no—
160C2×C22⋊C4—1062no—
161C2×C22⋊C4D21212yes—
162C2×C42—1062no—
163C2×C42—1061no—
164C2×C42C22×D5311yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—1062no—
168#100—42no—
169C22⋊C8—1062no—
170C4×C8—1062no—
171#95—531no—
172#95—532no—
173#95—531no—
174#95—1061no—
175#95—1062no—
176C24#1171992112yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—82no—
184C22×C4—1061no—
185C22×C4—1062no—
186C22×C4—1061no—
187C22×C4#1171992112yes—
188C22×C4C2×C4×D5311yescharacteristic
189C22×C4C2×D21211yescharacteristic
190C22×C4C2×C22⋊D5311yescharacteristic
191M16—42no—
192M16—42no—
193C2×C8—42no—
194C2×C8—2122no—
195C4⋊C4—22no—
196C4⋊C4—2122no—
197C4⋊C4—2122no—
198C22⋊C4—24no—
199C22⋊C4—4242no—
200C22⋊C4—2124no—
201C42—4242no—
202C42—2121no—
203C42—2121no—
204C42#1171994712yes—
205C42#1171994712yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×#1171992111yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—1062no—
221C2×C4—2122no—
222C2×C4—2121no—
223C2×C4—2122no—
224C2×C4—2121no—
225C2×C4—2122no—
226C2×C4—2121no—
227C2×C4—2121no—
228C2×C4—2121no—
229C2×C4#1313317311yescharacteristic
230C2×C4#1313317311yescharacteristic
231C2×C4#1313319812yes—
232C2×C4#1313320112yes—
233C2×C4#1313323712yes—
234C2×C4#1313324311yescharacteristic
235C2×C4#1313324811yescharacteristic
236C8—82no—
237C8—4242no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22#1681048411yescharacteristic
246C22#1681061711yescharacteristic
247C22#1681062211yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—4242no—
253C4—4241no—
254C4—2122no—
255C4#1681094412yes—
256C4#1681096112yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2027480711yescharacteristic
261C2#2027482011yescharacteristic
262C2#2027482411yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1800C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1800D53
hypocenterC53#8770
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20272616C2semidirect
4. #20272575C2semidirect
150. C53#8770semidirect

Maximal subgroups

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

subgroupindex
#202776882
#202730362
#202726162
#202725752
C53×#18002
#877053