Finite Group Databasebeta

← #27296899

Subgroups — #27296899

Order 13568.

456 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275879C211yesmaximal, characteristic
2#20273739C212yesmaximal, complemented
3#20272739C212yesmaximal, complemented
4#20272643C211yesmaximal, characteristic
5C53×#2842C211yesmaximal, characteristic
6#16811249C2211yescharacteristic
7#16810958—22no—
8#16810479—21no—
9#16810390C2211yescharacteristic
10#16810389—21no—
11#16810359—22no—
12#16810347C2212yes—
13C53×#399—21no—
14C53×#399C2212yes—
15C53×#396—22no—
16C53×#396—22no—
17C53×#379—21no—
18C53×#379C2211yescharacteristic
19#13133196—22no—
20#13133194—42no—
21C2×C22⋊Dic53—21no—
22C2×C22⋊Dic53—21no—
23C2×C22⋊Dic53—42no—
24#13133150—42no—
25#13133140—42no—
26C22⋊#88322—22no—
27C25×C53D411yescharacteristic
28C2×C4⋊C4×C53—22no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53D412yes—
32C2×C22⋊C4×C53—21no—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53C2311yescharacteristic
39#95×C53—21no—
40#95×C53D412yes—
41C24×C53—21no—
42C24×C53—21no—
43C24×C53—41no—
44C24×C53—41no—
45C24×C53—82no—
46C24×C53Q1612yes—
47C24×C53C2×D411yescharacteristic
48C22×C4×C53—22no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53C2×D412yes—
52C22×Dic53—22no—
53C22×Dic53—41no—
54C2×Dic106—22no—
55C4⋊C4×C53—42no—
56C4⋊C4×C53—82no—
57C4⋊C4×C53—82no—
58C4⋊C4×C53—81no—
59C22⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—82no—
62C22⋊C4×C53—82no—
63C22⋊C4×C53—82no—
64C22⋊C4×C53—81no—
65C22⋊C4×C53—81no—
66C22⋊Dic53—41no—
67C22⋊Dic53—41no—
68C22⋊Dic53—41no—
69C22⋊Dic53—41no—
70C22⋊Dic53—84no—
71C4⋊Dic53—22no—
72#11719919—42no—
73C2×#88322—42no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53—42no—
81C23×C53—44no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53—81no—
85C23×C53—82no—
86C23×C53—82no—
87C23×C53—82no—
88C23×C53—84no—
89C23×C53#12011yescharacteristic
90C23×C53C2×Q1611yescharacteristic
91C23×C53#13711yescharacteristic
92C2×C4×C53—22no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×Dic53—42no—
103C2×Dic53—82no—
104C2×Dic53—81no—
105C2×Dic53—82no—
106Dic106—82no—
107#88322—82no—
108#12218—531nomaximal, Sylow
109C22×C53—22no—
110C22×C53—21no—
111C22×C53—21no—
112C22×C53—41no—
113C22×C53—41no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—44no—
118C22×C53—42no—
119C22×C53—81no—
120C22×C53—82no—
121C22×C53—82no—
122C22×C53—84no—
123C22×C53—82no—
124C22×C53#45111yescharacteristic
125C22×C53#45511yescharacteristic
126C22×C53B(4,𝔽2)11yescharacteristic
127C4×C53—42no—
128C4×C53—81no—
129C4×C53—82no—
130C4×C53—82no—
131Dic53—82no—
132Dic53—161no—
133#2842D5311yescharacteristic
134#2029—531no—
135#1614—532no—
136#1352—532no—
137#1316—531no—
138C2×C53—21no—
139C2×C53—21no—
140C2×C53—41no—
141C2×C53—41no—
142C2×C53—82no—
143C2×C53#160211yescharacteristic
144C2×C53#220012yes—
145#543—531no—
146#484—1062no—
147#399—21no—
148#399C2×D5312yes—
149#396—22no—
150#396—22no—
151#379—21no—
152#379—1061no—
153#379C2×D5311yescharacteristic
154#342—531no—
155#342—1061no—
156#328—1062no—
157#323—532no—
158C53#1221811yescharacteristic, complemented, Sylow
159C25C22⋊D5311yescharacteristic
160C22⋊Q8—1062no—
161C22⋊Q8—2122no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊D5312yes—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—1061no—
172C2×C22⋊C4—1061no—
173C2×C22⋊C4—2122no—
174C2×C22⋊C4C22⋊D5311yescharacteristic
175C2×C22⋊C4C22×D5311yescharacteristic
176#107—2122no—
177#103—2122no—
178C22⋊C8—1062no—
179#95—21no—
180#95C22⋊D5312yes—
181C24—21no—
182C24—21no—
183C24—41no—
184C24—41no—
185C24—82no—
186C24#1171992512yes—
187C24C2×C22⋊D5311yescharacteristic
188C2×Q8—1062no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—1062no—
193C22×C4—2121no—
194C22×C4C2×C22⋊D5312yes—
195C2×C8—2122no—
196C4⋊C4—42no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C4⋊C4—81no—
200C4⋊C4—1062no—
201C4⋊C4—2122no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—82no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C22⋊C4—4244no—
210C22⋊C4—2121no—
211C22⋊C4—2121no—
212C22⋊C4—2121no—
213C22⋊C4—2121no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—44no—
222C23—41no—
223C23—41no—
224C23—81no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—84no—
229C23#1313318611yescharacteristic
230C23C2×#1171992511yescharacteristic
231C23#1313324811yescharacteristic
232Q8—4242no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—4241no—
244C2×C4—4242no—
245C2×C4—4242no—
246C2×C4—2122no—
247C8—4242no—
248C22—22no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22—84no—
262C22—82no—
263C22#1681077911yescharacteristic
264C22#1681080111yescharacteristic
265C22#1681081011yescharacteristic
266C4—42no—
267C4—81no—
268C4—82no—
269C4—82no—
270C4—8481no—
271C4—4242no—
272C2—21no—
273C2—21no—
274C2—41no—
275C2—41no—
276C2—82no—
277C2#2027367211yescharacteristic
278C2#2027672212yes—
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2842C2
socleC22×C53B(4,𝔽2)
prime radicalC25×C53D4
hypercenter#2842D53
hypocenterC53#12218
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2842

Complements

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

NH ≅ G/Ntype
2. #20273739C2semidirect
3. #20272739C2semidirect
158. C53#12218semidirect

Maximal subgroups

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

subgroupindex
#202758792
#202737392
#202727392
#202726432
C53×#28422
#1221853