Finite Group Databasebeta

← #27296353

Subgroups — #27296353

Order 13568.

398 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 59 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#20282703C211yesmaximal, characteristic, complemented
2#20273752C212yesmaximal
3#20272741C212yesmaximal, complemented
4#20272649C211yesmaximal, characteristic, complemented
5C53×#2007C211yesmaximal, characteristic
6#16810939—22no—
7#16810562C2212yes—
8#16810555—22no—
9#16810530—21no—
10#16810521—21no—
11#16810510—22no—
12#16810357—22no—
13#16810349C2212yes—
14C2×#137×C53—21no—
15C53×#537C2211yescharacteristic
16C2×#104×C53—21no—
17C53×#391C2211yescharacteristic
18C53×#343C2211yescharacteristic
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53D412yes—
23C2×#11719919—22no—
24C2×C4×Dic53—21no—
25#13133148—42no—
26#13133138—42no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—22no—
30#13133124—22no—
31#13133124D412yes—
32C2×#41×C53—21no—
33C22×Q8×C53D411yescharacteristic
34#137×C53—42no—
35#137×C53—42no—
36C2×Q16×C53—41no—
37C2×SD16×C53—41no—
38C2×M16×C53—21no—
39C22⋊Q8×C53—41no—
40#121×C53—22no—
41#121×C53—41no—
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C42×C53D411yescharacteristic
44#104×C53—42no—
45#95×C53—22no—
46#41×C53—41no—
47#41×C53—41no—
48#41×C53—41no—
49#41×C53—41no—
50C2×Q8×C53—21no—
51C2×Q8×C53—21no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53SD1612yes—
55C2×D4×C53—21no—
56C2×D4×C53—22no—
57C2×D4×C53—41no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—41no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D412yes—
63C22×Dic53—22no—
64C22×Dic53—22no—
65Q16×C53—44no—
66SD16×C53—44no—
67M16×C53—22no—
68M16×C53—41no—
69C2×C8×C53—41no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—42no—
73C42×C53—41no—
74C4⋊Dic53—42no—
75C4⋊Dic53—42no—
76C4⋊Dic53—81no—
77C4⋊Dic53—81no—
78#11719919—44no—
79C4×Dic53—81no—
80C2×#88322—42no—
81C23×C53—41no—
82C23×C53#12011yescharacteristic
83Q8×C53—42no—
84Q8×C53—41no—
85Q8×C53—41no—
86Q8×C53—81no—
87D4×C53—41no—
88D4×C53—41no—
89D4×C53—82no—
90D4×C53—81no—
91D4×C53—81no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—41no—
103C2×C4×C53—81no—
104C2×C4×C53—81no—
105C2×C4×C53C2×SD1611yescharacteristic
106C2×C4×C53#13711yescharacteristic
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C8×C53—42no—
114#88322—82no—
115#12131—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—81no—
119C22×C53—81no—
120C22×C53—81no—
121C22×C53#44811yescharacteristic
122C22×C53#45311yescharacteristic
123C22×C53B(4,𝔽2)11yescharacteristic
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—42no—
127C4×C53—42no—
128C4×C53—42no—
129C4×C53—81no—
130Dic53—82no—
131Dic53—82no—
132#2809—531no—
133#2007D5311yescharacteristic
134#1616—532no—
135#1353—532no—
136#1319—531no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—161no—
140C2×C53#159611yescharacteristic
141C2×C53#220111yescharacteristic
142C2×C53#220411yescharacteristic
143C2×#137—21no—
144#537C2×D5311yescharacteristic
145#481—1062no—
146C2×#104—21no—
147#400—532no—
148#398—1062no—
149#391—1061no—
150#391C2×D5311yescharacteristic
151#389—1061no—
152#387—1062no—
153#343C2×D5311yescharacteristic
154#327—1062no—
155#324—532no—
156C53#1213111yescharacteristic, complemented, Sylow
157C2×#41—21no—
158C22×Q8C22⋊D5311yescharacteristic
159#137—42no—
160#137—42no—
161C2×Q16—41no—
162C2×SD16—41no—
163C2×M16—21no—
164C22⋊Q8—41no—
165#121—22no—
166#121—41no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—1061no—
169C2×C4⋊C4—1061no—
170C2×C4⋊C4—1062no—
171C2×C4⋊C4—1061no—
172C2×C4⋊C4C22×D5311yescharacteristic
173C2×C42—1061no—
174C2×C42C22⋊D5311yescharacteristic
175#106—2122no—
176#104—42no—
177#102—2122no—
178C22⋊C8—1062no—
179#95—22no—
180#95—532no—
181#95—1062no—
182#95—1062no—
183#95—1062no—
184#41—41no—
185#41—41no—
186#41—41no—
187#41—41no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8#1171992312yes—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—41no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—41no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4D4×D5312yes—
202C22×C4C2×C22⋊D5311yescharacteristic
203Q16—44no—
204SD16—44no—
205M16—22no—
206M16—41no—
207C2×C8—41no—
208C2×C8—2122no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—4241no—
212C4⋊C4—4241no—
213C4⋊C4—2122no—
214C4⋊C4—2124no—
215C4⋊C4—2122no—
216C22⋊C4—42no—
217C42—41no—
218C42—4241no—
219C23—41no—
220C23#1313318511yescharacteristic
221Q8—42no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225D4—41no—
226D4—41no—
227D4—82no—
228D4—81no—
229D4—81no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—2122no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2122no—
249C2×C4C2×#1171992311yescharacteristic
250C2×C4#1313324711yescharacteristic
251C8—42no—
252C8—4242no—
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#1681076111yescharacteristic
259C22#1681079011yescharacteristic
260C22#1681080911yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—81no—
267C4—4242no—
268C4—4242no—
269C2—21no—
270C2—21no—
271C2—161no—
272C2#2027363711yescharacteristic
273C2#2027672711yescharacteristic
274C2#2027674011yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2007C2
socleC22×C53B(4,𝔽2)
prime radicalC53×#537C22
hypercenter#2007D53
hypocenterC53#12131
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2007

Complements

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

NH ≅ G/Ntype
1. #20282703C2semidirect
3. #20272741C2semidirect
4. #20272649C2semidirect
156. C53#12131semidirect

Maximal subgroups

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

subgroupindex
#202827032
#202737522
#202727412
#202726492
C53×#20072
#1213153