Finite Group Databasebeta

← #27273404

Subgroups — #27273404

Order 13568.

350 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 83 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
2#20276279C212yesmaximal
3#20276253C211yesmaximal, characteristic
4#20272539C212yesmaximal
5C53×#1272C211yesmaximal, characteristic
6C2×#13133150—21no—
7C2×#13133141—21no—
8#16810562—21no—
9#16810555—22no—
10#16810530—21no—
11#16810521C2212yes—
12#16810510—22no—
13#16810493C2211yescharacteristic
14#16810386—22no—
15#16810386—22no—
16#16810378C2211yescharacteristic
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×#88322—21no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53D412yes—
25C2×C4⋊Dic53D412yes—
26C2×#11719919—22no—
27C2×C4×Dic53—21no—
28#13133150—42no—
29#13133141—42no—
30#13133124—21no—
31#13133124—22no—
32#13133124—22no—
33#13133124—22no—
34C22×Q8×C53D411yescharacteristic
35C22×C8×C53—22no—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C42×C53C2311yescharacteristic
38#95×C53—22no—
39C2×Q8×C53—21no—
40C2×Q8×C53—41no—
41C2×Q8×C53—41no—
42C2×Q8×C53Q1612yes—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53C2×D411yescharacteristic
46C22×C4×C53C2×D412yes—
47C22×Dic53—22no—
48C22×Dic53—22no—
49C22×Dic53—41no—
50C2×C8×C53—42no—
51C2×C8×C53—42no—
52C2×C8×C53—42no—
53C4⋊C4×C53—41no—
54C42×C53—41no—
55C4⋊Dic53—22no—
56C4⋊Dic53—22no—
57C4⋊Dic53—42no—
58C4⋊Dic53—41no—
59C4⋊Dic53—41no—
60C4⋊Dic53—44no—
61C4⋊Dic53—81no—
62C4⋊Dic53—81no—
63#11719919—42no—
64#11719919—42no—
65C4×Dic53—81no—
66C2×#88322—22no—
67C2×#88322—41no—
68C2×#88322—41no—
69C23×C53#12011yescharacteristic
70Q8×C53—42no—
71Q8×C53—81no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—42no—
81C2×C4×C53C2×Q1611yescharacteristic
82C2×C4×C53#13512yes—
83C2×C4×C53#13712yes—
84C2×C4×C53#13711yescharacteristic
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—42no—
91C2×Dic53—81no—
92C2×Dic53—82no—
93C8×C53—82no—
94#88322—42no—
95#7831—531nomaximal, Sylow
96C22×C53#45111yescharacteristic
97C22×C53#45212yes—
98C22×C53#45311yescharacteristic
99C22×C53#45412yes—
100C22×C53#45811yescharacteristic
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—22no—
104C4×C53—22no—
105C4×C53—42no—
106C4×C53—81no—
107Dic53—82no—
108Dic53—82no—
109Dic53—161no—
110#2809—531no—
111#2100—532no—
112#2095—531no—
113#1272D5311yescharacteristic
114#1271—532no—
115C2×C53#161411yescharacteristic
116C2×C53#161512yes—
117C2×C53#163411yescharacteristic
118C2×C53#163612yes—
119C2×C53#165711yescharacteristic
120C2×#107—1061no—
121C2×#103—1061no—
122#400—1061no—
123#398—1062no—
124#391—1061no—
125#391C2×D5311yescharacteristic
126#389—532no—
127#387—1062no—
128#384—531no—
129#340—1062no—
130#340—1062no—
131#336—531no—
132#336C2×D5312yes—
133C53#783111yescharacteristic, complemented, Sylow
134C22×Q8D21211yescharacteristic
135C22×C8—22no—
136C22×C8—1061no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—1061no—
140C2×C4⋊C4—1061no—
141C2×C4⋊C4—1062no—
142C2×C4⋊C4—1061no—
143C2×C4⋊C4—1062no—
144C2×C4⋊C4D21211yescharacteristic
145C2×C42—1061no—
146C2×C42C22×D5311yescharacteristic
147#107—2122no—
148#103—2122no—
149#95—22no—
150#95—1061no—
151#95—1062no—
152#95—1062no—
153#95—1062no—
154C2×Q8—21no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8Dic21212yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1062no—
161C22×C4—1062no—
162C22×C4—2121no—
163C22×C4C2×D21211yescharacteristic
164C22×C4D4×D5312yes—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—1062no—
169C2×C8—2121no—
170C2×C8—2121no—
171C4⋊C4—41no—
172C4⋊C4—1062no—
173C4⋊C4—1062no—
174C4⋊C4—4241no—
175C4⋊C4—4241no—
176C4⋊C4—2121no—
177C4⋊C4—2121no—
178C4⋊C4—2124no—
179C4⋊C4—2122no—
180C4⋊C4—2122no—
181C4⋊C4—2122no—
182C42—41no—
183C42—4241no—
184C23#1313318711yescharacteristic
185Q8—42no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—4241no—
197C2×C4—4242no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4—2122no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4C2×Dic21211yescharacteristic
205C2×C4#1313323812yes—
206C2×C4#1313324912yes—
207C2×C4#1313325011yescharacteristic
208C8—82no—
209C8—2122no—
210C22#1681078111yescharacteristic
211C22#1681078812yes—
212C22#1681079311yescharacteristic
213C22#1681079712yes—
214C22#1681081411yescharacteristic
215C4—21no—
216C4—21no—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—81no—
221C4—8481no—
222C4—4242no—
223C4—4242no—
224C2#2027374211yescharacteristic
225C2#2027374912yes—
226C2#2027386111yescharacteristic
227C2#2027387612yes—
228C2#2027401411yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socle, prime radicalC23×C53#120
hypercenter#1272D53
hypocenterC53#7831
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
133. C53#7831semidirect

Maximal subgroups

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

subgroupindex
#202827032
#202762792
#202762532
#202725392
C53×#12722
#783153