Finite Group Databasebeta

← #27297170

Subgroups — #27297170

Order 13568.

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

normal subgroups only

This group has 342 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#20282987C211yesmaximal, characteristic
2#20273689C212yesmaximal
3#20272744C212yesmaximal
4#20272659C211yesmaximal, characteristic
5C53×#2061C211yesmaximal, characteristic
6#16810778—22no—
7#16810565—22no—
8#16810562—21no—
9#16810544—22no—
10#16810543C2212yes—
11#16810489—21no—
12#16810360—22no—
13#16810349C2212yes—
14C53×#544C2211yescharacteristic
15C53×#428—21no—
16C53×#382C2211yescharacteristic
17C53×#343C2211yescharacteristic
18C53×#339—21no—
19C22×Dic106D412yes—
20C2×#11719925—42no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×#11719919—22no—
25#13133140—42no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—22no—
29#13133124—21no—
30#13133124—22no—
31#13133124D412yes—
32C2×M16×C53—21no—
33C4⋊Q8×C53—41no—
34#125×C53—41no—
35C22⋊Q8×C53—22no—
36#117×C53—21no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53D411yescharacteristic
40#107×C53—41no—
41#106×C53—41no—
42#95×C53—22no—
43C2×Q8×C53—22no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C2×D411yescharacteristic
47C22×C4×C53C2×D412yes—
48C22×Dic53—22no—
49C22×Dic53—22no—
50C2×Dic106—42no—
51C2×Dic106—82no—
52M16×C53—22no—
53M16×C53—41no—
54C2×C8×C53—41no—
55C4⋊C4×C53—21no—
56C4⋊C4×C53—21no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—41no—
61C22⋊C4×C53—41no—
62C42×C53—21no—
63C42×C53—41no—
64C42×C53Q1612yescomplemented
65#11719925—84no—
66C4⋊Dic53—81no—
67C4⋊Dic53—81no—
68C4⋊Dic53—81no—
69#11719919—82no—
70C2×#88322—42no—
71C23×C53#12011yescharacteristic
72Q8×C53—82no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53C2×Q1611yescharacteristic
85C2×C4×C53#13711yescharacteristic
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92Dic106—44no—
93Dic106—82no—
94C8×C53—42no—
95#88322—82no—
96#12260—531nomaximal, Sylow
97C22×C53—21no—
98C22×C53—21no—
99C22×C53#45111yescharacteristic
100C22×C53#45611yescharacteristic
101C22×C53B(4,𝔽2)11yescharacteristic
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—42no—
105C4×C53—42no—
106C4×C53—81no—
107C4×C53—81no—
108C4×C53—81no—
109Dic53—82no—
110Dic53—82no—
111#2844—531no—
112#2061D5311yescharacteristic
113#1605—532no—
114#1354—532no—
115#1324—531no—
116C2×C53—21no—
117C2×C53—21no—
118C2×C53#160511yescharacteristic
119C2×C53#219911yescharacteristic
120C2×C53#220311yescharacteristic
121#544C2×D5311yescharacteristic
122#451—1062no—
123#428—21no—
124#400—1062no—
125#400—1061no—
126#395—532no—
127#395—1062no—
128#382—1061no—
129#382C2×D5311yescharacteristic
130#343C2×D5311yescharacteristic
131#339—21no—
132#328—1062no—
133#324—532no—
134C53#1226011yescharacteristic, complemented, Sylow
135C22×Q8—532no—
136C2×Q16—2122no—
137C2×M16—21no—
138C4⋊Q8—41no—
139#125—41no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—1061no—
143C2×C4⋊C4—1061no—
144C2×C4⋊C4—1061no—
145C2×C4⋊C4—1062no—
146C2×C4⋊C4C22⋊D5311yescharacteristic
147C2×C4⋊C4C22×D5311yescharacteristic
148C2×C42C22⋊D5311yescharacteristic
149#107—41no—
150#106—41no—
151#103—2122no—
152C22⋊C8—1062no—
153#95—22no—
154#95—532no—
155#95—1062no—
156#95—1062no—
157#95—1061no—
158#95—1062no—
159C2×Q8—22no—
160C2×Q8—4242no—
161C2×Q8—2122no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—1062no—
165C22×C4—1062no—
166C22×C4D4×D5312yes—
167C22×C4C2×C22⋊D5311yescharacteristic
168Q16—4244no—
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C2×C8—2122no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—4242no—
180C4⋊C4—4241no—
181C4⋊C4—4241no—
182C4⋊C4—4241no—
183C22⋊C4—41no—
184C42—21no—
185C42—41no—
186C42#1171992512yescomplemented
187C23#1313318511yescharacteristic
188Q8—82no—
189Q8—4242no—
190Q8—2124no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4C2×#1171992511yescharacteristic
209C2×C4#1313324711yescharacteristic
210C8—42no—
211C8—4242no—
212C22—21no—
213C22—21no—
214C22#1681077811yescharacteristic
215C22#1681080511yescharacteristic
216C22#1681080911yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4—4242no—
225C4—4242no—
226C2—21no—
227C2—21no—
228C2#2027368811yescharacteristic
229C2#2027671511yescharacteristic
230C2#2027673511yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2061C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2061D53
hypocenterC53#12260
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
64. C42×C53Q16semidirect
134. C53#12260semidirect
186. C42#11719925semidirect

Maximal subgroups

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

subgroupindex
#202829872
#202736892
#202727442
#202726592
C53×#20612
#1226053