Finite Group Databasebeta

← #27296589

Subgroups — #27296589

Order 13568.

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

normal subgroups only

This group has 330 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#20276648C212yesmaximal
2#20275983C211yesmaximal, characteristic
3#20272733C212yesmaximal
4#20272657C211yesmaximal, characteristic
5C53×#3261C211yesmaximal, characteristic
6C22×Dic212C2211yescharacteristic
7C2×#13133163—22no—
8#16810631—21no—
9C2×#13133135—21no—
10#16810426—22no—
11C22⋊#11719908C2212yes—
12#16810388C2211yescharacteristic
13C53×#541—21no—
14C53×#479—22no—
15C53×#470—21no—
16C53×#467C2212yes—
17C53×#462—22no—
18C53×#424C2211yescharacteristic
19C22×Dic106—21no—
20C2×Dic212—22no—
21C2×Dic212—42no—
22C2×#11719917—21no—
23#13133163—44no—
24C2×#11719908—22no—
25#13133140—42no—
26#13133135—42no—
27C2×Q16×C53—21no—
28C2×Q16×C53D412yes—
29C2×M16×C53D411yescharacteristic
30C22×C8×C53C2311yescharacteristic
31C4⋊Q8×C53—21no—
32#126×C53—42no—
33#125×C53—21no—
34#125×C53—42no—
35C22⋊Q8×C53—22no—
36C4×Q8×C53—22no—
37#117×C53D411yescharacteristic
38#107×C53D412yes—
39#106×C53—21no—
40C4⋊C8×C53—22no—
41#103×C53—22no—
42#103×C53—22no—
43C2×Q8×C53—22no—
44C22×C4×C53C2×D411yescharacteristic
45C22×Dic53—41no—
46C2×Dic106—22no—
47C2×Dic106—41no—
48C2×Dic106—41no—
49C2×Dic106—41no—
50C2×Dic106—41no—
51C2×Dic106—41no—
52Q16×C53—42no—
53Q16×C53—81no—
54M16×C53—41no—
55C2×C8×C53—21no—
56C2×C8×C53—41no—
57C2×C8×C53C2×D412yes—
58C4⋊C4×C53—22no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53D812yes—
64C22⋊C4×C53—21no—
65C22⋊C4×C53—42no—
66C42×C53—21no—
67C42×C53—42no—
68#11719917—41no—
69#11719917—41no—
70C2×#88322—41no—
71Dic212—24no—
72Dic212—42no—
73#11719908—42no—
74C23×C53AGL(1,ℤ/8)11yescharacteristic
75Q8×C53—42no—
76Q8×C53—42no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53#12011yescharacteristic
83C2×C4×C53C2×D811yescharacteristic
84C2×Dic53—42no—
85C2×Dic53—81no—
86C2×Dic53—81no—
87Dic106—42no—
88Dic106—42no—
89Dic106—81no—
90Dic106—81no—
91C8×C53—22no—
92C8×C53—41no—
93#88322—81no—
94#12168—531nomaximal, Sylow
95C22×C53—21no—
96C22×C53—22no—
97C22×C53#44711yescharacteristic
98C4×C53—21no—
99C4×C53—42no—
100C4×C53—82no—
101C4×C53—82no—
102C4×C53#45411yescharacteristic
103C4×C53B(4,𝔽2)11yescharacteristic
104Dic53—82no—
105#3261D5311yescharacteristic
106#2188—532no—
107#2047—531no—
108#1350—532no—
109#1323—531no—
110C2×C53—41no—
111C2×C53#159811yescharacteristic
112C2×C53#219512yes—
113C22×Q16—531no—
114#541—21no—
115C2×Q32—1062no—
116#479—22no—
117#470—21no—
118#467C2×D5312yes—
119#462—22no—
120#424C2×D5311yescharacteristic
121#419—1061no—
122C2×#101—1061no—
123#358—1062no—
124C22⋊C16—532no—
125#341—531no—
126C53#1216811yescharacteristic, complemented, Sylow
127C22×Q8—1061no—
128C2×Q16—21no—
129C2×Q16—1062no—
130C2×Q16—2122no—
131C2×Q16C22⋊D5312yes—
132C2×M16—1061no—
133C2×M16C22⋊D5311yescharacteristic
134C22×C8C22×D5311yescharacteristic
135C4⋊Q8—21no—
136#126—42no—
137#125—21no—
138#125—42no—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117C22⋊D5311yescharacteristic
142Q32—2124no—
143C2×C16—1062no—
144#107C22⋊D5312yes—
145#106—21no—
146C4⋊C8—22no—
147#103—22no—
148#103—22no—
149#103—2122no—
150#101—2122no—
151C2×Q8—22no—
152C2×Q8—1062no—
153C2×Q8—2121no—
154C2×Q8—2121no—
155C2×Q8—2121no—
156C2×Q8—2121no—
157C2×Q8—2121no—
158C22×C4—2121no—
159C22×C4C2×C22⋊D5311yescharacteristic
160Q16—42no—
161Q16—81no—
162Q16—1064no—
163Q16—2122no—
164M16—41no—
165M16—2121no—
166M16—2121no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—2121no—
170C2×C8C2×C22⋊D5312yes—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4#1171992212yes—
177C22⋊C4—21no—
178C22⋊C4—42no—
179C42—21no—
180C42—42no—
181C16—2122no—
182C23#1313324311yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—4241no—
186Q8—4241no—
187Q8—2122no—
188Q8—2122no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—4241no—
195C2×C4—4241no—
196C2×C4—2122no—
197C2×C4#1313318611yescharacteristic
198C2×C4C2×#1171992211yescharacteristic
199C8—22no—
200C8—41no—
201C8—4241no—
202C22—22no—
203C22—21no—
204C22#1681075711yescharacteristic
205C4—21no—
206C4—42no—
207C4—82no—
208C4—82no—
209C4—4242no—
210C4#1681079611yescharacteristic
211C4#1681081011yescharacteristic
212C2—41no—
213C2#2027365011yescharacteristic
214C2#2027669312yes—
subgroupG/H
centerC22#16810757
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#3261C2
socleC22×C53#447
prime radicalC23×C53AGL(1,ℤ/8)
hypercenter#3261D53
hypocenterC53#12168
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ #3261

Complements

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

NH ≅ G/Ntype
126. C53#12168semidirect

Maximal subgroups

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

subgroupindex
#202766482
#202759832
#202727332
#202726572
C53×#32612
#1216853