Finite Group Databasebeta

← #27273393

Subgroups — #27273393

Order 13568.

394 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 87 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#20281747C211yesmaximal, characteristic
2#20276279C212yesmaximal
3#20276253C211yesmaximal, characteristic
4#20272537C212yesmaximal
5C53×#1272C211yesmaximal, characteristic
6C2×#13133215—21no—
7C2×#13133150—21no—
8C2×#13133141—21no—
9#16810553—21no—
10#16810544—22no—
11#16810521C2212yes—
12#16810508—22no—
13#16810493C2211yescharacteristic
14#16810386—22no—
15#16810386—22no—
16#16810378C2211yescharacteristic
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106—21no—
20C22×#88322—21no—
21#13133215—42no—
22#13133215—42no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×C4×Dic53—21no—
31#13133150—42no—
32#13133141—42no—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36C22×Q8×C53D411yescharacteristic
37C22×C8×C53—22no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C42×C53C2311yescharacteristic
40#95×C53—22no—
41C2×Q8×C53—21no—
42C2×Q8×C53—41no—
43C2×Q8×C53—41no—
44C2×Q8×C53SD1612yes—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—41no—
52C2×Dic106—42no—
53C2×Dic106—81no—
54C2×Dic106—81no—
55C2×C8×C53—42no—
56C2×C8×C53—42no—
57C2×C8×C53—42no—
58C4⋊C4×C53—41no—
59C42×C53—41no—
60C4⋊Dic53—22no—
61C4⋊Dic53—42no—
62C4⋊Dic53—41no—
63C4⋊Dic53—41no—
64C4⋊Dic53—44no—
65C4⋊Dic53—81no—
66C4⋊Dic53Q1614yes—
67#11719919—44no—
68#11719919—42no—
69#11719919—42no—
70C4×Dic53—42no—
71C2×#88322—22no—
72C2×#88322—41no—
73C2×#88322—41no—
74C23×C53#12011yescharacteristic
75Q8×C53—42no—
76Q8×C53—81no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53C2×SD1611yescharacteristic
87C2×C4×C53C2×Q1612yes—
88C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C53#13712yes—
90C2×Dic53—24no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—81no—
97C2×Dic53—82no—
98Dic106—84no—
99C8×C53—82no—
100#88322—42no—
101#7829—531nomaximal, Sylow
102C22×C53#45011yescharacteristic
103C22×C53#45112yes—
104C22×C53#45311yescharacteristic
105C22×C53#45412yes—
106C22×C53#45811yescharacteristic
107C4×C53—21no—
108C4×C53—21no—
109C4×C53—22no—
110C4×C53—22no—
111C4×C53—42no—
112C4×C53—81no—
113Dic53—44no—
114Dic53—82no—
115Dic53—161no—
116#2734—531no—
117#2100—532no—
118#2095—531no—
119#1272D5311yescharacteristic
120#1271—532no—
121C2×C53#161311yescharacteristic
122C2×C53#161712yes—
123C2×C53#163311yescharacteristic
124C2×C53#163712yes—
125C2×C53#165711yescharacteristic
126C2×C4⋊Q8—1061no—
127C2×#107—1061no—
128C2×#103—1061no—
129#398—1061no—
130#395—1062no—
131#391C2×D5311yescharacteristic
132#389—532no—
133#387—1062no—
134#384—531no—
135#340—1062no—
136#340—1062no—
137#336—531no—
138#336C2×D5312yes—
139C53#782911yescharacteristic, complemented, Sylow
140C22×Q8—1061no—
141C22×Q8D21211yescharacteristic
142C22×C8—22no—
143C22×C8—1061no—
144C4⋊Q8—2122no—
145C4⋊Q8—2122no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—1061no—
149C2×C4⋊C4—1062no—
150C2×C4⋊C4—1062no—
151C2×C4⋊C4—1061no—
152C2×C4⋊C4—1062no—
153C2×C4⋊C4D21211yescharacteristic
154C2×C42—1061no—
155C2×C42C22×D5311yescharacteristic
156#107—2122no—
157#103—2122no—
158#95—22no—
159#95—1061no—
160#95—1062no—
161#95—1062no—
162C2×Q8—21no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—4241no—
166C2×Q8—4241no—
167C2×Q8—2122no—
168C2×Q8#1171991312yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1062no—
172C22×C4—1062no—
173C22×C4—2121no—
174C22×C4C2×D21211yescharacteristic
175C22×C4D4×D5312yes—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—1062no—
180C2×C8—2121no—
181C2×C8—2121no—
182C4⋊C4—41no—
183C4⋊C4—534no—
184C4⋊C4—1062no—
185C4⋊C4—4241no—
186C4⋊C4—2121no—
187C4⋊C4—2121no—
188C4⋊C4—2124no—
189C4⋊C4—2122no—
190C4⋊C4—2124no—
191C4⋊C4—2122no—
192C4⋊C4—2122no—
193C42—41no—
194C42—2122no—
195C23#1313318711yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—4244no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—1064no—
209C2×C4—4241no—
210C2×C4—4242no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4C2×#1171991311yescharacteristic
217C2×C4Q16×D5312yes—
218C2×C4#1313324511yescharacteristic
219C2×C4#1313324612yes—
220C8—82no—
221C8—2122no—
222C22#1681077611yescharacteristic
223C22#1681078012yes—
224C22#1681079311yescharacteristic
225C22#1681079712yes—
226C22#1681081411yescharacteristic
227C4—21no—
228C4—21no—
229C4—22no—
230C4—22no—
231C4—42no—
232C4—81no—
233C4—8481no—
234C4—4242no—
235C4—2124no—
236C2#2027373711yescharacteristic
237C2#2027375812yes—
238C2#2027385611yescharacteristic
239C2#2027388212yes—
240C2#2027401411yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socle, prime radicalC23×C53#120
hypercenter#1272D53
hypocenterC53#7829
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
139. C53#7829semidirect

Maximal subgroups

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

subgroupindex
#202817472
#202762792
#202762532
#202725372
C53×#12722
#782953