Finite Group Databasebeta

← #27277665

Subgroups — #27277665

Order 13568.

422 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 123 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#20277636C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272616C212yesmaximal
4#20272575C212yesmaximal
5C53×#1801C211yesmaximal, characteristic
6C2×Q8×Dic53—21no—
7#16810676C2212yes—
8#16810522—22no—
9#16810522—21no—
10#16810489C412yes—
11#16810489C2211yescharacteristic
12C4×C4⋊Dic53C412yes—
13C4×C4⋊Dic53C2211yescharacteristic
14#16810349—22no—
15C2×C4×Q8×C53C2211yescharacteristic
16C53×#423C2212yes—
17C2×#101×C53—21no—
18C53×#343—21no—
19Q8×Dic53—42no—
20Q8×Dic53—42no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4×Dic53—22no—
26C2×C4×Dic53—21no—
27C4⋊#88322—22no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30#13133124—22no—
31#13133124—21no—
32#13133124C2×C411yescharacteristic
33#13133124C2×C412yes—
34#13133124C2×C411yescharacteristic
35C22×Q8×C53D411yescharacteristic
36C2×M16×C53—22no—
37C4×Q8×C53—22no—
38C4×Q8×C53—22no—
39C2×C4⋊C4×C53D412yes—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C42×C53C2×C412yes—
42C2×C42×C53C2311yescharacteristic
43C4⋊C8×C53—22no—
44C4⋊C8×C53—22no—
45#101×C53—42no—
46C2×Q8×C53—22no—
47C2×Q8×C53—22no—
48C2×Q8×C53D812yes—
49C2×Q8×C53SD1612yes—
50C22×C4×C53C22⋊C412yes—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22×C411yescharacteristic
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53C2×D411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53—22no—
58M16×C53—42no—
59M16×C53—42no—
60C2×C8×C53—42no—
61C4⋊C4×C53—24no—
62C4⋊C4×C53—22no—
63C42×C53—22no—
64C42×C53—22no—
65C42×C53#4112yes—
66C42×C53#4112yes—
67C4⋊Dic53—42no—
68C4⋊Dic53—44no—
69C4⋊Dic53—44no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C4×Dic53—44no—
73C4×Dic53—41no—
74C4×Dic53—41no—
75C2×#88322—42no—
76C23×C53C2×C22⋊C411yescharacteristic
77Q8×C53—24no—
78Q8×C53—24no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—42no—
85C2×C4×C53#10214yes—
86C2×C4×C53#11711yescharacteristic
87C2×C4×C53#11711yescharacteristic
88C2×C4×C53#12312yes—
89C2×C4×C53#12312yes—
90C2×C4×C53C2×D811yescharacteristic
91C2×C4×C53C2×SD1611yescharacteristic
92C2×C4×C53#13712yes—
93C2×Dic53—22no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C2×Dic53—41no—
97C2×Dic53—42no—
98C2×Dic53—41no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—41no—
102C8×C53—82no—
103#88322—82no—
104#8766—531nomaximal, Sylow
105C22×C53—21no—
106C22×C53—21no—
107C22×C53#38111yescharacteristic
108C22×C53C2×#10211yescharacteristic
109C22×C53#41911yescharacteristic
110C4×C53—24no—
111C4×C53—22no—
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—42no—
115C4×C53#48012yes—
116C4×C53#48312yes—
117Dic53—42no—
118Dic53—82no—
119Dic53—81no—
120#2351—531no—
121#1801D5311yescharacteristic
122#1461—531no—
123#1303—532no—
124#1286—532no—
125C2×C53—21no—
126C2×C53—21no—
127C2×C53#182311yescharacteristic
128C2×C53#182711yescharacteristic
129C2×C53#183011yescharacteristic
130C2×C4×Q8—1061no—
131C2×C4×Q8C2×D5311yescharacteristic
132#431—532no—
133#423C2×D5312yes—
134C2×#101—21no—
135#389—1062no—
136#389—1061no—
137#382—532no—
138#382—531no—
139C4×C4⋊C4—532no—
140C4×C4⋊C4—531no—
141#343—21no—
142#324—1062no—
143C53#876611yescharacteristic, complemented, Sylow
144C22×Q8C22⋊D5311yescharacteristic
145C2×M16—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—2122no—
149C4×Q8—2122no—
150C2×C4⋊C4—1061no—
151C2×C4⋊C4—1061no—
152C2×C4⋊C4—1062no—
153C2×C4⋊C4—1062no—
154C2×C4⋊C4D21212yes—
155C2×C4⋊C4C22⋊D5311yescharacteristic
156C2×C42—1061no—
157C2×C42—1062no—
158C2×C42C4×D5312yes—
159C2×C42C22×D5311yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—1062no—
163#101—42no—
164C22⋊C8—1062no—
165C4×C8—1062no—
166#95—531no—
167#95—532no—
168#95—531no—
169#95—1061no—
170#95—1062no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8#1171992212yes—
174C2×Q8#1171992412yes—
175C22×C4—1061no—
176C22×C4—1062no—
177C22×C4—1061no—
178C22×C4#1171992112yes—
179C22×C4#1171992112yes—
180C22×C4C2×C4×D5311yescharacteristic
181C22×C4C2×D21211yescharacteristic
182C22×C4C2×C22⋊D5311yescharacteristic
183M16—42no—
184M16—42no—
185C2×C8—42no—
186C2×C8—2122no—
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—2122no—
190C4⋊C4—2124no—
191C4⋊C4—2124no—
192C4⋊C4—2121no—
193C4⋊C4—2121no—
194C42—22no—
195C42—22no—
196C42—2124no—
197C42—2121no—
198C42—2121no—
199C42#1171994712yes—
200C42#1171994712yes—
201C23C2×#1171992111yescharacteristic
202Q8—24no—
203Q8—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—1062no—
211C2×C4—2121no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4—2121no—
215C2×C4—2122no—
216C2×C4—2121no—
217C2×C4—2121no—
218C2×C4—2121no—
219C2×C4#1313313714yes—
220C2×C4#1313317311yescharacteristic
221C2×C4#1313317311yescharacteristic
222C2×C4#1313319812yes—
223C2×C4#1313320112yes—
224C2×C4C2×#1171992211yescharacteristic
225C2×C4C2×#1171992411yescharacteristic
226C2×C4#1313324712yes—
227C8—82no—
228C8—4242no—
229C22—21no—
230C22—21no—
231C22#1681048411yescharacteristic
232C22C2×#1313313711yescharacteristic
233C22#1681063111yescharacteristic
234C4—24no—
235C4—22no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—4242no—
240C4—4241no—
241C4—2122no—
242C4#1681093412yes—
243C4#1681095112yes—
244C2—21no—
245C2—21no—
246C2#2027479911yescharacteristic
247C2#2027482011yescharacteristic
248C2#2027482711yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1801C2
socleC22×C53#381
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1801D53
hypocenterC53#8766
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
143. C53#8766semidirect

Maximal subgroups

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

subgroupindex
#202776362
#202730392
#202726162
#202725752
C53×#18012
#876653