Finite Group Databasebeta

← #27281929

Subgroups — #27281929

Order 13568.

542 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 127 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#20283739C211yesmaximal, characteristic
2#20280069C211yesmaximal, characteristic
3#20273143C212yesmaximal, complemented
4#20272575C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7D4×#88322—24no—
8#16810676C2212yes—
9C2×C22⋊#88322—22no—
10#16810559C412yes—
11#16810535C412yes—
12#16810526—21no—
13#16810519—21no—
14#16810496—22no—
15#16810489C2211yescharacteristic
16#16810353—24no—
17C2×C4×D4×C53C2211yescharacteristic
18C53×#423C2212yes—
19C2×#100×C53—21no—
20C53×#343—21no—
21C22×Dic106C2×C411yescharacteristic
22C22×#88322—24no—
23C4×Dic106—22no—
24C4×Dic106—22no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C4×Dic53—21no—
32C4⋊#88322—22no—
33C22⋊#88322—24no—
34C22⋊#88322—22no—
35C4×#88322D414yes—
36#13133124—22no—
37#13133124—21no—
38#13133124C2×C412yes—
39C22×D4×C53D411yescharacteristic
40C23×C4×C53D412yes—
41C2×M16×C53—22no—
42C4×D4×C53—22no—
43C4×D4×C53—22no—
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53C2×C412yes—
46C2×C42×C53C2311yescharacteristic
47C4⋊C8×C53—22no—
48C4⋊C8×C53—22no—
49#100×C53—42no—
50C24×C53C22⋊C412yes—
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53SD1612yes—
54C2×D4×C53Q1612yes—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—82no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—22no—
64C22×Dic53—21no—
65C2×Dic106—22no—
66C2×Dic106—42no—
67M16×C53—42no—
68M16×C53—42no—
69C2×C8×C53—42no—
70C4⋊C4×C53—22no—
71C22⋊C4×C53—24no—
72C42×C53C2×D412yes—
73C42×C53#4112yes—
74C22⋊Dic53—44no—
75C22⋊Dic53—82no—
76C4⋊Dic53—22no—
77C4⋊Dic53—42no—
78#11719919—41no—
79#11719919—41no—
80#11719919—82no—
81C4×Dic53—42no—
82C2×#88322—24no—
83C2×#88322—84no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53#10314yes—
89C23×C53C2×C22⋊C411yescharacteristic
90D4×C53—24no—
91D4×C53—24no—
92C2×C4×C53—22no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—44no—
96C2×C4×C53—42no—
97C2×C4×C53—82no—
98C2×C4×C53C4×D412yes—
99C2×C4×C53#12011yescharacteristic
100C2×C4×C53#12112yes—
101C2×C4×C53#12311yescharacteristic
102C2×C4×C53C2×SD1611yescharacteristic
103C2×C4×C53C2×Q1611yescharacteristic
104C2×C4×C53#13512yes—
105C2×Dic53—22no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114Dic106—24no—
115Dic106—42no—
116C8×C53—82no—
117#88322—44no—
118#9483—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—24no—
122C22×C53—44no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—82no—
126C22×C53#38611yescharacteristic
127C22×C53C2×#10311yescharacteristic
128C22×C53#41611yescharacteristic
129C4×C53—22no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—42no—
133C4×C53#46512yes—
134C4×C53#46712yes—
135Dic53—42no—
136Dic53—82no—
137Dic53—81no—
138#2972—531no—
139#2580—531no—
140#1800D5311yescharacteristic
141#1487—532no—
142#1286—532no—
143C2×C53—21no—
144C2×C53—21no—
145C2×C53—44no—
146C2×C53#189111yescharacteristic
147C2×C53#189211yescharacteristic
148C2×C53#191111yescharacteristic
149C2×C4×Q8—531no—
150C2×C4×D4C2×D5311yescharacteristic
151C8×D4—1064no—
152#431—532no—
153#423C2×D5312yes—
154C2×#100—21no—
155C2×C22⋊C8—1062no—
156#399—532no—
157#393—532no—
158#390—1061no—
159#389—1061no—
160#385—1062no—
161#382—531no—
162#343—21no—
163Q8⋊C8—1064no—
164C53#948311yescharacteristic, complemented, Sylow
165C22×Q8—531no—
166C22×D4C22⋊D5311yescharacteristic
167C23×C4D21212yes—
168C2×M16—22no—
169C22×C8—1064no—
170C4×Q8—1062no—
171C4×Q8—1062no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—1062no—
176C2×C4⋊C4—1061no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4C22⋊D5311yescharacteristic
179C2×C22⋊C4—1062no—
180C2×C22⋊C4—1062no—
181C2×C22⋊C4C4×D5312yes—
182C2×C42—1061no—
183C2×C42C22×D5311yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—1062no—
187#100—42no—
188C22⋊C8—1064no—
189C22⋊C8—1062no—
190C4×C8—534no—
191#95—532no—
192#95—1061no—
193#95—1062no—
194C24#1171992112yes—
195C2×Q8—1062no—
196C2×Q8—2122no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4#1171992312yes—
200C2×D4#1171992512yes—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—82no—
204C22×C4—1061no—
205C22×C4—1062no—
206C22×C4—1061no—
207C22×C4#1171992112yes—
208C22×C4C2×C4×D5311yescharacteristic
209C22×C4C2×D21211yescharacteristic
210C22×C4C2×C22⋊D5311yescharacteristic
211M16—42no—
212M16—42no—
213C2×C8—42no—
214C2×C8—1064no—
215C2×C8—4244no—
216C4⋊C4—22no—
217C4⋊C4—1062no—
218C4⋊C4—4242no—
219C4⋊C4—2121no—
220C4⋊C4—2121no—
221C4⋊C4—2122no—
222C22⋊C4—24no—
223C22⋊C4—4242no—
224C22⋊C4—2124no—
225C42—2122no—
226C42D4×D5312yes—
227C42#1171994712yes—
228C23—22no—
229C23—22no—
230C23—42no—
231C23—42no—
232C23#1313314014yes—
233C23C2×#1171992111yescharacteristic
234Q8—1064no—
235Q8—2122no—
236D4—24no—
237D4—24no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—82no—
244C2×C4—1062no—
245C2×C4—2121no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2122no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4#1313317712yes—
254C2×C4#1313318711yescharacteristic
255C2×C4#1313318912yes—
256C2×C4#1313320111yescharacteristic
257C2×C4C2×#1171992311yescharacteristic
258C2×C4C2×#1171992511yescharacteristic
259C2×C4#1313323712yes—
260C8—82no—
261C8—2124no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—82no—
269C22#1681050011yescharacteristic
270C22C2×#1313314011yescharacteristic
271C22#1681061711yescharacteristic
272C4—21no—
273C4—21no—
274C4—22no—
275C4—42no—
276C4—4242no—
277C4—4241no—
278C4—2122no—
279C4#1681085012yes—
280C4#1681086212yes—
281C2—21no—
282C2—21no—
283C2—44no—
284C2#2027513111yescharacteristic
285C2#2027513611yescharacteristic
286C2#2027524511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1800C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1800D53
hypocenterC53#9483
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20273143C2semidirect
4. #20272575C2semidirect
164. C53#9483semidirect

Maximal subgroups

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

subgroupindex
#202837392
#202800692
#202731432
#202725752
C53×#18002
#948353