Finite Group Databasebeta

← #27281949

Subgroups — #27281949

Order 13568.

518 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283756C211yesmaximal, characteristic
2#20280069C211yesmaximal, characteristic
3#20273143C212yesmaximal, complemented
4#20272575C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7#16810701—24no—
8#16810676C2212yes—
9#16810587—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
22C2×#11719917—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—21no—
53C2×D4×C53—22no—
54C2×D4×C53—21no—
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—41no—
78C4⋊Dic53—41no—
79#11719919—41no—
80#11719919—41no—
81#11719919—82no—
82C4×Dic53—42no—
83#11719917—84no—
84C2×#88322—24no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53C2×C22⋊C411yescharacteristic
91D4×C53—42no—
92D4×C53—42no—
93C2×C4×C53—22no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—44no—
97C2×C4×C53—42no—
98C2×C4×C53—82no—
99C2×C4×C53C4×D412yes—
100C2×C4×C53#12011yescharacteristic
101C2×C4×C53#12112yes—
102C2×C4×C53#12311yescharacteristic
103C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
104C2×C4×C53#13712yes—
105C2×C4×C53#13711yescharacteristic
106C2×Dic53—22no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—42no—
115Dic106—24no—
116Dic106—42no—
117C8×C53—82no—
118#88322—44no—
119#9487—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—44no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—82no—
127C22×C53#38611yescharacteristic
128C22×C53#41711yescharacteristic
129C22×C53#41911yescharacteristic
130C4×C53—22no—
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—42no—
134C4×C53#46812yes—
135C4×C53#47012yes—
136Dic53—42no—
137Dic53—82no—
138Dic53—81no—
139#2975—531no—
140#2580—531no—
141#1800D5311yescharacteristic
142#1487—532no—
143#1286—532no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—82no—
147C2×C53#189111yescharacteristic
148C2×C53#189411yescharacteristic
149C2×C53#191111yescharacteristic
150C2×C4×Q8—531no—
151C2×C4×D4C2×D5311yescharacteristic
152#436—1064no—
153#431—532no—
154#423C2×D5312yes—
155C2×#100—21no—
156#407—1062no—
157#399—532no—
158#393—532no—
159#390—1061no—
160#389—1061no—
161#385—1062no—
162#382—531no—
163#343—21no—
164Q8⋊C8—1064no—
165C53#948711yescharacteristic, complemented, Sylow
166C22×Q8—531no—
167C22×D4C22⋊D5311yescharacteristic
168C23×C4D21212yes—
169C2×M16—22no—
170C2×M16—1064no—
171C4×Q8—1062no—
172C4×Q8—1062no—
173C4×D4—22no—
174C4×D4—22no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—1062no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4C22⋊D5311yescharacteristic
180C2×C22⋊C4—1062no—
181C2×C22⋊C4—1062no—
182C2×C22⋊C4C4×D5312yes—
183C2×C42—1061no—
184C2×C42C22×D5311yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—1062no—
188#100—42no—
189C22⋊C8—1064no—
190C22⋊C8—1062no—
191C4×C8—534no—
192#95—532no—
193#95—1061no—
194#95—1062no—
195C24#1171992112yes—
196C2×Q8—1062no—
197C2×Q8—2122no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—82no—
205C22×C4—1061no—
206C22×C4—1062no—
207C22×C4—1061no—
208C22×C4#1171992112yes—
209C22×C4C2×C4×D5311yescharacteristic
210C22×C4C2×D21211yescharacteristic
211C22×C4C2×C22⋊D5311yescharacteristic
212M16—42no—
213M16—42no—
214M16—4244no—
215C2×C8—42no—
216C2×C8—1064no—
217C4⋊C4—22no—
218C4⋊C4—1062no—
219C4⋊C4—4242no—
220C4⋊C4—2121no—
221C4⋊C4—2121no—
222C4⋊C4—2121no—
223C4⋊C4—2121no—
224C22⋊C4—24no—
225C22⋊C4—4242no—
226C22⋊C4—2124no—
227C42—2122no—
228C42D4×D5312yes—
229C42#1171994712yes—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—42no—
235C23C2×#1171992111yescharacteristic
236Q8—1064no—
237Q8—2122no—
238D4—42no—
239D4—42no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—1062no—
247C2×C4—2121no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2122no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4#1313317712yes—
256C2×C4#1313318711yescharacteristic
257C2×C4#1313318912yes—
258C2×C4#1313320111yescharacteristic
259C2×C4#1313324311yescharacteristic
260C2×C4#1313324712yes—
261C2×C4#1313324811yescharacteristic
262C8—82no—
263C8—2124no—
264C22—21no—
265C22—21no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22#1681050011yescharacteristic
272C22#1681062211yescharacteristic
273C22#1681063111yescharacteristic
274C4—21no—
275C4—21no—
276C4—22no—
277C4—42no—
278C4—4242no—
279C4—4241no—
280C4—2122no—
281C4#1681086712yes—
282C4#1681087912yes—
283C2—21no—
284C2—21no—
285C2—82no—
286C2#2027513111yescharacteristic
287C2#2027514611yescharacteristic
288C2#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#9487
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
165. C53#9487semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202800692
#202731432
#202725752
C53×#18002
#948753