Finite Group Databasebeta

← #27281952

Subgroups — #27281952

Order 13568.

518 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 119 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#20274707C211yesmaximal, characteristic
3#20273144C212yesmaximal, complemented
4#20272573C212yesmaximal, complemented
5C53×#2580C211yesmaximal, characteristic
6#16810701—24no—
7#16810676C2212yes—
8#16810647C2212yes—
9C2×#13133134—21no—
10#16810587—22no—
11#16810392—21no—
12Q8⋊#88322—24no—
13C2×C4×Q8×C53C2211yescharacteristic
14C2×C4×D4×C53C2211yescharacteristic
15C53×#399C412yes—
16C53×#393C412yes—
17C53×#390—21no—
18C53×#389—21no—
19C53×#385—22no—
20C53×#382C2211yescharacteristic
21C2×#11719917—24no—
22C2×#11719917—22no—
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26#13133134—42no—
27C22⋊#88322—24no—
28C22⋊#88322—22no—
29C4×#88322D414yes—
30C22×Q8×C53C2×C411yescharacteristic
31C22×D4×C53D411yescharacteristic
32C23×C4×C53D412yes—
33C4×Q8×C53—22no—
34C4×Q8×C53—22no—
35C4×D4×C53—22no—
36C4×D4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53C2×C412yes—
45C2×C42×C53—21no—
46C2×C42×C53C2311yescharacteristic
47#95×C53—22no—
48#95×C53—21no—
49#95×C53C2×C412yes—
50C24×C53C22⋊C412yes—
51C2×Q8×C53—22no—
52C2×Q8×C53—42no—
53C2×D4×C53—22no—
54C2×D4×C53—21no—
55C2×D4×C53—21no—
56C2×D4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—82no—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C4⋊C4×C53—22no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—82no—
74C22⋊C4×C53—24no—
75C22⋊C4×C53—44no—
76C22⋊C4×C53—82no—
77C42×C53—42no—
78C42×C53C2×D412yes—
79C42×C53#4112yes—
80#11719917—42no—
81#11719917—42no—
82#11719917—84no—
83C2×#88322—24no—
84C2×#88322—42no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53C2×C22⋊C411yescharacteristic
91Q8×C53—24no—
92Q8×C53—42no—
93D4×C53—42no—
94D4×C53—42no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—44no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—82no—
110C2×C4×C53C4×D412yes—
111C2×C4×C53#12011yescharacteristic
112C2×C4×C53#12112yes—
113C2×C4×C53#12311yescharacteristic
114C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C53#13711yescharacteristic
116C2×C4×C53#13712yes—
117#88322—44no—
118#88322—82no—
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—42no—
135C4×C53—82no—
136C4×C53—81no—
137C4×C53#46812yes—
138C4×C53#47012yes—
139#2975—531no—
140#2580D5311yescharacteristic
141#1800—531no—
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×Q8C2×D5311yescharacteristic
151C2×C4×D4C2×D5311yescharacteristic
152#436—1064no—
153#431—532no—
154#423—532no—
155C2×#100—1061no—
156#407—1062no—
157#399Dic5312yes—
158#393Dic5312yes—
159#390—21no—
160#389—21no—
161#385—22no—
162#382C2×D5311yescharacteristic
163#343—1061no—
164Q8⋊C8—1064no—
165C53#948711yescharacteristic, complemented, Sylow
166C22×Q8C2×Dic5311yescharacteristic
167C22×D4C22⋊D5311yescharacteristic
168C23×C4C22⋊D5312yes—
169C2×M16—1064no—
170C2×M16—1062no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×D4—22no—
174C4×D4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C2×Dic5311yescharacteristic
179C2×C4⋊C4C22⋊D5311yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C2×Dic5312yes—
183C2×C42—21no—
184C2×C42C22×D5311yescharacteristic
185C4⋊C8—1062no—
186C4⋊C8—1062no—
187C4⋊C8—1062no—
188#100—2122no—
189C22⋊C8—1064no—
190C22⋊C8—1062no—
191C4×C8—534no—
192#95—21no—
193#95—22no—
194#95C2×Dic5312yes—
195C24C22⋊Dic5312yes—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—82no—
208C22×C4C22⋊Dic5312yes—
209C22×C4C22×Dic5311yescharacteristic
210C22×C4C2×C22⋊D5311yescharacteristic
211C22×C4C2×C22⋊D5311yescharacteristic
212M16—4244no—
213M16—2122no—
214M16—2122no—
215C2×C8—1064no—
216C2×C8—2122no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—82no—
224C22⋊C4—24no—
225C22⋊C4—44no—
226C22⋊C4—82no—
227C42—42no—
228C42D4×D5312yes—
229C42#1171994712yes—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—42no—
235C23C2×C22⋊Dic5311yescharacteristic
236Q8—24no—
237Q8—42no—
238D4—42no—
239D4—42no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—44no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4D4×Dic5312yes—
256C2×C4#1313318511yescharacteristic
257C2×C4#1313318912yes—
258C2×C4#1313319811yescharacteristic
259C2×C4#1313324211yescharacteristic
260C2×C4#1313324711yescharacteristic
261C2×C4#1313324812yes—
262C8—4242no—
263C8—2124no—
264C22—21no—
265C22—21no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22#1681050311yescharacteristic
272C22#1681062411yescharacteristic
273C22#1681063211yescharacteristic
274C4—21no—
275C4—21no—
276C4—22no—
277C4—42no—
278C4—42no—
279C4—82no—
280C4—81no—
281C4#1681087112yes—
282C4#1681087912yes—
283C2—21no—
284C2—21no—
285C2—82no—
286C2#2027513411yescharacteristic
287C2#2027514911yescharacteristic
288C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2580C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2580D53
hypocenterC53#9487
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #20273144C2semidirect
4. #20272573C2semidirect
165. C53#9487semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202747072
#202731442
#202725732
C53×#25802
#948753