Finite Group Databasebeta

← #27300097

Subgroups — #27300097

Order 13568.

480 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 101 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#20277428C211yesmaximal, characteristic
2C2×#16810420C211yesmaximal, characteristic
3#20274709C212yesmaximal, complemented
4#20272841C212yesmaximal, complemented
5C2×C53×#355C211yesmaximal, characteristic
6C2×Q8×Dic53C2211yescharacteristic
7C2×D4×Dic53C411yescharacteristic
8#16810646—22no—
9C2×#13133136C2212yes—
10#16810530C2211yescharacteristic
11#16810526C411yescharacteristic
12#16810516—22no—
13#16810498—22no—
14#16810493C411yescharacteristic
15#16810469C411yescharacteristic
16#16810420—22no—
17#16810391—22no—
18C2×#124×C53C2211yescharacteristic
19C2×#101×C53C2212yes—
20C53×#355—22no—
21C23×Dic53—21no—
22C2×#11719917—22no—
23Q8×Dic53—42no—
24D4×Dic53—42no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53D412yes—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—22no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34#13133146—44no—
35#13133136—22no—
36#13133136—22no—
37#13133124—22no—
38#13133124—21no—
39#13133124—21no—
40C22×Q8×C53C2311yescharacteristic
41C22×D4×C53C2×C411yescharacteristic
42C2×M16×C53—22no—
43#124×C53—22no—
44#124×C53D412yes—
45#124×C53Q812yes—
46C2×C22⋊C4×C53—22no—
47C2×C42×C53C2×C411yescharacteristic
48#101×C53—22no—
49#101×C53—22no—
50C24×C53—21no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53C2×D411yescharacteristic
54C2×Q8×C53C2×Q811yescharacteristic
55C2×Q8×C53#4112yes—
56C2×D4×C53—21no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53C4⋊C412yes—
60C22×C4×C53—21no—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D412yes—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53—42no—
66C22×Dic53—82no—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C22⋊C412yes—
69C22×Dic53C22⋊C411yescharacteristic
70M16×C53—42no—
71M16×C53—42no—
72C2×C8×C53—42no—
73C22⋊C4×C53—24no—
74C22⋊C4×C53—24no—
75C42×C53—21no—
76C42×C53C4⋊C412yes—
77C22⋊Dic53—42no—
78C22⋊Dic53—82no—
79C4⋊Dic53—41no—
80C4⋊Dic53—42no—
81C4⋊Dic53—41no—
82C4⋊Dic53—82no—
83C4×Dic53—24no—
84C4×Dic53—22no—
85C4×Dic53—41no—
86#11719917—42no—
87#11719917—42no—
88C2×#88322—42no—
89C23×C53—22no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—41no—
96Q8×C53—41no—
97Q8×C53—42no—
98D4×C53—42no—
99D4×C53—42no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53C2×C4⋊C411yescharacteristic
108C2×C4×C53#11711yescharacteristic
109C2×C4×C53#12112yes—
110C2×C4×C53C22⋊Q812yes—
111C2×Dic53—21no—
112C2×Dic53—22no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—41no—
117C2×Dic53—42no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—84no—
122C2×Dic53#9912yes—
123C8×C53—82no—
124#88322—82no—
125#12803—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—44no—
129C22×C53—42no—
130C22×C53—82no—
131C22×C53—81no—
132C22×C53—81no—
133C22×C53#38011yescharacteristic
134C22×C53C2×#9911yescharacteristic
135C22×C53#41011yescharacteristic
136C4×C53—22no—
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—22no—
140C4×C53—42no—
141Dic53—22no—
142Dic53—42no—
143Dic53—41no—
144Dic53—81no—
145Dic53—81no—
146#2326—531no—
147C2×#355—531no—
148C2×#355D5311yescharacteristic
149#1801—532no—
150#1394—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—82no—
154C2×C53#179911yescharacteristic
155C2×C53#213311yescharacteristic
156C2×C53#213411yescharacteristic
157C2×#124C2×D5311yescharacteristic
158C2×C4×Q8—531no—
159C2×C4×D4—531no—
160#423—1062no—
161C2×#101—532no—
162C2×#101C2×D5312yes—
163#391—531no—
164#390—531no—
165#388—1062no—
166#385—1062no—
167#384—531no—
168#376—531no—
169#355—22no—
170#355—1062no—
171#343—1062no—
172C53#1280311yescharacteristic, complemented, Sylow
173C22×Q8C22×D5311yescharacteristic
174C22×D4C4×D5311yescharacteristic
175C23×C4—1061no—
176C2×M16—22no—
177C2×M16—1062no—
178#124—22no—
179#124Dic10612yes—
180#124C22⋊D5312yes—
181C4×Q8—2122no—
182C4×D4—2122no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—532no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—1062no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1062no—
189C2×C22⋊C4—1061no—
190C2×C42—531no—
191C2×C42—532no—
192C2×C42—531no—
193C2×C42C4×D5311yescharacteristic
194C4⋊C8—2124no—
195#101—22no—
196#101—22no—
197#101—1062no—
198#101—1062no—
199#95—1061no—
200#95—1061no—
201#95—1062no—
202C24—21no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C2×Dic10611yescharacteristic
206C2×Q8#1171994712yes—
207C2×Q8C2×C22⋊D5311yescharacteristic
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4#1171991912yes—
212C22×C4—21no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—532no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—4242no—
219C22×C4—2122no—
220C22×C4C2×C4×D5311yescharacteristic
221C22×C4D4×D5312yes—
222M16—42no—
223M16—42no—
224M16—2122no—
225M16—2122no—
226C2×C8—42no—
227C2×C8—2122no—
228C4⋊C4—4242no—
229C4⋊C4—2121no—
230C4⋊C4—2122no—
231C4⋊C4—2121no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—4242no—
235C22⋊C4—2122no—
236C42—21no—
237C42—1064no—
238C42—1062no—
239C42—2121no—
240C42#1171991912yes—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23C22⋊C4×D5311yescharacteristic
247Q8—41no—
248Q8—42no—
249Q8—41no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—532no—
260C2×C4—1062no—
261C2×C4—1061no—
262C2×C4—4244no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2122no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2122no—
270C2×C4—2121no—
271C2×C4C2×#1171991911yescharacteristic
272C2×C4#1313317311yescharacteristic
273C2×C4#1313318912yes—
274C2×C4#1313319412yes—
275C8—82no—
276C8—4242no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22#1681048111yescharacteristic
285C22#99×D5311yescharacteristic
286C22#1681059511yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—42no—
292C4—1062no—
293C4—4241no—
294C4—4241no—
295C4—2121no—
296C4—2122no—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2027470111yescharacteristic
301C2#2027640111yescharacteristic
302C2#2027640511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC2×C53×#355C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC22×Q8×C53C23
hypercenterC2×#355D53
hypocenterC53#12803
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #20274709C2semidirect
4. #20272841C2semidirect
172. C53#12803semidirect

Maximal subgroups

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

subgroupindex
#202774282
C2×#168104202
#202747092
#202728412
C2×C53×#3552
#1280353