Finite Group Databasebeta

← #27282084

Subgroups — #27282084

Order 13568.

478 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 95 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#20283353C211yesmaximal, characteristic
2#20278998C211yesmaximal, characteristic
3#20273044C212yesmaximal
4#20272624C212yesmaximal
5C53×#1822C211yesmaximal, characteristic
6C22×#11719919C2211yescharacteristic
7#16810684—24no—
8#16810647—21no—
9#16810587C2212yes—
10C4×#11719917—21no—
11#16810561—21no—
12#16810554—21no—
13#16810507—21no—
14#16810507—21no—
15#16810495C411yescharacteristic
16#16810495C411yescharacteristic
17#16810486C2211yescharacteristic
18#16810484C411yescharacteristic
19#16810484C411yescharacteristic
20#16810348—24no—
21C2×#117×C53C2211yescharacteristic
22C53×#407C2212yes—
23C53×#343—21no—
24C53×#343—21no—
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C2×#11719917—22no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—22no—
33C2×#11719919—22no—
34C2×C22⋊Dic53C2×C411yescharacteristic
35C2×C22⋊Dic53C2×C411yescharacteristic
36C4⋊#88322—24no—
37C22⋊#88322—22no—
38C22⋊#88322D414yes—
39#13133128—22no—
40C4×#88322—22no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2311yescharacteristic
50C2×M16×C53—22no—
51#117×C53—22no—
52#117×C53—22no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C22⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C24×C53C22×C411yescharacteristic
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D412yes—
71C22×C4×C53#4112yes—
72C22×Dic53—21no—
73C22×Dic53—22no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—22no—
78C22×Dic53—42no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81M16×C53—42no—
82M16×C53—42no—
83C2×C8×C53—42no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—22no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94#11719919—42no—
95#11719919—42no—
96#11719919—41no—
97#11719919—42no—
98#11719919—41no—
99#11719917—42no—
100#11719917—42no—
101C2×#88322—24no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107C23×C53C4×D412yes—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#12011yescharacteristic
123C2×C4×C53#12112yes—
124C2×C4×C53#12311yescharacteristic
125C2×Dic53—24no—
126C2×Dic53—22no—
127C2×Dic53—41no—
128C2×Dic53—42no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—82no—
140C8×C53—82no—
141#88322—44no—
142#9514—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53#38611yescharacteristic
151C22×C53#41011yescharacteristic
152C22×C53#41011yescharacteristic
153C4×C53—21no—
154C4×C53—22no—
155C4×C53—21no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160Dic53—42no—
161Dic53—81no—
162Dic53—81no—
163Dic53—81no—
164#2921—531no—
165#2484—531no—
166#1822D5311yescharacteristic
167#1463—532no—
168#1306—532no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—42no—
172C2×C53#189111yescharacteristic
173C2×C53#189911yescharacteristic
174C2×C53#191511yescharacteristic
175C2×#117C2×D5311yescharacteristic
176C22×C4⋊C4—531no—
177#433—1064no—
178#423—1061no—
179#407—532no—
180#407C2×D5312yes—
181C4×M16—1061no—
182#400—1061no—
183#398—1061no—
184#387—1061no—
185#387—1061no—
186#385—531no—
187#385—531no—
188#381—531no—
189#381—531no—
190#381—531no—
191#343—21no—
192#343—21no—
193#324—1064no—
194C53#951411yescharacteristic, complemented, Sylow
195C23×C4—531no—
196C23×C4—531no—
197C23×C4C22×D5311yescharacteristic
198C2×M16—22no—
199C2×M16—1062no—
200#117—22no—
201#117—22no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1062no—
204C2×C4⋊C4—1062no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4C22⋊D5311yescharacteristic
209C2×C4⋊C4C22⋊D5311yescharacteristic
210C2×C22⋊C4—531no—
211C2×C22⋊C4—531no—
212C2×C22⋊C4C4×D5311yescharacteristic
213C2×C22⋊C4C4×D5311yescharacteristic
214C2×C42D21211yescharacteristic
215C2×C42D21211yescharacteristic
216C4⋊C8—1064no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—534no—
220C22⋊C8—1062no—
221#97—1062no—
222C4×C8—1062no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231C24C2×C4×D5311yescharacteristic
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—1061no—
235C22×C4—1062no—
236C22×C4—1062no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—2122no—
241C22×C4—2121no—
242C22×C4—2121no—
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4#1171992111yescharacteristic
246C22×C4#1171992111yescharacteristic
247C22×C4C2×D21211yescharacteristic
248C22×C4D4×D5312yes—
249C22×C4#1171994712yes—
250C22×C4C2×C22⋊D5311yescharacteristic
251M16—42no—
252M16—42no—
253M16—2122no—
254M16—2122no—
255C2×C8—42no—
256C2×C8—1064no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—2122no—
260C4⋊C4—2122no—
261C4⋊C4—2121no—
262C4⋊C4—2122no—
263C4⋊C4—2121no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—2121no—
267C22⋊C4—2121no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×#1171992111yescharacteristic
277C23#1313317712yes—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—1062no—
293C2×C4—1064no—
294C2×C4—4242no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2122no—
306C2×C4—2121no—
307C2×C4#1313318711yescharacteristic
308C2×C4#1313318912yes—
309C2×C4#1313320111yescharacteristic
310C8—82no—
311C8—2124no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22#1681050011yescharacteristic
320C22#1681059611yescharacteristic
321C22#1681059611yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—4241no—
330C4—4241no—
331C4—4241no—
332C4—2122no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2027513111yescharacteristic
337C2#2027517111yescharacteristic
338C2#2027526511yescharacteristic
subgroupG/H
centerC22#16810500
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1822C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#1822D53
hypocenterC53#9514
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
194. C53#9514semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202789982
#202730442
#202726242
C53×#18222
#951453