Finite Group Databasebeta

← #27281969

Subgroups — #27281969

Order 13568.

544 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283134C211yesmaximal, characteristic
2#20280292C211yesmaximal, characteristic
3#20273161C212yesmaximal, complemented
4#20272581C212yesmaximal, complemented
5C53×#1464C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×C4×Dic106C2211yescharacteristic
8#16810579—24no—
9C4×#11719917—21no—
10C4×#11719917C2212yes—
11#16810553C411yescharacteristic
12#16810548C411yescharacteristic
13#16810543C411yescharacteristic
14#16810535C411yescharacteristic
15#16810511—21no—
16#16810507—21no—
17#16810504—21no—
18#16810496—21no—
19#16810491C2211yescharacteristic
20#16810353—24no—
21C2×#117×C53C2211yescharacteristic
22C53×#432C2212yes—
23C53×#324—21no—
24C53×#323—21no—
25C22×Dic106C2×C411yescharacteristic
26C2×#11719917—22no—
27C2×#11719917—24no—
28C2×#11719917—24no—
29C2×#11719917—22no—
30C22×#88322—22no—
31C4×Dic106—22no—
32C4×Dic106—22no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—21no—
40#13133128—22no—
41#13133128—22no—
42C4×#88322—22no—
43C4×#88322D414yes—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124C2×C411yescharacteristic
51#13133124C2×C411yescharacteristic
52C23×C4×C53D411yescharacteristic
53#117×C53—22no—
54#117×C53—22no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C22⋊C4×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53D411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—22no—
62C22⋊C8×C53—22no—
63#97×C53—22no—
64C24×C53C22⋊C411yescharacteristic
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C2×Dic106—22no—
80C2×Dic106—42no—
81C2×C8×C53—42no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53C2×D412yes—
89C42×C53#4112yes—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—22no—
93#11719919—42no—
94#11719919—42no—
95#11719919—41no—
96#11719919—41no—
97C4×Dic53—42no—
98#11719917—28no—
99#11719917—24no—
100#11719917—24no—
101C2×#88322—28no—
102C2×#88322—24no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#10412yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C2×C4×C53—24no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#10412yes—
123C2×C4×C53C4×D412yes—
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53#12112yes—
126C2×C4×C53#12311yescharacteristic
127C2×Dic53—22no—
128C2×Dic53—41no—
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—
139Dic106—24no—
140Dic106—42no—
141C8×C53—82no—
142#88322—28no—
143#9491—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—42no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—81no—
151C22×C53#38611yescharacteristic
152C22×C53C2×#10411yescharacteristic
153C22×C53C2×#10411yescharacteristic
154C4×C53—22no—
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—41no—
161Dic53—42no—
162Dic53—81no—
163Dic53—81no—
164Dic53—81no—
165#2873—531no—
166#2599—531no—
167#1491—532no—
168#1464D5311yescharacteristic
169#1288—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53#189111yescharacteristic
174C2×C53#189611yescharacteristic
175C2×C53#190911yescharacteristic
176C22×M16—1061no—
177C2×C4×Q8—531no—
178C2×#117C2×D5311yescharacteristic
179#432C2×D5312yes—
180#405—1064no—
181C4×M16—532no—
182C4×M16—1061no—
183#398—531no—
184#397—531no—
185#395—531no—
186#393—531no—
187#388—1061no—
188#387—1061no—
189#387—1061no—
190#385—1061no—
191#383—531no—
192Q8⋊C8—1064no—
193#324—21no—
194#323—21no—
195C53#949111yescharacteristic, complemented, Sylow
196C22×Q8—531no—
197C23×C4D21211yescharacteristic
198C2×M16—1062no—
199C2×M16—1064no—
200C2×M16—1064no—
201C2×M16—1062no—
202C22×C8—1062no—
203C4×Q8—1062no—
204C4×Q8—1062no—
205#117—22no—
206#117—22no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4C4×D5311yescharacteristic
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1061no—
215C2×C22⋊C4C4×D5311yescharacteristic
216C2×C22⋊C4C22⋊D5311yescharacteristic
217C2×C42—1061no—
218C2×C42D21211yescharacteristic
219C2×C42C22×D5311yescharacteristic
220C4⋊C8—22no—
221C22⋊C8—22no—
222#97—22no—
223#97—1062no—
224#97—1062no—
225C4×C8—534no—
226C4×C8—1062no—
227#95—531no—
228#95—531no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235C24#1171992111yescharacteristic
236C2×Q8—1062no—
237C2×Q8—2122no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4#1171992111yescharacteristic
247C22×C4#1171992111yescharacteristic
248C22×C4#1171992111yescharacteristic
249C22×C4C2×C4×D5311yescharacteristic
250C22×C4C2×D21211yescharacteristic
251C22×C4C2×C22⋊D5311yescharacteristic
252M16—1068no—
253M16—1064no—
254M16—1064no—
255C2×C8—42no—
256C2×C8—1068no—
257C2×C8—1064no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—1062no—
261C4⋊C4—2122no—
262C4⋊C4—2122no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—2122no—
268C22⋊C4—2122no—
269C42—21no—
270C42—21no—
271C42—2122no—
272C42D4×D5312yes—
273C42#1171994712yes—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23#1313314312yes—
279C23C2×#1171992111yescharacteristic
280Q8—1064no—
281Q8—2122no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—1062no—
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—2121no—
306C2×C4—2121no—
307C2×C4#1313314312yes—
308C2×C4#1313317712yes—
309C2×C4#1313318711yescharacteristic
310C2×C4#1313318912yes—
311C2×C4#1313320111yescharacteristic
312C8—82no—
313C8—1068no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1681050011yescharacteristic
322C22C2×#1313314311yescharacteristic
323C22C2×#1313314311yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—22no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—4241no—
332C4—4241no—
333C4—4241no—
334C4—2122no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2027513111yescharacteristic
339C2#2027515611yescharacteristic
340C2#2027523511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1464C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1464D53
hypocenterC53#9491
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #20273161C2semidirect
4. #20272581C2semidirect
195. C53#9491semidirect

Maximal subgroups

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

subgroupindex
#202831342
#202802922
#202731612
#202725812
C53×#14642
#949153