Finite Group Databasebeta

← #28299418

Subgroups — #28299418

Order 13568.

462 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C4×#13133284C211yesmaximal, characteristic
2#20290430C211yesmaximal, characteristic
3#20289535C212yesmaximal
4#20289534C212yesmaximal
5#20277190C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4×#88324—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×Q8×Dic53C411yescharacteristic
13#16810530C2211yescharacteristic
14#16810521C411yescharacteristic
15#16810519C2211yescharacteristic
16#16810508—21no—
17#16810506—21no—
18C4⋊C4×Dic53C411yescharacteristic
19#16810474—21no—
20#16810468—21no—
21C42×Dic53C2211yescharacteristic
22C53×#391C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—24no—
26C4×#88324—24no—
27C4×#88324—42no—
28Q8×Dic53—22no—
29Q8×Dic53C814yescomplemented
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C22×Q8×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53C2×Q8×C53—21no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×Q8×C53C2×C812yes—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×#88324—24no—
69C2×#88324—24no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C4⋊C4×C53—41no—
75C42×C53—41no—
76C4⋊Dic53—21no—
77C4⋊Dic53—24no—
78C4⋊Dic53—41no—
79C4⋊Dic53C2×C812yes—
80#11719919—42no—
81#11719919—42no—
82C4×Dic53—22no—
83C4×Dic53—24no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53—41no—
87C4×Dic53—42no—
88C4×Dic53C2×C812yes—
89C4×Dic53M1612yes—
90C4×Dic53SD1612yes—
91C4×Dic53Q1612yes—
92C23×C53C2×C22⋊C411yescharacteristic
93Q8×C53—42no—
94Q8×C53—42no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53#10312yes—
108C2×C4×C53#10412yes—
109C2×C4×C53C22×C811yescharacteristic
110C2×C4×C53C2×M1611yescharacteristic
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53C22⋊C812yes—
124C2×Dic53C22⋊C812yes—
125C2×Dic53#10312yes—
126C2×Dic53#10412yes—
127C2×Dic53C2×SD1611yescharacteristic
128C2×Dic53C2×Q1611yescharacteristic
129C2×Dic53#13512yes—
130#88324—44no—
131#88324—44no—
132#7562—531nomaximal, Sylow
133C22×C53C2×C22⋊C811yescharacteristic
134C22×C53#41011yescharacteristic
135C22×C53#41311yescharacteristic
136C22×C53C2×#10311yescharacteristic
137C22×C53#41611yescharacteristic
138C22×C53C2×#10411yescharacteristic
139C22×C53C2×#10411yescharacteristic
140C4×C53—22no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—22no—
144C4×C53—42no—
145C4×C53—81no—
146Dic53—22no—
147Dic53—22no—
148Dic53—21no—
149Dic53—21no—
150Dic53—42no—
151Dic53—81no—
152Dic53Q8⋊C814yes—
153#2295—531no—
154C4×C4⋊C8—531no—
155#1744—531no—
156#1272—532no—
157#1271—532no—
158C2×C53#145811yescharacteristic
159C2×C53C2×Q8⋊C811yescharacteristic
160C2×C53#147611yescharacteristic
161C2×C53#150011yescharacteristic
162C2×C53#150311yescharacteristic
163C2×C53#152411yescharacteristic
164C2×C53#154211yescharacteristic
165C2×C4×Q8—531no—
166C2×C4⋊C8—1061no—
167C2×C4×C8—1061no—
168C2×C4×C8—1061no—
169#391—531no—
170#391#615711yescharacteristic
171#389—531no—
172#389—531no—
173#387—1061no—
174#387—1061no—
175C4×C4⋊C4—531no—
176C4×C4⋊C4—1061no—
177#376—1061no—
178C43—531no—
179#336—532no—
180#336—532no—
181#336—1061no—
182C53#756211yescharacteristic, complemented, Sylow
183C22×Q8C2×#615711yescharacteristic
184C22×C8—1062no—
185C22×C8—1062no—
186C4×Q8—534no—
187C4×Q8—1062no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4C2×#615711yescharacteristic
194C2×C42—531no—
195C2×C42—531no—
196C2×C42—531no—
197C2×C42—531no—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—1061no—
201C2×C42—1061no—
202C2×C42C2×#615711yescharacteristic
203C4⋊C8—1064no—
204C4×C8—1064no—
205C4×C8—2122no—
206#95—21no—
207#95—21no—
208#95—1061no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213C2×Q8—21no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8D53⋊C812yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4C22⋊#615711yescharacteristic
226C22×C4C22⋊#615711yescharacteristic
227C22×C4C22×#615711yescharacteristic
228C2×C8—1064no—
229C2×C8—1064no—
230C2×C8—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C2×C8—2122no—
234C4⋊C4—41no—
235C4⋊C4—532no—
236C4⋊C4—1064no—
237C4⋊C4—1061no—
238C4⋊C4—2121no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C42—41no—
242C42—532no—
243C42—532no—
244C42—532no—
245C42—532no—
246C42—1062no—
247C42—1062no—
248C42—1064no—
249C42—1062no—
250C42—2121no—
251C42—2122no—
252C23C2×C22⋊#615711yescharacteristic
253Q8—42no—
254Q8—42no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—532no—
268C2×C4—532no—
269C2×C4—532no—
270C2×C4—532no—
271C2×C4—531no—
272C2×C4—531no—
273C2×C4—532no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1062no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4#1313328012yes—
287C2×C4#1313328212yes—
288C2×C4C2×D53⋊C811yescharacteristic
289C2×C4D53⋊M1611yescharacteristic
290C8—2124no—
291C8—2124no—
292C22D53⋊(C22⋊C8)11yescharacteristic
293C22#1681160311yescharacteristic
294C22#1681161011yescharacteristic
295C22C2×#1313328011yescharacteristic
296C22#1681161311yescharacteristic
297C22D53⋊#10411yescharacteristic
298C22C2×#1313328211yescharacteristic
299C4—22no—
300C4—21no—
301C4—21no—
302C4—22no—
303C4—42no—
304C4—81no—
305C4—534no—
306C4—1062no—
307C4—1061no—
308C4—1061no—
309C4—1062no—
310C4—4241no—
311C4—2122no—
312C2#2028994711yescharacteristic
313C2D53⋊(Q8⋊C8)11yescharacteristic
314C2#2029000911yescharacteristic
315C2#2029008711yescharacteristic
316C2#2029009911yescharacteristic
317C2#2029016811yescharacteristic
318C2#2029021711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#7562
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
29. Q8×Dic53C8semidirect
182. C53#7562semidirect

Maximal subgroups

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

subgroupindex
C4×#131332842
#202904302
#202895352
#202895342
#202771902
#756253