Finite Group Databasebeta

← #28298426

Subgroups — #28298426

Order 13568.

322 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 70 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#20290382C211yesmaximal, characteristic
2#20289949C211yesmaximal, characteristic
3#20273028C211yesmaximal, characteristic
4C22×#11719936C411yescharacteristic
5C22×D53⋊C8C411yescharacteristic
6#16811640—21no—
7#16811489—21no—
8#16811489—21no—
9#16811468C412yes—
10#16811467—21no—
11C2×#13133175C2211yescharacteristic
12#16810673—21no—
13#16810349C411yescharacteristic
14#16810346—21no—
15C53×#324C411yescharacteristic
16C2×#11719936—21no—
17C2×#11719936—41no—
18C2×#11719936—41no—
19C2×#11719936C812yescomplemented
20C2×D53⋊C8—21no—
21C2×D53⋊C8—41no—
22C2×D53⋊C8—41no—
23C2×D53⋊C8C812yescomplemented
24C22×#88324—21no—
25C22×#88324—21no—
26#13133284—41no—
27C22⋊#88324—22no—
28C4×#88324—41no—
29C22×C4×D53C2×C411yescharacteristic
30#13133175—41no—
31#13133175—41no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719921D411yescharacteristic
34C2×#11719921Q811yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37#13133146—41no—
38C22⋊#88322—21no—
39C8×Dic53—41no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C22⋊C8×C53—21no—
42C23×D53C4⋊C411yescharacteristic
43C22×C4×C53C4211yescharacteristic
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C4⋊C411yescharacteristic
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C22⋊C411yescharacteristic
48C22×Dic53C4⋊C411yescharacteristic
49C2×C4×D53—21no—
50C2×C4×D53—21no—
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53C2×C811yescharacteristic
54C2×C4×D53C2×C811yescharacteristic
55C2×C4×D53M1611yescharacteristic
56C2×C4×D53M1611yescharacteristic
57C2×#88324—22no—
58C2×#88324—22no—
59C2×#88324—41no—
60C2×#88324—41no—
61C2×#88324—41no—
62C2×#88324—41no—
63C2×#88324—42no—
64#11719936—42no—
65#11719936—42no—
66D53⋊C8—42no—
67D53⋊C8—42no—
68C2×C8×C53—41no—
69C4⋊C4×C53—41no—
70#11719921—41no—
71#11719921—41no—
72C4⋊Dic53—41no—
73C4×Dic53—21no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53—21no—
77C2×#88322—41no—
78C23×C53#9511yescharacteristic
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—41no—
82C22×D53—41no—
83C22×D53C4⋊C811yescharacteristic
84C22×D53C4⋊C811yescharacteristic
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53C4×C811yescharacteristic
92C2×C4×C53#9711yescharacteristic
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—41no—
96C2×Dic53—41no—
97C2×Dic53C22⋊C811yescharacteristic
98C2×Dic53C22⋊C811yescharacteristic
99C2×Dic53#9911yescharacteristic
100C2×Dic53#9911yescharacteristic
101C2×Dic53#10011yescharacteristic
102C2×Dic53#10111yescharacteristic
103C4×D53—21no—
104C4×D53—21no—
105C4×D53—21no—
106C4×D53—21no—
107C4×D53—21no—
108C4×D53—21no—
109C4×D53—21no—
110C4×D53—21no—
111#88324—42no—
112#88324—42no—
113#88324—82no—
114C8×C53—81no—
115#88322—81no—
116#7190—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—21no—
119C22×C53#33611yescharacteristic
120C22×C53#34211yescharacteristic
121C22×C53#34311yescharacteristic
122C2×D53—21no—
123C2×D53—21no—
124C2×D53—42no—
125C2×D53—41no—
126C2×D53—41no—
127C2×D53—41no—
128C2×D53—41no—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—41no—
132C4×C53—41no—
133Dic53—41no—
134Dic53—41no—
135Dic53#32311yescharacteristic
136Dic53#32311yescharacteristic
137Dic53#32411yescharacteristic
138Dic53#32411yescharacteristic
139#1729—531no—
140#1458—531no—
141#1458—531no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53#128011yescharacteristic
145C2×C53#139511yescharacteristic
146C2×C53#139511yescharacteristic
147D53—41no—
148D53—41no—
149C22×M16—531no—
150C23×C8—531no—
151C2×#117—531no—
152#431—1061no—
153#431—1061no—
154#336—1061no—
155#336—1061no—
156#324—532no—
157#324—531no—
158#324#615711yescharacteristic
159#323—1061no—
160#323—1061no—
161C53#719011yescharacteristic, complemented, Sylow
162C23×C4—531no—
163C2×M16—532no—
164C2×M16—1061no—
165C2×M16—2121no—
166C2×M16—2121no—
167C22×C8—532no—
168C22×C8—1061no—
169C22×C8—1061no—
170C22×C8—1061no—
171C22×C8—2121no—
172C22×C8—2121no—
173#117—2121no—
174#117—2121no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4C2×#615711yescharacteristic
177C2×C22⋊C4—531no—
178C2×C22⋊C4—531no—
179C2×C42—531no—
180C2×C42—531no—
181C4⋊C8—2121no—
182C4⋊C8—2121no—
183C22⋊C8—21no—
184C22⋊C8—1062no—
185C22⋊C8—1061no—
186C4×C8—2121no—
187C4×C8—2121no—
188C24—531no—
189C22×C4—531no—
190C22×C4—531no—
191C22×C4—531no—
192C22×C4—531no—
193C22×C4—531no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—1061no—
197C22×C4—1061no—
198C22×C4—1061no—
199C22×C4—1061no—
200C22×C4C4×#615711yescharacteristic
201C22×C4C4⋊#615711yescharacteristic
202C22×C4C22⋊#615711yescharacteristic
203M16—2122no—
204M16—2122no—
205C2×C8—41no—
206C2×C8—1062no—
207C2×C8—1062no—
208C2×C8—2121no—
209C2×C8—2122no—
210C2×C8—2121no—
211C2×C8—2121no—
212C2×C8—2121no—
213C2×C8—2121no—
214C2×C8—2122no—
215C2×C8—2122no—
216C4⋊C4—41no—
217C4⋊C4—2121no—
218C22⋊C4—2121no—
219C22⋊C4—2121no—
220C42—1061no—
221C42—1061no—
222C42—1061no—
223C42—1061no—
224C23—531no—
225C23—531no—
226C23—1061no—
227C23—1061no—
228C23—2121no—
229C23—2121no—
230C23(C2×#6157)⋊C411yescharacteristic
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—531no—
238C2×C4—531no—
239C2×C4—531no—
240C2×C4—531no—
241C2×C4—531no—
242C2×C4—531no—
243C2×C4—1061no—
244C2×C4—1061no—
245C2×C4—1061no—
246C2×C4—1061no—
247C2×C4—1061no—
248C2×C4—1061no—
249C2×C4—1061no—
250C2×C4—1061no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4C8×#615711yescharacteristic
256C2×C4#1313327111yescharacteristic
257C8—81no—
258C8—4241no—
259C8—4242no—
260C8—2122no—
261C8—2122no—
262C22—21no—
263C22—21no—
264C22—1061no—
265C22—1061no—
266C22—2121no—
267C22—2121no—
268C22—2122no—
269C22—2121no—
270C22—2121no—
271C22(C2×#6157)⋊C811yescharacteristic
272C22#1681150011yescharacteristic
273C22#1681150211yescharacteristic
274C4—21no—
275C4—21no—
276C4—41no—
277C4—41no—
278C4—531no—
279C4—531no—
280C4—531no—
281C4—531no—
282C4—2121no—
283C4—2121no—
284C2—21no—
285C2—21no—
286C2—2121no—
287C2—2121no—
288C2#2028955211yescharacteristic
289C2#2028981411yescharacteristic
290C2#2028981711yescharacteristic
subgroupG/H
centerC22#16811500
commutatorC2×C4×C53C4×C8
FrattiniC2×C4⋊C4C2×#6157
FittingC53×#324C4
socleC22×C53#342
prime radicalC23×D53C4⋊C4
cosocleC2×#13133175C22
hypercenter#324#6157
hypocenterC53#7190
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
19. C2×#11719936C8semidirect
23. C2×D53⋊C8C8semidirect
161. C53#7190semidirect

Maximal subgroups

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

subgroupindex
#202903822
#202899492
#202730282
#719053