Finite Group Databasebeta

← #28308121

Subgroups — #28308121

Order 13568.

390 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 92 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#20291077C211yesmaximal, characteristic, complemented
2#20291073C211yesmaximal, characteristic, complemented
3#20289636C212yesmaximal, complemented
4#20289635C212yesmaximal, complemented
5C2×#16810699C211yesmaximal, characteristic
6#16811544—21no—
7#16811544—21no—
8C22⋊#11719910C2212yes—
9C22⋊#11719910C2212yes—
10#16811512—21no—
11#16811509—21no—
12C22×#11719912C411yescharacteristic
13C2×C4×C22⋊D53C411yescharacteristic
14#16810699—21no—
15#16810699—21no—
16#16810699—21no—
17#16810699—21no—
18C2×#13133146C411yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20C2×#13133129C2211yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22C22×M16×C53C411yescharacteristic
23C2×#11719910—22no—
24C2×#11719910—22no—
25C22×C22⋊D53C811yescharacteristic
26C22×C4×D53C2×C411yescharacteristic
27C2×#11719912—21no—
28C2×#11719912—21no—
29C2×#11719912—42no—
30C22×#88322C2×C411yescharacteristic
31C22×#88322C2311yescharacteristic
32C4×C22⋊D53—21no—
33C4×C22⋊D53—21no—
34C4×C22⋊D53—21no—
35C4×C22⋊D53—21no—
36C2×#11719919C811yescharacteristic
37C2×C22⋊Dic53C811yescharacteristic
38C2×#11719921C811yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40#13133146—21no—
41#13133146—21no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44#13133129—21no—
45#13133129—21no—
46C8×Dic53D411yescharacteristic
47C8×Dic53D411yescharacteristic
48C8×Dic53D411yescharacteristic
49C8×Dic53D411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C2×M16×C53—21no—
52C2×M16×C53—21no—
53C2×M16×C53—42no—
54C22×C8×C53C2×C411yescharacteristic
55C22×C8×C53C2×C411yescharacteristic
56C24×C53C2×C811yescharacteristic
57C23×D53C2×C811yescharacteristic
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53C2×C811yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—42no—
67C22×Dic53C2×C811yescharacteristic
68C22×Dic53C2×C811yescharacteristic
69C22×Dic53C2×C811yescharacteristic
70C2×C4×D53—21no—
71C2×C4×D53—21no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74M16×C53—42no—
75M16×C53—42no—
76C2×C8×C53—22no—
77C2×C8×C53—21no—
78C2×C8×C53—41no—
79C2×C8×C53C22⋊C411yescharacteristic
80C2×C8×C53C22⋊C411yescharacteristic
81C22⋊Dic53—21no—
82C22⋊Dic53—21no—
83#11719921—21no—
84#11719921—21no—
85#11719919—21no—
86#11719919—21no—
87C4×Dic53C22⋊C411yescharacteristic
88C4×Dic53C22⋊C411yescharacteristic
89C4×Dic53M1611yescharacteristic
90C4×Dic53M1611yescharacteristic
91C2×#88322—21no—
92C2×#88322—41no—
93C2×#88322—41no—
94C2×#88322C2×D411yescharacteristic
95C2×#88322C2×D411yescharacteristic
96#11719912—42no—
97#11719912—42no—
98#11719910—42no—
99#11719910—42no—
100C23×C53—21no—
101C23×C53—22no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C22×C811yescharacteristic
105C22×D53—21no—
106C22×D53—22no—
107C22×D53—41no—
108C22×D53—41no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53C2×C22⋊C411yescharacteristic
117C2×C4×C53C2×M1611yescharacteristic
118C22⋊D53—42no—
119C22⋊D53—42no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—21no—
123C2×Dic53—41no—
124C2×Dic53C22⋊C811yescharacteristic
125C2×Dic53C22⋊C811yescharacteristic
126C2×Dic53C22⋊C811yescharacteristic
127C2×Dic53C22⋊C811yescharacteristic
128C4×D53—42no—
129C4×D53—41no—
130C4×D53—41no—
131C8×C53—21no—
132C8×C53—41no—
133C8×C53#10012yes—
134#88322—21no—
135#88322—21no—
136#88322—41no—
137#11489—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—82no—
144C22×C53C2×C22⋊C811yescharacteristic
145C22×C53#50412yes—
146C2×D53—41no—
147C2×D53—41no—
148C2×D53—41no—
149C2×D53—82no—
150C4×C53—21no—
151C4×C53—41no—
152C4×C53#41011yescharacteristic
153C4×C53C2×#10011yescharacteristic
154Dic53—21no—
155Dic53—21no—
156Dic53—21no—
157Dic53—41no—
158Dic53#34912yes—
159C2×#436—531no—
160#2163—531no—
161#2162—531no—
162#1314—532no—
163#1314—532no—
164C2×C53—22no—
165C2×C53—81no—
166C2×C53#146011yescharacteristic
167C2×C53#211311yescharacteristic
168C2×C53C2×#34911yescharacteristic
169D53—81no—
170C22×M16—531no—
171C22×M16#615711yescharacteristic
172C2×C4×D4—531no—
173#436—1061no—
174#436—1061no—
175#436—1061no—
176#436—1061no—
177C2×C4⋊C8—531no—
178C2×C22⋊C8—531no—
179C2×C22⋊C8—531no—
180C2×C4×C8—531no—
181C4⋊C16—1061no—
182C4⋊C16—1061no—
183C22⋊C16—532no—
184C22⋊C16—532no—
185#346—1061no—
186C4×C16—1061no—
187C53#1148911yescharacteristic, complemented, Sylow
188C22×D4—531no—
189C23×C4—531no—
190C23×C4C2×#615711yescharacteristic
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—42no—
194C2×M16—1061no—
195C2×M16—1061no—
196C2×M16—2122no—
197C22×C8—531no—
198C22×C8—531no—
199C22×C8C2×#615711yescharacteristic
200C22×C8C2×#615711yescharacteristic
201C4×D4—1061no—
202C4×D4—1061no—
203C4×D4—1061no—
204C4×D4—1061no—
205C2×C4⋊C4—531no—
206C2×C22⋊C4—531no—
207C2×C22⋊C4—531no—
208C2×C42—531no—
209C2×C16—1062no—
210C2×C16—1062no—
211C4⋊C8—1061no—
212C4⋊C8—1061no—
213C22⋊C8—1061no—
214C22⋊C8—1061no—
215C22⋊C8—1061no—
216C22⋊C8—1061no—
217C4×C8—531no—
218C4×C8—531no—
219C4×C8—531no—
220C4×C8—531no—
221C24—531no—
222C24D53⋊C811yescharacteristic
223C2×D4—1061no—
224C2×D4—1061no—
225C2×D4—2122no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4—2121no—
236C22×C4—2121no—
237C22×C4D53⋊C811yescharacteristic
238C22×C4C22×#615711yescharacteristic
239M16—42no—
240M16—42no—
241M16—2122no—
242M16—2122no—
243C2×C8—22no—
244C2×C8—21no—
245C2×C8—41no—
246C2×C8—531no—
247C2×C8—531no—
248C2×C8—1061no—
249C2×C8—2121no—
250C2×C8—2121no—
251C2×C8C22⋊#615711yescharacteristic
252C2×C8C22⋊#615711yescharacteristic
253C4⋊C4—1061no—
254C4⋊C4—1061no—
255C22⋊C4—1061no—
256C22⋊C4—1061no—
257C22⋊C4—1061no—
258C22⋊C4—1061no—
259C42—531no—
260C42—531no—
261C42—531no—
262C42—531no—
263C16—2122no—
264C16—2122no—
265C23—21no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—1062no—
270C23—1061no—
271C23—2121no—
272C23—2121no—
273C23C2×D53⋊C811yescharacteristic
274D4—2122no—
275D4—2122no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—531no—
284C2×C4—531no—
285C2×C4—531no—
286C2×C4—531no—
287C2×C4—1062no—
288C2×C4—1062no—
289C2×C4—1061no—
290C2×C4—2122no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4C2×C22⋊#615711yescharacteristic
295C2×C4D53⋊M1611yescharacteristic
296C8—21no—
297C8—41no—
298C8—1061no—
299C8—1061no—
300C8—2121no—
301C8#1313327712yes—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—82no—
308C22—4242no—
309C22—2121no—
310C22—2121no—
311C22—2121no—
312C22D53⋊(C22⋊C8)11yescharacteristic
313C22#1681168212yes—
314C4—21no—
315C4—41no—
316C4—532no—
317C4—1061no—
318C4—1061no—
319C4—1061no—
320C4—2121no—
321C4#1681160211yescharacteristic
322C4C2×#1313327711yescharacteristic
323C2—22no—
324C2—81no—
325C2—4241no—
326C2#2028995411yescharacteristic
327C2#2029094611yescharacteristic
328C2D53⋊#34911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC23×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC22×M16×C53C4
socleC22×C53C2×C22⋊C8
prime radicalC22×C22⋊D53C8
cosocleC22×#88322C23
hypercenterC22×M16#6157
hypocenterC53#11489
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. #20291077C2semidirect
2. #20291073C2semidirect
3. #20289636C2semidirect
4. #20289635C2semidirect
187. C53#11489semidirect

Maximal subgroups

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

subgroupindex
#202910772
#202910732
#202896362
#202896352
C2×#168106992
#1148953