Finite Group Databasebeta

← #28308630

Subgroups — #28308630

Order 13568.

356 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 84 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#20291079C211yesmaximal, characteristic, complemented
2#20291078C211yesmaximal, characteristic, complemented
3#20289668C212yesmaximal, complemented
4#20289667C212yesmaximal, complemented
5#20283422C211yesmaximal, characteristic
6#16811544—21no—
7C4⋊#11719910C2212yes—
8#16811518—21no—
9#16811518—21no—
10#16811512—21no—
11#16811511C2212yes—
12C2×#13133223—21no—
13#16811170C411yescharacteristic
14#16810701C2211yescharacteristic
15#16810700C2211yescharacteristic
16#16810698—21no—
17C8×D212C411yescharacteristic
18C8×C22⋊D53—21no—
19#16810645C411yescharacteristic
20#16810588—21no—
21C4×C8×D53C2211yescharacteristic
22C8×D4×C53C411yescharacteristic
23C2×#11719910—22no—
24C2×#11719910—22no—
25C2×#11719945—21no—
26#13133223—44no—
27C2×#11719917—21no—
28C2×#11719912—21no—
29C2×#11719912—21no—
30C2×C8×D53—21no—
31C2×C8×D53D411yescharacteristic
32C2×C8×D53D411yescharacteristic
33#13133216C811yescharacteristic
34#13133213C811yescharacteristic
35#13133203—21no—
36#13133192C811yescharacteristic
37#13133190—21no—
38#13133188C811yescharacteristic
39C4×D212C2×C411yescharacteristic
40C4×C22⋊D53—21no—
41C42×D53C2×C411yescharacteristic
42C4⋊#88322C2×C411yescharacteristic
43#13133146—21no—
44C22⋊#88322—21no—
45#13133129—21no—
46#13133129—21no—
47C4×#88322C2311yescharacteristic
48C8×Dic53D411yescharacteristic
49C8×Dic53D411yescharacteristic
50C22×C8×C53—21no—
51C4×D4×C53C2×C411yescharacteristic
52C4⋊C8×C53C2×C411yescharacteristic
53C22⋊C8×C53—21no—
54C4×C8×C53C2×C411yescharacteristic
55C2×D4×C53C2×C811yescharacteristic
56C22×C4×C53—21no—
57C2×C22⋊D53—21no—
58C2×C22⋊D53—21no—
59C2×C22⋊D53—21no—
60#11719945—42no—
61#11719945—42no—
62C2×D212—21no—
63C2×D212C2×C811yescharacteristic
64C2×C4×D53—21no—
65C2×C4×D53C22⋊C411yescharacteristic
66C2×C4×D53C2×C811yescharacteristic
67C2×C4×D53M1611yescharacteristic
68C2×Dic106—21no—
69C2×C8×C53—21no—
70C2×C8×C53—42no—
71C2×C8×C53C22⋊C411yescharacteristic
72C2×C8×C53C22⋊C411yescharacteristic
73C4⋊C4×C53C2×C811yescharacteristic
74C22⋊C4×C53—21no—
75C42×C53C22×C411yescharacteristic
76C22⋊Dic53—21no—
77#11719921—21no—
78#11719921—21no—
79C4⋊Dic53C2×C811yescharacteristic
80#11719919—21no—
81C4×Dic53C22⋊C411yescharacteristic
82C4×Dic53C2×C811yescharacteristic
83C4×Dic53M1611yescharacteristic
84#11719917—42no—
85C2×#88322—21no—
86C2×#88322C2×D411yescharacteristic
87C2×#88322C2×D411yescharacteristic
88#11719912—42no—
89#11719912—42no—
90C8×D53—22no—
91C8×D53—41no—
92C8×D53—42no—
93#11719910—42no—
94#11719910—42no—
95C23×C53—21no—
96C22×D53—21no—
97C22×D53C22⋊C811yescharacteristic
98D4×C53—41no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53C2×C22⋊C411yescharacteristic
103C2×C4×C53C22×C811yescharacteristic
104C2×C4×C53C2×M1611yescharacteristic
105C22⋊D53—42no—
106C22⋊D53—42no—
107C22⋊D53—42no—
108C2×Dic53—21no—
109C2×Dic53C22⋊C811yescharacteristic
110C2×Dic53C22⋊C811yescharacteristic
111C2×Dic53C22⋊C811yescharacteristic
112D212—41no—
113D212—42no—
114C4×D53—21no—
115C4×D53—21no—
116C4×D53—41no—
117C4×D53—41no—
118C4×D53—41no—
119C4×D53—41no—
120Dic106—42no—
121C8×C53—21no—
122C8×C53—21no—
123C8×C53—41no—
124#88322—21no—
125#88322—41no—
126#88322AGL(1,ℤ/8)12yes—
127#11635—531nomaximal, Sylow
128C22×C53—41no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53C2×C22⋊C811yescharacteristic
132C2×D53—21no—
133C2×D53—21no—
134C2×D53—21no—
135C2×D53—41no—
136C2×D53—41no—
137C2×D53—41no—
138C4×C53—21no—
139C4×C53—21no—
140C4×C53—41no—
141C4×C53#41811yescharacteristic
142C4×C53#42111yescharacteristic
143Dic53—21no—
144Dic53—21no—
145Dic53—21no—
146Dic53—41no—
147#2931—531no—
148#2163—531no—
149#2163—531no—
150#1330—532no—
151#1330—532no—
152C2×C53—81no—
153C2×C53#147811yescharacteristic
154C2×C53#211511yescharacteristic
155C2×C53#211711yescharacteristic
156D53—41no—
157D53—81no—
158C2×#131—1061no—
159#532—531no—
160#436—531no—
161#436—531no—
162#436—1061no—
163C8×D4—531no—
164C8×D4—1061no—
165C8×D4#615711yescharacteristic
166#423—531no—
167#408—1061no—
168C2×C4×C8—531no—
169C4⋊C16—532no—
170C4⋊C16—1061no—
171C22⋊C16—1061no—
172C22⋊C16—1061no—
173#346—532no—
174#346—1061no—
175C53#1163511yescharacteristic, complemented, Sylow
176C2×#41—1061no—
177#131—2124no—
178C2×M16—1061no—
179C2×M16—1061no—
180C2×M16—1061no—
181C22×C8—21no—
182C22×C8—531no—
183C22×C8—531no—
184C22×C8—1061no—
185C4⋊Q8—531no—
186#127—531no—
187#124—1061no—
188#121—531no—
189#121—531no—
190#121—1061no—
191C4×D4—531no—
192C4×D4—1061no—
193C4×D4C2×#615711yescharacteristic
194C2×C42—531no—
195C2×C16—1062no—
196C2×C16—1062no—
197C4⋊C8—531no—
198C4⋊C8—1061no—
199C4⋊C8C2×#615711yescharacteristic
200C22⋊C8—21no—
201C22⋊C8—1061no—
202C22⋊C8—1061no—
203C22⋊C8—1061no—
204C4×C8—531no—
205C4×C8—531no—
206C4×C8—531no—
207C4×C8C2×#615711yescharacteristic
208#41—2122no—
209#41—2122no—
210C2×Q8—1061no—
211C2×D4—531no—
212C2×D4—1061no—
213C2×D4—1061no—
214C2×D4—1061no—
215C2×D4—1061no—
216C2×D4D53⋊C811yescharacteristic
217C22×C4—21no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—1061no—
222M16—2122no—
223M16—2122no—
224M16—2122no—
225C2×C8—21no—
226C2×C8—42no—
227C2×C8—531no—
228C2×C8—531no—
229C2×C8—1062no—
230C2×C8—1061no—
231C2×C8—2121no—
232C2×C8—2122no—
233C2×C8C22⋊#615711yescharacteristic
234C2×C8C22⋊#615711yescharacteristic
235C4⋊C4—531no—
236C4⋊C4—1061no—
237C4⋊C4D53⋊C811yescharacteristic
238C22⋊C4—21no—
239C22⋊C4—1061no—
240C22⋊C4—1061no—
241C22⋊C4—1061no—
242C42—531no—
243C42—531no—
244C42—531no—
245C42C22×#615711yescharacteristic
246C16—2122no—
247C16—2122no—
248C23—21no—
249C23—531no—
250C23—1061no—
251Q8—2122no—
252D4—41no—
253D4—2121no—
254D4—2122no—
255D4—2122no—
256D4—2122no—
257D4—2122no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—531no—
262C2×C4—531no—
263C2×C4—531no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4C2×C22⋊#615711yescharacteristic
272C2×C4C2×D53⋊C811yescharacteristic
273C2×C4D53⋊M1611yescharacteristic
274C8—21no—
275C8—21no—
276C8—41no—
277C8—532no—
278C8—1061no—
279C8—2121no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—1061no—
284C22—1061no—
285C22—1061no—
286C22—2121no—
287C22—2121no—
288C22—2121no—
289C22D53⋊(C22⋊C8)11yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—1061no—
294C4—1061no—
295C4—1061no—
296C4—2121no—
297C4#1681161511yescharacteristic
298C4#1681162011yescharacteristic
299C2—81no—
300C2—4241no—
301C2—2121no—
302C2#2029001511yescharacteristic
303C2#2029095211yescharacteristic
304C2#2029095811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC42C22×#6157
FittingC8×D4×C53C4
socleC22×C53C2×C22⋊C8
prime radical#16811170C4
cosocleC4×#88322C23
hypercenterC8×D4#6157
hypocenterC53#11635
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20291079C2semidirect
2. #20291078C2semidirect
3. #20289668C2semidirect
4. #20289667C2semidirect
175. C53#11635semidirect

Maximal subgroups

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

subgroupindex
#202910792
#202910782
#202896682
#202896672
#202834222
#1163553