Finite Group Databasebeta

← #27293310

Subgroups — #27293310

Order 13568.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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#20283358C211yesmaximal, characteristic
2#20276515C211yesmaximal, characteristic, complemented
3#20276511C211yesmaximal, characteristic, complemented
4#20272657C211yesmaximal, characteristic, complemented
5#20272640C211yesmaximal, characteristic
6#20272640C211yesmaximal, characteristic
7C53×#1323C211yesmaximal, characteristic, complemented
8#16811091C411yescharacteristic
9#16810683—21no—
10#16810683—21no—
11#16810675—21no—
12#16810675—21no—
13#16810650C411yescharacteristic
14#16810591C411yescharacteristic
15#16810589C2211yescharacteristic
16M16×Dic53C411yescharacteristic
17C2×C8×Dic53C2211yescharacteristic
18#16810441—21no—
19#16810441—21no—
20C22⋊#11719908C2211yescharacteristic
21C22⋊#11719908C2211yescharacteristic
22#16810398—21no—
23C16×Dic53—21no—
24C53×#424C2211yescharacteristic
25C53×C22⋊C16C2211yescharacteristic
26C53×C22⋊C16C2211yescharacteristic
27C2×#11719945C2×C411yescharacteristic
28C2×#11719917C2×C411yescharacteristic
29C22×#88322C2×C411yescharacteristic
30#13133182C811yescharacteristic
31#13133182C811yescharacteristic
32#13133179C811yescharacteristic
33#13133179C811yescharacteristic
34#13133177—21no—
35#13133177—21no—
36#13133174—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C42×D53—21no—
39C2×#11719908—21no—
40C2×#11719908—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43#13133129—21no—
44#13133129—21no—
45#13133129—21no—
46#13133129—21no—
47#13133127—21no—
48C8×Dic53—21no—
49C8×Dic53D411yescharacteristic
50C8×Dic53D411yescharacteristic
51C8×Dic53D411yescharacteristic
52C8×Dic53D411yescharacteristic
53C2×M16×C53C2×C411yescharacteristic
54C22×C8×C53C2311yescharacteristic
55#117×C53C2×C411yescharacteristic
56C2×C16×C53—21no—
57C2×C16×C53—21no—
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C22×C4×C53C22×C411yescharacteristic
61C2×C22⋊D53—21no—
62C22×Dic53M1611yescharacteristic
63C22×Dic53M1611yescharacteristic
64#11719945—41no—
65#11719945—41no—
66C2×D212C2×C811yescharacteristic
67C2×C4×D53—21no—
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×Dic106C2×C811yescharacteristic
71M16×C53—41no—
72C2×C8×C53—21no—
73C2×C8×C53—41no—
74C2×C8×C53C2×D411yescharacteristic
75C2×C8×C53C2×D411yescharacteristic
76C4⋊C4×C53C2×C811yescharacteristic
77C4⋊C4×C53C2×C811yescharacteristic
78C22⋊C4×C53—21no—
79C42×C53—21no—
80#11719921—21no—
81#11719921—21no—
82#11719919—21no—
83#11719919—21no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C4×Dic53C22⋊C411yescharacteristic
87C4×Dic53C22⋊C411yescharacteristic
88C4×Dic53C2×C811yescharacteristic
89C4×Dic53C2×C811yescharacteristic
90#11719917—41no—
91C2×#88322—21no—
92C2×#88322—41no—
93C2×#88322C22⋊C411yescharacteristic
94C2×#88322C22⋊C411yescharacteristic
95C16×C53—41no—
96C16×C53—41no—
97#11719908—41no—
98#11719908—41no—
99C23×C53C2×M1611yescharacteristic
100C22×D53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53C22×C811yescharacteristic
107C22⋊D53—42no—
108C2×Dic53—22no—
109C2×Dic53—21no—
110C2×Dic53—41no—
111C2×Dic53C22⋊C811yescharacteristic
112C2×Dic53C22⋊C811yescharacteristic
113C2×Dic53C22⋊C811yescharacteristic
114C2×Dic53C22⋊C811yescharacteristic
115D212—41no—
116C4×D53—42no—
117C4×D53—41no—
118C4×D53—41no—
119C4×D53—42no—
120Dic106—41no—
121C8×C53—21no—
122C8×C53—21no—
123C8×C53—41no—
124#88322—21no—
125#88322—21no—
126#88322—41no—
127#11464—531nomaximal, Sylow
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53C2×C22⋊C811yescharacteristic
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—41no—
135C4×C53—21no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53#41011yescharacteristic
139C4×C53#41311yescharacteristic
140Dic53—21no—
141Dic53—21no—
142Dic53—21no—
143Dic53—41no—
144Dic53#35012yes—
145#2922—531no—
146#2163—531no—
147#2162—531no—
148#1323—531no—
149#1323D5311yescharacteristic, complemented
150#1315—531no—
151#1315—531no—
152C2×C53—41no—
153C2×C53#145811yescharacteristic
154C2×C53C2×#35011yescharacteristic
155C2×C53#211711yescharacteristic
156D53—81no—
157C4×#41—531no—
158#433—1061no—
159#433—1061no—
160#431—1061no—
161#431—1061no—
162#424—531no—
163#424C2×D5311yescharacteristic
164#408—531no—
165#408—531no—
166C4×M16—531no—
167C2×C4×C8—531no—
168C4⋊C16—1061no—
169C4⋊C16—1061no—
170C22⋊C16—531no—
171C22⋊C16—531no—
172C22⋊C16C2×D5311yescharacteristic
173C22⋊C16C2×D5311yescharacteristic
174#346—1061no—
175C4×C16—1061no—
176C53#1146411yescharacteristic, complemented, Sylow
177C2×#41—531no—
178C2×M16—531no—
179C2×M16C4×D5311yescharacteristic
180C22×C8—531no—
181C22×C8C22×D5311yescharacteristic
182C4×Q8—531no—
183C4×Q8—531no—
184C4×D4—531no—
185C4×D4—531no—
186C4×D4—1061no—
187C4×D4—1061no—
188#117—1061no—
189#117C4×D5311yescharacteristic
190C2×C42—531no—
191C2×C42—1061no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—1061no—
195C2×C16—1061no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—1061no—
199C4⋊C8—1061no—
200C22⋊C8—1061no—
201C22⋊C8—1061no—
202C22⋊C8—1061no—
203C22⋊C8—1061no—
204#97—1061no—
205C4×C8—531no—
206C4×C8—531no—
207C4×C8—531no—
208C4×C8—531no—
209C4×C8—1061no—
210#41—2121no—
211#41—2121no—
212C2×Q8—531no—
213C2×D4—531no—
214C2×D4—1061no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4C2×C4×D5311yescharacteristic
221M16—41no—
222M16—2121no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—531no—
226C2×C8—531no—
227C2×C8—1061no—
228C2×C8—2121no—
229C2×C8D4×D5311yescharacteristic
230C2×C8D4×D5311yescharacteristic
231C4⋊C4—1061no—
232C4⋊C4—1061no—
233C4⋊C4C8×D5311yescharacteristic
234C4⋊C4C8×D5311yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—1061no—
237C22⋊C4—1061no—
238C42—21no—
239C42—531no—
240C42—531no—
241C42—531no—
242C42—531no—
243C42—1061no—
244C42—1061no—
245C16—41no—
246C16—41no—
247C16—2121no—
248C16—2121no—
249C23—1061no—
250C23M16×D5311yescharacteristic
251Q8—2121no—
252D4—2121no—
253D4—2122no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—531no—
259C2×C4—531no—
260C2×C4—531no—
261C2×C4—531no—
262C2×C4—1062no—
263C2×C4—1061no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4C22⋊C4×D5311yescharacteristic
270C2×C4C2×C8×D5311yescharacteristic
271C8—21no—
272C8—21no—
273C8—41no—
274C8—1061no—
275C8—1061no—
276C8—2121no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—2121no—
281C22—2121no—
282C22—2121no—
283C22C22⋊C8×D5311yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—532no—
288C4—1061no—
289C4—1061no—
290C4—1061no—
291C4—2121no—
292C4#1681059511yescharacteristic
293C4#1681060611yescharacteristic
294C2—41no—
295C2—4241no—
296C2#2027302811yescharacteristic
297C2#350×D5311yescharacteristic
298C2#2027634111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1323C2
socleC22×C53C2×C22⋊C8
prime radicalC2×#11719945C2×C4
cosocleC22×C8×C53C23
hypercenter#1323D53
hypocenterC53#11464
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #20276515C2semidirect
3. #20276511C2semidirect
4. #20272657C2semidirect
7. C53×#1323C2semidirect
149. #1323D53semidirect
176. C53#11464semidirect

Maximal subgroups

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

subgroupindex
#202833582
#202765152
#202765112
#202726572
#202726402
C53×#13232
#1146453