Finite Group Databasebeta

← #20275016

Subgroups — #20275016

Order 6784.

376 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×M16×D53C211yesmaximal, characteristic, complemented
2#16811135C211yesmaximal, characteristic
3C2×#13133145C211yesmaximal, characteristic, complemented
4C2×#13133143C211yesmaximal, characteristic, complemented
5C4×#11719921C211yesmaximal, characteristic, complemented
6#16810382C211yesmaximal, characteristic, complemented
7C53×#339C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×C4×D53C2211yescharacteristic
10C2×#11719917C2211yescharacteristic
11C2×#11719912—21no—
12M16×D53—22no—
13M16×D53—22no—
14C2×C8×D53—21no—
15#13133201—21no—
16#13133197C411yescharacteristic, complemented
17#13133197C411yescharacteristic, complemented
18#13133192C411yescharacteristic, complemented
19#13133192C411yescharacteristic, complemented
20#13133187—21no—
21C4×D212—21no—
22C4×D212—21no—
23C2×#11719921—21no—
24C2×C4×Dic53C2211yescharacteristic
25#13133145—22no—
26#13133143—22no—
27#13133124—21no—
28C2×M16×C53C2211yescharacteristic
29#117×C53C2211yescharacteristic
30C2×C42×C53C2211yescharacteristic
31C23×D53D411yescharacteristic
32C22×C4×C53—21no—
33C22×C4×C53C2311yescharacteristic
34C2×C22⋊D53—21no—
35C22×Dic53—21no—
36C22×Dic53D411yescharacteristic
37#11719945—21no—
38#11719945—21no—
39#11719945—21no—
40#11719945—21no—
41C2×D212—21no—
42C2×D212—21no—
43C2×D212C2×C411yescharacteristic
44C2×C4×D53—21no—
45C2×C4×D53—21no—
46C2×C4×D53—21no—
47C2×C4×D53—21no—
48C2×C4×D53—21no—
49C2×C4×D53C2×C411yescharacteristic
50C2×C4×D53C2×C411yescharacteristic
51C2×C4×D53D411yescharacteristic
52C2×C4×D53D411yescharacteristic
53C2×Dic106C2×C411yescharacteristic
54M16×C53—21no—
55M16×C53D412yescomplemented
56C2×C8×C53—21no—
57C4⋊C4×C53C2×C411yescharacteristic
58C4⋊C4×C53C2×C411yescharacteristic
59C22⋊C4×C53—21no—
60C42×C53—21no—
61C42×C53—21no—
62C42×C53—21no—
63#11719921—24no—
64#11719921—21no—
65#11719921—21no—
66#11719921—21no—
67#11719921—21no—
68C4⋊Dic53—21no—
69C4⋊Dic53—21no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72#11719917—21no—
73#11719917D412yescomplemented
74C2×#88322—21no—
75#11719912—24no—
76C8×D53—24no—
77C23×C53C2×D411yescharacteristic
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—22no—
81C22×D53—21no—
82C22×D53—41no—
83C22×D53C22⋊C411yescharacteristic
84C22×D53C22⋊C411yescharacteristic
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53C22×C411yescharacteristic
91C2×C4×C53C2×D411yescharacteristic
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53#4111yescharacteristic
95C2×C4×C53#4111yescharacteristic
96C22⋊D53—41no—
97C22⋊D53—41no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53C22⋊C411yescharacteristic
105C2×Dic53C22⋊C411yescharacteristic
106D212—21no—
107D212—21no—
108D212—42no—
109D212—42no—
110C4×D53—21no—
111C4×D53—21no—
112C4×D53—21no—
113C4×D53—21no—
114C4×D53—21no—
115C4×D53—21no—
116C4×D53—21no—
117C4×D53—21no—
118C4×D53—41no—
119C4×D53—41no—
120Dic106—21no—
121Dic106—21no—
122C8×C53—22no—
123#88322—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53C2×C22⋊C411yescharacteristic
127C22×C53#12111yescharacteristic
128C22×C53#12111yescharacteristic
129C2×D53—21no—
130C2×D53—21no—
131C2×D53—21no—
132C2×D53—21no—
133C2×D53—21no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—41no—
137C2×D53—41no—
138C2×D53—41no—
139C2×D53#10412yescomplemented
140C4×C53—42no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53C4×D411yescharacteristic
144C4×C53C4×D411yescharacteristic
145C4×C53#12011yescharacteristic
146C4×C53#12311yescharacteristic
147Dic53—21no—
148Dic53—21no—
149Dic53—42no—
150Dic53—41no—
151#1871—531nomaximal, Sylow
152C2×C53—21no—
153C2×C53—21no—
154C2×C53#38611yescharacteristic
155C2×C53C2×#10411yescharacteristic
156C2×C53#42111yescharacteristic
157D53—22no—
158D53—41no—
159D53—81no—
160C22×M16—531no—
161#525—531no—
162C2×#104—531no—
163C2×#104—531no—
164C4×C22⋊C4—531no—
165#339—531no—
166#339D5311yescharacteristic, complemented
167C53#187111yescharacteristic, complemented, Sylow
168C2×#41—531no—
169C23×C4—531no—
170C2×M16—531no—
171C2×M16—1061no—
172C2×M16—1062no—
173C2×M16—1062no—
174C2×M16C2×D5311yescharacteristic
175C22×C8—1061no—
176#123—1061no—
177C22⋊Q8—531no—
178C22⋊Q8—531no—
179#121—531no—
180#121—531no—
181#120—1061no—
182C4×D4—1061no—
183C4×D4—1061no—
184#117C2×D5311yescharacteristic
185C2×C22⋊C4—1061no—
186C2×C42—531no—
187C2×C42C2×D5311yescharacteristic
188#104—1062no—
189#104—1062no—
190#95—1061no—
191C24—531no—
192#41—1061no—
193#41—1061no—
194#41—1061no—
195#41—1061no—
196C2×Q8—531no—
197C2×D4—531no—
198C2×D4—1061no—
199C2×D4—1061no—
200C2×D4—1061no—
201C22×C4—21no—
202C22×C4—531no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—531no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4—1061no—
211C22×C4—1061no—
212C22×C4—1061no—
213C22×C4C22×D5311yescharacteristic
214M16—21no—
215M16—532no—
216M16—1064no—
217M16—1061no—
218M16D21212yescomplemented
219C2×C8—21no—
220C2×C8—1064no—
221C2×C8—1061no—
222C4⋊C4—1061no—
223C4⋊C4—1061no—
224C4⋊C4C4×D5311yescharacteristic
225C4⋊C4C4×D5311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1061no—
228C22⋊C4—1061no—
229C22⋊C4—1061no—
230C22⋊C4—1061no—
231C22⋊C4—1064no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—1061no—
236C42—1061no—
237C23—531no—
238C23—531no—
239C23—1062no—
240C23—1061no—
241C23—1061no—
242C23—1061no—
243C23—2121no—
244C23D4×D5311yescharacteristic
245Q8—1061no—
246Q8—1061no—
247D4—1061no—
248D4—1061no—
249D4—2122no—
250D4—2122no—
251D4—2121no—
252D4—2121no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—531no—
259C2×C4—531no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1062no—
271C2×C4—1061no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4C2×C4×D5311yescharacteristic
277C2×C4C2×D21211yescharacteristic
278C2×C4#1171994511yescharacteristic
279C2×C4D4×D5311yescharacteristic
280C2×C4D4×D5311yescharacteristic
281C2×C4#1171994911yescharacteristic
282C8—22no—
283C8—1062no—
284C22—21no—
285C22—21no—
286C22—532no—
287C22—1061no—
288C22—1061no—
289C22—1061no—
290C22—1061no—
291C22—1061no—
292C22—2122no—
293C22—2122no—
294C22—2121no—
295C22—2121no—
296C22—2121no—
297C22C22⋊C4×D5311yescharacteristic
298C22#1313319011yescharacteristic
299C22#1313319211yescharacteristic
300C4—42no—
301C4—41no—
302C4—41no—
303C4—1061no—
304C4—1061no—
305C4—2122no—
306C4—2121no—
307C4#1313317911yescharacteristic
308C4C4×D21211yescharacteristic
309C4#1313318711yescharacteristic
310C4#1313320211yescharacteristic
311C2—21no—
312C2—21no—
313C2—1062no—
314C2—4241no—
315C2—2121no—
316C2#1681050111yescharacteristic
317C2#104×D5311yescharacteristic
318C2#1681063811yescharacteristic
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#339C2
socleC22×C53C2×C22⋊C4
hypercenter#339D53
hypocenterC53#1871
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×M16×D53C2semidirect
3. C2×#13133145C2semidirect
4. C2×#13133143C2semidirect
5. C4×#11719921C2semidirect
6. #16810382C2semidirect
7. C53×#339C2semidirect
16. #13133197C4semidirect
17. #13133197C4semidirect
18. #13133192C4semidirect
19. #13133192C4semidirect
55. M16×C53D4semidirect
73. #11719917D4semidirect
139. C2×D53#104semidirect
166. #339D53semidirect
167. C53#1871semidirect
218. M16D212semidirect