Finite Group Databasebeta

← #20277458

Subgroups — #20277458

Order 6784.

460 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133215C211yesmaximal, characteristic
2#16810530C211yesmaximal, characteristic
3#16810527C211yesmaximal, characteristic
4#16810527C211yesmaximal, characteristic
5#16810520C211yesmaximal, characteristic
6#16810518C211yesmaximal, characteristic
7#16810510C211yesmaximal, characteristic
8#16810507C211yesmaximal, characteristic
9#16810505C211yesmaximal, characteristic
10#16810504C211yesmaximal, characteristic
11#16810487C211yesmaximal, characteristic
12C4×#11719919C211yesmaximal, characteristic
13#16810474C211yesmaximal, characteristic
14#16810468C211yesmaximal, characteristic
15C53×#391C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17#13133215C418yescomplemented
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×C4×Dic53C2211yescharacteristic
28C2×C4×Dic53C2211yescharacteristic
29C2×C4×Dic53C2211yescharacteristic
30C2×C4×Dic53C2211yescharacteristic
31C2×C4×Dic53C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33#13133124C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C22×Q8×C53C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C42×C53C2211yescharacteristic
48#95×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53C2×C414yes—
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C2×Dic106—41no—
71C2×Dic106—41no—
72C2×Dic106C2×C414yes—
73C4⋊C4×C53—41no—
74C42×C53—41no—
75C4⋊Dic53—41no—
76#11719919—22no—
77#11719919—22no—
78#11719919—41no—
79#11719919—41no—
80#11719919C2×C414yes—
81#11719919C2×C414yes—
82#11719919C2×C414yes—
83#11719919C2×C414yes—
84C4×Dic53—22no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C4×Dic53C2×C414yes—
89C4×Dic53Q814yescomplemented
90C23×C53C2411yescharacteristic
91Q8×C53—44no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53C22×C412yes—
110C2×C4×C53C22×C412yes—
111C2×C4×C53C22×C412yes—
112C2×C4×C53C2×Q812yes—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53C22×C412yes—
131C2×Dic53C22×C412yes—
132C2×Dic53C22×C412yes—
133C2×Dic53C22×C412yes—
134C2×Dic53C2×Q812yes—
135C2×Dic53C2×Q812yes—
136C2×Dic53#4112yes—
137Dic106—44no—
138C22×C53C23×C411yescharacteristic
139C22×C53C22×Q811yescharacteristic
140C22×C53C2×#4111yescharacteristic
141C22×C53H(5,𝔽2)11yescharacteristic
142C22×C53H(5,𝔽2)11yescharacteristic
143C22×C532−1+411yescharacteristic
144C22×C532−1+411yescharacteristic
145C4×C53—22no—
146C4×C53—22no—
147C4×C53—22no—
148C4×C53—22no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152Dic53—22no—
153Dic53—22no—
154Dic53—22no—
155Dic53—22no—
156Dic53—41no—
157Dic53—41no—
158Dic53—41no—
159Dic53C4×Q814yes—
160#2331—531nomaximal, Sylow
161C2×C53C2×C4×Q811yescharacteristic
162C2×C53#51811yescharacteristic
163C2×C53#51911yescharacteristic
164C2×C53#54211yescharacteristic
165C2×C53#54211yescharacteristic
166C2×C53#54411yescharacteristic
167C2×C53#54411yescharacteristic
168C2×C4⋊Q8—531no—
169#391—531no—
170#391—531no—
171#391—531no—
172#391D5311yescharacteristic
173#389—531no—
174#389—531no—
175#387—531no—
176#387—531no—
177#387—531no—
178#387—531no—
179#382—531no—
180C4×C4⋊C4—531no—
181C4×C4⋊C4—531no—
182#376—531no—
183C53#233111yescharacteristic, complemented, Sylow
184C22×Q8—531no—
185C22×Q8C2×D5311yescharacteristic
186C4⋊Q8—538no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—531no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4C2×D5311yescharacteristic
197C2×C42—531no—
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42C2×D5311yescharacteristic
204#95—531no—
205#95—531no—
206#95—531no—
207#95—531no—
208#95—531no—
209#95—531no—
210#95—531no—
211#95—531no—
212#95—531no—
213#95—531no—
214#95—531no—
215#95—531no—
216#95C2×D5311yescharacteristic
217#95C2×D5311yescharacteristic
218#95C2×D5311yescharacteristic
219#95C2×D5311yescharacteristic
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—534no—
223C2×Q8—2121no—
224C2×Q8—2121no—
225C2×Q8C4×D5314yes—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4C22×D5311yescharacteristic
235C22×C4C22×D5311yescharacteristic
236C22×C4C22×D5311yescharacteristic
237C22×C4C22×D5311yescharacteristic
238C22×C4C22×D5311yescharacteristic
239C22×C4C22×D5311yescharacteristic
240C22×C4C22×D5311yescharacteristic
241C4⋊C4—41no—
242C4⋊C4—534no—
243C4⋊C4—534no—
244C4⋊C4—534no—
245C4⋊C4—534no—
246C4⋊C4—1062no—
247C4⋊C4—1062no—
248C4⋊C4—2121no—
249C4⋊C4—2121no—
250C4⋊C4—2121no—
251C42—41no—
252C42—534no—
253C42—534no—
254C42—1062no—
255C42—2121no—
256C42—2121no—
257C42—2121no—
258C23C23×D5311yescharacteristic
259Q8—44no—
260Q8—2124no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—532no—
279C2×C4—532no—
280C2×C4—532no—
281C2×C4—532no—
282C2×C4—532no—
283C2×C4—532no—
284C2×C4—532no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4C2×C4×D5312yes—
303C2×C4C2×C4×D5312yes—
304C2×C4C2×C4×D5312yes—
305C2×C4Q8×D5312yes—
306C22C22×C4×D5311yescharacteristic
307C22C2×Q8×D5311yescharacteristic
308C22#41×D5311yescharacteristic
309C22#1313326211yescharacteristic
310C22#1313326211yescharacteristic
311C22#1313326411yescharacteristic
312C22#1313326511yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—534no—
321C4—1062no—
322C4—1062no—
323C4—1062no—
324C4—1062no—
325C4—2121no—
326C4—2121no—
327C4—2121no—
328C2C4×Q8×D5311yescharacteristic
329C2#1681109611yescharacteristic
330C2#1681109911yescharacteristic
331C2#1681123711yescharacteristic
332C2#1681124311yescharacteristic
333C2#1681125011yescharacteristic
334C2#1681125211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#391C2
socleC23×C53C24
hypercenter#391D53
hypocenterC53#2331
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
17. #13133215C4semidirect
89. C4×Dic53Q8semidirect
183. C53#2331semidirect