Finite Group Databasebeta

← #20280093

Subgroups — #20280093

Order 6784.

544 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133195C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic
3#16810557C211yesmaximal, characteristic, complemented
4#16810556C211yesmaximal, characteristic, complemented
5#16810544C211yesmaximal, characteristic, complemented
6#16810543C211yesmaximal, characteristic, complemented
7#16810521C211yesmaximal, characteristic, complemented
8#16810518C211yesmaximal, characteristic, complemented
9#16810518C211yesmaximal, characteristic, complemented
10#16810497C211yesmaximal, characteristic
11#16810497C211yesmaximal, characteristic
12#16810494C211yesmaximal, characteristic
13#16810494C211yesmaximal, characteristic
14C4×#11719921C211yesmaximal, characteristic
15C53×#389C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133195—24no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C4⋊C4×D53—24no—
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×C4×Dic53C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic
50C2×C42×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic
53C23×D53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53D414yes—
77C2×Dic106—22no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—41no—
84C42×C53—41no—
85#11719921—22no—
86#11719921—22no—
87#11719921Q814yescomplemented
88#11719921Q814yescomplemented
89C4⋊Dic53—22no—
90C4⋊Dic53—22no—
91C4⋊Dic53—41no—
92#11719919—22no—
93#11719919—22no—
94#11719919—22no—
95#11719919—22no—
96#11719919—22no—
97#11719919—22no—
98#11719919—41no—
99#11719919—41no—
100C4×Dic53—41no—
101C23×C53C2411yescharacteristic
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53C2×D412yes—
107C22×D53C2×Q812yes—
108C22×D53C2×Q812yes—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53C2×D412yes—
127C2×C4×C53C2×Q812yes—
128C2×C4×C53C2×Q812yes—
129C2×C4×C53#4112yes—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53C2×D412yes—
148C2×Dic53C2×Q812yes—
149C2×Dic53C2×Q812yes—
150C2×Dic53#4112yes—
151C4×D53—28no—
152C4×D53—44no—
153C4×D53—44no—
154Dic106—44no—
155C22×C53C22×D411yescharacteristic
156C22×C53C22×Q811yescharacteristic
157C22×C53C22×Q811yescharacteristic
158C22×C53C2×#4111yescharacteristic
159C22×C53C2×#4111yescharacteristic
160C22×C53H(5,𝔽2)11yescharacteristic
161C22×C53H(5,𝔽2)11yescharacteristic
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—22no—
167C2×D53—22no—
168C2×D53C4⋊Q814yes—
169C4×C53—22no—
170C4×C53—22no—
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176Dic53—22no—
177Dic53—22no—
178Dic53—22no—
179Dic53—22no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#2583—531nomaximal, Sylow
184C2×C53#52511yescharacteristic
185C2×C53C2×C4⋊Q811yescharacteristic
186C2×C53#53411yescharacteristic
187C2×C53#55411yescharacteristic
188C2×C53#55411yescharacteristic
189C2×C53#55411yescharacteristic
190C2×C53#55411yescharacteristic
191D53—24no—
192C2×C22⋊Q8—531no—
193C22×C4⋊C4—531no—
194#398—531no—
195#398—531no—
196#395—531no—
197#395—531no—
198#389—531no—
199#389—531no—
200#389—531no—
201#389D5311yescharacteristic, complemented
202#385—531no—
203#385—531no—
204#385—531no—
205#385—531no—
206C4×C22⋊C4—531no—
207C53#258311yescharacteristic, complemented, Sylow
208C22×Q8—531no—
209C23×C4—531no—
210C23×C4—531no—
211C23×C4—531no—
212C22⋊Q8—1064no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1064no—
225C2×C4⋊C4C2×D5311yescharacteristic
226C2×C4⋊C4C2×D5311yescharacteristic
227C2×C4⋊C4C2×D5311yescharacteristic
228C2×C4⋊C4C2×D5311yescharacteristic
229C2×C22⋊C4—531no—
230C2×C22⋊C4—531no—
231C2×C22⋊C4—531no—
232C2×C22⋊C4—531no—
233C2×C42—531no—
234C2×C42C2×D5311yescharacteristic
235#95—531no—
236#95—531no—
237#95—531no—
238#95—531no—
239#95—531no—
240#95—531no—
241#95—531no—
242#95—531no—
243#95C2×D5311yescharacteristic
244#95C2×D5311yescharacteristic
245C24—531no—
246C2×Q8—1062no—
247C2×Q8—2121no—
248C2×Q8—2121no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—534no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—1062no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4C22×D5311yescharacteristic
266C22×C4C22×D5311yescharacteristic
267C22×C4C22×D5311yescharacteristic
268C22×C4C22×D5311yescharacteristic
269C22×C4C22×D5311yescharacteristic
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—1062no—
277C4⋊C4—1062no—
278C4⋊C4—1062no—
279C4⋊C4—1062no—
280C4⋊C4—1062no—
281C4⋊C4—1062no—
282C4⋊C4—1062no—
283C4⋊C4—1062no—
284C4⋊C4—2121no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C22⋊C4—534no—
288C22⋊C4—534no—
289C22⋊C4—1062no—
290C22⋊C4—1062no—
291C42—41no—
292C42—2121no—
293C23—532no—
294C23—532no—
295C23—532no—
296C23—1061no—
297C23—1061no—
298C23—1061no—
299C23—1061no—
300C23C23×D5311yescharacteristic
301Q8—2124no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—532no—
320C2×C4—532no—
321C2×C4—532no—
322C2×C4—532no—
323C2×C4—1068no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—2124no—
342C2×C4—2124no—
343C2×C4#1171994512yes—
344C2×C4D4×D5312yes—
345C2×C4Q8×D5312yes—
346C2×C4Q8×D5312yes—
347C22—534no—
348C22—1062no—
349C22—1062no—
350C22—1062no—
351C22—1062no—
352C22—1062no—
353C22—1062no—
354C22C2×D4×D5311yescharacteristic
355C22C2×Q8×D5311yescharacteristic
356C22C2×Q8×D5311yescharacteristic
357C22#41×D5311yescharacteristic
358C22C2×#1171994511yescharacteristic
359C22#1313326211yescharacteristic
360C22#1313326311yescharacteristic
361C4—22no—
362C4—22no—
363C4—22no—
364C4—22no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—1062no—
369C4—1062no—
370C4—1062no—
371C4—1062no—
372C4—2121no—
373C4—2121no—
374C4—2121no—
375C2—1064no—
376C2#1681113211yescharacteristic
377C2C4⋊Q8×D5311yescharacteristic
378C2#1681118111yescharacteristic
379C2#1681132711yescharacteristic
380C2#1681132811yescharacteristic
381C2#1681132911yescharacteristic
382C2#1681133311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#389C2
hypercenter#389D53
hypocenterC53#2583
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #16810557C2semidirect
4. #16810556C2semidirect
5. #16810544C2semidirect
6. #16810543C2semidirect
7. #16810521C2semidirect
8. #16810518C2semidirect
9. #16810518C2semidirect
15. C53×#389C2semidirect
87. #11719921Q8semidirect
88. #11719921Q8semidirect
201. #389D53semidirect
207. C53#2583semidirect