Finite Group Databasebeta

← #20280324

Subgroups — #20280324

Order 6784.

582 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133199C211yesmaximal, characteristic, complemented
2C2×#13133177C211yesmaximal, characteristic, complemented
3#16810549C211yesmaximal, characteristic, complemented
4#16810539C211yesmaximal, characteristic, complemented
5#16810539C211yesmaximal, characteristic, complemented
6#16810531C211yesmaximal, characteristic
7#16810521C211yesmaximal, characteristic, complemented
8#16810517C211yesmaximal, characteristic, complemented
9#16810517C211yesmaximal, characteristic, complemented
10#16810516C211yesmaximal, characteristic, complemented
11#16810509C211yesmaximal, characteristic, complemented
12#16810503C211yesmaximal, characteristic, complemented
13#16810498C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic, complemented
15C53×#390C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133199—24no—
21#13133177—24no—
22C2×C4⋊Dic53C2211yescharacteristic
23C2×C4⋊Dic53C2211yescharacteristic
24C2×C4⋊Dic53C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×C4×Dic53C2211yescharacteristic
37C2×C4×Dic53C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46C22×D4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C42×C53C2211yescharacteristic
53C24×C53C2311yescharacteristic
54C24×C53C2311yescharacteristic
55C23×D53C2311yescharacteristic
56C2×D4×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—41no—
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C2×C22⋊D53—22no—
65C2×C22⋊D53—22no—
66C2×C22⋊D53—22no—
67C2×C22⋊D53—22no—
68C2×C22⋊D53—41no—
69C2×C22⋊D53—41no—
70C22×Dic53—22no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53D414yescomplemented
83C4⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—22no—
88C42×C53—41no—
89C22⋊Dic53—22no—
90C22⋊Dic53—22no—
91C22⋊Dic53—22no—
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—22no—
98C4⋊Dic53—22no—
99C4⋊Dic53—22no—
100C4⋊Dic53—41no—
101#11719919—22no—
102#11719919—22no—
103C4×Dic53—22no—
104C4×Dic53—41no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53#4112yes—
119C23×C53C2411yescharacteristic
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53C2×D412yes—
127D4×C53—44no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53C2×D412yes—
141C2×C4×C53#4112yes—
142C2×C4×C53#4112yes—
143C22⋊D53—28no—
144C22⋊D53—44no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—44no—
165C2×Dic53C2×D412yes—
166C2×Dic53#4112yes—
167C4×D53—44no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53C22×D411yescharacteristic
183C22×C53C2×#4111yescharacteristic
184C22×C53C2×#4111yescharacteristic
185C22×C53C2×#4111yescharacteristic
186C22×C53C2×#4111yescharacteristic
187C22×C53H(5,𝔽2)11yescharacteristic
188C22×C53H(5,𝔽2)11yescharacteristic
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—22no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—41no—
195C2×D53—41no—
196C4×C53—22no—
197C4×C53—22no—
198C4×C53—22no—
199C4×C53—41no—
200C4×C53—41no—
201Dic53—22no—
202Dic53—22no—
203Dic53—41no—
204Dic53—41no—
205Dic53—41no—
206Dic53—41no—
207Dic53—41no—
208#2602—531nomaximal, Sylow
209C2×C53—42no—
210C2×C53—81no—
211C2×C53#52511yescharacteristic
212C2×C53#53211yescharacteristic
213C2×C53#53411yescharacteristic
214C2×C53#55111yescharacteristic
215C2×C53#55311yescharacteristic
216C2×C53#55311yescharacteristic
217C2×C53#55911yescharacteristic
218D53—42no—
219C2×#123—531no—
220C2×C4×D4—531no—
221#397—531no—
222#394—531no—
223#394—531no—
224#392—531no—
225#390D5311yescharacteristic, complemented
226#389—531no—
227#388—531no—
228#388—531no—
229#388—531no—
230#387—531no—
231#386—531no—
232#385—531no—
233C4×C22⋊C4—531no—
234C53#260211yescharacteristic, complemented, Sylow
235C22×D4—531no—
236C22×D4—531no—
237C22×D4C2×D5311yescharacteristic
238C23×C4—531no—
239C23×C4—531no—
240#123—1064no—
241C4×D4—1064no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4C2×D5311yescharacteristic
248C2×C22⋊C4—531no—
249C2×C22⋊C4—531no—
250C2×C22⋊C4—531no—
251C2×C22⋊C4—531no—
252C2×C22⋊C4—531no—
253C2×C22⋊C4—531no—
254C2×C22⋊C4—531no—
255C2×C22⋊C4—531no—
256C2×C22⋊C4—531no—
257C2×C22⋊C4C2×D5311yescharacteristic
258C2×C22⋊C4C2×D5311yescharacteristic
259C2×C22⋊C4C2×D5311yescharacteristic
260C2×C22⋊C4C2×D5311yescharacteristic
261C2×C42—531no—
262C2×C42—531no—
263C2×C42C2×D5311yescharacteristic
264#95—531no—
265#95—531no—
266#95—531no—
267#95—531no—
268#95—531no—
269#95—531no—
270#95—531no—
271#95—531no—
272C24—531no—
273C24C22×D5311yescharacteristic
274C24C22×D5311yescharacteristic
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—1062no—
279C2×D4—1062no—
280C2×D4—1062no—
281C2×D4—1062no—
282C2×D4—2121no—
283C2×D4—2121no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—534no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—1062no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4C22×D5311yescharacteristic
298C22×C4C22×D5311yescharacteristic
299C22×C4C22×D5311yescharacteristic
300C22×C4C22×D5311yescharacteristic
301C22×C4C22×D5311yescharacteristic
302C4⋊C4—22no—
303C4⋊C4—1062no—
304C4⋊C4—1062no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4—2121no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—1062no—
313C22⋊C4—1062no—
314C22⋊C4—1062no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—2121no—
320C22⋊C4—2121no—
321C42—41no—
322C42—1062no—
323C42—2121no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—532no—
338C23—1061no—
339C23—1061no—
340C23—1061no—
341C23—1061no—
342C23—1061no—
343C23—1061no—
344C23#1171994512yes—
345C23C23×D5311yescharacteristic
346D4—44no—
347D4—1068no—
348D4—2124no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—532no—
362C2×C4—532no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—1061no—
373C2×C4—1061no—
374C2×C4—1061no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—2124no—
383C2×C4—2124no—
384C2×C4D4×D5312yes—
385C2×C4#1171994712yes—
386C2×C4#1171994712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—1062no—
402C22—1062no—
403C22—1062no—
404C22—2121no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22C2×D4×D5311yescharacteristic
409C22#41×D5311yescharacteristic
410C22C2×#1171994711yescharacteristic
411C22C2×#1171994711yescharacteristic
412C22C2×#1171994511yescharacteristic
413C22#1313326211yescharacteristic
414C22#1313326311yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—1062no—
421C4—1062no—
422C4—2121no—
423C4—2121no—
424C4—2121no—
425C4—2121no—
426C4—2121no—
427C2—42no—
428C2—81no—
429C2—2122no—
430C2#1681113211yescharacteristic
431C2#1681117111yescharacteristic
432C2#1681118111yescharacteristic
433C2#1681129911yescharacteristic
434C2#1681131611yescharacteristic
435C2#1681131911yescharacteristic
436C2#1681135511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2602
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13133199C2semidirect
2. C2×#13133177C2semidirect
3. #16810549C2semidirect
4. #16810539C2semidirect
5. #16810539C2semidirect
7. #16810521C2semidirect
8. #16810517C2semidirect
9. #16810517C2semidirect
10. #16810516C2semidirect
11. #16810509C2semidirect
12. #16810503C2semidirect
13. #16810498C2semidirect
14. C4×#11719921C2semidirect
15. C53×#390C2semidirect
82. C2×C4×D53D4semidirect
225. #390D53semidirect
234. C53#2602semidirect