Finite Group Databasebeta

← #20279673

Subgroups — #20279673

Order 6784.

724 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133189C211yesmaximal, characteristic, complemented
2C2×#13133189C211yesmaximal, characteristic, complemented
3C2×D4×Dic53C211yesmaximal, characteristic, complemented
4#16810556C211yesmaximal, characteristic, complemented
5#16810551C211yesmaximal, characteristic, complemented
6#16810539C211yesmaximal, characteristic, complemented
7#16810531C211yesmaximal, characteristic, complemented
8#16810517C211yesmaximal, characteristic, complemented
9#16810513C211yesmaximal, characteristic, complemented
10#16810503C211yesmaximal, characteristic, complemented
11#16810503C211yesmaximal, characteristic, complemented
12#16810502C211yesmaximal, characteristic, complemented
13#16810500C211yesmaximal, characteristic, complemented
14C2×#13133124C211yesmaximal, characteristic, complemented
15C2×#121×C53C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20C23×Dic53C2211yescharacteristic
21C23×Dic53C2211yescharacteristic
22#13133189—24no—
23#13133189—24no—
24D4×Dic53—24no—
25C2×C4⋊Dic53C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C22×D4×C53C2211yescharacteristic
48C22×D4×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50C23×C4×C53C2211yescharacteristic
51#121×C53—24no—
52C2×C4⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C24×C53C2311yescharacteristic
57C24×C53C2311yescharacteristic
58C23×D53C2311yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—22no—
64C2×D4×C53—22no—
65C2×D4×C53—22no—
66C2×D4×C53—41no—
67C2×D4×C53D414yescomplemented
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—41no—
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C2×C22⋊D53—22no—
76C2×C22⋊D53—22no—
77C2×C22⋊D53—22no—
78C2×C22⋊D53—22no—
79C2×C22⋊D53—22no—
80C2×C22⋊D53—22no—
81C2×C22⋊D53—41no—
82C2×C22⋊D53—41no—
83C2×C22⋊D53—41no—
84C22×Dic53—22no—
85C22×Dic53—22no—
86C22×Dic53—22no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53D414yescomplemented
93C22×Dic53C2311yescharacteristic
94C22×Dic53C2311yescharacteristic
95C22×Dic53C2311yescharacteristic
96C22×Dic53C2311yescharacteristic
97C22×Dic53C2311yescharacteristic
98C22×Dic53C2311yescharacteristic
99C22×Dic53C2311yescharacteristic
100C4⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—22no—
103C22⋊Dic53—22no—
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106C22⋊Dic53—22no—
107C22⋊Dic53—22no—
108C22⋊Dic53—41no—
109#11719921—22no—
110#11719921—22no—
111#11719921—22no—
112#11719921—41no—
113C4⋊Dic53—22no—
114#11719919—22no—
115#11719919—22no—
116C4×Dic53—22no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53C2×D412yes—
135C23×C53C2×D412yes—
136C23×C53C2×D412yes—
137C23×C53#4112yes—
138C23×C53C2411yescharacteristic
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53—21no—
146D4×C53—28no—
147D4×C53—44no—
148D4×C53—44no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—44no—
160C2×C4×C53C2×D412yes—
161C2×C4×C53#4112yes—
162C22⋊D53—44no—
163C22⋊D53—44no—
164C22⋊D53—44no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—44no—
184C2×Dic53—44no—
185C2×Dic53—44no—
186C2×Dic53C2×D412yes—
187C2×Dic53C2×D412yes—
188C2×Dic53#4112yes—
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—22no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—22no—
196C22×C53—22no—
197C22×C53—22no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53#12114yes—
210C22×C53C22×D411yescharacteristic
211C22×C53C22×D411yescharacteristic
212C22×C53C2×#4111yescharacteristic
213C22×C53C2×#4111yescharacteristic
214C22×C53C2×#4111yescharacteristic
215C22×C53H(5,𝔽2)11yescharacteristic
216C22×C53H(5,𝔽2)11yescharacteristic
217C2×D53—41no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53—41no—
221C2×D53—41no—
222C2×D53—41no—
223C2×D53—41no—
224C4×C53—22no—
225C4×C53—22no—
226C4×C53—41no—
227C4×C53—41no—
228Dic53—22no—
229Dic53—22no—
230Dic53—22no—
231Dic53—41no—
232Dic53—41no—
233Dic53—41no—
234Dic53—41no—
235#2543—531nomaximal, Sylow
236C2×C53—24no—
237C2×C53—42no—
238C2×C53—81no—
239C2×C53C2×#12111yescharacteristic
240C2×C53#53211yescharacteristic
241C2×C53#53411yescharacteristic
242C2×C53#54611yescharacteristic
243C2×C53#54611yescharacteristic
244C2×C53#55311yescharacteristic
245C2×C53#55311yescharacteristic
246D53—81no—
247C2×#121—531no—
248C2×#121—531no—
249C2×#121D5311yescharacteristic, complemented
250C2×C4×D4—531no—
251#398—531no—
252#397—531no—
253#394—531no—
254#392—531no—
255#388—531no—
256#388—531no—
257#386—531no—
258#386—531no—
259#386—531no—
260#386—531no—
261C2×#95—531no—
262C53#254311yescharacteristic, complemented, Sylow
263C22×D4—531no—
264C22×D4—531no—
265C22×D4—531no—
266C22×D4C2×D5311yescharacteristic
267C22×D4C2×D5311yescharacteristic
268C22×D4C2×D5311yescharacteristic
269C23×C4—531no—
270C23×C4—531no—
271C23×C4—531no—
272C23×C4C2×D5311yescharacteristic
273#121—24no—
274#121—1064no—
275#121—1064no—
276C4×D4—1064no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4C2×D5311yescharacteristic
281C2×C22⋊C4—531no—
282C2×C22⋊C4—531no—
283C2×C22⋊C4—531no—
284C2×C22⋊C4—531no—
285C2×C22⋊C4—531no—
286C2×C22⋊C4—531no—
287C2×C22⋊C4—531no—
288C2×C22⋊C4—531no—
289C2×C22⋊C4—531no—
290C2×C22⋊C4—531no—
291C2×C22⋊C4C2×D5311yescharacteristic
292C2×C22⋊C4C2×D5311yescharacteristic
293C2×C42—531no—
294#95—531no—
295#95—531no—
296#95—531no—
297#95—531no—
298#95—531no—
299#95—531no—
300#95—531no—
301#95—531no—
302C24—531no—
303C24C22×D5311yescharacteristic
304C24C22×D5311yescharacteristic
305C24C22×D5311yescharacteristic
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—41no—
314C2×D4—1062no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—1062no—
318C2×D4—1062no—
319C2×D4—1062no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4C22⋊D5314yescomplemented
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—531no—
330C22×C4—534no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—531no—
335C22×C4—1062no—
336C22×C4—1062no—
337C22×C4—1062no—
338C22×C4—2121no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4—2121no—
342C22×C4—2121no—
343C22×C4C22×D5311yescharacteristic
344C22×C4C22×D5311yescharacteristic
345C22×C4C22×D5311yescharacteristic
346C22×C4C22×D5311yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—1062no—
349C4⋊C4—1062no—
350C4⋊C4—1062no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—1062no—
354C22⋊C4—1062no—
355C22⋊C4—1062no—
356C22⋊C4—1062no—
357C22⋊C4—1062no—
358C22⋊C4—1062no—
359C22⋊C4—1062no—
360C22⋊C4—1062no—
361C22⋊C4—2121no—
362C22⋊C4—2121no—
363C42—1062no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—1061no—
382C23—1061no—
383C23—1061no—
384C23—1061no—
385C23—1061no—
386C23—1061no—
387C23—1061no—
388C23D4×D5312yes—
389C23#1171994712yes—
390C23C2×C22⋊D5312yes—
391C23C2×C22⋊D5312yes—
392C23C23×D5311yescharacteristic
393D4—28no—
394D4—44no—
395D4—44no—
396D4—2124no—
397D4—2124no—
398D4—2124no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—44no—
410C2×C4—532no—
411C2×C4—532no—
412C2×C4—532no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—1061no—
417C2×C4—1061no—
418C2×C4—1061no—
419C2×C4—1061no—
420C2×C4—1061no—
421C2×C4—1061no—
422C2×C4—1061no—
423C2×C4—1061no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—2124no—
432C2×C4—2124no—
433C2×C4—2124no—
434C2×C4#1171994712yes—
435C2×C4C2×C22⋊D5312yes—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—2121no—
457C22—2121no—
458C22—2121no—
459C22—2121no—
460C22—2121no—
461C22—2121no—
462C22—2121no—
463C22#1313318914yes—
464C22C2×D4×D5311yescharacteristic
465C22C22×C22⋊D5311yescharacteristic
466C22#41×D5311yescharacteristic
467C22C2×#1171994711yescharacteristic
468C22C2×#1171994711yescharacteristic
469C22#1313326211yescharacteristic
470C22#1313326311yescharacteristic
471C4—22no—
472C4—22no—
473C4—41no—
474C4—41no—
475C4—1062no—
476C4—1062no—
477C4—1062no—
478C4—2121no—
479C4—2121no—
480C4—2121no—
481C4—2121no—
482C2—24no—
483C2—42no—
484C2—81no—
485C2—4241no—
486C2C2×#1313318911yescharacteristic
487C2#1681116811yescharacteristic
488C2#1681118111yescharacteristic
489C2#1681126111yescharacteristic
490C2#1681126711yescharacteristic
491C2#1681131611yescharacteristic
492C2#1681132511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2543
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. C2×#13133189C2semidirect
3. C2×D4×Dic53C2semidirect
4. #16810556C2semidirect
5. #16810551C2semidirect
6. #16810539C2semidirect
7. #16810531C2semidirect
8. #16810517C2semidirect
9. #16810513C2semidirect
10. #16810503C2semidirect
11. #16810503C2semidirect
12. #16810502C2semidirect
13. #16810500C2semidirect
14. C2×#13133124C2semidirect
15. C2×#121×C53C2semidirect
67. C2×D4×C53D4semidirect
92. C22×Dic53D4semidirect
249. C2×#121D53semidirect
262. C53#2543semidirect
323. C2×D4C22⋊D53semidirect