Finite Group Databasebeta

← #27278696

Subgroups — #27278696

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D424C211yesmaximal, characteristic
2C2×#16810700C211yesmaximal, characteristic, complemented
3#20275508C211yesmaximal, characteristic, complemented
4C2×#16810350C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic, complemented
6#20272542C211yesmaximal, characteristic, complemented
7C53×#1277C211yesmaximal, characteristic, complemented
8C22×D424C411yescharacteristic
9C22×#11719912—21no—
10C2×C4×D212C2211yescharacteristic
11C2×C4×D212C2211yescharacteristic
12C4×D424—22no—
13C4×D424—22no—
14#16810700—22no—
15#16810700—22no—
16C2×#13133151C411yescharacteristic
17C2×#13133139C411yescharacteristic
18C2×#13133139C411yescharacteristic
19C2×#13133129—21no—
20C2×#13133129—21no—
21C2×#13133129—21no—
22C2×C4×#88322C2211yescharacteristic
23C2×C4×#88322C2211yescharacteristic
24#16810350—22no—
25#16810350—22no—
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×D212C2×C411yescharacteristic
31C22×C4×D53—21no—
32C22×C4×D53—21no—
33C2×D424—22no—
34C2×D424—22no—
35C2×D424—22no—
36C2×#11719912—24no—
37C2×#11719912—42no—
38C2×#11719912—42no—
39C22×#88322—21no—
40C22×#88322D411yescharacteristic
41C22×#88322D411yescharacteristic
42C4×D212—21no—
43C4×D212—21no—
44C4×D212—21no—
45C4×D212—21no—
46C4×D212—21no—
47C4×D212—21no—
48C4×D212D412yes—
49C4×D212D412yes—
50C2×C4⋊Dic53C2×C411yescharacteristic
51C2×C4⋊Dic53C2×C411yescharacteristic
52C2×#11719921—21no—
53C2×#11719921—21no—
54#13133151—22no—
55#13133139—22no—
56#13133139—22no—
57#13133129—24no—
58#13133129—42no—
59#13133129—42no—
60C4×#88322—21no—
61C4×#88322—21no—
62C4×#88322—21no—
63C4×#88322—21no—
64C22×C8×C53—21no—
65C22×C8×C53—21no—
66C22×C8×C53C2×C411yescharacteristic
67C22×C8×C53C2×C411yescharacteristic
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53D412yes—
73C4×C8×C53—21no—
74C4×C8×C53—21no—
75C23×D53—21no—
76C23×D53—21no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212—21no—
91C2×D212—21no—
92C2×D212C22⋊C412yes—
93C2×D212M1612yescomplemented
94C2×C4×D53—22no—
95C2×C4×D53—22no—
96C2×C4×D53—41no—
97C2×C4×D53—41no—
98C2×C4×D53—41no—
99C2×C4×D53—41no—
100C2×C4×D53—41no—
101C2×C4×D53—41no—
102C2×C4×D53—41no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C4×D53—41no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53—22no—
110C2×C8×C53—21no—
111C2×C8×C53—21no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C2×C8×C53—41no—
117C2×C8×C53M1612yescomplemented
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122#11719921—22no—
123#11719921—22no—
124#11719921—41no—
125#11719921—41no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129C4⋊Dic53C22⋊C412yes—
130C2×#88322—21no—
131C2×#88322—21no—
132C2×#88322—22no—
133C2×#88322—21no—
134C2×#88322—21no—
135C2×#88322—21no—
136C2×#88322—41no—
137C2×#88322—41no—
138C2×#88322D812yescomplemented
139D424—28no—
140#11719912—48no—
141C23×C53C4×D411yescharacteristic
142C22×D53—22no—
143C22×D53—22no—
144C22×D53—41no—
145C22×D53—41no—
146C22×D53—41no—
147C22×D53—41no—
148C22×D53—41no—
149C22×D53—41no—
150C22×D53—41no—
151C22×D53—41no—
152C22×D53—41no—
153C22×D53—41no—
154C22×D53—41no—
155C22×D53—41no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53C2×C22⋊C411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12311yescharacteristic
164C2×C4×C53C2×M1611yescharacteristic
165C2×C4×C53C2×M1611yescharacteristic
166C2×C4×C53#13111yescharacteristic
167C2×C4×C53#13111yescharacteristic
168C2×C4×C53C2×D811yescharacteristic
169C2×C4×C53#13511yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
172C2×Dic53—22no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178D212—22no—
179D212—22no—
180D212—22no—
181D212—22no—
182D212—41no—
183D212—41no—
184D212—41no—
185D212—41no—
186C4×D53—42no—
187C4×D53—42no—
188C4×D53—42no—
189C4×D53—42no—
190C4×D53—81no—
191C4×D53—81no—
192C4×D53—81no—
193C4×D53—81no—
194C8×C53—22no—
195C8×C53—41no—
196C8×C53—42no—
197C8×C53—81no—
198#88322—22no—
199#88322—42no—
200#88322—41no—
201#8951—531nomaximal, Sylow
202C22×C53#38611yescharacteristic
203C22×C53#43511yescharacteristic
204C22×C53#43611yescharacteristic
205C22×C53C4×D811yescharacteristic
206C22×C53#44211yescharacteristic
207C22×C53#44311yescharacteristic
208C22×C53#44411yescharacteristic
209C2×D53—42no—
210C2×D53—42no—
211C2×D53—42no—
212C2×D53—42no—
213C2×D53—42no—
214C2×D53—42no—
215C2×D53—81no—
216C2×D53—81no—
217C2×D53—81no—
218C2×D53—81no—
219C2×D53—81no—
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53#40711yescharacteristic
226C4×C53#40811yescharacteristic
227C4×C53#43511yescharacteristic
228C4×C53#43611yescharacteristic
229C4×C53#44711yescharacteristic
230C4×C53#44911yescharacteristic
231C4×C53#45911yescharacteristic
232C4×C53#46311yescharacteristic
233Dic53—42no—
234Dic53—81no—
235C2×C4×D8—531no—
236C2×#436—531no—
237#1954—531no—
238C2×D4⋊C8—531no—
239C2×D4⋊C8—531no—
240#1277D5311yescharacteristic, complemented
241#1273—531no—
242C2×C53#158111yescharacteristic
243C2×C53#158611yescharacteristic
244C2×C53#158911yescharacteristic
245C2×C53#159411yescharacteristic
246C2×C53#187011yescharacteristic
247C2×C53#187611yescharacteristic
248C2×C53#187911yescharacteristic
249D53—82no—
250D53—82no—
251C22×D8—531no—
252C22×M16—1061no—
253C2×C4×D4—531no—
254C2×C4×D4—531no—
255C4×D8—1062no—
256C4×D8—1062no—
257#436—1062no—
258#436—1062no—
259C2×#107—531no—
260C2×C4⋊C8C2×D5311yescharacteristic
261C2×C4⋊C8C2×D5311yescharacteristic
262C2×#102—531no—
263C2×#102—531no—
264C2×C22⋊C8—1061no—
265C2×C22⋊C8—1061no—
266C2×C22⋊C8—1061no—
267C2×C4×C8—531no—
268C2×C4×C8—531no—
269C2×C4×C8C2×D5311yescharacteristic
270D4⋊C8—1062no—
271D4⋊C8—1062no—
272C53#895111yescharacteristic, complemented, Sylow
273C22×D4—531no—
274C22×D4—531no—
275C23×C4—1061no—
276C23×C4—1061no—
277C2×D8—1062no—
278C2×D8—1062no—
279C2×D8—1062no—
280C2×M16—1064no—
281C2×M16—2122no—
282C2×M16—2122no—
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—531no—
286C22×C8—531no—
287C22×C8—1061no—
288C22×C8C4×D5311yescharacteristic
289C22×C8C4×D5311yescharacteristic
290C4×D4—532no—
291C4×D4—532no—
292C4×D4—1061no—
293C4×D4—1061no—
294C4×D4—1061no—
295C4×D4—1061no—
296C4×D4—1061no—
297C4×D4—1061no—
298C2×C4⋊C4—531no—
299C2×C4⋊C4—531no—
300C2×C22⋊C4—1061no—
301C2×C22⋊C4—1061no—
302C2×C42C22×D5311yescharacteristic
303#107—1062no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8D21212yes—
308#102—1062no—
309#102—1062no—
310C22⋊C8—1064no—
311C22⋊C8—2122no—
312C22⋊C8—2122no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—1061no—
316C4×C8—1061no—
317C4×C8—1061no—
318C4×C8—1061no—
319C24—1061no—
320C24—1061no—
321C2×D4—532no—
322C2×D4—532no—
323C2×D4—1061no—
324C2×D4—1061no—
325C2×D4—1061no—
326C2×D4—1061no—
327C2×D4—1061no—
328C2×D4—1061no—
329C2×D4—1061no—
330C2×D4—1061no—
331C2×D4—1061no—
332C2×D4—1061no—
333C22×C4—1061no—
334C22×C4—1062no—
335C22×C4—1062no—
336C22×C4—1061no—
337C22×C4—2121no—
338C22×C4—2121no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4—2121no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4—2121no—
345C22×C4—2121no—
346C22×C4—2121no—
347C22×C4C2×C4×D5311yescharacteristic
348C22×C4D4×D5311yescharacteristic
349C22×C4#1171994911yescharacteristic
350D8—1068no—
351M16—2128no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—532no—
364C2×C8—1061no—
365C2×C8—1061no—
366C2×C8—1062no—
367C2×C8—1061no—
368C2×C8—1061no—
369C2×C8—1061no—
370C2×C8—2121no—
371C2×C8—2121no—
372C2×C8#1171991212yescomplemented
373C4⋊C4—532no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C4⋊C4—1061no—
377C22⋊C4—1062no—
378C22⋊C4—1062no—
379C22⋊C4—2121no—
380C22⋊C4—2121no—
381C42C2×D21211yescharacteristic
382C42#1171994511yescharacteristic
383C42D4×D5311yescharacteristic
384C42D4×D5311yescharacteristic
385C23—1062no—
386C23—1062no—
387C23—2121no—
388C23—2121no—
389C23—2121no—
390C23—2121no—
391C23—2121no—
392C23—2121no—
393C23—2121no—
394C23—2121no—
395C23—2121no—
396C23—2121no—
397C23—2121no—
398C23—2121no—
399C23#1313317911yescharacteristic
400D4—1062no—
401D4—1062no—
402D4—1062no—
403D4—1062no—
404D4—2121no—
405D4—2121no—
406D4—2121no—
407D4—2121no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—1062no—
411C2×C4—4241no—
412C2×C4—4241no—
413C2×C4—4241no—
414C2×C4—4241no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2122no—
421C2×C4—2122no—
422C2×C4—2122no—
423C2×C4—2122no—
424C2×C4C22⋊C4×D5311yescharacteristic
425C2×C4C4×D21211yescharacteristic
426C2×C4#1313318711yescharacteristic
427C2×C4#1313319011yescharacteristic
428C2×C4#1313319211yescharacteristic
429C2×C4#1313320211yescharacteristic
430C2×C4M16×D5311yescharacteristic
431C2×C4C2×#1171991211yescharacteristic
432C2×C4#1313322311yescharacteristic
433C2×C4#1313322511yescharacteristic
434C2×C4D8×D5311yescharacteristic
435C2×C4#1313324011yescharacteristic
436C2×C4#1313324411yescharacteristic
437C2×C4#1313324411yescharacteristic
438C8—22no—
439C8—42no—
440C8—41no—
441C8—81no—
442C8—1062no—
443C8—2122no—
444C8—2121no—
445C22—4241no—
446C22—4241no—
447C22—4241no—
448C22—4241no—
449C22—4241no—
450C22—4241no—
451C22—4241no—
452C22—4241no—
453C22—2122no—
454C22—2122no—
455C22—2122no—
456C22—2122no—
457C22—2122no—
458C22—2122no—
459C22#1681050111yescharacteristic
460C22#1681069311yescharacteristic
461C22#1681070011yescharacteristic
462C22#1681070511yescharacteristic
463C22#1681073411yescharacteristic
464C22#1681073911yescharacteristic
465C22#1681074411yescharacteristic
466C4—21no—
467C4—21no—
468C4—4241no—
469C4—2122no—
470C4#1681058511yescharacteristic
471C4#1681058811yescharacteristic
472C4#1681069611yescharacteristic
473C4#1681070211yescharacteristic
474C4#1681075811yescharacteristic
475C4#1681077111yescharacteristic
476C4#1681081711yescharacteristic
477C4#1681084111yescharacteristic
478C2—4242no—
479C2—4242no—
480C2#2027354311yescharacteristic
481C2#2027357811yescharacteristic
482C2#2027359911yescharacteristic
483C2#2027362811yescharacteristic
484C2#2027501111yescharacteristic
485C2#2027504311yescharacteristic
486C2#2027506411yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1277C2
socleC23×C53C4×D4
prime radicalC22×D424C4
cosocleC2×C42×C53C23
hypercenter#1277D53
hypocenterC53#8951
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. C2×#16810700C2semidirect
3. #20275508C2semidirect
4. C2×#16810350C2semidirect
5. C2×#16810350C2semidirect
6. #20272542C2semidirect
7. C53×#1277C2semidirect
93. C2×D212M16semidirect
117. C2×C8×C53M16semidirect
138. C2×#88322D8semidirect
240. #1277D53semidirect
272. C53#8951semidirect
372. C2×C8#11719912semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C4×D4242
C2×#168107002
#202755082
C2×#168103502
#202725422
C53×#12772
#895153