Finite Group Databasebeta

← #34183051

Subgroups — #34183051

Order 20224.

692 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098758C211yesmaximal, characteristic
2#23764327C211yesmaximal, characteristic, complemented
3#23764205C211yesmaximal, characteristic, complemented
4C2×D4⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
5#23761283C211yesmaximal, characteristic, complemented
6#23761241C211yesmaximal, characteristic, complemented
7C79×#2354C211yesmaximal, characteristic, complemented
8C22×#11929609—21no—
9C2×#14579158C2211yescharacteristic
10#19098758—22no—
11#19098758C414yescomplemented
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579081—21no—
15C2×#14579081—21no—
16C2×#14579081—21no—
17C2×#14579081—22no—
18C2×C4×C79⋊C8C2211yescharacteristic
19D4⋊(C79⋊C8)—22no—
20C2×C4×D4×C79C2211yescharacteristic
21C2×#117×C79—21no—
22#390×C79—21no—
23#388×C79—22no—
24#384×C79C2211yescharacteristic
25#380×C79—21no—
26C4×C4⋊C4×C79—21no—
27C4×C4⋊C4×C79C2211yescharacteristic
28C22×D316—21no—
29C22×D316—21no—
30C22×D316—21no—
31C2×#11929609—24no—
32C2×#11929609—42no—
33C2×#11929609—42no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36C22×C79⋊C8—21no—
37#14579158—21no—
38#14579158—21no—
39#14579158—21no—
40#14579158C2×C412yes—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)C2×C412yes—
45#14579081—24no—
46#14579081—44no—
47#14579081—42no—
48#14579081—42no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8—21no—
51C22×D4×C79D411yescharacteristic
52C23×C4×C79—21no—
53C4×D4×C79—21no—
54C4×D4×C79—21no—
55C4×D4×C79C2×C412yes—
56C4×D4×C79D412yes—
57#117×C79—42no—
58#117×C79—42no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79D411yescharacteristic
62C2×C4⋊C4×C79D411yescharacteristic
63C2×C4⋊C4×C79D411yescharacteristic
64C2×C4⋊C4×C79D412yes—
65C2×C22⋊C4×C79—21no—
66C2×C22⋊C4×C79—22no—
67C2×C22⋊C4×C79—22no—
68C2×C42×C79—22no—
69C2×C42×C79—21no—
70C2×C42×C79C2311yescharacteristic
71#95×C79—22no—
72#95×C79—21no—
73C23×D79—41no—
74C2×D316—22no—
75C2×D316—22no—
76C2×D316—22no—
77C2×D316—42no—
78C2×D316—42no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316—41no—
82C2×D316—41no—
83C2×D316—41no—
84C2×D316—41no—
85C2×D316—41no—
86C2×D316—41no—
87C2×D316—41no—
88C2×D316—41no—
89C2×D316—41no—
90#11929609—48no—
91C2×C79⋊C8—22no—
92C2×C79⋊C8—22no—
93C2×C79⋊C8—41no—
94C2×C79⋊C8—41no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—41no—
100C24×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79C22⋊C412yes—
107C22×C4×C79—21no—
108C22×C4×C79—21no—
109C22×C4×C79—22no—
110C22×C4×C79—22no—
111C22×C4×C79—21no—
112C22×C4×C79—22no—
113C22×C4×C79—41no—
114C22×C4×C79—41no—
115C22×C4×C79—41no—
116C22×C4×C79—41no—
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79C2×D411yescharacteristic
120C4⋊C4×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—21no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79—42no—
125C4⋊C4×C79—41no—
126C4⋊C4×C79—41no—
127C4⋊C4×C79—41no—
128C4⋊C4×C79—41no—
129C4⋊C4×C79C22⋊C412yes—
130C4⋊C4×C79D812yescomplemented
131C22⋊C4×C79—22no—
132C22⋊C4×C79—22no—
133C22⋊C4×C79—44no—
134C22⋊C4×C79—82no—
135C42×C79—44no—
136C42×C79—41no—
137C42×C79—41no—
138C42×C79C22×C411yescharacteristic
139C42×C79C2×D411yescharacteristic
140C42×C79#4111yescharacteristic
141C42×C79#4111yescharacteristic
142C23×C79—22no—
143C23×C79—41no—
144C23×C79—41no—
145C23×C79—41no—
146C23×C79—41no—
147C23×C79—41no—
148C23×C79—41no—
149C23×C79#12011yescharacteristic
150C22×D79—42no—
151C22×D79—81no—
152C22×D79—81no—
153C22×D79—81no—
154C22×D79—81no—
155C22×D79—81no—
156C22×D79—81no—
157D4×C79—22no—
158D4×C79—22no—
159D4×C79—41no—
160D4×C79—41no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—22no—
164C2×C4×C79—22no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—42no—
170C2×C4×C79—42no—
171C2×C4×C79—42no—
172C2×C4×C79—42no—
173C2×C4×C79—42no—
174C2×C4×C79—41no—
175C2×C4×C79—41no—
176C2×C4×C79—42no—
177C2×C4×C79—41no—
178C2×C4×C79—42no—
179C2×C4×C79C2×C22⋊C411yescharacteristic
180C2×C4×C79C4×D411yescharacteristic
181C2×C4×C79C4×D411yescharacteristic
182C2×C4×C79#12111yescharacteristic
183C2×C4×C79#12111yescharacteristic
184C2×C4×C79#12311yescharacteristic
185C2×C4×C79C2×D811yescharacteristic
186C2×C4×C79#13511yescharacteristic
187C2×C4×C79#13511yescharacteristic
188C2×C4×C79#13511yescharacteristic
189C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
193D316—44no—
194D316—42no—
195D316—42no—
196D316—42no—
197D316—42no—
198D316—82no—
199D316—81no—
200D316—81no—
201D316—81no—
202D316—81no—
203C79⋊C8—42no—
204C79⋊C8—42no—
205C79⋊C8—81no—
206C22×C79—42no—
207C22×C79—42no—
208C22×C79—42no—
209C22×C79—81no—
210C22×C79—81no—
211C22×C79—81no—
212C22×C79—81no—
213C22×C79#38611yescharacteristic
214C22×C79#44711yescharacteristic
215C22×C79#44911yescharacteristic
216C22×C79#44911yescharacteristic
217C22×C79#44911yescharacteristic
218C22×C79#45311yescharacteristic
219C22×C79#45411yescharacteristic
220C2×D79—82no—
221C2×D79—82no—
222C2×D79—82no—
223C2×D79—161no—
224C2×D79—161no—
225C2×D79—161no—
226C2×D79—161no—
227C4×C79—21no—
228C4×C79—21no—
229C4×C79—42no—
230C4×C79—42no—
231C4×C79—82no—
232C4×C79—81no—
233C4×C79#41611yescharacteristic
234C4×C79#41811yescharacteristic
235C4×C79C4×D811yescharacteristic
236C4×C79#44211yescharacteristic
237C4×C79#45911yescharacteristic
238C4×C79#46311yescharacteristic
239C4×C79#48211yescharacteristic
240C4×C79#48211yescharacteristic
241#8980—791nomaximal, Sylow
242C2×C79—82no—
243C2×C79#162111yescharacteristic
244C2×C79#162411yescharacteristic
245C2×C79#162511yescharacteristic
246C2×C79#162811yescharacteristic
247C2×C79#187511yescharacteristic
248C2×C79#187911yescharacteristic
249C2×C79#188311yescharacteristic
250D79—162no—
251C2×#459—791no—
252#2354D7911yescharacteristic, complemented
253#1982—791no—
254#1957—791no—
255C2×D4⋊C8—791no—
256#1296—791no—
257#1279—791no—
258C79#898011yescharacteristic, complemented, Sylow
259C22×D8—1581no—
260C2×#127—791no—
261C2×C4×D4C2×D7911yescharacteristic
262C2×#117—21no—
263#459—794no—
264#459—1582no—
265C2×C4⋊C8—791no—
266C2×C4⋊C8—791no—
267C2×#102—1582no—
268C2×#102—1581no—
269C2×#102—1581no—
270C2×#102—1581no—
271C2×C4×C8—791no—
272#390—21no—
273#388—22no—
274#384C2×D7911yescharacteristic
275#380—21no—
276C4×C4⋊C4—21no—
277C4×C4⋊C4C2×D7911yescharacteristic
278D4⋊C8—1582no—
279C22×D4—1581no—
280C22×D4—1581no—
281C22×D4—1581no—
282C22×D4C22⋊D7911yescharacteristic
283C23×C4—21no—
284C2×D8—3162no—
285C2×D8—3162no—
286C2×D8—1584no—
287C22×C8—1581no—
288C22×C8—1581no—
289C22×C8—1581no—
290#127—792no—
291#127—1581no—
292#127—1581no—
293#127—1581no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4C4×D7912yes—
297C4×D4D31612yes—
298#117—42no—
299#117—42no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4C22⋊D7911yescharacteristic
303C2×C4⋊C4C22⋊D7911yescharacteristic
304C2×C4⋊C4C22⋊D7911yescharacteristic
305C2×C4⋊C4C22⋊D7912yes—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42C22×D7911yescharacteristic
312C4⋊C8—792no—
313C4⋊C8—1581no—
314C4⋊C8—1581no—
315C4⋊C8—1581no—
316#102—3162no—
317#102—3164no—
318#102—3162no—
319#102—1584no—
320C4×C8—1581no—
321C4×C8—1581no—
322#95—22no—
323#95—21no—
324C24—21no—
325C24—3161no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—3162no—
332C2×D4—3162no—
333C2×D4—3161no—
334C2×D4—3161no—
335C2×D4—3161no—
336C2×D4—3161no—
337C2×D4—3161no—
338C2×D4—3161no—
339C2×D4—3161no—
340C2×D4—3161no—
341C2×D4—3161no—
342C2×D4—3161no—
343C2×D4—3161no—
344C2×D4—1582no—
345C2×D4—1582no—
346C2×D4—1582no—
347C2×D4(C2×D79)⋊C412yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4C2×C22⋊D7911yescharacteristic
359C22×C4C2×C22⋊D7911yescharacteristic
360C22×C4C2×C22⋊D7911yescharacteristic
361D8—3168no—
362C2×C8—3161no—
363C2×C8—3161no—
364C2×C8—3161no—
365C2×C8—3161no—
366C2×C8—3161no—
367C2×C8—3161no—
368C2×C8—3161no—
369C2×C8—1582no—
370C2×C8—1582no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4(C2×D79)⋊C412yes—
381C4⋊C4#1192960912yescomplemented
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—44no—
385C22⋊C4—82no—
386C42—44no—
387C42—41no—
388C42—41no—
389C42C2×C4×D7911yescharacteristic
390C42C2×D31611yescharacteristic
391C42#1192962411yescharacteristic
392C42#1192962411yescharacteristic
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—3162no—
401C23—6321no—
402C23—6321no—
403C23—6321no—
404C23—6321no—
405C23—6321no—
406C23—6321no—
407C23#1457913011yescharacteristic
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412D4—3164no—
413D4—3162no—
414D4—3162no—
415D4—3162no—
416D4—3162no—
417D4—6322no—
418D4—6321no—
419D4—6321no—
420D4—6321no—
421D4—6321no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4C2×(C2×D79)⋊C411yescharacteristic
441C2×C4C4×C22⋊D7911yescharacteristic
442C2×C4C4×C22⋊D7911yescharacteristic
443C2×C4#1457913511yescharacteristic
444C2×C4#1457913511yescharacteristic
445C2×C4#1457914411yescharacteristic
446C2×C4C2×#1192960911yescharacteristic
447C2×C4#1457918111yescharacteristic
448C2×C4#1457918111yescharacteristic
449C2×C4#1457918111yescharacteristic
450C2×C4#1457918611yescharacteristic
451C2×C4#1457918611yescharacteristic
452C2×C4#1457918611yescharacteristic
453C2×C4#1457918711yescharacteristic
454C8—3162no—
455C8—3162no—
456C8—6321no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—6322no—
465C22—6322no—
466C22—6322no—
467C22—12641no—
468C22—12641no—
469C22—12641no—
470C22—12641no—
471C22#1909844211yescharacteristic
472C22#1909869711yescharacteristic
473C22#1909870811yescharacteristic
474C22#1909870811yescharacteristic
475C22#1909870811yescharacteristic
476C22#1909873111yescharacteristic
477C22#1909873611yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—81no—
484C4#1909855711yescharacteristic
485C4#1909856711yescharacteristic
486C4C4×#1192960911yescharacteristic
487C4#1909867311yescharacteristic
488C4#1909875811yescharacteristic
489C4#1909877811yescharacteristic
490C4#1909888711yescharacteristic
491C4#1909888711yescharacteristic
492C2—82no—
493C2—12642no—
494C2#2376246511yescharacteristic
495C2#2376248311yescharacteristic
496C2#2376248911yescharacteristic
497C2#2376251011yescharacteristic
498C2#2376372211yescharacteristic
499C2#2376374811yescharacteristic
500C2#2376377611yescharacteristic
subgroupG/H
centerC23#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2354C2
socleC23×C79#120
prime radicalC2×#19098758C2
cosocleC2×C42×C79C23
hypercenter#2354D79
hypocenterC79#8980
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #23764327C2semidirect
3. #23764205C2semidirect
4. C2×D4⋊(C79⋊C8)C2semidirect
5. #23761283C2semidirect
6. #23761241C2semidirect
7. C79×#2354C2semidirect
11. #19098758C4semidirect
130. C4⋊C4×C79D8semidirect
252. #2354D79semidirect
258. C79#8980semidirect
381. C4⋊C4#11929609semidirect

Maximal subgroups

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

subgroupindex
C2×#190987582
#237643272
#237642052
C2×D4⋊(C79⋊C8)2
#237612832
#237612412
C79×#23542
#898079