Finite Group Databasebeta

← #34190572

Subgroups — #34190572

Order 20224.

744 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098763C211yesmaximal, characteristic, complemented
2C2×#19098758C211yesmaximal, characteristic
3#23764205C211yesmaximal, characteristic, complemented
4C2×Q8⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
5C2×D4⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
6#23761283C211yesmaximal, characteristic, complemented
7C79×#2538C211yesmaximal, characteristic, complemented
8C22×#11929611—21no—
9C22×#11929609—21no—
10C2×#14579158C2211yescharacteristic
11#19098763—22no—
12#19098763—22no—
13#19098758—22no—
14#19098758—22no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×C4⋊(C79⋊C8)C2211yescharacteristic
17C2×#14579081—21no—
18C2×#14579081—21no—
19C2×#14579081—22no—
20C2×C4×C79⋊C8C2211yescharacteristic
21Q8⋊(C79⋊C8)—22no—
22D4⋊(C79⋊C8)—22no—
23C2×C22⋊Q8×C79—21no—
24C2×C4×Q8×C79C2211yescharacteristic
25C2×C4×D4×C79C2211yescharacteristic
26#396×C79—22no—
27#390×C79—21no—
28#389×C79—21no—
29#384×C79C2211yescharacteristic
30#380×C79—21no—
31C22×D316—21no—
32C22×D316—21no—
33C22×D316—21no—
34C2×#11929611—24no—
35C2×#11929611—42no—
36C2×#11929611—42no—
37C2×#11929609—24no—
38C2×#11929609—42no—
39C2×#11929609—42no—
40C22×C79⋊C8—21no—
41C22×C79⋊C8—21no—
42C22×C79⋊C8—21no—
43#14579158—21no—
44#14579158—21no—
45#14579158—21no—
46#14579158D412yes—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)—21no—
49C4⋊(C79⋊C8)D412yes—
50C4⋊(C79⋊C8)D412yes—
51#14579081—24no—
52#14579081—24no—
53#14579081—44no—
54C4×C79⋊C8—21no—
55C4×C79⋊C8—21no—
56C22×Q8×C79D411yescharacteristic
57C22×D4×C79D411yescharacteristic
58C23×C4×C79—21no—
59C22⋊Q8×C79—42no—
60C22⋊Q8×C79—42no—
61C4×Q8×C79—21no—
62C4×Q8×C79—21no—
63C4×Q8×C79—21no—
64C4×Q8×C79—21no—
65C4×D4×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79—21no—
68C4×D4×C79—21no—
69C2×C4⋊C4×C79—22no—
70C2×C4⋊C4×C79—21no—
71C2×C4⋊C4×C79—21no—
72C2×C4⋊C4×C79—21no—
73C2×C4⋊C4×C79D411yescharacteristic
74C2×C4⋊C4×C79D411yescharacteristic
75C2×C4⋊C4×C79D412yes—
76C2×C22⋊C4×C79—22no—
77C2×C22⋊C4×C79—21no—
78C2×C22⋊C4×C79—22no—
79C2×C42×C79—21no—
80C2×C42×C79C2311yescharacteristic
81#95×C79—22no—
82C23×D79—41no—
83C2×D316—22no—
84C2×D316—22no—
85C2×D316—22no—
86C2×D316—42no—
87C2×D316—42no—
88C2×D316—41no—
89C2×D316—41no—
90C2×D316—41no—
91C2×D316—41no—
92C2×D316—41no—
93C2×D316—41no—
94C2×D316—41no—
95C2×D316—41no—
96C2×D316—41no—
97C2×D316—41no—
98C2×D316—41no—
99#11929611—48no—
100#11929609—48no—
101C2×C79⋊C8—22no—
102C2×C79⋊C8—22no—
103C2×C79⋊C8—22no—
104C2×C79⋊C8—41no—
105C2×C79⋊C8—41no—
106C2×C79⋊C8—41no—
107C2×C79⋊C8—41no—
108C2×C79⋊C8—41no—
109C2×C79⋊C8—41no—
110C24×C79—21no—
111C2×Q8×C79—21no—
112C2×Q8×C79—21no—
113C2×Q8×C79—21no—
114C2×Q8×C79—21no—
115C2×Q8×C79—21no—
116C2×Q8×C79D812yescomplemented
117C2×D4×C79—21no—
118C2×D4×C79—21no—
119C2×D4×C79—21no—
120C2×D4×C79—21no—
121C2×D4×C79—21no—
122C2×D4×C79SD1612yescomplemented
123C22×C4×C79—21no—
124C22×C4×C79—21no—
125C22×C4×C79—21no—
126C22×C4×C79—22no—
127C22×C4×C79—41no—
128C22×C4×C79—41no—
129C22×C4×C79—41no—
130C22×C4×C79—41no—
131C22×C4×C79—41no—
132C22×C4×C79—41no—
133C22×C4×C79C2×D411yescharacteristic
134C22×C4×C79C2×D411yescharacteristic
135C22×C4×C79C2×D411yescharacteristic
136C4⋊C4×C79—21no—
137C4⋊C4×C79—21no—
138C4⋊C4×C79—21no—
139C4⋊C4×C79—21no—
140C4⋊C4×C79—44no—
141C4⋊C4×C79—42no—
142C4⋊C4×C79—41no—
143C4⋊C4×C79—41no—
144C4⋊C4×C79—41no—
145C4⋊C4×C79—41no—
146C4⋊C4×C79—41no—
147C4⋊C4×C79—41no—
148C22⋊C4×C79—44no—
149C22⋊C4×C79—41no—
150C22⋊C4×C79—41no—
151C22⋊C4×C79—82no—
152C42×C79—41no—
153C42×C79—41no—
154C42×C79C2×D411yescharacteristic
155C42×C79C2×D411yescharacteristic
156C42×C79C2×D411yescharacteristic
157C42×C79#4111yescharacteristic
158C23×C79—22no—
159C23×C79—41no—
160C23×C79—41no—
161C23×C79—41no—
162C23×C79—41no—
163C23×C79—41no—
164C23×C79—41no—
165C23×C79#12011yescharacteristic
166C22×D79—42no—
167C22×D79—81no—
168C22×D79—81no—
169C22×D79—81no—
170C22×D79—81no—
171C22×D79—81no—
172C22×D79—81no—
173Q8×C79—22no—
174Q8×C79—22no—
175Q8×C79—41no—
176Q8×C79—41no—
177D4×C79—22no—
178D4×C79—22no—
179D4×C79—41no—
180D4×C79—41no—
181C2×C4×C79—21no—
182C2×C4×C79—21no—
183C2×C4×C79—22no—
184C2×C4×C79—41no—
185C2×C4×C79—41no—
186C2×C4×C79—41no—
187C2×C4×C79—41no—
188C2×C4×C79—42no—
189C2×C4×C79—41no—
190C2×C4×C79—42no—
191C2×C4×C79—41no—
192C2×C4×C79—41no—
193C2×C4×C79—42no—
194C2×C4×C79—41no—
195C2×C4×C79—81no—
196C2×C4×C79—81no—
197C2×C4×C79—81no—
198C2×C4×C79—81no—
199C2×C4×C79#12011yescharacteristic
200C2×C4×C79#12011yescharacteristic
201C2×C4×C79#12111yescharacteristic
202C2×C4×C79#12111yescharacteristic
203C2×C4×C79#12111yescharacteristic
204C2×C4×C79#12711yescharacteristic
205C2×C4×C79C2×D811yescharacteristic
206C2×C4×C79C2×SD1611yescharacteristic
207C2×C4×C79#13511yescharacteristic
208C2×C4×C79#13511yescharacteristic
209C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
210C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
211C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
212C2×C4×C79#13711yescharacteristic
213D316—44no—
214D316—42no—
215D316—42no—
216D316—42no—
217D316—42no—
218D316—82no—
219D316—81no—
220D316—81no—
221D316—81no—
222D316—81no—
223C79⋊C8—42no—
224C79⋊C8—42no—
225C79⋊C8—42no—
226C22×C79—42no—
227C22×C79—42no—
228C22×C79—42no—
229C22×C79—81no—
230C22×C79—81no—
231C22×C79—81no—
232C22×C79—81no—
233C22×C79#39211yescharacteristic
234C22×C79#44711yescharacteristic
235C22×C79#44811yescharacteristic
236C22×C79#44911yescharacteristic
237C22×C79#44911yescharacteristic
238C22×C79#45311yescharacteristic
239C22×C79#45411yescharacteristic
240C2×D79—82no—
241C2×D79—82no—
242C2×D79—82no—
243C2×D79—161no—
244C2×D79—161no—
245C2×D79—161no—
246C2×D79—161no—
247C4×C79—21no—
248C4×C79—21no—
249C4×C79—42no—
250C4×C79—82no—
251C4×C79—81no—
252C4×C79—81no—
253C4×C79#44711yescharacteristic
254C4×C79#44911yescharacteristic
255C4×C79#45011yescharacteristic
256C4×C79#45211yescharacteristic
257C4×C79#46511yescharacteristic
258C4×C79#46611yescharacteristic
259C4×C79#46811yescharacteristic
260C4×C79#46911yescharacteristic
261#10229—791nomaximal, Sylow
262C2×C79—82no—
263C2×C79#162111yescharacteristic
264C2×C79#162211yescharacteristic
265C2×C79#162711yescharacteristic
266C2×C79#162811yescharacteristic
267C2×C79#199911yescharacteristic
268C2×C79#200011yescharacteristic
269C2×C79#200611yescharacteristic
270D79—162no—
271C2×#460—791no—
272C2×#459—791no—
273#2538D7911yescharacteristic, complemented
274#1957—791no—
275C2×Q8⋊C8—791no—
276C2×D4⋊C8—791no—
277#1296—791no—
278C79#1022911yescharacteristic, complemented, Sylow
279C22×SD16—1581no—
280C22×D8—1581no—
281C2×#127—791no—
282C2×C22⋊Q8—21no—
283C2×C4×Q8C2×D7911yescharacteristic
284C2×C4×D4C2×D7911yescharacteristic
285#460—1582no—
286#460—1582no—
287#459—1582no—
288#459—1582no—
289C2×C4⋊C8—791no—
290C2×C4⋊C8—791no—
291C2×#102—1582no—
292C2×#102—1581no—
293C2×#102—1581no—
294C2×C4×C8—791no—
295#396—22no—
296#390—21no—
297#389—21no—
298#384C2×D7911yescharacteristic
299#380—21no—
300Q8⋊C8—1582no—
301D4⋊C8—1582no—
302C22×Q8C22⋊D7911yescharacteristic
303C22×D4—1581no—
304C22×D4—1581no—
305C22×D4—1581no—
306C22×D4C22⋊D7911yescharacteristic
307C23×C4—21no—
308C2×SD16—3162no—
309C2×SD16—3162no—
310C2×SD16—1584no—
311C2×D8—3162no—
312C2×D8—3162no—
313C2×D8—1584no—
314C22×C8—1581no—
315C22×C8—1581no—
316C22×C8—1581no—
317#127—792no—
318#127—1581no—
319#127—1581no—
320#127—1581no—
321C22⋊Q8—42no—
322C22⋊Q8—42no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4C22⋊D7911yescharacteristic
336C2×C4⋊C4C22⋊D7911yescharacteristic
337C2×C4⋊C4C22⋊D7912yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C42—21no—
342C2×C42C22×D7911yescharacteristic
343C4⋊C8—792no—
344C4⋊C8—792no—
345C4⋊C8—1581no—
346C4⋊C8—1581no—
347#102—3164no—
348#102—1584no—
349#102—1584no—
350C4×C8—1581no—
351C4×C8—1581no—
352#95—22no—
353C24—21no—
354C24—3161no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8#1192960912yescomplemented
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—3162no—
367C2×D4—3162no—
368C2×D4—3161no—
369C2×D4—3161no—
370C2×D4—3161no—
371C2×D4—3161no—
372C2×D4—3161no—
373C2×D4—3161no—
374C2×D4—3161no—
375C2×D4—3161no—
376C2×D4—3161no—
377C2×D4—3161no—
378C2×D4—3161no—
379C2×D4—1582no—
380C2×D4—1582no—
381C2×D4—1582no—
382C2×D4#1192961112yescomplemented
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4C2×C22⋊D7911yescharacteristic
394C22×C4C2×C22⋊D7911yescharacteristic
395C22×C4C2×C22⋊D7911yescharacteristic
396SD16—3168no—
397D8—3168no—
398C2×C8—3161no—
399C2×C8—3161no—
400C2×C8—3161no—
401C2×C8—3161no—
402C2×C8—3161no—
403C2×C8—3161no—
404C2×C8—1582no—
405C2×C8—1582no—
406C2×C8—1582no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C22⋊C4—44no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—82no—
423C42—41no—
424C42—41no—
425C42D4×D7911yescharacteristic
426C42D4×D7911yescharacteristic
427C42D4×D7911yescharacteristic
428C42#1192962311yescharacteristic
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—3162no—
437C23—6321no—
438C23—6321no—
439C23—6321no—
440C23—6321no—
441C23—6321no—
442C23—6321no—
443C23#1457913011yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448D4—22no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—3164no—
453D4—3162no—
454D4—3162no—
455D4—3162no—
456D4—3162no—
457D4—6322no—
458D4—6321no—
459D4—6321no—
460D4—6321no—
461D4—6321no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4#1457912911yescharacteristic
481C2×C4#1457912911yescharacteristic
482C2×C4#1457913211yescharacteristic
483C2×C4#1457913311yescharacteristic
484C2×C4#1457913311yescharacteristic
485C2×C4#1457915711yescharacteristic
486C2×C4C2×#1192960911yescharacteristic
487C2×C4C2×#1192961111yescharacteristic
488C2×C4#1457918111yescharacteristic
489C2×C4#1457918111yescharacteristic
490C2×C4#1457918611yescharacteristic
491C2×C4#1457918611yescharacteristic
492C2×C4#1457918711yescharacteristic
493C2×C4#1457919211yescharacteristic
494C8—3162no—
495C8—3162no—
496C8—3162no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—6322no—
505C22—6322no—
506C22—6322no—
507C22—12641no—
508C22—12641no—
509C22—12641no—
510C22—12641no—
511C22#1909847111yescharacteristic
512C22#1909869711yescharacteristic
513C22#1909870211yescharacteristic
514C22#1909870811yescharacteristic
515C22#1909870811yescharacteristic
516C22#1909873111yescharacteristic
517C22#1909873611yescharacteristic
518C4—21no—
519C4—21no—
520C4—42no—
521C4—82no—
522C4—81no—
523C4—81no—
524C4#1909869611yescharacteristic
525C4#1909870611yescharacteristic
526C4#1909871311yescharacteristic
527C4#1909872311yescharacteristic
528C4#1909879411yescharacteristic
529C4#1909879711yescharacteristic
530C4#1909881111yescharacteristic
531C4#1909881411yescharacteristic
532C2—82no—
533C2—12642no—
534C2#2376246511yescharacteristic
535C2#2376247211yescharacteristic
536C2#2376250311yescharacteristic
537C2#2376251011yescharacteristic
538C2#2376439611yescharacteristic
539C2#2376440111yescharacteristic
540C2#2376444211yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2538C2
socleC23×C79#120
prime radicalC2×#19098758C2
hypercenter#2538D79
hypocenterC79#10229
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#19098763C2semidirect
3. #23764205C2semidirect
4. C2×Q8⋊(C79⋊C8)C2semidirect
5. C2×D4⋊(C79⋊C8)C2semidirect
6. #23761283C2semidirect
7. C79×#2538C2semidirect
116. C2×Q8×C79D8semidirect
122. C2×D4×C79SD16semidirect
273. #2538D79semidirect
278. C79#10229semidirect
360. C2×Q8#11929609semidirect
382. C2×D4#11929611semidirect

Maximal subgroups

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

subgroupindex
C2×#190987632
C2×#190987582
#237642052
C2×Q8⋊(C79⋊C8)2
C2×D4⋊(C79⋊C8)2
#237612832
C79×#25382
#1022979