Finite Group Databasebeta

← #34443296

Subgroups — #34443296

Order 20224.

554 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14579183C211yesmaximal, characteristic, complemented
2#23765605C211yesmaximal, characteristic, complemented
3#23765588C211yesmaximal, characteristic, complemented
4#23765561C211yesmaximal, characteristic, complemented
5#23765554C211yesmaximal, characteristic, complemented
6#23765459C211yesmaximal, characteristic, complemented
7#23765452C211yesmaximal, characteristic, complemented
8#23765338C211yesmaximal, characteristic, complemented
9#23765321C211yesmaximal, characteristic
10#23765274C211yesmaximal, characteristic, complemented
11#23765269C211yesmaximal, characteristic, complemented
12#23765264C211yesmaximal, characteristic, complemented
13#23765262C211yesmaximal, characteristic, complemented
14#23765203C211yesmaximal, characteristic, complemented
15C79×#3168C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C4×#11929624—21no—
18C2×#14579107C2211yescharacteristic
19C2×#14579105C2211yescharacteristic
20C2×#14579104C2211yescharacteristic
21C2×#14579102C2211yescharacteristic
22C4×Dic316C2211yescharacteristic
23C4×Dic316C2211yescharacteristic
24C4×#11929613—21no—
25C4×#11929613—21no—
26C4×D632C2211yescharacteristic
27C4×D632C2211yescharacteristic
28#19098601C2211yescharacteristic
29#19098560—21no—
30#19098390C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098388C2211yescharacteristic
33#19098388C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35C4⋊(C79⋊C16)C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098366C2211yescharacteristic
39#19098366C2211yescharacteristic
40#19098364C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098363C2211yescharacteristic
43#19098363C2211yescharacteristic
44C22⋊(C79⋊C16)C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46#19098339C2211yescharacteristic
47#19098339C2211yescharacteristic
48#529×C79—21no—
49#499×C79C2211yescharacteristic
50#495×C79C2211yescharacteristic
51#488×C79—21no—
52#467×C79C2211yescharacteristic
53#466×C79C2211yescharacteristic
54#465×C79—21no—
55C4×Q16×C79C2211yescharacteristic
56C4×SD16×C79—21no—
57C4×D8×C79C2211yescharacteristic
58C2×C4×C8×C79C2211yescharacteristic
59C2×#11929624—21no—
60#14579183—22no—
61#14579183—22no—
62C2×Dic316C2311yescharacteristic
63C2×#11929613—21no—
64C2×D632C2311yescharacteristic
65C4×Dic158—21no—
66C4×Dic158—21no—
67C4×D316—21no—
68C4×D316—21no—
69C4×C22⋊D79—41no—
70C4×C22⋊D79—41no—
71#14579120—21no—
72#14579118—41no—
73C42×D79—41no—
74#14579107—22no—
75#14579105—22no—
76#14579104—22no—
77#14579102—22no—
78C2×C79⋊C16C2311yescharacteristic
79C2×C79⋊C16C2311yescharacteristic
80C2×C79⋊C16C2311yescharacteristic
81C2×C79⋊C16C2311yescharacteristic
82#14579095C2311yescharacteristic
83#14579095C2311yescharacteristic
84#14579093—21no—
85#14579086—21no—
86#14579086—21no—
87#14579083—21no—
88#14579083—21no—
89C2×Q16×C79C2311yescharacteristic
90C2×SD16×C79—21no—
91C2×D8×C79C2311yescharacteristic
92C22×C8×C79D411yescharacteristic
93C22×C8×C79C2311yescharacteristic
94#126×C79—41no—
95#125×C79—21no—
96#124×C79—21no—
97#123×C79—41no—
98C22⋊Q8×C79—21no—
99#121×C79—21no—
100C4×Q8×C79—21no—
101C4×D4×C79—21no—
102C4×D4×C79—41no—
103#117×C79—41no—
104C2×C42×C79D411yescharacteristic
105#107×C79C2311yescharacteristic
106#107×C79C2311yescharacteristic
107#106×C79—21no—
108#103×C79—21no—
109#103×C79—21no—
110#102×C79—21no—
111#102×C79—21no—
112C4×C8×C79D411yescharacteristic
113C4×C8×C79D411yescharacteristic
114C4×C8×C79C2311yescharacteristic
115C4×C8×C79C2311yescharacteristic
116#11929624—24no—
117#11929624—42no—
118C2×Dic158—21no—
119C2×D316—21no—
120C2×C22⋊D79—41no—
121C2×C4×D79—41no—
122C2×C4×D79—41no—
123C2×C4×D79—41no—
124Dic316—21no—
125Dic316#4112yes—
126#11929613—24no—
127D632—21no—
128D632#4112yes—
129C4⋊Dic79—21no—
130C4⋊Dic79—21no—
131Dic79⋊C4—41no—
132Dic79⋊C4—41no—
133C22⋊Dic79—41no—
134(C2×D79)⋊C4—41no—
135(C2×D79)⋊C4—41no—
136C4×Dic79—41no—
137C4×Dic79—41no—
138C4×Dic79—41no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C79⋊C16—21no—
142C79⋊C16—21no—
143C2×Q8×C79—21no—
144C2×D4×C79—21no—
145C2×D4×C79—41no—
146C22×C4×C79—41no—
147C22×C4×C79D811yescharacteristic
148C22×C4×C79D811yescharacteristic
149C22×C4×C79C2×D411yescharacteristic
150Q16×C79—21no—
151Q16×C79—21no—
152SD16×C79—42no—
153D8×C79—21no—
154D8×C79—21no—
155C2×C8×C79—21no—
156C2×C8×C79—21no—
157C2×C8×C79—41no—
158C2×C8×C79C2×D411yescharacteristic
159C2×C8×C79C2×D411yescharacteristic
160C2×C8×C79C2×D411yescharacteristic
161C2×C8×C79C2411yescharacteristic
162C4⋊C4×C79—21no—
163C4⋊C4×C79—21no—
164C4⋊C4×C79—41no—
165C4⋊C4×C79—41no—
166C4⋊C4×C79—41no—
167C4⋊C4×C79—41no—
168C22⋊C4×C79—41no—
169C22⋊C4×C79—41no—
170C22⋊C4×C79—41no—
171C22⋊C4×C79—41no—
172C22⋊C4×C79—41no—
173C42×C79—41no—
174C42×C79D811yescharacteristic
175C42×C79D811yescharacteristic
176C42×C79C2×D411yescharacteristic
177C42×C79C2×D411yescharacteristic
178C23×C79—41no—
179C23×C79C2×D811yescharacteristic
180C22×D79—41no—
181Q8×C79—41no—
182Q8×C79—41no—
183D4×C79—41no—
184D4×C79—41no—
185D4×C79—82no—
186C2×C4×C79—21no—
187C2×C4×C79—21no—
188C2×C4×C79—41no—
189C2×C4×C79—41no—
190C2×C4×C79—41no—
191C2×C4×C79—41no—
192C2×C4×C79—41no—
193C2×C4×C79—82no—
194C2×C4×C79C2×D811yescharacteristic
195C2×C4×C79C2×D811yescharacteristic
196C2×C4×C79C2×D811yescharacteristic
197C2×C4×C79C2×D811yescharacteristic
198C2×C4×C79C2×D811yescharacteristic
199C2×C4×C79C22×D411yescharacteristic
200C22⋊D79—44no—
201C2×Dic79—41no—
202C2×Dic79—41no—
203C2×Dic79—41no—
204D316—22no—
205D316—41no—
206C4×D79—44no—
207C4×D79—82no—
208C4×D79—82no—
209Dic158—22no—
210Dic158—41no—
211C8×C79—21no—
212C8×C79—21no—
213C8×C79—21no—
214C8×C79C2×#4111yescharacteristic
215C8×C79C2×#4111yescharacteristic
216C22×C79—21no—
217C22×C79—21no—
218C22×C79—21no—
219C22×C79—81no—
220C22×C79—81no—
221C22×C79—81no—
222C22×C79C22×D811yescharacteristic
223C2×D79—42no—
224C2×D79—81no—
225C2×D79—81no—
226C4×C79—21no—
227C4×C79—21no—
228C4×C79—21no—
229C4×C79—21no—
230C4×C79—21no—
231C4×C79—81no—
232C4×C79—81no—
233C4×C79—81no—
234C4×C79#52511yescharacteristic
235C4×C79#57511yescharacteristic
236Dic79—42no—
237Dic79—81no—
238Dic79—81no—
239#33671—791nomaximal, Sylow
240C2×C79—41no—
241C2×C79—161no—
242C2×C79#310011yescharacteristic
243C2×C79#341411yescharacteristic
244C2×C79#341411yescharacteristic
245D79—82no—
246#3168D7911yescharacteristic, complemented
247C4×#135—791no—
248#2237—791no—
249#2234—791no—
250#2229—791no—
251#2228—791no—
252#2211—791no—
253#2210—791no—
254#2189—791no—
255#2186—791no—
256#2177—791no—
257#2176—791no—
258#2175—791no—
259#2175—791no—
260#2163—791no—
261C79#3367111yescharacteristic, complemented, Sylow
262C2×#135—791no—
263#529—21no—
264C4×#41—1581no—
265C2×Q32—791no—
266C2×SD32—791no—
267C2×SD32—791no—
268C2×D16—791no—
269#499C2×D7911yescharacteristic
270#495C2×D7911yescharacteristic
271#488—21no—
272#467C2×D7911yescharacteristic
273#466C2×D7911yescharacteristic
274#465—21no—
275C4×Q16—791no—
276C4×Q16—791no—
277C4×Q16C2×D7911yescharacteristic
278C4×SD16—21no—
279C4×SD16—1581no—
280C4×SD16—1581no—
281C4×D8—791no—
282C4×D8—791no—
283C4×D8C2×D7911yescharacteristic
284#427—791no—
285#416—1581no—
286C2×C4×C8C2×D7911yescharacteristic
287#367—791no—
288#367—791no—
289#366—791no—
290#366—791no—
291C4⋊C16—791no—
292C4⋊C16—791no—
293#358—791no—
294#358—791no—
295#358—791no—
296#358—791no—
297#357—791no—
298#357—791no—
299#357—791no—
300#357—791no—
301C22⋊C16—791no—
302C22⋊C16—791no—
303#346—791no—
304#346—791no—
305C2×#41—1581no—
306#135—1582no—
307#135—1582no—
308C2×Q16—791no—
309C2×Q16C22×D7911yescharacteristic
310C2×SD16—21no—
311C2×SD16—1581no—
312C2×D8—791no—
313C2×D8C22×D7911yescharacteristic
314C22×C8C22⋊D7911yescharacteristic
315C22×C8C22×D7911yescharacteristic
316#126—41no—
317#125—21no—
318#124—21no—
319#123—41no—
320C22⋊Q8—21no—
321#121—21no—
322C4×Q8—21no—
323C4×Q8—1581no—
324C4×Q8—1581no—
325C4×D4—21no—
326C4×D4—41no—
327C4×D4—3161no—
328C4×D4—3161no—
329C4×D4—1581no—
330C4×D4—1581no—
331#117—41no—
332#117—3161no—
333#117—1581no—
334C2×C42—3161no—
335C2×C42C22⋊D7911yescharacteristic
336Q32—1582no—
337SD32—1582no—
338SD32—1582no—
339D16—1582no—
340C2×C16—791no—
341C2×C16—791no—
342C2×C16—791no—
343C2×C16—791no—
344#107—791no—
345#107—791no—
346#107C22×D7911yescharacteristic
347#107C22×D7911yescharacteristic
348#106—21no—
349#106—1581no—
350#103—21no—
351#103—21no—
352#103—1581no—
353#103—1581no—
354#102—21no—
355#102—21no—
356#102—1581no—
357#102—1581no—
358C4×C8C22⋊D7911yescharacteristic
359C4×C8C22⋊D7911yescharacteristic
360C4×C8C22×D7911yescharacteristic
361C4×C8C22×D7911yescharacteristic
362#41—3162no—
363#41—1584no—
364C2×Q8—21no—
365C2×Q8—1581no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—3161no—
369C2×D4—1581no—
370C22×C4—41no—
371C22×C4—3161no—
372C22×C4—3161no—
373C22×C4—3161no—
374C22×C4#1192960911yescharacteristic
375C22×C4#1192960911yescharacteristic
376C22×C4C2×C22⋊D7911yescharacteristic
377Q16—21no—
378Q16—21no—
379Q16—792no—
380Q16—1581no—
381SD16—42no—
382SD16—1584no—
383D8—21no—
384D8—21no—
385D8—792no—
386D8—1581no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—41no—
390C2×C8C2×C22⋊D7911yescharacteristic
391C2×C8C2×C22⋊D7911yescharacteristic
392C2×C8C2×C22⋊D7911yescharacteristic
393C2×C8C23×D7911yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—3161no—
401C4⋊C4—3161no—
402C4⋊C4—1581no—
403C4⋊C4—1581no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—3161no—
410C22⋊C4—3161no—
411C22⋊C4—3161no—
412C42—41no—
413C42—3161no—
414C42—3161no—
415C42—3161no—
416C42#1192960911yescharacteristic
417C42#1192960911yescharacteristic
418C42C2×C22⋊D7911yescharacteristic
419C42C2×C22⋊D7911yescharacteristic
420C16—1581no—
421C16—1581no—
422C16—1581no—
423C16—1581no—
424C23—41no—
425C23—3161no—
426C23C2×#1192960911yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—3161no—
430Q8—1582no—
431D4—41no—
432D4—41no—
433D4—82no—
434D4—3164no—
435D4—3161no—
436D4—1582no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—82no—
445C2×C4—3164no—
446C2×C4—3161no—
447C2×C4—3161no—
448C2×C4—3161no—
449C2×C4—6322no—
450C2×C4—6322no—
451C2×C4C2×#1192960911yescharacteristic
452C2×C4C2×#1192960911yescharacteristic
453C2×C4C2×#1192960911yescharacteristic
454C2×C4C2×#1192960911yescharacteristic
455C2×C4C2×#1192960911yescharacteristic
456C2×C4C22×C22⋊D7911yescharacteristic
457C8—21no—
458C8—21no—
459C8—21no—
460C8#41×D7911yescharacteristic
461C8#41×D7911yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—3162no—
469C22—6321no—
470C22—6321no—
471C22C22×#1192960911yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—3162no—
481C4—6321no—
482C4—6321no—
483C4#1909907011yescharacteristic
484C4#1909936111yescharacteristic
485C2—41no—
486C2—161no—
487C2—6322no—
488C2#2377339611yescharacteristic
489C2#2377653411yescharacteristic
490C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3168C2
socleC22×C79C22×D8
prime radical#23765321C2
hypercenter#3168D79
hypocenterC79#33671
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. C4×#14579183C2semidirect
2. #23765605C2semidirect
3. #23765588C2semidirect
4. #23765561C2semidirect
5. #23765554C2semidirect
6. #23765459C2semidirect
7. #23765452C2semidirect
8. #23765338C2semidirect
10. #23765274C2semidirect
11. #23765269C2semidirect
12. #23765264C2semidirect
13. #23765262C2semidirect
14. #23765203C2semidirect
15. C79×#3168C2semidirect
246. #3168D79semidirect
261. C79#33671semidirect

Maximal subgroups

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

subgroupindex
C4×#145791832
#237656052
#237655882
#237655612
#237655542
#237654592
#237654522
#237653382
#237653212
#237652742
#237652692
#237652642
#237652622
#237652032
C79×#31682
#3367179