Finite Group Databasebeta

← #34446041

Subgroups — #34446041

Order 20224.

480 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776292C211yesmaximal, characteristic, complemented
2#23765641C211yesmaximal, characteristic
3#23765636C211yesmaximal, characteristic, complemented
4#23765631C211yesmaximal, characteristic, complemented
5#23765564C211yesmaximal, characteristic, complemented
6#23765557C211yesmaximal, characteristic, complemented
7#23765547C211yesmaximal, characteristic, complemented
8#23765459C211yesmaximal, characteristic, complemented
9#23765459C211yesmaximal, characteristic, complemented
10#23765437C211yesmaximal, characteristic
11#23765309C211yesmaximal, characteristic, complemented
12#23765269C211yesmaximal, characteristic, complemented
13#23765264C211yesmaximal, characteristic, complemented
14C4×#14579105C211yesmaximal, characteristic, complemented
15C79×#3396C211yesmaximal, characteristic
16#19099291—21no—
17C2×#14579107C2211yescharacteristic
18C2×#14579105C2211yescharacteristic
19C2×#14579105C2211yescharacteristic
20#19098968C2211yescharacteristic
21#19098965C2211yescharacteristic
22#19098965C2211yescharacteristic
23#19098871—21no—
24#19098871—21no—
25#19098849—21no—
26C4×Dic316C2211yescharacteristic
27C4×Dic316C2211yescharacteristic
28C4×Dic316C2211yescharacteristic
29#19098390C2211yescharacteristic
30#19098390C2211yescharacteristic
31#19098388C2211yescharacteristic
32C4⋊(C79⋊C16)C2211yescharacteristic
33C4⋊(C79⋊C16)C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098367C2211yescharacteristic
36#19098367C2211yescharacteristic
37#19098367C2211yescharacteristic
38#19098366C2211yescharacteristic
39#19098364C2211yescharacteristic
40#19098364C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098364C2211yescharacteristic
45#19098339C2211yescharacteristic
46#19098339C2211yescharacteristic
47C4×C79⋊C16C2211yescharacteristic
48#559×C79—21no—
49#495×C79C2211yescharacteristic
50#495×C79C2211yescharacteristic
51#493×C79C2211yescharacteristic
52#463×C79—21no—
53#463×C79—21no—
54#460×C79—21no—
55C8×Q8×C79C2211yescharacteristic
56C4×Q16×C79C2211yescharacteristic
57C4×D8×C79C2211yescharacteristic
58C4×D8×C79C2211yescharacteristic
59C2×Dic316C2311yescharacteristic
60#14579161—21no—
61#14579159—41no—
62#14579153—21no—
63#14579153—21no—
64#14579151—41no—
65#14579151—41no—
66C4×Dic158—21no—
67C4×Dic158—21no—
68C4×Dic158—21no—
69#14579126—41no—
70#14579107—22no—
71#14579105—22no—
72#14579105—22no—
73C2×C79⋊C16C2311yescharacteristic
74C2×C79⋊C16C2311yescharacteristic
75C2×C79⋊C16C2311yescharacteristic
76C2×C79⋊C16C2311yescharacteristic
77#14579095—21no—
78#14579095C2311yescharacteristic
79#14579095C2311yescharacteristic
80#14579095C2311yescharacteristic
81#14579093—21no—
82#14579093—21no—
83#14579086—21no—
84#14579086—21no—
85#14579086—21no—
86C2×Q16×C79C2311yescharacteristic
87C2×SD16×C79—21no—
88C2×SD16×C79—21no—
89C2×D8×C79—21no—
90C2×D8×C79C2311yescharacteristic
91C2×D8×C79C2311yescharacteristic
92#127×C79—21no—
93#124×C79—21no—
94#124×C79—21no—
95#124×C79—41no—
96#123×C79—41no—
97#123×C79—41no—
98C4×Q8×C79—21no—
99C4×Q8×C79D411yescharacteristic
100C4×D4×C79—21no—
101C4×D4×C79—21no—
102C4×D4×C79—41no—
103#107×C79C2311yescharacteristic
104C4⋊C8×C79D411yescharacteristic
105C4⋊C8×C79D411yescharacteristic
106C4⋊C8×C79D411yescharacteristic
107#103×C79—21no—
108#102×C79—21no—
109#102×C79—21no—
110C4×C8×C79C2311yescharacteristic
111C4×C8×C79C2311yescharacteristic
112C4×C8×C79C2311yescharacteristic
113C2×Dic158—21no—
114C2×Dic158—41no—
115Dic316—21no—
116Dic316#4112yes—
117C4⋊Dic79—21no—
118C4⋊Dic79—21no—
119C4⋊Dic79—21no—
120C4⋊Dic79—41no—
121Dic79⋊C4—41no—
122Dic79⋊C4—41no—
123Dic79⋊C4—41no—
124Dic79⋊C4—41no—
125Dic79⋊C4—41no—
126Dic79⋊C4—41no—
127Dic79⋊C4—41no—
128C4×Dic79—41no—
129C4×Dic79—41no—
130C4×Dic79—41no—
131C79⋊C16—21no—
132C79⋊C16—21no—
133C79⋊C16—21no—
134C79⋊C16—21no—
135C2×Q8×C79—21no—
136C2×Q8×C79D811yescharacteristic
137C2×D4×C79—21no—
138C2×D4×C79—21no—
139C2×D4×C79—41no—
140C2×D4×C79—41no—
141C22×C4×C79—41no—
142C22×C4×C79—41no—
143Q16×C79—21no—
144Q16×C79—21no—
145SD16×C79—81no—
146SD16×C79—81no—
147D8×C79—21no—
148D8×C79—21no—
149D8×C79—21no—
150D8×C79—21no—
151D8×C79—81no—
152C2×C8×C79C2×D411yescharacteristic
153C2×C8×C79C2×D411yescharacteristic
154C2×C8×C79C2×D411yescharacteristic
155C2×C8×C79C2411yescharacteristic
156C4⋊C4×C79—21no—
157C4⋊C4×C79—41no—
158C4⋊C4×C79D811yescharacteristic
159C4⋊C4×C79D811yescharacteristic
160C4⋊C4×C79D811yescharacteristic
161C22⋊C4×C79—41no—
162C22⋊C4×C79—41no—
163C22⋊C4×C79—41no—
164C22⋊C4×C79—41no—
165C22⋊C4×C79—41no—
166C22⋊C4×C79—41no—
167C42×C79—41no—
168C42×C79C2×D411yescharacteristic
169C42×C79C2×D411yescharacteristic
170C42×C79C2×D411yescharacteristic
171C23×C79—41no—
172C23×C79—41no—
173Q8×C79—41no—
174Q8×C79—41no—
175Q8×C79—41no—
176D4×C79—41no—
177D4×C79—41no—
178D4×C79—41no—
179D4×C79—41no—
180D4×C79—82no—
181D4×C79—161no—
182C2×C4×C79—41no—
183C2×C4×C79—41no—
184C2×C4×C79—161no—
185C2×C4×C79—161no—
186C2×C4×C79C2×D811yescharacteristic
187C2×C4×C79C2×D811yescharacteristic
188C2×C4×C79C2×D811yescharacteristic
189C2×C4×C79C2×D811yescharacteristic
190C2×C4×C79C2×D811yescharacteristic
191C2×C4×C79C2×D811yescharacteristic
192C2×C4×C79C22×D411yescharacteristic
193C2×Dic79—41no—
194C2×Dic79—41no—
195C2×Dic79—41no—
196C2×Dic79—41no—
197Dic158—22no—
198Dic158—41no—
199Dic158—82no—
200C8×C79—21no—
201C8×C79—21no—
202C8×C79—21no—
203C8×C79C2×#4111yescharacteristic
204C8×C792−1+411yescharacteristic
205C22×C79—81no—
206C22×C79—81no—
207C22×C79—81no—
208C22×C79—81no—
209C22×C79—81no—
210C22×C79—81no—
211C22×C79C22×D811yescharacteristic
212C4×C79—21no—
213C4×C79—21no—
214C4×C79—21no—
215C4×C79—21no—
216C4×C79—21no—
217C4×C79—21no—
218C4×C79—81no—
219C4×C79—81no—
220C4×C79#54811yescharacteristic
221C4×C79#57711yescharacteristic
222Dic79—42no—
223Dic79—81no—
224Dic79—81no—
225Dic79—81no—
226#33938—791nomaximal, Sylow
227C2×C79—161no—
228C2×C79—161no—
229C2×C79#335811yescharacteristic
230C2×C79#341411yescharacteristic
231C2×C79#341611yescharacteristic
232#3396D7911yescharacteristic
233#3383—791no—
234#2244—791no—
235#2243—791no—
236#2242—791no—
237#2229—791no—
238#2228—791no—
239#2226—791no—
240#2211—791no—
241#2211—791no—
242#2207—791no—
243#2183—791no—
244#2176—791no—
245#2175—791no—
246C4×SD32—791no—
247C79#3393811yescharacteristic, complemented, Sylow
248#559—21no—
249#557—1581no—
250C2×Q32—791no—
251C2×SD32—791no—
252C2×SD32—791no—
253#500—791no—
254#499—791no—
255#499—791no—
256#495C2×D7911yescharacteristic
257#495C2×D7911yescharacteristic
258#493C2×D7911yescharacteristic
259#479—1581no—
260#479—1581no—
261#475—1581no—
262#463—21no—
263#463—21no—
264#460—21no—
265C8×Q8C2×D7911yescharacteristic
266C4×Q16—791no—
267C4×Q16—791no—
268C4×Q16—791no—
269C4×Q16C2×D7911yescharacteristic
270C4×D8C2×D7911yescharacteristic
271C4×D8C2×D7911yescharacteristic
272#367—791no—
273#367—791no—
274#366—791no—
275C4⋊C16—791no—
276C4⋊C16—791no—
277C4⋊C16—791no—
278#358—791no—
279#358—791no—
280#358—791no—
281#358—791no—
282#357—791no—
283#357—791no—
284#357—791no—
285#357—791no—
286#357—791no—
287#357—791no—
288#346—791no—
289#346—791no—
290C4×C16—791no—
291C2×Q16—791no—
292C2×Q16C22×D7911yescharacteristic
293C2×SD16—21no—
294C2×SD16—21no—
295C2×D8—21no—
296C2×D8C22×D7911yescharacteristic
297C2×D8C22×D7911yescharacteristic
298C4⋊Q8—3161no—
299C4⋊Q8—1581no—
300#127—21no—
301#125—3161no—
302#125—3161no—
303#125—1581no—
304#125—1581no—
305#124—21no—
306#124—21no—
307#124—41no—
308#123—41no—
309#123—41no—
310C4×Q8—21no—
311C4×Q8—3161no—
312C4×Q8—1581no—
313C4×Q8—1581no—
314C4×Q8—1581no—
315C4×Q8C22⋊D7911yescharacteristic
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—41no—
319Q32—1582no—
320SD32—1582no—
321SD32—1582no—
322C2×C16—791no—
323C2×C16—791no—
324C2×C16—791no—
325C2×C16—791no—
326#107—791no—
327#107—791no—
328#107—791no—
329#107—1581no—
330#107C22×D7911yescharacteristic
331#106—1581no—
332#106—1581no—
333C4⋊C8C22⋊D7911yescharacteristic
334C4⋊C8C22⋊D7911yescharacteristic
335C4⋊C8C22⋊D7911yescharacteristic
336#103—21no—
337#103—1581no—
338#103—1581no—
339#103—1581no—
340#102—21no—
341#102—21no—
342C4×C8C22×D7911yescharacteristic
343C4×C8C22×D7911yescharacteristic
344C4×C8C22×D7911yescharacteristic
345C2×Q8—21no—
346C2×Q8—3161no—
347C2×Q8—1581no—
348C2×Q8#1192960911yescharacteristic
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—41no—
353C22×C4—41no—
354C22×C4—41no—
355Q16—21no—
356Q16—21no—
357Q16—792no—
358Q16—1581no—
359SD16—81no—
360SD16—81no—
361D8—21no—
362D8—21no—
363D8—21no—
364D8—21no—
365D8—81no—
366C2×C8C2×C22⋊D7911yescharacteristic
367C2×C8C2×C22⋊D7911yescharacteristic
368C2×C8C2×C22⋊D7911yescharacteristic
369C2×C8C23×D7911yescharacteristic
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—3161no—
373C4⋊C4—3161no—
374C4⋊C4—3161no—
375C4⋊C4—3161no—
376C4⋊C4—3161no—
377C4⋊C4—3161no—
378C4⋊C4—3161no—
379C4⋊C4—3161no—
380C4⋊C4—1581no—
381C4⋊C4—1581no—
382C4⋊C4—1581no—
383C4⋊C4#1192960911yescharacteristic
384C4⋊C4#1192960911yescharacteristic
385C4⋊C4#1192960911yescharacteristic
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C42—41no—
393C42—3161no—
394C42—3161no—
395C42—3161no—
396C42C2×C22⋊D7911yescharacteristic
397C42C2×C22⋊D7911yescharacteristic
398C42C2×C22⋊D7911yescharacteristic
399C16—1581no—
400C16—1581no—
401C16—1581no—
402C16—1581no—
403C23—41no—
404C23—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408Q8—3161no—
409Q8—6322no—
410Q8—1582no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—82no—
416D4—161no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—161no—
420C2×C4—161no—
421C2×C4—3161no—
422C2×C4—3161no—
423C2×C4—3161no—
424C2×C4—3161no—
425C2×C4C2×#1192960911yescharacteristic
426C2×C4C2×#1192960911yescharacteristic
427C2×C4C2×#1192960911yescharacteristic
428C2×C4C2×#1192960911yescharacteristic
429C2×C4C2×#1192960911yescharacteristic
430C2×C4C2×#1192960911yescharacteristic
431C2×C4C22×C22⋊D7911yescharacteristic
432C8—21no—
433C8—21no—
434C8—21no—
435C8#41×D7911yescharacteristic
436C8#1457920811yescharacteristic
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22C22×#1192960911yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—81no—
451C4—81no—
452C4—3162no—
453C4—6321no—
454C4—6321no—
455C4—6321no—
456C4#1909922011yescharacteristic
457C4#1909937411yescharacteristic
458C2—161no—
459C2—161no—
460C2#2377603111yescharacteristic
461C2#2377653411yescharacteristic
462C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3396C2
socleC22×C79C22×D8
prime radical#493×C79C22
hypercenter#3396D79
hypocenterC79#33938
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
1. #23776292C2semidirect
3. #23765636C2semidirect
4. #23765631C2semidirect
5. #23765564C2semidirect
6. #23765557C2semidirect
7. #23765547C2semidirect
8. #23765459C2semidirect
9. #23765459C2semidirect
11. #23765309C2semidirect
12. #23765269C2semidirect
13. #23765264C2semidirect
14. C4×#14579105C2semidirect
247. C79#33938semidirect

Maximal subgroups

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

subgroupindex
#237762922
#237656412
#237656362
#237656312
#237655642
#237655572
#237655472
#237654592
#237654372
#237653092
#237652692
#237652642
C4×#145791052
C79×#33962
#3393879