Finite Group Databasebeta

← #34446032

Subgroups — #34446032

Order 20224.

494 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776413C211yesmaximal, characteristic, complemented
2#23765643C211yesmaximal, characteristic, complemented
3#23765639C211yesmaximal, characteristic, complemented
4#23765634C211yesmaximal, characteristic, complemented
5#23765561C211yesmaximal, characteristic, complemented
6#23765554C211yesmaximal, characteristic, complemented
7#23765534C211yesmaximal, characteristic, complemented
8#23765455C211yesmaximal, characteristic, complemented
9#23765455C211yesmaximal, characteristic, complemented
10#23765435C211yesmaximal, characteristic
11#23765309C211yesmaximal, characteristic, complemented
12#23765274C211yesmaximal, characteristic, complemented
13#23765264C211yesmaximal, characteristic, complemented
14C4×#14579102C211yesmaximal, characteristic, complemented
15C79×#3380C211yesmaximal, characteristic, complemented
16#19099298—21no—
17C2×#14579105C2211yescharacteristic
18C2×#14579102C2211yescharacteristic
19C2×#14579102C2211yescharacteristic
20#19098947C2211yescharacteristic
21#19098947C2211yescharacteristic
22#19098940C2211yescharacteristic
23#19098779—21no—
24#19098779—21no—
25#19098764—21no—
26C4×Dic316C2211yescharacteristic
27C4×D632C2211yescharacteristic
28C4×D632C2211yescharacteristic
29#19098390C2211yescharacteristic
30#19098388C2211yescharacteristic
31#19098388C2211yescharacteristic
32C4⋊(C79⋊C16)C2211yescharacteristic
33C4⋊(C79⋊C16)C2211yescharacteristic
34C4⋊(C79⋊C16)C2211yescharacteristic
35#19098366C2211yescharacteristic
36#19098366C2211yescharacteristic
37#19098366C2211yescharacteristic
38#19098364C2211yescharacteristic
39#19098363C2211yescharacteristic
40#19098363C2211yescharacteristic
41#19098363C2211yescharacteristic
42#19098363C2211yescharacteristic
43#19098363C2211yescharacteristic
44#19098363C2211yescharacteristic
45#19098339C2211yescharacteristic
46#19098339C2211yescharacteristic
47C4×C79⋊C16C2211yescharacteristic
48#554×C79—21no—
49#500×C79C2211yescharacteristic
50#499×C79C2211yescharacteristic
51#499×C79C2211yescharacteristic
52#478×C79—21no—
53#478×C79—21no—
54#476×C79—21no—
55C8×Q8×C79C2211yescharacteristic
56C4×D8×C79C2211yescharacteristic
57C4×D8×C79C2211yescharacteristic
58C4×D8×C79C2211yescharacteristic
59C2×Dic316C2311yescharacteristic
60C2×#11929613—21no—
61C2×#11929613—21no—
62C2×D632—21no—
63C2×D632C2311yescharacteristic
64C2×D632C2311yescharacteristic
65#14579158—21no—
66#14579150—41no—
67#14579149—21no—
68#14579149—21no—
69#14579146—41no—
70#14579146—41no—
71C4×Dic158—21no—
72C4×D316—21no—
73C4×D316—21no—
74#14579123—41no—
75#14579105—22no—
76#14579102—22no—
77#14579102—22no—
78C2×C79⋊C16C2311yescharacteristic
79C2×C79⋊C16C2311yescharacteristic
80C2×C79⋊C16C2311yescharacteristic
81C2×C79⋊C16C2311yescharacteristic
82#14579095C2311yescharacteristic
83#14579086—21no—
84#14579083—21no—
85#14579083—21no—
86C2×D8×C79C2311yescharacteristic
87C4⋊Q8×C79—21no—
88#125×C79—21no—
89#125×C79—21no—
90C22⋊Q8×C79—41no—
91C22⋊Q8×C79—41no—
92C22⋊Q8×C79—41no—
93C4×Q8×C79D411yescharacteristic
94C4×D4×C79—21no—
95C4×D4×C79—21no—
96C4×D4×C79—21no—
97C2×C4⋊C4×C79—41no—
98#107×C79—21no—
99#107×C79C2311yescharacteristic
100#107×C79C2311yescharacteristic
101#107×C79C2311yescharacteristic
102#106×C79—21no—
103#106×C79—21no—
104C4⋊C8×C79D411yescharacteristic
105C4⋊C8×C79D411yescharacteristic
106C4⋊C8×C79D411yescharacteristic
107#102×C79—21no—
108#102×C79—21no—
109#102×C79—21no—
110C4×C8×C79C2311yescharacteristic
111C4×C8×C79C2311yescharacteristic
112C4×C8×C79C2311yescharacteristic
113C2×Dic158—21no—
114C2×D316—21no—
115C2×D316—21no—
116C2×D316—41no—
117C2×D316—41no—
118C2×C4×D79—41no—
119C2×C4×D79—41no—
120Dic316—21no—
121Dic316—21no—
122#11929613—81no—
123#11929613—81no—
124D632—21no—
125D632—21no—
126D632—21no—
127D632—21no—
128D632—81no—
129C4⋊Dic79—21no—
130Dic79⋊C4—41no—
131(C2×D79)⋊C4—41no—
132(C2×D79)⋊C4—41no—
133(C2×D79)⋊C4—41no—
134(C2×D79)⋊C4—41no—
135(C2×D79)⋊C4—41no—
136(C2×D79)⋊C4—41no—
137C4×Dic79—41no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C79⋊C16—21no—
142C2×Q8×C79—41no—
143C2×Q8×C79D811yescharacteristic
144C2×D4×C79—21no—
145C22×C4×C79—41no—
146C22×C4×C79—41no—
147C22×C4×C79—41no—
148D8×C79—21no—
149D8×C79#4112yes—
150C2×C8×C79C2×D411yescharacteristic
151C2×C8×C79C2×D411yescharacteristic
152C2×C8×C79C2×D411yescharacteristic
153C2×C8×C79C2411yescharacteristic
154C4⋊C4×C79—21no—
155C4⋊C4×C79—21no—
156C4⋊C4×C79—21no—
157C4⋊C4×C79—41no—
158C4⋊C4×C79—41no—
159C4⋊C4×C79—41no—
160C4⋊C4×C79—41no—
161C4⋊C4×C79—41no—
162C4⋊C4×C79D811yescharacteristic
163C4⋊C4×C79D811yescharacteristic
164C4⋊C4×C79D811yescharacteristic
165C22⋊C4×C79—41no—
166C22⋊C4×C79—41no—
167C22⋊C4×C79—41no—
168C42×C79C2×D411yescharacteristic
169C42×C79C2×D411yescharacteristic
170C42×C79C2×D411yescharacteristic
171C23×C79—41no—
172C22×D79—41no—
173C22×D79—41no—
174Q8×C79—41no—
175Q8×C79—161no—
176D4×C79—22no—
177D4×C79—41no—
178C2×C4×C79—41no—
179C2×C4×C79—41no—
180C2×C4×C79—41no—
181C2×C4×C79—82no—
182C2×C4×C79—161no—
183C2×C4×C79—161no—
184C2×C4×C79C2×D811yescharacteristic
185C2×C4×C79C2×D811yescharacteristic
186C2×C4×C79C2×D811yescharacteristic
187C2×C4×C79C2×D811yescharacteristic
188C2×C4×C79C2×D811yescharacteristic
189C2×C4×C79C2×D811yescharacteristic
190C2×C4×C79C22×D411yescharacteristic
191C2×Dic79—41no—
192C2×Dic79—41no—
193D316—41no—
194D316—41no—
195D316—41no—
196D316—41no—
197D316—82no—
198D316—161no—
199C4×D79—161no—
200C4×D79—161no—
201Dic158—41no—
202Dic158—41no—
203C8×C79—21no—
204C8×C79—21no—
205C8×C79—21no—
206C8×C79C2×#4111yescharacteristic
207C8×C792−1+411yescharacteristic
208C22×C79—42no—
209C22×C79—81no—
210C22×C79—81no—
211C22×C79C22×D811yescharacteristic
212C2×D79—81no—
213C2×D79—81no—
214C2×D79—81no—
215C2×D79—81no—
216C2×D79—81no—
217C2×D79—81no—
218C4×C79—21no—
219C4×C79—21no—
220C4×C79—21no—
221C4×C79—21no—
222C4×C79—21no—
223C4×C79—21no—
224C4×C79—81no—
225C4×C79—81no—
226C4×C79—81no—
227C4×C79#54811yescharacteristic
228C4×C79#57711yescharacteristic
229Dic79—81no—
230Dic79—81no—
231#33937—791nomaximal, Sylow
232C2×C79—82no—
233C2×C79#335811yescharacteristic
234C2×C79#341411yescharacteristic
235C2×C79#341511yescharacteristic
236D79—161no—
237D79—161no—
238#3396—791no—
239#3380D7911yescharacteristic, complemented
240#2244—791no—
241#2243—791no—
242#2242—791no—
243#2229—791no—
244#2228—791no—
245#2224—791no—
246#2210—791no—
247#2210—791no—
248#2206—791no—
249#2183—791no—
250#2177—791no—
251#2175—791no—
252C4×D16—791no—
253C79#3393711yescharacteristic, complemented, Sylow
254#559—1581no—
255#554—21no—
256C2×SD32—791no—
257C2×D16—791no—
258C2×D16—791no—
259#500C2×D7911yescharacteristic
260#499C2×D7911yescharacteristic
261#499C2×D7911yescharacteristic
262#495—791no—
263#495—791no—
264#493—791no—
265#478—21no—
266#478—21no—
267#476—21no—
268#463—1581no—
269#463—1581no—
270#460—1581no—
271C8×Q8C2×D7911yescharacteristic
272C4×Q16—791no—
273C4×D8—791no—
274C4×D8—791no—
275C4×D8C2×D7911yescharacteristic
276C4×D8C2×D7911yescharacteristic
277C4×D8C2×D7911yescharacteristic
278#367—791no—
279#366—791no—
280#366—791no—
281C4⋊C16—791no—
282C4⋊C16—791no—
283C4⋊C16—791no—
284#358—791no—
285#358—791no—
286#358—791no—
287#357—791no—
288#357—791no—
289#357—791no—
290#357—791no—
291#357—791no—
292#357—791no—
293#357—791no—
294#346—791no—
295#346—791no—
296C4×C16—791no—
297C2×Q16—791no—
298C2×SD16—1581no—
299C2×SD16—1581no—
300C2×D8—791no—
301C2×D8—791no—
302C2×D8—1581no—
303C2×D8C22×D7911yescharacteristic
304C4⋊Q8—21no—
305#127—1581no—
306#125—21no—
307#125—21no—
308#124—3161no—
309#124—1581no—
310#124—1581no—
311#123—3161no—
312#123—3161no—
313C22⋊Q8—41no—
314C22⋊Q8—41no—
315C22⋊Q8—41no—
316C4×Q8—1581no—
317C4×Q8C22⋊D7911yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—3161no—
322C4×D4—1581no—
323C4×D4—1581no—
324C2×C4⋊C4—41no—
325SD32—1582no—
326D16—1582no—
327D16—1582no—
328C2×C16—791no—
329C2×C16—791no—
330C2×C16—791no—
331C2×C16—791no—
332#107—21no—
333#107—791no—
334#107C22×D7911yescharacteristic
335#107C22×D7911yescharacteristic
336#107C22×D7911yescharacteristic
337#106—21no—
338#106—21no—
339C4⋊C8C22⋊D7911yescharacteristic
340C4⋊C8C22⋊D7911yescharacteristic
341C4⋊C8C22⋊D7911yescharacteristic
342#103—1581no—
343#102—21no—
344#102—21no—
345#102—21no—
346#102—1581no—
347#102—1581no—
348C4×C8C22×D7911yescharacteristic
349C4×C8C22×D7911yescharacteristic
350C4×C8C22×D7911yescharacteristic
351C2×Q8—41no—
352C2×Q8—1581no—
353C2×Q8#1192960911yescharacteristic
354C2×D4—21no—
355C2×D4—3161no—
356C2×D4—3161no—
357C2×D4—1581no—
358C2×D4—1581no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—3161no—
363C22×C4—3161no—
364Q16—1581no—
365Q16—1581no—
366SD16—6321no—
367SD16—6321no—
368D8—21no—
369D8—6321no—
370D8—1581no—
371D8—1581no—
372D8—1581no—
373D8—1581no—
374D8#1192962512yes—
375C2×C8C2×C22⋊D7911yescharacteristic
376C2×C8C2×C22⋊D7911yescharacteristic
377C2×C8C2×C22⋊D7911yescharacteristic
378C2×C8C23×D7911yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—3161no—
388C4⋊C4—1581no—
389C4⋊C4#1192960911yescharacteristic
390C4⋊C4#1192960911yescharacteristic
391C4⋊C4#1192960911yescharacteristic
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—3161no—
396C22⋊C4—3161no—
397C22⋊C4—3161no—
398C22⋊C4—3161no—
399C22⋊C4—3161no—
400C22⋊C4—3161no—
401C42—3161no—
402C42C2×C22⋊D7911yescharacteristic
403C42C2×C22⋊D7911yescharacteristic
404C42C2×C22⋊D7911yescharacteristic
405C16—1581no—
406C16—1581no—
407C16—1581no—
408C16—1581no—
409C23—41no—
410C23—3161no—
411C23—3161no—
412Q8—41no—
413Q8—161no—
414Q8—3161no—
415Q8—3161no—
416D4—22no—
417D4—41no—
418D4—3161no—
419D4—3161no—
420D4—3161no—
421D4—3161no—
422D4—6322no—
423D4—12641no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—82no—
428C2×C4—161no—
429C2×C4—161no—
430C2×C4—3161no—
431C2×C4—3161no—
432C2×C4—12641no—
433C2×C4—12641no—
434C2×C4C2×#1192960911yescharacteristic
435C2×C4C2×#1192960911yescharacteristic
436C2×C4C2×#1192960911yescharacteristic
437C2×C4C2×#1192960911yescharacteristic
438C2×C4C2×#1192960911yescharacteristic
439C2×C4C2×#1192960911yescharacteristic
440C2×C4C22×C22⋊D7911yescharacteristic
441C8—21no—
442C8—21no—
443C8—21no—
444C8C2×#1192962511yescharacteristic
445C8#1457920911yescharacteristic
446C22—42no—
447C22—81no—
448C22—81no—
449C22—6321no—
450C22—6321no—
451C22—6321no—
452C22—6321no—
453C22—6321no—
454C22—6321no—
455C22C22×#1192960911yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—6321no—
466C4—6321no—
467C4#1909922411yescharacteristic
468C4#1909937111yescharacteristic
469C2—82no—
470C2—12641no—
471C2—12641no—
472C2#2377603511yescharacteristic
473C2#2377653411yescharacteristic
474C2#2377653911yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3380C2
socleC22×C79C22×D8
prime radical#23765435C2
hypercenter#3380D79
hypocenterC79#33937
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #23776413C2semidirect
2. #23765643C2semidirect
3. #23765639C2semidirect
4. #23765634C2semidirect
5. #23765561C2semidirect
6. #23765554C2semidirect
7. #23765534C2semidirect
8. #23765455C2semidirect
9. #23765455C2semidirect
11. #23765309C2semidirect
12. #23765274C2semidirect
13. #23765264C2semidirect
14. C4×#14579102C2semidirect
15. C79×#3380C2semidirect
239. #3380D79semidirect
253. C79#33937semidirect

Maximal subgroups

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

subgroupindex
#237764132
#237656432
#237656392
#237656342
#237655612
#237655542
#237655342
#237654552
#237654352
#237653092
#237652742
#237652642
C4×#145791022
C79×#33802
#3393779