Finite Group Databasebeta

← #34272104

Subgroups — #34272104

Order 20224.

656 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773960C211yesmaximal, characteristic
2#23772872C211yesmaximal, characteristic
3C2×Q8×C79⋊C8C211yesmaximal, characteristic, complemented
4#23762828C212yesmaximal, complemented
5#23762818C212yesmaximal, complemented
6#23762784C212yesmaximal, complemented
7#23762750C212yesmaximal, complemented
8#23762155C211yesmaximal, characteristic, complemented
9#23762090C211yesmaximal, characteristic, complemented
10#23762078C211yesmaximal, characteristic
11C79×#3078C211yesmaximal, characteristic, complemented
12C2×#14579192—21no—
13#19099113C2211yescharacteristic
14#19098959C2211yescharacteristic
15#19098937C2211yescharacteristic
16#19098913C2211yescharacteristic
17#19098911C2211yescharacteristic
18#19098794—21no—
19#19098794—21no—
20#19098785—22no—
21#19098773C2212yes—
22#19098763C2212yes—
23Q8×C79⋊C8—21no—
24Q8×C79⋊C8—21no—
25Q8×C79⋊C8C2212yes—
26Q8×C79⋊C8C2212yes—
27C4×#11929611—21no—
28C4×#11929611—21no—
29#19098587C2211yescharacteristic
30C2×C4⋊(C79⋊C8)—21no—
31C2×C4⋊(C79⋊C8)C2211yescharacteristic
32#19098562—21no—
33C2×C4×C79⋊C8—21no—
34C2×C4×C79⋊C8C2211yescharacteristic
35#19098313C2212yes—
36#19098313C2212yes—
37#19098312C2212yes—
38#19098310C2212yes—
39#19098309C2212yes—
40#19098306C2212yes—
41#533×C79C2211yescharacteristic
42C2×C4×Q8×C79C2211yescharacteristic
43#479×C79—22no—
44#477×C79C2212yes—
45#475×C79C2212yes—
46#428×C79—21no—
47#423×C79C2211yescharacteristic
48#419×C79—21no—
49C2×#11929624—21no—
50#14579192—44no—
51C2×#11929614—22no—
52C2×#11929611—22no—
53C2×#11929611—21no—
54C2×#11929611—21no—
55C2×#11929608—21no—
56C22×C79⋊C8—21no—
57C22×C79⋊C8D411yescharacteristic
58C22×C79⋊C8D411yescharacteristic
59#14579161C2311yescharacteristic
60#14579158C2311yescharacteristic
61#14579149—21no—
62#14579135—21no—
63#14579135—21no—
64C4×D316—21no—
65C4×D316—21no—
66#14579092—21no—
67#14579092—21no—
68C4⋊(C79⋊C8)—21no—
69C4⋊(C79⋊C8)—21no—
70C4⋊(C79⋊C8)—22no—
71C4⋊(C79⋊C8)—22no—
72C4⋊(C79⋊C8)C2312yes—
73C4⋊(C79⋊C8)C2312yes—
74#14579085—21no—
75#14579085—21no—
76#14579084—22no—
77#14579081—22no—
78#14579081—21no—
79#14579081—21no—
80C4×C79⋊C8—22no—
81C4×C79⋊C8—22no—
82C4×C79⋊C8D411yescharacteristic
83C4×C79⋊C8D411yescharacteristic
84C4×C79⋊C8C2311yescharacteristic
85C4×C79⋊C8C2311yescharacteristic
86C22×Q8×C79D411yescharacteristic
87C2×M16×C79—21no—
88C4⋊Q8×C79C2311yescharacteristic
89C4⋊Q8×C79C2311yescharacteristic
90#125×C79—21no—
91C22⋊Q8×C79—21no—
92C22⋊Q8×C79—21no—
93C4×Q8×C79—21no—
94C4×Q8×C79—21no—
95C4×Q8×C79—21no—
96C4×Q8×C79—21no—
97C4×Q8×C79D412yes—
98C4×Q8×C79C2312yes—
99#117×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79D411yescharacteristic
102C2×C42×C79—21no—
103C2×C42×C79C2311yescharacteristic
104#107×C79—22no—
105#106×C79—22no—
106C4⋊C8×C79—21no—
107C4⋊C8×C79C2312yes—
108#103×C79—22no—
109#103×C79—22no—
110#11929624—42no—
111#11929624—42no—
112C2×Dic158—21no—
113C2×D316—21no—
114C2×D316—21no—
115C2×D316—21no—
116C2×C22⋊D79—41no—
117C2×C4×D79—41no—
118#11929614—44no—
119#11929611—42no—
120#11929611—42no—
121#11929611—44no—
122#11929608—42no—
123C2×C79⋊C8—22no—
124C2×C79⋊C8—22no—
125C2×C79⋊C8—22no—
126C2×C79⋊C8—41no—
127C2×C79⋊C8—42no—
128C2×C79⋊C8C2×D411yescharacteristic
129C2×C79⋊C8C2×D411yescharacteristic
130C2×C79⋊C8C2×D411yescharacteristic
131C2×C79⋊C8C2×D411yescharacteristic
132C4⋊Dic79—21no—
133C4⋊Dic79—21no—
134(C2×D79)⋊C4—41no—
135(C2×D79)⋊C4—41no—
136C2×Q8×C79—21no—
137C2×Q8×C79—22no—
138C2×Q8×C79—21no—
139C2×Q8×C79—21no—
140C2×Q8×C79SD1614yescomplemented
141C2×Q8×C79C2×D412yes—
142C22×C4×C79—21no—
143C22×C4×C79—21no—
144C22×C4×C79C2×D411yescharacteristic
145C22×C4×C79C2×D411yescharacteristic
146C22×C4×C79#4111yescharacteristic
147M16×C79—42no—
148C2×C8×C79—22no—
149C4⋊C4×C79—22no—
150C4⋊C4×C79—21no—
151C4⋊C4×C79—22no—
152C4⋊C4×C79—21no—
153C4⋊C4×C79—21no—
154C4⋊C4×C79—21no—
155C4⋊C4×C79—21no—
156C4⋊C4×C79—41no—
157C4⋊C4×C79—41no—
158C4⋊C4×C79C2×D412yes—
159C22⋊C4×C79—41no—
160C42×C79—22no—
161C42×C79—22no—
162C42×C79—41no—
163C42×C79C2×D411yescharacteristic
164C42×C79C2×D411yescharacteristic
165C42×C79#4111yescharacteristic
166C42×C79C2411yescharacteristic
167C23×C79#12111yescharacteristic
168C22×D79—41no—
169Q8×C79—22no—
170Q8×C79—24no—
171Q8×C79—41no—
172Q8×C79—41no—
173Q8×C79—81no—
174Q8×C79C2×SD1614yes—
175C2×C4×C79—22no—
176C2×C4×C79—24no—
177C2×C4×C79—21no—
178C2×C4×C79—21no—
179C2×C4×C79—22no—
180C2×C4×C79—41no—
181C2×C4×C79—42no—
182C2×C4×C79—41no—
183C2×C4×C79—41no—
184C2×C4×C79#12111yescharacteristic
185C2×C4×C79#12111yescharacteristic
186C2×C4×C79#12111yescharacteristic
187C2×C4×C79C2×SD1612yes—
188C2×C4×C79#13512yes—
189C2×C4×C79C22×D411yescharacteristic
190C2×C4×C79C22×D411yescharacteristic
191C2×C4×C79C2×#4111yescharacteristic
192C22⋊D79—82no—
193C2×Dic79—41no—
194D316—41no—
195D316—41no—
196D316—41no—
197D316—41no—
198D316—81no—
199C4×D79—82no—
200Dic158—41no—
201Dic158—41no—
202C8×C79—42no—
203C79⋊C8—21no—
204C79⋊C8—21no—
205C79⋊C8—21no—
206C79⋊C8—42no—
207C79⋊C8—42no—
208C79⋊C8#13712yescomplemented
209C22×C79—21no—
210C22×C79—21no—
211C22×C79#46512yes—
212C22×C79C2×#12111yescharacteristic
213C2×D79—81no—
214C2×D79—81no—
215C2×D79—81no—
216C4×C79—21no—
217C4×C79—24no—
218C4×C79—21no—
219C4×C79—21no—
220C4×C79—42no—
221C4×C79—81no—
222C4×C79—81no—
223C4×C79#46512yes—
224C4×C79C22×SD1611yescharacteristic
225C4×C79C2×#13511yescharacteristic
226C4×C79C2×#13711yescharacteristic
227C4×C79#57511yescharacteristic
228Dic79—81no—
229#19260—791nomaximal, Sylow
230C2×C79—22no—
231C2×C79#303711yescharacteristic
232C2×C79C2×#46511yescharacteristic
233C2×C79#306711yescharacteristic
234D79—161no—
235#3167—791no—
236#3078D7911yescharacteristic, complemented
237#3037—791no—
238C2×C8×Q8—791no—
239#1680—792no—
240#1678—792no—
241#1672—792no—
242#1666—792no—
243#1566—791no—
244#1550—791no—
245#1547—791no—
246C79#1926011yescharacteristic, complemented, Sylow
247C2×#137—1581no—
248#533C2×D7911yescharacteristic
249#532—791no—
250C2×C4×Q8C2×D7911yescharacteristic
251#498—791no—
252#492—791no—
253#487—791no—
254#486—791no—
255#479—22no—
256#477C2×D7912yes—
257#475C2×D7912yes—
258#465—1581no—
259#465—1581no—
260#464—1582no—
261#462—792no—
262#460—792no—
263C8×Q8—792no—
264C8×Q8—792no—
265C8×Q8—1581no—
266C8×Q8—1581no—
267C4×SD16—1581no—
268C4×SD16—1581no—
269#428—21no—
270#423—791no—
271#423C2×D7911yescharacteristic
272C2×C4⋊C8—791no—
273C2×C4⋊C8—1581no—
274#419—21no—
275#417—1581no—
276C2×C4×C8—791no—
277C2×C4×C8—1581no—
278#334—792no—
279#334—792no—
280#333—792no—
281#332—792no—
282#332—792no—
283#331—792no—
284C2×#41—1581no—
285C22×Q8C22⋊D7911yescharacteristic
286#137—3164no—
287C2×Q16—1582no—
288C2×SD16—1581no—
289C2×SD16—1581no—
290C2×SD16—1582no—
291C2×M16—21no—
292C2×M16—1581no—
293C22×C8—791no—
294C22×C8—791no—
295C22×C8—1581no—
296C4⋊Q8—791no—
297C4⋊Q8C22×D7911yescharacteristic
298C4⋊Q8C22×D7911yescharacteristic
299#127—791no—
300#125—21no—
301#124—1581no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—1581no—
305#121—1581no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8C22⋊D7912yes—
311C4×Q8C22×D7912yes—
312C4×D4—1581no—
313C4×D4—1581no—
314#117—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4C22⋊D7911yescharacteristic
317C2×C42—21no—
318C2×C42C22×D7911yescharacteristic
319#107—22no—
320#106—22no—
321#106—1581no—
322#106—1581no—
323C4⋊C8—21no—
324C4⋊C8—792no—
325C4⋊C8—792no—
326C4⋊C8—1582no—
327C4⋊C8—1582no—
328C4⋊C8—1581no—
329C4⋊C8—1581no—
330C4⋊C8C22×D7912yes—
331#103—22no—
332#103—22no—
333#103—1581no—
334#103—1581no—
335#103—1582no—
336#102—1581no—
337#102—1581no—
338#102—1582no—
339C4×C8—791no—
340C4×C8—791no—
341C4×C8—791no—
342C4×C8—791no—
343C4×C8—1582no—
344C4×C8—1582no—
345#41—3162no—
346#41—3162no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—1581no—
352C2×Q8#1192961114yescomplemented
353C2×Q8C2×C22⋊D7912yes—
354C2×D4—3161no—
355C2×D4—1581no—
356C2×D4—1581no—
357C2×D4—1581no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—3161no—
361C22×C4D4×D7911yescharacteristic
362C22×C4C2×C22⋊D7911yescharacteristic
363C22×C4#1192962311yescharacteristic
364Q16—3164no—
365SD16—3164no—
366SD16—3162no—
367SD16—3162no—
368M16—42no—
369M16—3162no—
370C2×C8—22no—
371C2×C8—3162no—
372C2×C8—3161no—
373C2×C8—791no—
374C2×C8—791no—
375C2×C8—791no—
376C2×C8—791no—
377C2×C8—1582no—
378C2×C8—1582no—
379C2×C8—1582no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—1581no—
390C4⋊C4—1581no—
391C4⋊C4C2×C22⋊D7912yes—
392C22⋊C4—41no—
393C22⋊C4—3161no—
394C22⋊C4—3161no—
395C42—22no—
396C42—22no—
397C42—41no—
398C42D4×D7911yescharacteristic
399C42C2×C22⋊D7911yescharacteristic
400C42#1192962311yescharacteristic
401C42C23×D7911yescharacteristic
402C23—3161no—
403C23#1457913311yescharacteristic
404Q8—24no—
405Q8—22no—
406Q8—41no—
407Q8—41no—
408Q8—81no—
409Q8—3161no—
410Q8—3161no—
411Q8C2×#1192961114yes—
412D4—3161no—
413D4—3161no—
414D4—3161no—
415D4—3161no—
416D4—6321no—
417D4—6322no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—3161no—
428C2×C4—6322no—
429C2×C4#1457913311yescharacteristic
430C2×C4#1457913311yescharacteristic
431C2×C4#1457913311yescharacteristic
432C2×C4C2×#1192961112yes—
433C2×C4#1457918112yes—
434C2×C4C2×D4×D7911yescharacteristic
435C2×C4C22×C22⋊D7911yescharacteristic
436C2×C4C2×#1192962311yescharacteristic
437C8—42no—
438C8—3162no—
439C8—3162no—
440C8—792no—
441C8—1581no—
442C8—1581no—
443C8—1581no—
444C22—21no—
445C22—21no—
446C22—6321no—
447C22—6321no—
448C22—6321no—
449C22#1909879412yes—
450C22C2×#1457913311yescharacteristic
451C4—24no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—81no—
458C4—6321no—
459C4#1909879412yes—
460C4C22×#1192961111yescharacteristic
461C4C2×#1457918111yescharacteristic
462C4#137×D7911yescharacteristic
463C4#1909936211yescharacteristic
464C2—22no—
465C2—12641no—
466C2#2377286911yescharacteristic
467C2C2×#1909879411yescharacteristic
468C2#2377311711yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#3078C2
socleC22×C79C2×#121
prime radical#19099113C22
hypercenter#3078D79
hypocenterC79#19260
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
3. C2×Q8×C79⋊C8C2semidirect
4. #23762828C2semidirect
5. #23762818C2semidirect
6. #23762784C2semidirect
7. #23762750C2semidirect
8. #23762155C2semidirect
9. #23762090C2semidirect
11. C79×#3078C2semidirect
140. C2×Q8×C79SD16semidirect
208. C79⋊C8#137semidirect
236. #3078D79semidirect
246. C79#19260semidirect
352. C2×Q8#11929611semidirect

Maximal subgroups

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

subgroupindex
#237739602
#237728722
C2×Q8×C79⋊C82
#237628282
#237628182
#237627842
#237627502
#237621552
#237620902
#237620782
C79×#30782
#1926079