Finite Group Databasebeta

← #34442911

Subgroups — #34442911

Order 20224.

546 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#23773912C211yesmaximal, characteristic, complemented
2#135×Dic79C211yesmaximal, characteristic
3#23765566C211yesmaximal, characteristic, complemented
4#23765562C211yesmaximal, characteristic, complemented
5#23765559C211yesmaximal, characteristic, complemented
6#23765555C211yesmaximal, characteristic, complemented
7#23765550C211yesmaximal, characteristic, complemented
8#23765545C211yesmaximal, characteristic, complemented
9#23765540C211yesmaximal, characteristic, complemented
10#23765535C211yesmaximal, characteristic, complemented
11#23765184C211yesmaximal, characteristic, complemented
12#23765182C211yesmaximal, characteristic, complemented
13C2×Dic79⋊C16C211yesmaximal, characteristic, complemented
14#23765107C211yesmaximal, characteristic
15C79×#2138C211yesmaximal, characteristic
16#19099116—21no—
17#41×Dic79—21no—
18C22×C79⋊C16C2211yescharacteristic
19#19098967C2211yescharacteristic
20#19098967C2211yescharacteristic
21#19098964C2211yescharacteristic
22#19098964C2211yescharacteristic
23#19098963—21no—
24#19098960—21no—
25Q16×Dic79C2211yescharacteristic
26Q16×Dic79C2211yescharacteristic
27SD16×Dic79—21no—
28SD16×Dic79—21no—
29D8×Dic79C2211yescharacteristic
30D8×Dic79C2211yescharacteristic
31#19098601C2211yescharacteristic
32#19098601C2211yescharacteristic
33#19098599—21no—
34#19098559—21no—
35C2×C8×Dic79C2211yescharacteristic
36#19098391C2211yescharacteristic
37#19098391C2211yescharacteristic
38#19098390C2211yescharacteristic
39#19098390C2211yescharacteristic
40#19098389C2211yescharacteristic
41#19098389C2211yescharacteristic
42#19098388C2211yescharacteristic
43#19098388C2211yescharacteristic
44Dic79⋊C16C2211yescharacteristic
45Dic79⋊C16C2211yescharacteristic
46Dic79⋊C16C2211yescharacteristic
47Dic79⋊C16C2211yescharacteristic
48#19098367C2211yescharacteristic
49#19098367C2211yescharacteristic
50#19098364C2211yescharacteristic
51#19098364C2211yescharacteristic
52C2×#135×C79C2211yescharacteristic
53C22×C16×C79C2211yescharacteristic
54#427×C79C2211yescharacteristic
55#358×C79C2211yescharacteristic
56#358×C79C2211yescharacteristic
57#357×C79C2211yescharacteristic
58#357×C79C2211yescharacteristic
59C22×C79⋊C8D411yescharacteristic
60#14579160—21no—
61#14579160—21no—
62#14579152—21no—
63#14579152—21no—
64#14579138—41no—
65#14579138—41no—
66Q8×Dic79—21no—
67Q8×Dic79—21no—
68C4×Dic158—41no—
69C4×Dic158—41no—
70D4×Dic79—21no—
71D4×Dic79—21no—
72D4×Dic79—41no—
73D4×Dic79—41no—
74#14579120—21no—
75#14579120—21no—
76#14579120—41no—
77C2×C4×Dic79—41no—
78C2×C4×Dic79D411yescharacteristic
79C2×C79⋊C16—22no—
80C2×C79⋊C16C2311yescharacteristic
81C2×C79⋊C16C2311yescharacteristic
82#14579095C2311yescharacteristic
83#14579095C2311yescharacteristic
84#14579095C2311yescharacteristic
85#14579095C2311yescharacteristic
86#14579094—21no—
87#14579094—21no—
88#14579093—21no—
89#14579093—21no—
90#14579092—21no—
91#14579092—21no—
92#14579085—21no—
93#14579085—21no—
94#14579082—21no—
95#14579082—21no—
96C8×Dic79D411yescharacteristic
97C8×Dic79D411yescharacteristic
98C8×Dic79C2311yescharacteristic
99C8×Dic79C2311yescharacteristic
100C2×#41×C79—21no—
101#135×C79—22no—
102#135×C79Q814yescomplemented
103C2×Q16×C79C2311yescharacteristic
104C2×SD16×C79—21no—
105C2×D8×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107#117×C79—21no—
108C2×C16×C79—22no—
109C2×C16×C79C2311yescharacteristic
110C2×C16×C79C2311yescharacteristic
111#107×C79C2311yescharacteristic
112#107×C79C2311yescharacteristic
113#106×C79—21no—
114C2×Dic158—41no—
115C2×Dic158—41no—
116C22×Dic79—41no—
117C22×Dic79—41no—
118C22×Dic79D811yescharacteristic
119C22×Dic79D811yescharacteristic
120C2×C79⋊C8—22no—
121C2×C79⋊C8C2×D411yescharacteristic
122C2×C79⋊C8C2×D411yescharacteristic
123C4⋊Dic79—21no—
124C4⋊Dic79—21no—
125C4⋊Dic79—21no—
126C4⋊Dic79—21no—
127C4⋊Dic79—41no—
128C4⋊Dic79—41no—
129C4⋊Dic79—41no—
130C4⋊Dic79—41no—
131Dic79⋊C4—41no—
132Dic79⋊C4—41no—
133Dic79⋊C4—41no—
134Dic79⋊C4—41no—
135C22⋊Dic79—41no—
136C22⋊Dic79—41no—
137C22⋊Dic79—41no—
138C22⋊Dic79—41no—
139C4×Dic79—41no—
140C4×Dic79—41no—
141C4×Dic79—41no—
142C4×Dic79—41no—
143C4×Dic79D811yescharacteristic
144C4×Dic79D811yescharacteristic
145C4×Dic79C2×D411yescharacteristic
146C4×Dic79C2×D411yescharacteristic
147C79⋊C16—21no—
148C79⋊C16—21no—
149#41×C79—24no—
150#41×C79—42no—
151C2×Q8×C79—21no—
152C2×D4×C79—21no—
153C2×D4×C79—41no—
154C22×C4×C79—41no—
155C22×C4×C79C2×D411yescharacteristic
156Q16×C79—21no—
157Q16×C79C2×Q812yes—
158SD16×C79—24no—
159D8×C79—21no—
160D8×C79C2×Q812yes—
161C2×C8×C79C2×D411yescharacteristic
162C2×C8×C79C2×Q812yes—
163C2×C8×C79#4112yes—
164C2×C8×C79C2411yescharacteristic
165C4⋊C4×C79—21no—
166C4⋊C4×C79—21no—
167C22⋊C4×C79—41no—
168C42×C79—41no—
169C16×C79—21no—
170C16×C79—21no—
171C23×C79—41no—
172C23×C79C2×D811yescharacteristic
173Q8×C79—22no—
174Q8×C79—41no—
175D4×C79—22no—
176D4×C79—44no—
177D4×C79—41no—
178C2×C4×C79—41no—
179C2×C4×C79—44no—
180C2×C4×C79—41no—
181C2×C4×C79—41no—
182C2×C4×C79C22⋊Q812yes—
183C2×C4×C79C2×D811yescharacteristic
184C2×C4×C79#13712yes—
185C2×C4×C79C22×D411yescharacteristic
186C2×Dic79—22no—
187C2×Dic79—41no—
188C2×Dic79—41no—
189C2×Dic79—41no—
190C2×Dic79—41no—
191C2×Dic79—41no—
192C2×Dic79—82no—
193C2×Dic79—82no—
194C2×Dic79C2×D811yescharacteristic
195C2×Dic79C2×D811yescharacteristic
196C2×Dic79C2×D811yescharacteristic
197C2×Dic79C2×D811yescharacteristic
198Dic158—82no—
199Dic158—82no—
200C8×C79C22⋊Q812yes—
201C8×C79C22×Q811yescharacteristic
202C8×C79C2×#4111yescharacteristic
203C79⋊C8—21no—
204C79⋊C8—21no—
205C22×C79—21no—
206C22×C79—21no—
207C22×C79—42no—
208C22×C79—81no—
209C22×C79—81no—
210C22×C79#47412yes—
211C22×C79C22×D811yescharacteristic
212C4×C79—42no—
213C4×C79—81no—
214C4×C79—81no—
215C4×C79#47412yes—
216C4×C79C2×C22⋊Q811yescharacteristic
217C4×C79C2×#13711yescharacteristic
218Dic79—21no—
219Dic79—21no—
220Dic79—21no—
221Dic79—81no—
222Dic79—81no—
223Dic79—81no—
224Dic79—81no—
225Dic79#50812yes—
226#33626—791nomaximal, Sylow
227C2×C79—22no—
228C2×C79—82no—
229C2×C79C2×#47411yescharacteristic
230C2×C79C2×#50811yescharacteristic
231C2×C79#341411yescharacteristic
232#3160—791no—
233C4×#135—791no—
234#2229—791no—
235#2229—791no—
236#2228—791no—
237#2228—791no—
238#2227—791no—
239#2226—791no—
240#2225—791no—
241#2224—791no—
242#2158—791no—
243#2158—791no—
244C2×C4⋊C16—791no—
245#2138—791no—
246#2138D7911yescharacteristic
247C79#3362611yescharacteristic, complemented, Sylow
248C2×#135C2×D7911yescharacteristic
249#533—1581no—
250C4×#41—1581no—
251C22×C16—791no—
252C22×C16C2×D7911yescharacteristic
253#500—791no—
254#500—791no—
255#499—791no—
256#499—791no—
257#499—1581no—
258#498—1581no—
259C4×Q16—791no—
260C4×Q16—791no—
261C4×SD16—1581no—
262C4×SD16—1581no—
263C4×D8—791no—
264C4×D8—791no—
265#427—791no—
266#427—791no—
267#427—1581no—
268#427C2×D7911yescharacteristic
269#416—1581no—
270C2×C4×C8—791no—
271#367—791no—
272#367—791no—
273#367—791no—
274#367—791no—
275#366—791no—
276#366—791no—
277#366—791no—
278#366—791no—
279C4⋊C16—791no—
280C4⋊C16—791no—
281C4⋊C16—791no—
282C4⋊C16—791no—
283#358—791no—
284#358—791no—
285#358C2×D7911yescharacteristic
286#358C2×D7911yescharacteristic
287#357—791no—
288#357—791no—
289#357C2×D7911yescharacteristic
290#357C2×D7911yescharacteristic
291C2×#41—21no—
292#135—22no—
293#135Dic15814yescomplemented
294C2×Q16C22×D7911yescharacteristic
295C2×SD16—21no—
296C2×D8C22×D7911yescharacteristic
297C22×C8—791no—
298C22×C8C22×D7911yescharacteristic
299C4⋊Q8—1581no—
300C4⋊Q8—1581no—
301#125—1581no—
302#125—1581no—
303C22⋊Q8—3161no—
304C22⋊Q8—3161no—
305C4×Q8—3161no—
306C4×Q8—3161no—
307C4×Q8—1581no—
308C4×Q8—1581no—
309C4×D4—3161no—
310C4×D4—3161no—
311C4×D4—1581no—
312C4×D4—1581no—
313#117—21no—
314#117—3161no—
315#117—1581no—
316#117—1581no—
317C2×C42—3161no—
318C2×C42—791no—
319C2×C16—22no—
320C2×C16—791no—
321C2×C16—791no—
322C2×C16—1582no—
323C2×C16C22×D7911yescharacteristic
324C2×C16C22×D7911yescharacteristic
325#107—791no—
326#107—791no—
327#107—791no—
328#107—791no—
329#107—1581no—
330#107—1581no—
331#107C22×D7911yescharacteristic
332#107C22×D7911yescharacteristic
333#106—21no—
334#106—1581no—
335#106—1581no—
336#106—1581no—
337#106—1581no—
338#103—1581no—
339#103—1581no—
340#102—1581no—
341#102—1581no—
342C4×C8—791no—
343C4×C8—791no—
344C4×C8—791no—
345C4×C8—791no—
346#41—24no—
347#41—42no—
348C2×Q8—21no—
349C2×Q8—3161no—
350C2×Q8—3161no—
351C2×D4—21no—
352C2×D4—41no—
353C22×C4—41no—
354C22×C4—3161no—
355C22×C4—3161no—
356C22×C4—791no—
357C22×C4—791no—
358C22×C4D4×D7911yescharacteristic
359Q16—21no—
360Q16C2×Dic15812yes—
361SD16—24no—
362D8—21no—
363D8C2×Dic15812yes—
364C2×C8—791no—
365C2×C8—791no—
366C2×C8—1582no—
367C2×C8D4×D7911yescharacteristic
368C2×C8C2×Dic15812yes—
369C2×C8#1192962312yes—
370C2×C8C23×D7911yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—3161no—
374C4⋊C4—3161no—
375C4⋊C4—3161no—
376C4⋊C4—3161no—
377C4⋊C4—3161no—
378C4⋊C4—3161no—
379C4⋊C4—3161no—
380C4⋊C4—3161no—
381C4⋊C4—1581no—
382C4⋊C4—1581no—
383C4⋊C4—1581no—
384C4⋊C4—1581no—
385C22⋊C4—41no—
386C22⋊C4—3161no—
387C22⋊C4—3161no—
388C22⋊C4—3161no—
389C22⋊C4—3161no—
390C42—41no—
391C42—3161no—
392C42—3161no—
393C42—3161no—
394C42—3161no—
395C42—791no—
396C42—791no—
397C42—791no—
398C42—791no—
399C16—21no—
400C16—21no—
401C16—1581no—
402C16—1581no—
403C23—41no—
404C23D8×D7911yescharacteristic
405Q8—22no—
406Q8—41no—
407Q8—6322no—
408Q8—6322no—
409D4—22no—
410D4—44no—
411D4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—3161no—
417C2×C4—3161no—
418C2×C4—3161no—
419C2×C4—3161no—
420C2×C4—3161no—
421C2×C4—791no—
422C2×C4—791no—
423C2×C4—791no—
424C2×C4—791no—
425C2×C4—6322no—
426C2×C4—6322no—
427C2×C4—1582no—
428C2×C4#1457913812yes—
429C2×C4D8×D7911yescharacteristic
430C2×C4#1457919012yes—
431C2×C4C2×D4×D7911yescharacteristic
432C8—1581no—
433C8—1581no—
434C8#1457913812yes—
435C8C22×Dic15811yescharacteristic
436C8C2×#1192962311yescharacteristic
437C22—21no—
438C22—21no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22#1909884112yes—
443C22C2×D8×D7911yescharacteristic
444C4—42no—
445C4—81no—
446C4—81no—
447C4—792no—
448C4—6321no—
449C4—6321no—
450C4—6321no—
451C4—6321no—
452C4—1581no—
453C4—1581no—
454C4—1581no—
455C4#1909884112yes—
456C4C2×#1457913811yescharacteristic
457C4C2×#1457919011yescharacteristic
458C2—22no—
459C2—82no—
460C2C2×#1909884111yescharacteristic
461C2#508×D7911yescharacteristic
462C2#2377653311yescharacteristic
subgroupG/H
centerC2×C4D8×D79
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#2138C2
socleC22×C79C22×D8
prime radicalC2×#135×C79C22
hypercenter#2138D79
hypocenterC79#33626
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #23773912C2semidirect
3. #23765566C2semidirect
4. #23765562C2semidirect
5. #23765559C2semidirect
6. #23765555C2semidirect
7. #23765550C2semidirect
8. #23765545C2semidirect
9. #23765540C2semidirect
10. #23765535C2semidirect
11. #23765184C2semidirect
12. #23765182C2semidirect
13. C2×Dic79⋊C16C2semidirect
102. #135×C79Q8semidirect
247. C79#33626semidirect
293. #135Dic158semidirect

Maximal subgroups

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

subgroupindex
#237739122
#135×Dic792
#237655662
#237655622
#237655592
#237655552
#237655502
#237655452
#237655402
#237655352
#237651842
#237651822
C2×Dic79⋊C162
#237651072
C79×#21382
#3362679