Finite Group Databasebeta

← #34187413

Subgroups — #34187413

Order 20224.

650 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771701C211yesmaximal, characteristic
2#23763380C211yesmaximal, characteristic, complemented
3C2×#19098299C212yesmaximal, complemented
4C2×#19098296C212yesmaximal
5C79×#1434C211yesmaximal, characteristic, complemented
6C2×#14579149C411yescharacteristic, complemented
7C2×#14579140C2211yescharacteristic
8C2×#14579135C2211yescharacteristic
9C2×C4⋊(C79⋊C8)—21no—
10C2×C22⋊(C79⋊C8)C2212yes—
11#19098469C411yescharacteristic, complemented
12#19098465C411yescharacteristic, complemented
13#19098454—21no—
14#19098454—21no—
15#19098442—21no—
16#19098442—21no—
17#19098429C411yescharacteristic, complemented
18#19098318—21no—
19#19098299—24no—
20#19098296—24no—
21C22×C42×C79C2211yescharacteristic
22C2×#95×C79—21no—
23C2×#95×C79—21no—
24C2×#95×C79C2212yes—
25C22×Dic158C2×C411yescharacteristic
26C22×D316C2×C411yescharacteristic
27C22×C22⋊D79—21no—
28C22×C79⋊C8—22no—
29#14579149—42no—
30#14579140—22no—
31#14579140—22no—
32#14579135—22no—
33#14579135—22no—
34C2×C4⋊Dic79C2×C411yescharacteristic
35C2×C4⋊Dic79C2×C411yescharacteristic
36C2×Dic79⋊C4—21no—
37C2×C22⋊Dic79—21no—
38C2×(C2×D79)⋊C4—21no—
39C2×(C2×D79)⋊C4—21no—
40C2×(C2×D79)⋊C4—21no—
41C2×(C2×D79)⋊C4—21no—
42C2×(C2×D79)⋊C4—21no—
43C4⋊(C79⋊C8)—42no—
44C22⋊(C79⋊C8)—22no—
45C22⋊(C79⋊C8)—22no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C79⋊#95—21no—
50C23×C4×C79—22no—
51C23×C4×C79D411yescharacteristic
52C23×C4×C79D411yescharacteristic
53C23×C4×C79C2311yescharacteristic
54C2×C42×C79—42no—
55C2×C42×C79C2×C411yescharacteristic
56C2×C42×C79C2×C411yescharacteristic
57C2×C42×C79D411yescharacteristic
58C2×C42×C79D411yescharacteristic
59#95×C79—22no—
60#95×C79—42no—
61#95×C79—42no—
62#95×C79D414yes—
63C23×D79—21no—
64C2×Dic158—22no—
65C2×Dic158—41no—
66C2×Dic158—41no—
67C2×D316—22no—
68C2×D316—41no—
69C2×D316—41no—
70C2×C22⋊D79—42no—
71C2×C22⋊D79—42no—
72C2×C22⋊D79—41no—
73C2×C22⋊D79—41no—
74C22×Dic79—21no—
75C22×Dic79—21no—
76C22×Dic79—21no—
77C2×C79⋊C8—42no—
78C2×C79⋊C8—42no—
79C2×C79⋊C8—42no—
80C4⋊Dic79—22no—
81C4⋊Dic79—22no—
82Dic79⋊C4—41no—
83Dic79⋊C4—41no—
84C22⋊Dic79—42no—
85(C2×D79)⋊C4—42no—
86(C2×D79)⋊C4—42no—
87(C2×D79)⋊C4—42no—
88(C2×D79)⋊C4—42no—
89(C2×D79)⋊C4—41no—
90(C2×D79)⋊C4—41no—
91C24×C79C2×D411yescharacteristic
92C22×C4×C79—22no—
93C22×C4×C79—21no—
94C22×C4×C79—21no—
95C22×C4×C79—22no—
96C22×C4×C79—24no—
97C22×C4×C79—41no—
98C22×C4×C79—41no—
99C22×C4×C79—41no—
100C22×C4×C79—41no—
101C22×C4×C79—41no—
102C22×C4×C79—42no—
103C22×C4×C79—42no—
104C22×C4×C79—42no—
105C22×C4×C79—42no—
106C22×C4×C79—42no—
107C22×C4×C79—42no—
108C22×C4×C79C22⋊C411yescharacteristic
109C22×C4×C79C22⋊C411yescharacteristic
110C22×C4×C79C22⋊C411yescharacteristic
111C22×C4×C79C22⋊C411yescharacteristic
112C22×C4×C79C22×C411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D412yes—
115C22×C4×C79#4112yes—
116C42×C79—22no—
117C42×C79—22no—
118C42×C79—41no—
119C42×C79—41no—
120C23×C79—21no—
121C23×C79—21no—
122C23×C79—21no—
123C23×C79—21no—
124C23×C79C2×C22⋊C411yescharacteristic
125C23×C79#12112yes—
126C23×C79#13512yes—
127C23×C79AGL(1,ℤ/8)11yescharacteristic
128C23×C79#13711yescharacteristic
129C22×D79—41no—
130C22×D79—41no—
131C22×D79—41no—
132C22×D79—41no—
133C22×D79—41no—
134C22×D79—41no—
135C22×D79—41no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—42no—
152C2×C4×C79—42no—
153C2×C4×C79—44no—
154C2×C4×C79—42no—
155C2×C4×C79—42no—
156C2×C4×C79—44no—
157C2×C4×C79—44no—
158C2×C4×C79—44no—
159C2×C4×C79—82no—
160C2×C4×C79—82no—
161C2×C4×C79—82no—
162C2×C4×C79—82no—
163C2×C4×C79#9912yes—
164C2×C4×C79C4×D412yes—
165C2×C4×C79#12411yescharacteristic
166C2×C4×C79#12711yescharacteristic
167C22⋊D79—82no—
168C22⋊D79—82no—
169C2×Dic79—41no—
170C2×Dic79—41no—
171C2×Dic79—41no—
172C2×Dic79—41no—
173C2×Dic79—41no—
174C2×Dic79—41no—
175C2×Dic79—41no—
176C2×Dic79—41no—
177C2×Dic79—41no—
178D316—82no—
179Dic158—82no—
180C79⋊C8—82no—
181C22×C79—22no—
182C22×C79—22no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—21no—
187C22×C79—22no—
188C22×C79—22no—
189C22×C79#39011yescharacteristic
190C22×C79C2×#9911yescharacteristic
191C22×C79#41011yescharacteristic
192C22×C79#41611yescharacteristic
193C22×C79#41711yescharacteristic
194C22×C79#42111yescharacteristic
195C22×C79#42111yescharacteristic
196C22×C79#46312yes—
197C22×C79#46412yes—
198C2×D79—81no—
199C2×D79—81no—
200C2×D79—81no—
201C2×D79—81no—
202C2×D79—81no—
203C2×D79—81no—
204C2×D79—81no—
205C4×C79—22no—
206C4×C79—21no—
207C4×C79—21no—
208C4×C79—22no—
209C4×C79—41no—
210C4×C79—41no—
211C4×C79—41no—
212C4×C79—84no—
213Dic79—81no—
214Dic79—81no—
215Dic79—81no—
216#9716—791nomaximal, Sylow
217C2×C79—22no—
218C2×C79—22no—
219C2×C79#151211yescharacteristic
220C2×C79#151311yescharacteristic
221C2×C79#151611yescharacteristic
222C2×C79#152111yescharacteristic
223C2×C79#196311yescharacteristic
224C2×C79#196411yescharacteristic
225C2×C79#198311yescharacteristic
226D79—161no—
227#2850—791no—
228#1797—791no—
229C2×#328—792no—
230C2×#327—792no—
231#1434D7911yescharacteristic, complemented
232C79#971611yescharacteristic, complemented, Sylow
233C2×#124—791no—
234C2×C22⋊Q8—791no—
235C2×#121—791no—
236C22×C42C2×D7911yescharacteristic
237C2×C4⋊C8—1581no—
238C2×C22⋊C8—792no—
239#391—791no—
240#390—791no—
241#388—1581no—
242#388—1581no—
243#386—1581no—
244#386—1581no—
245#382—791no—
246C2×#95—21no—
247C2×#95—21no—
248C2×#95C2×D7912yes—
249#336—1581no—
250#328—1584no—
251#327—1584no—
252C22×Q8—791no—
253C22×D4—791no—
254C22×D4—1581no—
255C23×C4—22no—
256C23×C4C22⋊D7911yescharacteristic
257C23×C4C22⋊D7911yescharacteristic
258C23×C4C22×D7911yescharacteristic
259C22×C8—1582no—
260#124—3162no—
261C22⋊Q8—1582no—
262C22⋊Q8—1582no—
263#121—1582no—
264#121—1582no—
265C2×C4⋊C4—791no—
266C2×C4⋊C4—791no—
267C2×C4⋊C4—1581no—
268C2×C22⋊C4—1581no—
269C2×C22⋊C4—1581no—
270C2×C22⋊C4—1581no—
271C2×C22⋊C4—1581no—
272C2×C22⋊C4—1581no—
273C2×C22⋊C4—1581no—
274C2×C42—42no—
275C2×C42C4×D7911yescharacteristic
276C2×C42C4×D7911yescharacteristic
277C2×C42D31611yescharacteristic
278C2×C42D31611yescharacteristic
279C4⋊C8—3162no—
280C22⋊C8—1582no—
281C22⋊C8—1582no—
282#95—22no—
283#95—42no—
284#95—42no—
285#95—1581no—
286#95—1581no—
287#95—1581no—
288#95—1581no—
289#95D31614yes—
290C24—1581no—
291C24C2×C22⋊D7911yescharacteristic
292C2×Q8—3161no—
293C2×Q8—3161no—
294C2×Q8—1582no—
295C2×D4—3161no—
296C2×D4—3161no—
297C2×D4—3162no—
298C2×D4—3162no—
299C2×D4—3161no—
300C2×D4—3161no—
301C2×D4—1582no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—1581no—
319C22×C4—1581no—
320C22×C4—1581no—
321C22×C4(C2×D79)⋊C411yescharacteristic
322C22×C4(C2×D79)⋊C411yescharacteristic
323C22×C4(C2×D79)⋊C411yescharacteristic
324C22×C4(C2×D79)⋊C411yescharacteristic
325C22×C4C2×C4×D7911yescharacteristic
326C22×C4C2×D31611yescharacteristic
327C22×C4C2×D31612yes—
328C22×C4#1192962412yes—
329C2×C8—3162no—
330C2×C8—3162no—
331C2×C8—3162no—
332C4⋊C4—3161no—
333C4⋊C4—3161no—
334C4⋊C4—1582no—
335C4⋊C4—1582no—
336C22⋊C4—3162no—
337C22⋊C4—3162no—
338C22⋊C4—3162no—
339C22⋊C4—3162no—
340C22⋊C4—3162no—
341C22⋊C4—3161no—
342C22⋊C4—3161no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—3161no—
352C23—3161no—
353C23—3161no—
354C23—3161no—
355C23—3161no—
356C23—3161no—
357C23—3161no—
358C23C2×(C2×D79)⋊C411yescharacteristic
359C23#1457913512yes—
360C23#1457918112yes—
361C23#1457918711yescharacteristic
362C23#1457919211yescharacteristic
363Q8—6322no—
364D4—6322no—
365D4—6322no—
366D4—6322no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—3161no—
395C2×C4—3161no—
396C2×C4—3161no—
397C2×C4—3161no—
398C2×C4—3161no—
399C2×C4—3161no—
400C2×C4—3161no—
401C2×C4—3161no—
402C2×C4—3161no—
403C2×C4#1457907512yes—
404C2×C4C4×D31612yes—
405C2×C4#1457914911yescharacteristic
406C2×C4#1457915811yescharacteristic
407C8—6322no—
408C22—22no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—6321no—
417C22—6321no—
418C22—6321no—
419C22—6321no—
420C22—6321no—
421C22—6321no—
422C22—6321no—
423C22#1909846511yescharacteristic
424C22C2×#1457907511yescharacteristic
425C22#1909853611yescharacteristic
426C22#1909855711yescharacteristic
427C22#1909856211yescharacteristic
428C22#1909858111yescharacteristic
429C22#1909858111yescharacteristic
430C22#1909877812yes—
431C22#1909878512yes—
432C4—21no—
433C4—22no—
434C4—21no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—84no—
440C4—6321no—
441C4—6321no—
442C4—6321no—
443C2—22no—
444C2—22no—
445C2—12641no—
446C2#2376193611yescharacteristic
447C2#2376194111yescharacteristic
448C2#2376195411yescharacteristic
449C2#2376197511yescharacteristic
450C2#2376423211yescharacteristic
451C2#2376423611yescharacteristic
452C2#2376433211yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1434C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenter#1434D79
hypocenterC79#9716
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. #23763380C2semidirect
3. C2×#19098299C2semidirect
5. C79×#1434C2semidirect
6. C2×#14579149C4semidirect
11. #19098469C4semidirect
12. #19098465C4semidirect
17. #19098429C4semidirect
231. #1434D79semidirect
232. C79#9716semidirect

Maximal subgroups

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

subgroupindex
#237717012
#237633802
C2×#190982992
C2×#190982962
C79×#14342
#971679