Finite Group Databasebeta

← #34184508

Subgroups — #34184508

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098641C211yesmaximal, characteristic, complemented
2#23764207C211yesmaximal, characteristic, complemented
3#23763400C211yesmaximal, characteristic, complemented
4C2×#19098307C211yesmaximal, characteristic, complemented
5#23761284C211yesmaximal, characteristic, complemented
6#23761234C211yesmaximal, characteristic, complemented
7C2×#459×C79C211yesmaximal, characteristic
8C22×#11929608—21no—
9#19098641—22no—
10#19098641—22no—
11C2×C4⋊(C79⋊C8)C2211yescharacteristic
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×#14579082—21no—
14C2×#14579082—21no—
15C2×#14579082—21no—
16C2×C22⋊(C79⋊C8)—21no—
17C2×C4×C79⋊C8C2211yescharacteristic
18#19098433C2211yescharacteristic
19#19098325—21no—
20#19098307—22no—
21C22×D8×C79—21no—
22C2×#127×C79C2211yescharacteristic
23C2×C4×D4×C79C2211yescharacteristic
24#459×C79—22no—
25#459×C79C414yescomplemented
26C2×C4⋊C8×C79C2211yescharacteristic
27C2×#102×C79—21no—
28C2×#11929608—24no—
29C2×#11929608—42no—
30C2×#11929608—42no—
31C22×C79⋊C8—21no—
32C22×C79⋊C8—21no—
33C22×C79⋊C8D411yescharacteristic
34C22×C79⋊C8D411yescharacteristic
35C2×C4⋊Dic79—21no—
36C2×C4⋊Dic79—21no—
37C2×C4⋊Dic79—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)D412yes—
42#14579082—42no—
43#14579082—42no—
44#14579082—42no—
45C22⋊(C79⋊C8)—24no—
46C4×C79⋊C8—21no—
47C4×C79⋊C8—21no—
48C22×D4×C79—21no—
49C22×D4×C79—21no—
50C22×D4×C79—21no—
51C22×D4×C79D411yescharacteristic
52C23×C4×C79—21no—
53C2×D8×C79—24no—
54C2×D8×C79—42no—
55C2×D8×C79—42no—
56C22×C8×C79—21no—
57#127×C79—21no—
58#127×C79—21no—
59#127×C79—21no—
60#127×C79C2×C412yes—
61C4×D4×C79—21no—
62C4×D4×C79—21no—
63C4×D4×C79C2×C412yes—
64C4×D4×C79D412yes—
65C2×C4⋊C4×C79D411yescharacteristic
66C2×C22⋊C4×C79—21no—
67C2×C42×C79C2311yescharacteristic
68C4⋊C8×C79—21no—
69C4⋊C8×C79C2×C412yes—
70#102×C79—24no—
71C22×Dic79—41no—
72#11929608—48no—
73C2×C79⋊C8—21no—
74C2×C79⋊C8—21no—
75C2×C79⋊C8—22no—
76C2×C79⋊C8—21no—
77C2×C79⋊C8—21no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—41no—
80C2×C79⋊C8—41no—
81C2×C79⋊C8—41no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8D812yescomplemented
85C4⋊Dic79—41no—
86C4⋊Dic79—41no—
87C4⋊Dic79—41no—
88C4⋊Dic79—41no—
89C4⋊Dic79—81no—
90C24×C79—21no—
91C24×C79—41no—
92C2×D4×C79—22no—
93C2×D4×C79—22no—
94C2×D4×C79—22no—
95C2×D4×C79—21no—
96C2×D4×C79—21no—
97C2×D4×C79—21no—
98C2×D4×C79—21no—
99C2×D4×C79—21no—
100C2×D4×C79—41no—
101C2×D4×C79—41no—
102C2×D4×C79—41no—
103C2×D4×C79—41no—
104C2×D4×C79—41no—
105C2×D4×C79—41no—
106C2×D4×C79—41no—
107C2×D4×C79—41no—
108C2×D4×C79—41no—
109C2×D4×C79—41no—
110C2×D4×C79—41no—
111C2×D4×C79—81no—
112C2×D4×C79—81no—
113C2×D4×C79C22⋊C412yes—
114C22×C4×C79—21no—
115C22×C4×C79—22no—
116C22×C4×C79—22no—
117C22×C4×C79—41no—
118C22×C4×C79—41no—
119C22×C4×C79—41no—
120C22×C4×C79—41no—
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79C2×D411yescharacteristic
123C22×C4×C79#4111yescharacteristic
124D8×C79—48no—
125C2×C8×C79—22no—
126C2×C8×C79—41no—
127C2×C8×C79—41no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79C22⋊C412yes—
130C22⋊C4×C79—22no—
131C22⋊C4×C79—22no—
132C42×C79C22×C411yescharacteristic
133C42×C79C2×D411yescharacteristic
134C42×C79C2×D411yescharacteristic
135C42×C79#4111yescharacteristic
136C23×C79—22no—
137C23×C79—41no—
138C23×C79—41no—
139C23×C79—41no—
140C23×C79—41no—
141C23×C79—41no—
142C23×C79—41no—
143C23×C79—42no—
144C23×C79—81no—
145C23×C79—81no—
146C23×C79—81no—
147C23×C79—81no—
148C23×C79—81no—
149C23×C79—81no—
150C23×C79#12111yescharacteristic
151D4×C79—22no—
152D4×C79—22no—
153D4×C79—42no—
154D4×C79—42no—
155D4×C79—42no—
156D4×C79—42no—
157D4×C79—41no—
158D4×C79—41no—
159D4×C79—82no—
160D4×C79—82no—
161D4×C79—81no—
162D4×C79—81no—
163D4×C79—81no—
164D4×C79—81no—
165C2×C4×C79—21no—
166C2×C4×C79—21no—
167C2×C4×C79—22no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—42no—
171C2×C4×C79—42no—
172C2×C4×C79—42no—
173C2×C4×C79—42no—
174C2×C4×C79C2×C22⋊C411yescharacteristic
175C2×C4×C79C4×D411yescharacteristic
176C2×C4×C79C4×D411yescharacteristic
177C2×C4×C79#12011yescharacteristic
178C2×C4×C79#12111yescharacteristic
179C2×C4×C79#12311yescharacteristic
180C2×C4×C79C2×D811yescharacteristic
181C2×C4×C79#13511yescharacteristic
182C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C79#13711yescharacteristic
188C2×Dic79—81no—
189C2×Dic79—81no—
190C2×Dic79—81no—
191C8×C79—42no—
192C79⋊C8—22no—
193C79⋊C8—42no—
194C79⋊C8—41no—
195C79⋊C8—81no—
196C22×C79—42no—
197C22×C79—42no—
198C22×C79—42no—
199C22×C79—81no—
200C22×C79—81no—
201C22×C79—81no—
202C22×C79—81no—
203C22×C79—82no—
204C22×C79—82no—
205C22×C79—82no—
206C22×C79—161no—
207C22×C79—161no—
208C22×C79—161no—
209C22×C79—161no—
210C22×C79#38611yescharacteristic
211C22×C79#45911yescharacteristic
212C22×C79#46311yescharacteristic
213C22×C79#46811yescharacteristic
214C22×C79#46911yescharacteristic
215C22×C79#47111yescharacteristic
216C22×C79#47211yescharacteristic
217C4×C79—21no—
218C4×C79—21no—
219C4×C79—42no—
220C4×C79#41711yescharacteristic
221C4×C79#41811yescharacteristic
222C4×C79C4×D811yescharacteristic
223C4×C79#44211yescharacteristic
224C4×C79#44711yescharacteristic
225C4×C79#44911yescharacteristic
226C4×C79#46811yescharacteristic
227C4×C79#46911yescharacteristic
228Dic79—161no—
229#9229—791nomaximal, Sylow
230C2×C79—82no—
231C2×C79—162no—
232C2×C79#168511yescharacteristic
233C2×C79#168711yescharacteristic
234C2×C79#169711yescharacteristic
235C2×C79#170111yescharacteristic
236C2×C79#187911yescharacteristic
237C2×C79#189511yescharacteristic
238C2×C79#190711yescharacteristic
239C2×#459D7911yescharacteristic
240C2×#436—791no—
241#1957—791no—
242#1802—791no—
243C2×#331—791no—
244#1296—791no—
245#1277—791no—
246C79#922911yescharacteristic, complemented, Sylow
247C22×D8—21no—
248C22×M16—1581no—
249C2×#127C2×D7911yescharacteristic
250C2×C4×D4C2×D7911yescharacteristic
251#459—22no—
252#459Dic7914yescomplemented
253#436—1582no—
254#436—1582no—
255C2×C4⋊C8—791no—
256C2×C4⋊C8—791no—
257C2×C4⋊C8C2×D7911yescharacteristic
258C2×#102—21no—
259C2×#102—1581no—
260C2×#102—1581no—
261C2×#102—1581no—
262C2×C22⋊C8—1581no—
263C2×C4×C8—791no—
264#384—791no—
265#340—1581no—
266#331—1582no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4C22⋊D7911yescharacteristic
271C23×C4—21no—
272C2×D8—24no—
273C2×D8—42no—
274C2×D8—42no—
275C2×M16—3162no—
276C2×M16—3162no—
277C2×M16—1584no—
278C22×C8—21no—
279C22×C8—791no—
280C22×C8—791no—
281C22×C8—1581no—
282C22×C8—1581no—
283#127—21no—
284#127—21no—
285#127—21no—
286#127C2×Dic7912yes—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C2×Dic7912yes—
290C4×D4C22⋊D7912yes—
291C2×C4⋊C4—1581no—
292C2×C4⋊C4—1581no—
293C2×C4⋊C4—1581no—
294C2×C4⋊C4C22⋊D7911yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C22×D7911yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—792no—
299C4⋊C8—1581no—
300C4⋊C8—1581no—
301C4⋊C8—1581no—
302C4⋊C8C2×Dic7912yes—
303#102—24no—
304#102—3162no—
305#102—3162no—
306#102—3162no—
307C22⋊C8—1584no—
308C4×C8—1581no—
309C4×C8—1581no—
310C24—21no—
311C24—41no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—81no—
332C2×D4—81no—
333C2×D4C22⋊Dic7912yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—3161no—
342C22×C4D4×D7911yescharacteristic
343C22×C4C2×C22⋊D7911yescharacteristic
344C22×C4#1192962311yescharacteristic
345D8—48no—
346M16—3168no—
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—3161no—
351C2×C8—3161no—
352C2×C8—3161no—
353C2×C8—3161no—
354C2×C8—3161no—
355C2×C8—792no—
356C2×C8—1581no—
357C2×C8—1582no—
358C2×C8—1581no—
359C2×C8—1581no—
360C2×C8—1581no—
361C2×C8—1581no—
362C4⋊C4—21no—
363C4⋊C4—3161no—
364C4⋊C4—3161no—
365C4⋊C4—3161no—
366C4⋊C4—3161no—
367C4⋊C4—6321no—
368C4⋊C4C22⋊Dic7912yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C42C22×Dic7911yescharacteristic
372C42D4×D7911yescharacteristic
373C42C2×C22⋊D7911yescharacteristic
374C42#1192962311yescharacteristic
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23#1457913311yescharacteristic
390D4—22no—
391D4—22no—
392D4—42no—
393D4—42no—
394D4—41no—
395D4—41no—
396D4—42no—
397D4—42no—
398D4—82no—
399D4—82no—
400D4—81no—
401D4—81no—
402D4—81no—
403D4—81no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—6321no—
414C2×C4—6321no—
415C2×C4—6321no—
416C2×C4C2×C22⋊Dic7911yescharacteristic
417C2×C4D4×Dic7911yescharacteristic
418C2×C4D4×Dic7911yescharacteristic
419C2×C4#1457912911yescharacteristic
420C2×C4#1457913311yescharacteristic
421C2×C4#1457914211yescharacteristic
422C2×C4D8×D7911yescharacteristic
423C2×C4#1457918011yescharacteristic
424C2×C4#1457918511yescharacteristic
425C2×C4#1457918511yescharacteristic
426C2×C4#1457918611yescharacteristic
427C2×C4#1457918711yescharacteristic
428C2×C4#1457918711yescharacteristic
429C2×C4#1457919111yescharacteristic
430C8—42no—
431C8—3162no—
432C8—3161no—
433C8—6321no—
434C8—1582no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22#1909844311yescharacteristic
450C22#1909875611yescharacteristic
451C22#1909877611yescharacteristic
452C22#1909880711yescharacteristic
453C22#1909881411yescharacteristic
454C22#1909882411yescharacteristic
455C22#1909883111yescharacteristic
456C4—21no—
457C4—21no—
458C4—42no—
459C4—12641no—
460C4#1909856411yescharacteristic
461C4#1909856811yescharacteristic
462C4D8×Dic7911yescharacteristic
463C4#1909867511yescharacteristic
464C4#1909869511yescharacteristic
465C4#1909870511yescharacteristic
466C4#1909880711yescharacteristic
467C4#1909881411yescharacteristic
468C2—82no—
469C2—162no—
470C2#2376286311yescharacteristic
471C2#2376287711yescharacteristic
472C2#2376293111yescharacteristic
473C2#2376295911yescharacteristic
474C2#2376375011yescharacteristic
475C2#2376383911yescharacteristic
476C2#2376391011yescharacteristic
subgroupG/H
centerC23#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC2×#459×C79C2
socleC23×C79#121
cosocleC2×C42×C79C23
hypercenterC2×#459D79
hypocenterC79#9229
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#19098641C2semidirect
2. #23764207C2semidirect
3. #23763400C2semidirect
4. C2×#19098307C2semidirect
5. #23761284C2semidirect
6. #23761234C2semidirect
25. #459×C79C4semidirect
84. C2×C79⋊C8D8semidirect
246. C79#9229semidirect
252. #459Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190986412
#237642072
#237634002
C2×#190983072
#237612842
#237612342
C2×#459×C792
#922979