Finite Group Databasebeta

← #34190614

Subgroups — #34190614

Order 20224.

628 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
1C2×#19098850C211yesmaximal, characteristic
2#23768318C211yesmaximal, characteristic
3#23764203C211yesmaximal, characteristic, complemented
4C2×Q8⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
5C2×#19098290C211yesmaximal, characteristic
6#23761284C211yesmaximal, characteristic, complemented
7C2×#477×C79C211yesmaximal, characteristic, complemented
8C2×#14579137—21no—
9C2×C4×D316C2211yescharacteristic
10#19098850—22no—
11#19098850—22no—
12C2×#14579092—21no—
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579085—22no—
15C2×#14579081—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17#19098485—22no—
18#19098465—21no—
19#19098461—21no—
20#19098433C2211yescharacteristic
21#19098420—21no—
22Q8⋊(C79⋊C8)—22no—
23#19098290—22no—
24C2×C4⋊Q8×C79C2211yescharacteristic
25C2×C4×Q8×C79C2211yescharacteristic
26#477×C79—22no—
27#477×C79—22no—
28C2×#107×C79—21no—
29C2×C4⋊C8×C79C2211yescharacteristic
30C2×#103×C79—21no—
31C22×D316D411yescharacteristic
32C22×C4×D79—21no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8—21no—
35#14579137—42no—
36#14579137—42no—
37C4×D316—21no—
38C4×D316—21no—
39C4×D316—21no—
40C4×D316—21no—
41C2×C4⋊Dic79—21no—
42C2×C4⋊Dic79—21no—
43C2×C4⋊Dic79D411yescharacteristic
44C2×C4⋊Dic79D412yes—
45C2×Dic79⋊C4—22no—
46C2×(C2×D79)⋊C4—22no—
47C2×(C2×D79)⋊C4—21no—
48C2×(C2×D79)⋊C4—22no—
49#14579092—24no—
50C4⋊(C79⋊C8)—21no—
51C4⋊(C79⋊C8)D412yes—
52#14579085—44no—
53#14579081—24no—
54C4×C79⋊C8—21no—
55C4×C79⋊C8—21no—
56C79⋊#95—22no—
57C22×Q8×C79—21no—
58C22×Q8×C79D411yescharacteristic
59C22×C8×C79—21no—
60C4⋊Q8×C79—21no—
61C4⋊Q8×C79—21no—
62C4⋊Q8×C79—21no—
63C4⋊Q8×C79D412yes—
64C4×Q8×C79—21no—
65C4×Q8×C79—21no—
66C4×Q8×C79—21no—
67C4×Q8×C79—21no—
68C2×C4⋊C4×C79—21no—
69C2×C4⋊C4×C79—21no—
70C2×C4⋊C4×C79—21no—
71C2×C4⋊C4×C79D411yescharacteristic
72C2×C42×C79—21no—
73C2×C42×C79C2311yescharacteristic
74#107×C79—24no—
75C4⋊C8×C79—21no—
76C4⋊C8×C79D412yes—
77#103×C79—24no—
78C23×D79—21no—
79C2×D316—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×D316—21no—
83C2×D316—21no—
84C2×D316Q1612yes—
85C22×Dic79—21no—
86C22×Dic79—22no—
87C2×C4×D79—41no—
88C2×C4×D79—41no—
89C2×C4×D79—41no—
90C2×C4×D79—41no—
91C2×C4×D79—41no—
92C2×C4×D79—41no—
93C2×C79⋊C8—22no—
94C2×C79⋊C8—22no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99C4⋊Dic79—21no—
100C4⋊Dic79—21no—
101C4⋊Dic79—42no—
102C4⋊Dic79—41no—
103C4⋊Dic79—41no—
104C4⋊Dic79—41no—
105C4⋊Dic79—41no—
106Dic79⋊C4—44no—
107(C2×D79)⋊C4—44no—
108(C2×D79)⋊C4—41no—
109(C2×D79)⋊C4—41no—
110(C2×D79)⋊C4—82no—
111C2×Q8×C79—21no—
112C2×Q8×C79—21no—
113C2×Q8×C79—21no—
114C2×Q8×C79—22no—
115C2×Q8×C79—21no—
116C2×Q8×C79—21no—
117C2×Q8×C79—42no—
118C2×Q8×C79—42no—
119C2×Q8×C79—41no—
120C2×Q8×C79SD1612yes—
121C22×C4×C79—21no—
122C22×C4×C79—21no—
123C22×C4×C79—41no—
124C22×C4×C79C2×D411yescharacteristic
125C22×C4×C79C2×D411yescharacteristic
126C22×C4×C79C2×D411yescharacteristic
127C2×C8×C79—22no—
128C2×C8×C79—41no—
129C2×C8×C79—41no—
130C4⋊C4×C79—22no—
131C4⋊C4×C79—21no—
132C4⋊C4×C79—22no—
133C4⋊C4×C79—21no—
134C4⋊C4×C79—41no—
135C4⋊C4×C79—41no—
136C4⋊C4×C79—41no—
137C4⋊C4×C79—41no—
138C42×C79—41no—
139C42×C79—41no—
140C42×C79C2×D411yescharacteristic
141C42×C79C2×D411yescharacteristic
142C42×C79C2×D411yescharacteristic
143C42×C79#4111yescharacteristic
144C23×C79#12011yescharacteristic
145C22×D79—22no—
146C22×D79—41no—
147C22×D79—41no—
148C22×D79—41no—
149C22×D79—41no—
150C22×D79—41no—
151C22×D79—41no—
152Q8×C79—22no—
153Q8×C79—22no—
154Q8×C79—44no—
155Q8×C79—41no—
156Q8×C79—41no—
157Q8×C79—82no—
158C2×C4×C79—21no—
159C2×C4×C79—21no—
160C2×C4×C79—22no—
161C2×C4×C79—41no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—42no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—81no—
168C2×C4×C79—81no—
169C2×C4×C79#12011yescharacteristic
170C2×C4×C79#12011yescharacteristic
171C2×C4×C79#12111yescharacteristic
172C2×C4×C79#12111yescharacteristic
173C2×C4×C79#12111yescharacteristic
174C2×C4×C79#12711yescharacteristic
175C2×C4×C79C2×SD1611yescharacteristic
176C2×C4×C79C2×Q1611yescharacteristic
177C2×C4×C79#13511yescharacteristic
178C2×C4×C79#13511yescharacteristic
179C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C79#13711yescharacteristic
181C2×C4×C79#13711yescharacteristic
182C2×C4×C79#13711yescharacteristic
183C2×Dic79—41no—
184C2×Dic79—42no—
185C2×Dic79—41no—
186C2×Dic79—42no—
187C2×Dic79—41no—
188C2×Dic79—42no—
189D316—22no—
190D316—22no—
191D316—41no—
192D316—41no—
193C4×D79—81no—
194C4×D79—81no—
195C4×D79—81no—
196C4×D79—81no—
197C8×C79—42no—
198C79⋊C8—42no—
199C79⋊C8—42no—
200C22×C79#39211yescharacteristic
201C22×C79#45011yescharacteristic
202C22×C79#45111yescharacteristic
203C22×C79#45211yescharacteristic
204C22×C79#45211yescharacteristic
205C22×C79#45311yescharacteristic
206C22×C79#45411yescharacteristic
207C2×D79—42no—
208C2×D79—42no—
209C2×D79—42no—
210C2×D79—81no—
211C2×D79—81no—
212C2×D79—81no—
213C2×D79—81no—
214C4×C79—21no—
215C4×C79—21no—
216C4×C79—42no—
217C4×C79—81no—
218C4×C79—82no—
219C4×C79#44811yescharacteristic
220C4×C79#44911yescharacteristic
221C4×C79#45111yescharacteristic
222C4×C79#45211yescharacteristic
223C4×C79#46511yescharacteristic
224C4×C79#46711yescharacteristic
225C4×C79#46811yescharacteristic
226C4×C79#47011yescharacteristic
227Dic79—82no—
228Dic79—81no—
229#10236—791nomaximal, Sylow
230C2×C79#163111yescharacteristic
231C2×C79#163211yescharacteristic
232C2×C79#163711yescharacteristic
233C2×C79#163811yescharacteristic
234C2×C79#200011yescharacteristic
235C2×C79#200111yescharacteristic
236C2×C79#200611yescharacteristic
237D79—82no—
238C2×#477D7911yescharacteristic, complemented
239C2×#476—791no—
240#2538—791no—
241#1956—791no—
242C2×Q8⋊C8—791no—
243C2×D4⋊C8—791no—
244#1296—791no—
245C79#1023611yescharacteristic, complemented, Sylow
246C2×C4⋊Q8C2×D7911yescharacteristic
247C2×C22⋊Q8—1581no—
248C2×C4×Q8C2×D7911yescharacteristic
249C2×C4×D4—791no—
250#477—22no—
251#477—22no—
252#476—1582no—
253#476—1582no—
254C2×#107—21no—
255C2×#106—1581no—
256C2×C4⋊C8—791no—
257C2×C4⋊C8C2×D7911yescharacteristic
258C2×#103—21no—
259C2×#103—1582no—
260C2×#102—1581no—
261C2×C4×C8—791no—
262#396—1582no—
263#390—1581no—
264#389—1581no—
265#384—791no—
266#380—1581no—
267Q8⋊C8—1582no—
268D4⋊C8—1582no—
269C22×Q8—21no—
270C22×Q8C22⋊D7911yescharacteristic
271C22×D4—791no—
272C23×C4—1581no—
273C22×C8—21no—
274C22×C8—1581no—
275C22×C8—1581no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8C22⋊D7912yes—
280C22⋊Q8—3162no—
281C22⋊Q8—3162no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×D4—1581no—
287C4×D4—1581no—
288C4×D4—1581no—
289C4×D4—1581no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—792no—
294C2×C4⋊C4—791no—
295C2×C4⋊C4—1582no—
296C2×C4⋊C4—1581no—
297C2×C4⋊C4—1581no—
298C2×C4⋊C4C22⋊D7911yescharacteristic
299C2×C22⋊C4—1582no—
300C2×C22⋊C4—1581no—
301C2×C22⋊C4—1582no—
302C2×C42—21no—
303C2×C42C22×D7911yescharacteristic
304#107—24no—
305#106—1584no—
306C4⋊C8—21no—
307C4⋊C8—792no—
308C4⋊C8—1581no—
309C4⋊C8C22⋊D7912yes—
310#103—24no—
311#103—3164no—
312#102—1584no—
313C4×C8—1581no—
314C4×C8—1581no—
315#95—1582no—
316C24—1581no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—22no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×Q8#1192961112yes—
327C2×D4—792no—
328C2×D4—1581no—
329C2×D4—1581no—
330C2×D4—1581no—
331C2×D4—1581no—
332C2×D4—1581no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—41no—
336C22×C4—3161no—
337C22×C4—3161no—
338C22×C4—3161no—
339C22×C4—3161no—
340C22×C4—3161no—
341C22×C4—3161no—
342C22×C4—1581no—
343C22×C4—1582no—
344C22×C4D4×D7911yescharacteristic
345C22×C4D4×D7911yescharacteristic
346C22×C4C2×C22⋊D7911yescharacteristic
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—3161no—
351C2×C8—3161no—
352C2×C8—3161no—
353C2×C8—3161no—
354C2×C8—1582no—
355C2×C8—1582no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—3162no—
365C4⋊C4—3164no—
366C4⋊C4—3161no—
367C4⋊C4—3161no—
368C4⋊C4—3161no—
369C4⋊C4—3161no—
370C4⋊C4—1581no—
371C4⋊C4—1581no—
372C22⋊C4—3164no—
373C22⋊C4—3161no—
374C22⋊C4—3161no—
375C22⋊C4—6322no—
376C42—41no—
377C42—41no—
378C42D4×D7911yescharacteristic
379C42C2×C22⋊D7911yescharacteristic
380C42C2×C22⋊D7911yescharacteristic
381C42#1192962311yescharacteristic
382C23—3161no—
383C23—3161no—
384C23—3161no—
385C23—3161no—
386C23—3161no—
387C23—3161no—
388C23—1582no—
389C23#1457912911yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—44no—
393Q8—41no—
394Q8—41no—
395Q8—82no—
396D4—3161no—
397D4—3161no—
398D4—1582no—
399D4—1582no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—3162no—
412C2×C4—3161no—
413C2×C4—3161no—
414C2×C4—3162no—
415C2×C4—3161no—
416C2×C4—3162no—
417C2×C4—6321no—
418C2×C4—6321no—
419C2×C4—6321no—
420C2×C4—6321no—
421C2×C4#1457912911yescharacteristic
422C2×C4#1457913011yescharacteristic
423C2×C4#1457913211yescharacteristic
424C2×C4#1457913311yescharacteristic
425C2×C4#1457913311yescharacteristic
426C2×C4#1457915711yescharacteristic
427C2×C4C2×#1192961111yescharacteristic
428C2×C4Q16×D7911yescharacteristic
429C2×C4#1457918111yescharacteristic
430C2×C4#1457918411yescharacteristic
431C2×C4#1457918611yescharacteristic
432C2×C4#1457919211yescharacteristic
433C2×C4#1457919311yescharacteristic
434C2×C4#1457919311yescharacteristic
435C8—42no—
436C8—3162no—
437C8—3162no—
438C22—3162no—
439C22—3162no—
440C22—3162no—
441C22—6321no—
442C22—6321no—
443C22—6321no—
444C22—6321no—
445C22#1909847111yescharacteristic
446C22#1909871311yescharacteristic
447C22#1909871711yescharacteristic
448C22#1909872311yescharacteristic
449C22#1909872411yescharacteristic
450C22#1909872911yescharacteristic
451C22#1909873411yescharacteristic
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—82no—
457C4—6322no—
458C4—6321no—
459C4#1909870211yescharacteristic
460C4#1909870811yescharacteristic
461C4#1909871711yescharacteristic
462C4#1909872411yescharacteristic
463C4#1909879411yescharacteristic
464C4#1909880111yescharacteristic
465C4#1909881111yescharacteristic
466C4#1909881811yescharacteristic
467C2—6322no—
468C2#2376252511yescharacteristic
469C2#2376253311yescharacteristic
470C2#2376256311yescharacteristic
471C2#2376257111yescharacteristic
472C2#2376440111yescharacteristic
473C2#2376440711yescharacteristic
474C2#2376444011yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC2×#477×C79C2
socleC23×C79#120
prime radicalC22×D316D4
hypercenterC2×#477D79
hypocenterC79#10236
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #23764203C2semidirect
4. C2×Q8⋊(C79⋊C8)C2semidirect
6. #23761284C2semidirect
7. C2×#477×C79C2semidirect
238. C2×#477D79semidirect
245. C79#10236semidirect

Maximal subgroups

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

subgroupindex
C2×#190988502
#237683182
#237642032
C2×Q8⋊(C79⋊C8)2
C2×#190982902
#237612842
C2×#477×C792
#1023679