Finite Group Databasebeta

← #34183515

Subgroups — #34183515

Order 20224.

690 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
1C2×#19098911C211yesmaximal, characteristic
2#23766464C211yesmaximal, characteristic
3#23764353C211yesmaximal, characteristic, complemented
4C2×#19098306C211yesmaximal, characteristic
5#23761241C211yesmaximal, characteristic, complemented
6#23761239C211yesmaximal, characteristic, complemented
7C79×#1981C211yesmaximal, characteristic, complemented
8C2×#14579158C2211yescharacteristic
9C2×C4×D316—21no—
10C2×#14579123—21no—
11#19098911—22no—
12#19098911C414yescomplemented
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579081—22no—
15C2×C4×C79⋊C8C2211yescharacteristic
16#19098465—21no—
17#19098464—21no—
18#19098441—22no—
19C4×C4⋊Dic79C2211yescharacteristic
20#19098325—21no—
21#19098325—21no—
22#19098306—22no—
23C2×C4⋊Q8×C79C2211yescharacteristic
24C2×C4⋊C8×C79C2211yescharacteristic
25C2×#103×C79—21no—
26C2×#103×C79—21no—
27C4×C4⋊C4×C79C2211yescharacteristic
28C22×D316—21no—
29C22×D316—21no—
30C22×D316D412yes—
31C22×C4×D79—22no—
32C22×C79⋊C8—21no—
33C22×C79⋊C8—21no—
34#14579158—21no—
35#14579158—21no—
36#14579158C2×C412yes—
37#14579158D412yes—
38C4×D316—42no—
39C4×D316—42no—
40#14579123—42no—
41#14579123—42no—
42C2×C4⋊Dic79D411yescharacteristic
43C2×C4⋊Dic79D411yescharacteristic
44C2×(C2×D79)⋊C4—22no—
45C2×(C2×D79)⋊C4—22no—
46C2×(C2×D79)⋊C4—22no—
47C2×C4×Dic79—21no—
48C4⋊(C79⋊C8)—21no—
49C4⋊(C79⋊C8)—21no—
50#14579081—28no—
51C4×C79⋊C8—21no—
52C4×C79⋊C8C2×C412yescomplemented
53C79⋊#95—21no—
54C22×Q8×C79—21no—
55C22×C8×C79—21no—
56C4⋊Q8×C79—21no—
57C4⋊Q8×C79—21no—
58C4⋊Q8×C79—21no—
59C4⋊Q8×C79C2×C412yescomplemented
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79D411yescharacteristic
63C2×C4⋊C4×C79D411yescharacteristic
64C2×C42×C79—21no—
65C2×C42×C79C2311yescharacteristic
66C4⋊C8×C79—21no—
67C4⋊C8×C79—21no—
68#103×C79—42no—
69#103×C79—42no—
70#95×C79—21no—
71C23×D79—22no—
72C2×D316—22no—
73C2×D316—22no—
74C2×D316—22no—
75C2×D316—22no—
76C2×D316—22no—
77C2×D316—42no—
78C2×D316—42no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316—41no—
82C2×D316—41no—
83C2×D316—41no—
84C2×D316—41no—
85C2×D316—41no—
86C2×D316—41no—
87C2×D316C22⋊C414yes—
88C22×Dic79—21no—
89C22×Dic79—21no—
90C2×C4×D79—44no—
91C2×C4×D79—44no—
92C2×C4×D79—82no—
93C2×C79⋊C8—22no—
94C2×C79⋊C8—41no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99C4⋊Dic79—21no—
100C4⋊Dic79—21no—
101C4⋊Dic79—21no—
102C4⋊Dic79SD1612yes—
103(C2×D79)⋊C4—44no—
104(C2×D79)⋊C4—44no—
105(C2×D79)⋊C4—82no—
106C4×Dic79—41no—
107C4×Dic79—41no—
108C2×Q8×C79—22no—
109C2×Q8×C79—41no—
110C2×Q8×C79—41no—
111C2×Q8×C79—41no—
112C2×Q8×C79—41no—
113C2×Q8×C79—41no—
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79—41no—
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79C2×D411yescharacteristic
120C2×C8×C79—41no—
121C2×C8×C79—41no—
122C2×C8×C79—41no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79—22no—
125C4⋊C4×C79—22no—
126C4⋊C4×C79—21no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79—41no—
129C4⋊C4×C79—41no—
130C4⋊C4×C79D812yes—
131C42×C79—41no—
132C42×C79—41no—
133C42×C79C22×C411yescharacteristic
134C42×C79C2×D411yescharacteristic
135C42×C79#4111yescharacteristic
136C42×C79#4111yescharacteristic
137C23×C79#12011yescharacteristic
138C22×D79—24no—
139C22×D79—42no—
140C22×D79—42no—
141C22×D79—42no—
142C22×D79—42no—
143C22×D79—42no—
144C22×D79—42no—
145Q8×C79—42no—
146Q8×C79—42no—
147Q8×C79—81no—
148Q8×C79—81no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—22no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—42no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—81no—
159C2×C4×C79—81no—
160C2×C4×C79C2×C22⋊C411yescharacteristic
161C2×C4×C79C4×D411yescharacteristic
162C2×C4×C79C4×D411yescharacteristic
163C2×C4×C79#12111yescharacteristic
164C2×C4×C79#12111yescharacteristic
165C2×C4×C79#12311yescharacteristic
166C2×C4×C79C2×D811yescharacteristic
167C2×C4×C79C2×SD1611yescharacteristic
168C2×C4×C79#13511yescharacteristic
169C2×C4×C79#13511yescharacteristic
170C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C79#13711yescharacteristic
174C2×Dic79—22no—
175C2×Dic79—41no—
176C2×Dic79—41no—
177C2×Dic79—41no—
178C2×Dic79—41no—
179C2×Dic79—41no—
180D316—44no—
181D316—42no—
182D316—42no—
183D316—42no—
184D316—42no—
185D316—42no—
186D316—42no—
187D316—42no—
188D316—42no—
189D316—82no—
190C4×D79—88no—
191C8×C79—81no—
192C79⋊C8—42no—
193C79⋊C8—81no—
194C22×C79#38611yescharacteristic
195C22×C79#44711yescharacteristic
196C22×C79#44911yescharacteristic
197C22×C79#45011yescharacteristic
198C22×C79#45211yescharacteristic
199C22×C79#45511yescharacteristic
200C22×C79#45511yescharacteristic
201C2×D79—44no—
202C2×D79—44no—
203C2×D79—44no—
204C2×D79—82no—
205C2×D79—82no—
206C2×D79—82no—
207C2×D79—82no—
208C4×C79—21no—
209C4×C79—21no—
210C4×C79—42no—
211C4×C79—82no—
212C4×C79—81no—
213C4×C79C4×D811yescharacteristic
214C4×C79C4×SD1611yescharacteristic
215C4×C79#44011yescharacteristic
216C4×C79#44211yescharacteristic
217C4×C79#45911yescharacteristic
218C4×C79#46111yescharacteristic
219C4×C79#46311yescharacteristic
220C4×C79#46311yescharacteristic
221Dic79—42no—
222Dic79—81no—
223#9057—791nomaximal, Sylow
224C2×C79#163911yescharacteristic
225C2×C79#164011yescharacteristic
226C2×C79#164111yescharacteristic
227C2×C79#164211yescharacteristic
228C2×C79#187711yescharacteristic
229C2×C79#187911yescharacteristic
230C2×C79#188511yescharacteristic
231D79—84no—
232C2×#486—791no—
233#2366—791no—
234#1988—791no—
235#1981D7911yescharacteristic, complemented
236C2×#331—791no—
237#1279—791no—
238#1279—791no—
239C79#905711yescharacteristic, complemented, Sylow
240C2×C4⋊Q8C2×D7911yescharacteristic
241C2×#127—791no—
242C2×C4×D4—1581no—
243C2×C4×D4—1581no—
244#486—794no—
245#486—1582no—
246C2×C4⋊C8—791no—
247C2×C4⋊C8C2×D7911yescharacteristic
248C2×#103—21no—
249C2×#103—21no—
250C2×#102—1582no—
251C2×C4×C8—791no—
252#390—1581no—
253#390—1581no—
254#386—1582no—
255C4×C4⋊C4—791no—
256C4×C4⋊C4C2×D7911yescharacteristic
257#340—1581no—
258#340—1581no—
259#331—1582no—
260C22×Q8—21no—
261C22×D4—792no—
262C22×D4—1581no—
263C22×D4—1581no—
264C23×C4—1582no—
265C22×C8—21no—
266C22×C8—1581no—
267C22×C8—1581no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8C4×D7912yescomplemented
272#127—792no—
273#127—792no—
274#127—1581no—
275#127—1581no—
276C4×D4—3162no—
277C4×D4—3162no—
278C4×D4—3162no—
279C4×D4—3162no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—791no—
283C2×C4⋊C4—791no—
284C2×C4⋊C4C22⋊D7911yescharacteristic
285C2×C4⋊C4C22⋊D7911yescharacteristic
286C2×C22⋊C4—1582no—
287C2×C22⋊C4—1582no—
288C2×C22⋊C4—1582no—
289C2×C42—21no—
290C2×C42—1581no—
291C2×C42C22×D7911yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—1581no—
295C4⋊C8—1581no—
296#103—42no—
297#103—42no—
298#102—1588no—
299C4×C8—792no—
300C4×C8—1581no—
301#95—21no—
302#95—1581no—
303C24—1582no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×D4—3162no—
311C2×D4—3162no—
312C2×D4—3161no—
313C2×D4—3161no—
314C2×D4—3161no—
315C2×D4—3161no—
316C2×D4—3161no—
317C2×D4—3161no—
318C2×D4—3161no—
319C2×D4—3161no—
320C2×D4—794no—
321C2×D4—1582no—
322C2×D4—1582no—
323C2×D4—1582no—
324C2×D4—1582no—
325C2×D4—1582no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—3164no—
330C22×C4—3164no—
331C22×C4—6322no—
332C22×C4—1581no—
333C22×C4—1581no—
334C22×C4D4×D7911yescharacteristic
335C22×C4D4×D7911yescharacteristic
336C22×C4C2×C22⋊D7911yescharacteristic
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—3161no—
341C2×C8—3161no—
342C2×C8—3161no—
343C2×C8—3161no—
344C2×C8—3161no—
345C2×C8—1582no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—792no—
354C4⋊C4—1581no—
355C4⋊C4—1581no—
356C4⋊C4—1581no—
357C4⋊C4#1192960912yes—
358C22⋊C4—3164no—
359C22⋊C4—3164no—
360C22⋊C4—6322no—
361C42—41no—
362C42—41no—
363C42—3161no—
364C42—3161no—
365C42C2×C4×D7911yescharacteristic
366C42D4×D7911yescharacteristic
367C42#1192962311yescharacteristic
368C42#1192962411yescharacteristic
369C23—3162no—
370C23—3162no—
371C23—3162no—
372C23—3162no—
373C23—3162no—
374C23—3162no—
375C23—1584no—
376C23#1457912911yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
381D4—3164no—
382D4—3162no—
383D4—3162no—
384D4—3162no—
385D4—3162no—
386D4—3162no—
387D4—3162no—
388D4—3162no—
389D4—3162no—
390D4—6322no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—3161no—
403C2×C4—3161no—
404C2×C4—3161no—
405C2×C4—3161no—
406C2×C4—3161no—
407C2×C4—6328no—
408C2×C4—1582no—
409C2×C4C22⋊C4×D7911yescharacteristic
410C2×C4(C22⋊D79)⋊C411yescharacteristic
411C2×C4C4×C22⋊D7911yescharacteristic
412C2×C4#1457913211yescharacteristic
413C2×C4#1457913411yescharacteristic
414C2×C4#1457914311yescharacteristic
415C2×C4C2×#1192960911yescharacteristic
416C2×C4SD16×D7911yescharacteristic
417C2×C4#1457918111yescharacteristic
418C2×C4#1457918411yescharacteristic
419C2×C4#1457918611yescharacteristic
420C2×C4#1457918711yescharacteristic
421C2×C4#1457918811yescharacteristic
422C2×C4#1457919311yescharacteristic
423C8—81no—
424C8—3162no—
425C8—6321no—
426C22—3164no—
427C22—3164no—
428C22—3164no—
429C22—6322no—
430C22—6322no—
431C22—6322no—
432C22—6322no—
433C22#1909843911yescharacteristic
434C22#1909869611yescharacteristic
435C22#1909870611yescharacteristic
436C22#1909871211yescharacteristic
437C22#1909872411yescharacteristic
438C22#1909873911yescharacteristic
439C22#1909873911yescharacteristic
440C4—21no—
441C4—21no—
442C4—42no—
443C4—82no—
444C4—81no—
445C4—3162no—
446C4—6321no—
447C4C4×#1192960911yescharacteristic
448C4#1909864811yescharacteristic
449C4#1909866611yescharacteristic
450C4#1909867311yescharacteristic
451C4#1909875511yescharacteristic
452C4#1909876711yescharacteristic
453C4#1909877511yescharacteristic
454C4#1909878111yescharacteristic
455C2—6324no—
456C2#2376257711yescharacteristic
457C2#2376258411yescharacteristic
458C2#2376259111yescharacteristic
459C2#2376259811yescharacteristic
460C2#2376373111yescharacteristic
461C2#2376375111yescharacteristic
462C2#2376378511yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1981C2
socleC23×C79#120
prime radicalC2×#14579158C22
cosocleC2×C42×C79C23
hypercenter#1981D79
hypocenterC79#9057
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. #23764353C2semidirect
5. #23761241C2semidirect
6. #23761239C2semidirect
7. C79×#1981C2semidirect
12. #19098911C4semidirect
52. C4×C79⋊C8C2×C4semidirect
59. C4⋊Q8×C79C2×C4semidirect
235. #1981D79semidirect
239. C79#9057semidirect
271. C4⋊Q8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989112
#237664642
#237643532
C2×#190983062
#237612412
#237612392
C79×#19812
#905779