Finite Group Databasebeta

← #34204014

Subgroups — #34204014

Order 20224.

876 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#23765149C211yesmaximal, characteristic, complemented
2C2×#19098379C211yesmaximal, characteristic, complemented
3#23765136C211yesmaximal, characteristic, complemented
4#23765120C211yesmaximal, characteristic, complemented
5#23765028C211yesmaximal, characteristic, complemented
6#23763454C211yesmaximal, characteristic, complemented
7C2×#578×C79C211yesmaximal, characteristic
8C2×#14579101C2211yescharacteristic
9C2×#14579101C2211yescharacteristic
10C2×#14579097C2211yescharacteristic
11#19098601C2211yescharacteristic
12#19098580—21no—
13#19098379—22no—
14#19098379—21no—
15#19098376—21no—
16#19098373—22no—
17#19098367—22no—
18#19098347—22no—
19#19098335—21no—
20C2×2−1+4×C79—21no—
21#578×C79—22no—
22#578×C79—22no—
23#578×C79—22no—
24#578×C79C414yescomplemented
25C2×#137×C79—22no—
26C2×#137×C79—21no—
27C2×#135×C79—21no—
28C2×#135×C79C2211yescharacteristic
29C22×Q16×C79—21no—
30C22×Q16×C79C2211yescharacteristic
31C2×#131×C79C2211yescharacteristic
32C2×#11929608—21no—
33#14579120—21no—
34#14579101—21no—
35#14579101—21no—
36#14579101—21no—
37#14579101—21no—
38C2×C79⋊C16—21no—
39C2×C79⋊C16—21no—
40#14579097—21no—
41#14579097—21no—
42#14579095—21no—
43#14579093—21no—
44#14579088—42no—
452−1+4×C79—24no—
462−1+4×C79—24no—
472−1+4×C79—42no—
482−1+4×C79—42no—
49C2×#41×C79—22no—
50C2×#41×C79—21no—
51C2×#41×C79—42no—
52C2×#41×C79—41no—
53C2×#41×C79D411yescharacteristic
54C22×Q8×C79—22no—
55C22×Q8×C79—21no—
56C22×Q8×C79—41no—
57#137×C79—28no—
58#137×C79—24no—
59#137×C79—28no—
60#137×C79—24no—
61#135×C79—24no—
62#135×C79—24no—
63#135×C79—21no—
64#135×C79—21no—
65#135×C79C2×C412yes—
66#135×C79D412yes—
67C2×Q16×C79—22no—
68C2×Q16×C79—24no—
69C2×Q16×C79—24no—
70C2×Q16×C79—21no—
71C2×Q16×C79—21no—
72C2×Q16×C79—21no—
73C2×Q16×C79C2×C412yes—
74C2×Q16×C79D411yescharacteristic
75C2×Q16×C79D412yes—
76C2×Q16×C79D412yes—
77C2×SD16×C79—22no—
78C2×SD16×C79—21no—
79C2×D8×C79D411yescharacteristic
80#131×C79—21no—
81#131×C79—21no—
82#131×C79C2×C412yes—
83#131×C79D412yes—
84C2×M16×C79—21no—
85C2×M16×C79D411yescharacteristic
86C22×C8×C79—21no—
87C22×C8×C79C2311yescharacteristic
88#11929608—41no—
89#11929608—41no—
90C2×C79⋊C8—41no—
91C4⋊Dic79—41no—
92C22⋊Dic79—41no—
93C4×Dic79—41no—
94C79⋊C16—41no—
95C79⋊C16—41no—
96#41×C79—21no—
97#41×C79—21no—
98#41×C79—28no—
99#41×C79—22no—
100#41×C79—22no—
101#41×C79—44no—
102#41×C79—48no—
103#41×C79—48no—
104#41×C79—44no—
105#41×C79—44no—
106#41×C79—41no—
107#41×C79—41no—
108#41×C79C22⋊C412yes—
109#41×C79D812yes—
110C2×Q8×C79—28no—
111C2×Q8×C79—22no—
112C2×Q8×C79—22no—
113C2×Q8×C79—21no—
114C2×Q8×C79—22no—
115C2×Q8×C79—42no—
116C2×Q8×C79—44no—
117C2×Q8×C79—44no—
118C2×Q8×C79—44no—
119C2×Q8×C79—41no—
120C2×Q8×C79—41no—
121C2×Q8×C79—41no—
122C2×Q8×C79SD1611yescharacteristic
123C2×D4×C79—21no—
124C2×D4×C79—21no—
125C2×D4×C79—42no—
126C2×D4×C79—41no—
127C2×D4×C79SD1611yescharacteristic
128C22×C4×C79—21no—
129C22×C4×C79—42no—
130C22×C4×C79—42no—
131C22×C4×C79—41no—
132C22×C4×C79—41no—
133C22×C4×C79C2×D411yescharacteristic
134Q16×C79—28no—
135Q16×C79—22no—
136Q16×C79—21no—
137Q16×C79—22no—
138Q16×C79—22no—
139Q16×C79C22⋊C412yes—
140Q16×C79C22⋊C414yes—
141SD16×C79—28no—
142SD16×C79—22no—
143SD16×C79—22no—
144D8×C79—21no—
145D8×C79C22⋊C412yes—
146M16×C79—22no—
147M16×C79—22no—
148M16×C79—21no—
149M16×C79C22⋊C412yes—
150C2×C8×C79—22no—
151C2×C8×C79—22no—
152C2×C8×C79—21no—
153C2×C8×C79C22×C411yescharacteristic
154C2×C8×C79C2×D411yescharacteristic
155C2×C8×C79C2×D411yescharacteristic
156C2×C8×C79C2×D411yescharacteristic
157C2×C8×C79C2×D411yescharacteristic
158C2×C8×C79C2×D411yescharacteristic
159C23×C79—21no—
160C23×C79—41no—
161C23×C79AGL(1,ℤ/8)11yescharacteristic
162Q8×C79—21no—
163Q8×C79—24no—
164Q8×C79—22no—
165Q8×C79—48no—
166Q8×C79—42no—
167Q8×C79—41no—
168Q8×C79—42no—
169Q8×C79—42no—
170Q8×C79#10212yes—
171D4×C79—22no—
172D4×C79—22no—
173D4×C79—22no—
174D4×C79—21no—
175D4×C79—48no—
176D4×C79—42no—
177D4×C79—42no—
178D4×C79—41no—
179D4×C79#10212yes—
180C2×C4×C79—21no—
181C2×C4×C79—22no—
182C2×C4×C79—22no—
183C2×C4×C79—48no—
184C2×C4×C79—48no—
185C2×C4×C79—42no—
186C2×C4×C79—42no—
187C2×C4×C79—42no—
188C2×C4×C79—41no—
189C2×C4×C79—42no—
190C2×C4×C79—42no—
191C2×C4×C79C2×C22⋊C411yescharacteristic
192C2×C4×C79#12011yescharacteristic
193C2×C4×C79#12011yescharacteristic
194C2×C4×C79C2×D811yescharacteristic
195C2×C4×C79C2×SD1611yescharacteristic
196C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
197C2×Dic79—81no—
198C8×C79—22no—
199C8×C79C2×C22⋊C411yescharacteristic
200C8×C79C2×C22⋊C411yescharacteristic
201C8×C79#12011yescharacteristic
202C8×C79#12011yescharacteristic
203C79⋊C8—81no—
204C22×C79—22no—
205C22×C79—41no—
206C22×C79—42no—
207C22×C79—42no—
208C22×C79—82no—
209C22×C79#41811yescharacteristic
210C22×C79#44711yescharacteristic
211C22×C79#45011yescharacteristic
212C4×C79—22no—
213C4×C79—44no—
214C4×C79—42no—
215C4×C79#37911yescharacteristic
216C4×C79C2×#10211yescharacteristic
217C4×C79#44711yescharacteristic
218C4×C79#45011yescharacteristic
219Dic79—161no—
220#12751—791nomaximal, Sylow
221C2×C79—21no—
222C2×C79—21no—
223C2×C79—21no—
224C2×C79—42no—
225C2×C79—82no—
226C2×C79#178611yescharacteristic
227C2×#578D7911yescharacteristic
228#2148—791no—
229C2×#362—791no—
230#2145—791no—
231#2141—791no—
232#2117—791no—
233#1814—791no—
234C79#1275111yescharacteristic, complemented, Sylow
235C2×2−1+4—21no—
236#578—22no—
237#578—22no—
238#578—22no—
239#578Dic7914yescomplemented
240C2×#137—22no—
241C2×#137—21no—
242C2×#135—21no—
243C2×#135C2×D7911yescharacteristic
244C22×Q16—21no—
245C22×Q16C2×D7911yescharacteristic
246C2×#131C2×D7911yescharacteristic
247C2×M32—791no—
248C2×M32—791no—
249C2×#108—791no—
250#427—791no—
251#421—1581no—
252#362—1582no—
253#362—1581no—
254#361—1581no—
255#360—1582no—
256#358—1582no—
257#350—1582no—
258#344—1581no—
2592−1+4—24no—
2602−1+4—24no—
2612−1+4—42no—
2622−1+4—42no—
263C2×#41—22no—
264C2×#41—21no—
265C2×#41—42no—
266C2×#41—41no—
267C2×#41C22⋊D7911yescharacteristic
268C22×Q8—22no—
269C22×Q8—21no—
270C22×Q8—41no—
271#137—24no—
272#137—28no—
273#137—28no—
274#137—24no—
275#135—24no—
276#135—24no—
277#135—21no—
278#135—21no—
279#135C2×Dic7912yes—
280#135C22⋊D7912yes—
281C2×Q16—22no—
282C2×Q16—24no—
283C2×Q16—24no—
284C2×Q16—21no—
285C2×Q16—21no—
286C2×Q16—21no—
287C2×Q16C2×Dic7912yes—
288C2×Q16C22⋊D7911yescharacteristic
289C2×Q16C22⋊D7912yes—
290C2×Q16C22⋊D7912yes—
291C2×SD16—22no—
292C2×SD16—21no—
293C2×D8C22⋊D7911yescharacteristic
294#131—21no—
295#131—21no—
296#131C2×Dic7912yes—
297#131C22⋊D7912yes—
298C2×M16—21no—
299C2×M16—1581no—
300C2×M16C22⋊D7911yescharacteristic
301C22×C8—21no—
302C22×C8C22×D7911yescharacteristic
303#117—1581no—
304M32—1581no—
305M32—1581no—
306M32—1581no—
307M32—1581no—
308C2×C16—1581no—
309C2×C16—1581no—
310#108—1581no—
311#108—1581no—
312#107—1581no—
313#106—1581no—
314#104—3162no—
315#41—21no—
316#41—21no—
317#41—28no—
318#41—22no—
319#41—22no—
320#41—44no—
321#41—48no—
322#41—48no—
323#41—44no—
324#41—44no—
325#41—41no—
326#41—41no—
327#41C22⋊Dic7912yes—
328#41#1192960912yes—
329C2×Q8—28no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—42no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8#1192961211yescharacteristic
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4#1192961211yescharacteristic
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C2×C22⋊D7911yescharacteristic
353Q16—28no—
354Q16—22no—
355Q16—21no—
356Q16—22no—
357Q16—22no—
358Q16C22⋊Dic7912yes—
359Q16C22⋊Dic7914yes—
360SD16—28no—
361SD16—22no—
362SD16—22no—
363D8—21no—
364D8C22⋊Dic7912yes—
365M16—21no—
366M16—22no—
367M16—22no—
368M16—3161no—
369M16—3161no—
370M16C22⋊Dic7912yes—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—3161no—
375C2×C8C22×Dic7911yescharacteristic
376C2×C8C2×C22⋊D7911yescharacteristic
377C2×C8C2×C22⋊D7911yescharacteristic
378C2×C8C2×C22⋊D7911yescharacteristic
379C2×C8C2×C22⋊D7911yescharacteristic
380C2×C8C2×C22⋊D7911yescharacteristic
381C4⋊C4—3161no—
382C22⋊C4—3161no—
383C42—3161no—
384C16—3161no—
385C16—3161no—
386C23—21no—
387C23—41no—
388C23#1457918711yescharacteristic
389Q8—21no—
390Q8—24no—
391Q8—22no—
392Q8—48no—
393Q8—41no—
394Q8—42no—
395Q8—42no—
396Q8—42no—
397Q8#1457908212yes—
398D4—22no—
399D4—22no—
400D4—22no—
401D4—21no—
402D4—48no—
403D4—42no—
404D4—42no—
405D4—41no—
406D4#1457908212yes—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—6321no—
419C2×C4C2×C22⋊Dic7911yescharacteristic
420C2×C4#1457913011yescharacteristic
421C2×C4#1457913011yescharacteristic
422C2×C4C2×#1192960911yescharacteristic
423C2×C4C2×#1192961211yescharacteristic
424C2×C4#1457918711yescharacteristic
425C8—22no—
426C8—6321no—
427C8C2×C22⋊Dic7911yescharacteristic
428C8C2×C22⋊Dic7911yescharacteristic
429C8#1457913011yescharacteristic
430C8#1457913011yescharacteristic
431C22—22no—
432C22—41no—
433C22—42no—
434C22—42no—
435C22—82no—
436C22#1909856811yescharacteristic
437C22#1909869711yescharacteristic
438C22#1909871411yescharacteristic
439C4—22no—
440C4—44no—
441C4—42no—
442C4—12641no—
443C4#1909841911yescharacteristic
444C4C2×#1457908211yescharacteristic
445C4#1909869711yescharacteristic
446C4#1909871411yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—42no—
451C2—82no—
452C2#2376333111yescharacteristic
subgroupG/H
centerC2#23763331
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC2×#578×C79C2
socleC2×C79#1786
cosocleC22×C8×C79C23
hypercenterC2×#578D79
hypocenterC79#12751
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. #23765149C2semidirect
2. C2×#19098379C2semidirect
3. #23765136C2semidirect
4. #23765120C2semidirect
5. #23765028C2semidirect
6. #23763454C2semidirect
24. #578×C79C4semidirect
234. C79#12751semidirect
239. #578Dic79semidirect

Maximal subgroups

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

subgroupindex
#237651492
C2×#190983792
#237651362
#237651202
#237650282
#237634542
C2×#578×C792
#1275179