Finite Group Databasebeta

← #34441601

Subgroups — #34441601

Order 20224.

674 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099002C211yesmaximal, characteristic
2C4×#14579180C211yesmaximal, characteristic
3C4×Q16×D79C211yesmaximal, characteristic, complemented
4#23765272C211yesmaximal, characteristic, complemented
5#23765269C211yesmaximal, characteristic, complemented
6#23765264C211yesmaximal, characteristic, complemented
7#23765258C211yesmaximal, characteristic, complemented
8#23765256C211yesmaximal, characteristic, complemented
9#23765255C211yesmaximal, characteristic, complemented
10SD32×Dic79C211yesmaximal, characteristic, complemented
11#23765185C211yesmaximal, characteristic, complemented
12#23765118C211yesmaximal, characteristic, complemented
13#23765109C211yesmaximal, characteristic
14#23764993C211yesmaximal, characteristic, complemented
15C79×#2175C211yesmaximal, characteristic, complemented
16C2×#14579180C2211yescharacteristic
17C2×Q16×D79C2211yescharacteristic
18C4×#11929623—21no—
19C4×Q8×D79—21no—
20#19099002C418yescomplemented
21C2×Dic632C2211yescharacteristic
22C2×#14579107C2211yescharacteristic
23C2×#14579105C2211yescharacteristic
24C2×#14579100C2211yescharacteristic
25C2×#14579100C2211yescharacteristic
26C4×Dic316C2211yescharacteristic
27C4×Dic316C2211yescharacteristic
28Q16×Dic79C2211yescharacteristic
29#19098657C2211yescharacteristic
30#19098657C2211yescharacteristic
31C4×#11929614—21no—
32#19098655—21no—
33#19098654—21no—
34C4×#11929612—21no—
35D8×Dic79C2211yescharacteristic
36#19098600C2211yescharacteristic
37#107×D79C2211yescharacteristic
38#19098556—21no—
39#103×D79—21no—
40C4×C8×D79C2211yescharacteristic
41#19098391C2211yescharacteristic
42#19098390C2211yescharacteristic
43#19098388C2211yescharacteristic
44#19098368C2211yescharacteristic
45#19098368C2211yescharacteristic
46#19098367C2211yescharacteristic
47#19098366C2211yescharacteristic
48#19098366C2211yescharacteristic
49#19098364C2211yescharacteristic
50#19098339C2211yescharacteristic
51C16×Dic79C2211yescharacteristic
52C16×Dic79C2211yescharacteristic
53C2×SD32×C79C2211yescharacteristic
54C4×Q16×C79C2211yescharacteristic
55C4×D8×C79C2211yescharacteristic
56#366×C79C2211yescharacteristic
57#358×C79C2211yescharacteristic
58#357×C79C2211yescharacteristic
59#346×C79C2211yescharacteristic
60C2×#11929623—21no—
61C2×Q8×D79—21no—
62#14579180—22no—
63#14579180C2×C414yes—
64C2×Dic316C2311yescharacteristic
65C2×Dic316C2311yescharacteristic
66C2×#11929614—21no—
67Q16×D79—22no—
68Q16×D79C2×C414yescomplemented
69C2×#11929612—21no—
70C2×C8×D79D411yescharacteristic
71C2×C8×D79C2311yescharacteristic
72Q8×Dic79—21no—
73C4×Dic158—21no—
74C4×Dic158—21no—
75C4×Dic158—41no—
76#14579126—21no—
77#14579126—21no—
78#14579126—41no—
79D4×Dic79—21no—
80C4×C22⋊D79—41no—
81(C22⋊D79)⋊C4—41no—
82#14579119—21no—
83#14579117—41no—
84C4⋊C4×D79—21no—
85C4⋊C4×D79—41no—
86C2×C4×Dic79—41no—
87C42×D79—41no—
88C42×D79D411yescharacteristic
89Dic632C2×C414yescomplemented
90#14579107C2×C414yescomplemented
91#14579105C2×C414yescomplemented
92#14579100—22no—
93#14579100C2×C414yescomplemented
94C2×C79⋊C16C2311yescharacteristic
95C2×C79⋊C16C2311yescharacteristic
96#14579095C2311yescharacteristic
97#14579095C2311yescharacteristic
98#14579094—21no—
99#14579092—21no—
100#14579086—21no—
101#14579086—21no—
102#14579085—21no—
103#14579084—21no—
104#14579084—21no—
105#14579082—21no—
106C4×C79⋊C8D411yescharacteristic
107C8×Dic79D411yescharacteristic
108C8×Dic79C2311yescharacteristic
109C2×Q16×C79C2311yescharacteristic
110C2×D8×C79C2311yescharacteristic
111C4×Q8×C79—21no—
112C4×D4×C79—21no—
113SD32×C79C2×C414yescomplemented
114C2×C16×C79C2311yescharacteristic
115C2×C16×C79C2311yescharacteristic
116#107×C79C2311yescharacteristic
117#107×C79C2311yescharacteristic
118#103×C79—21no—
119#102×C79—21no—
120C4×C8×C79C2311yescharacteristic
121#11929623—24no—
122#11929623—42no—
123C2×Dic158—21no—
124C2×Dic158—21no—
125C2×Dic158—41no—
126Q8×D79—24no—
127Q8×D79—42no—
128C2×C22⋊D79—41no—
129C22×Dic79—41no—
130C22×Dic79—41no—
131C2×C4×D79—41no—
132C2×C4×D79—41no—
133C2×C4×D79D811yescharacteristic
134C2×C4×D79D811yescharacteristic
135C2×C4×D79C2×D411yescharacteristic
136Dic316—21no—
137Dic316—21no—
138Dic316C22×C412yes—
139Dic316C22×C412yes—
140#11929614—24no—
141#11929612—24no—
142C2×C79⋊C8C2×D411yescharacteristic
143C2×C79⋊C8C2×D411yescharacteristic
144C8×D79—22no—
145C8×D79C22×C412yes—
146C8×D79#4112yes—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—41no—
150Dic79⋊C4—41no—
151Dic79⋊C4—41no—
152Dic79⋊C4—41no—
153Dic79⋊C4—41no—
154Dic79⋊C4—41no—
155C22⋊Dic79—41no—
156(C2×D79)⋊C4—41no—
157C4×Dic79—41no—
158C4×Dic79—41no—
159C4×Dic79—41no—
160C4×Dic79—41no—
161C4×Dic79—41no—
162C4×Dic79—41no—
163C4×Dic79D811yescharacteristic
164C4×Dic79D811yescharacteristic
165C4×Dic79C2×D411yescharacteristic
166C79⋊C16—21no—
167C79⋊C16C22×C412yes—
168C2×Q8×C79—21no—
169C2×D4×C79—21no—
170C22×C4×C79—41no—
171Q16×C79—21no—
172Q16×C79C22×C412yes—
173D8×C79—21no—
174D8×C79C22×C412yes—
175C2×C8×C79C2×D411yescharacteristic
176C2×C8×C79C2411yescharacteristic
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C42×C79—41no—
182C42×C79C2×D411yescharacteristic
183C16×C79—21no—
184C16×C79C22×C412yes—
185C23×C79—41no—
186C22×D79C2×D811yescharacteristic
187Q8×C79—22no—
188Q8×C79—41no—
189D4×C79—22no—
190D4×C79—41no—
191C2×C4×C79—41no—
192C2×C4×C79—41no—
193C2×C4×C79—41no—
194C2×C4×C79—82no—
195C2×C4×C79C2×D811yescharacteristic
196C2×C4×C79C2×D811yescharacteristic
197C2×C4×C79C22×D411yescharacteristic
198C22⋊D79—44no—
199C2×Dic79—41no—
200C2×Dic79—41no—
201C2×Dic79—44no—
202C2×Dic79—41no—
203C2×Dic79—41no—
204C2×Dic79—82no—
205C2×Dic79C2×D811yescharacteristic
206C2×Dic79C2×D811yescharacteristic
207C2×Dic79C2×D811yescharacteristic
208C4×D79—22no—
209C4×D79—41no—
210C4×D79—44no—
211C4×D79—82no—
212C4×D79C4×D412yes—
213C4×D79#13512yes—
214Dic158—22no—
215Dic158—22no—
216Dic158—41no—
217Dic158—41no—
218Dic158—44no—
219C8×C79—21no—
220C8×C79C23×C411yescharacteristic
221C8×C79C2×#4111yescharacteristic
222C79⋊C8—21no—
223C79⋊C8C4×D412yes—
224C22×C79—42no—
225C22×C79—81no—
226C22×C79—81no—
227C22×C79C22×D811yescharacteristic
228C2×D79—21no—
229C2×D79—21no—
230C2×D79C4×D812yes—
231C4×C79—21no—
232C4×C79—21no—
233C4×C79—42no—
234C4×C79—81no—
235C4×C79—81no—
236C4×C79C2×C4×D411yescharacteristic
237C4×C79C2×#13511yescharacteristic
238Dic79—21no—
239Dic79—42no—
240Dic79—42no—
241Dic79—81no—
242Dic79—81no—
243Dic79C4×D812yes—
244Dic79#51012yescomplemented
245C4×#510—791nomaximal, Sylow
246C2×C79—82no—
247C2×C79C2×C4×D811yescharacteristic
248C2×C79C2×#51011yescharacteristic
249C2×C79#341411yescharacteristic
250D79—22no—
251C2×#510—791no—
252C4×#135—791no—
253C2×C4×Q16—791no—
254#2176—791no—
255#2176—791no—
256#2175—791no—
257#2175D7911yescharacteristic, complemented
258C4×Q32—791no—
259C4×Q32—791no—
260C4×SD32—791no—
261C4×SD32—791no—
262#2159—791no—
263#2141—791no—
264#2139—791no—
265C4×M32—791no—
266C79C4×#51011yescharacteristic, complemented, Sylow
267C2×#135—791no—
268C22×Q16—791no—
269C4×#41—1581no—
270C2×C4×Q8—1581no—
271#510—798no—
272C2×Q32—791no—
273C2×Q32—791no—
274C2×SD32—791no—
275C2×SD32C2×D7911yescharacteristic
276C2×M32—791no—
277C2×M32—791no—
278C4×Q16—791no—
279C4×Q16—791no—
280C4×Q16—791no—
281C4×Q16—791no—
282C4×Q16—791no—
283C4×Q16—1581no—
284C4×Q16—1581no—
285C4×Q16C2×D7911yescharacteristic
286C4×SD16—1581no—
287C4×SD16—1581no—
288C4×D8—791no—
289C4×D8C2×D7911yescharacteristic
290#427—791no—
291C2×#107—791no—
292#416—1581no—
293C2×#103—1581no—
294C2×C4×C8—791no—
295#367—791no—
296#367—791no—
297#366—791no—
298#366C2×D7911yescharacteristic
299#358—791no—
300#358—791no—
301#358—791no—
302#358—791no—
303#358—791no—
304#358C2×D7911yescharacteristic
305#357—791no—
306#357C2×D7911yescharacteristic
307#346—791no—
308#346C2×D7911yescharacteristic
309C4×C16—791no—
310C4×C16—791no—
311C2×#41—1581no—
312C22×Q8—1581no—
313#135—794no—
314#135—1582no—
315C2×Q16—794no—
316C2×Q16—791no—
317C2×Q16—791no—
318C2×Q16—1582no—
319C2×Q16—1581no—
320C2×Q16C22×D7911yescharacteristic
321C2×SD16—1581no—
322C2×D8C22×D7911yescharacteristic
323C22×C8—791no—
324C22×C8—791no—
325C4×Q8—21no—
326C4×Q8—3161no—
327C4×Q8—3161no—
328C4×Q8—1581no—
329C4×Q8—1581no—
330C4×Q8—1581no—
331C4×Q8—1581no—
332C4×Q8—1581no—
333C4×D4—21no—
334C4×D4—3161no—
335C4×D4—3161no—
336C4×D4—1581no—
337#117—3161no—
338#117—1581no—
339C2×C4⋊C4—3161no—
340C2×C4⋊C4—1581no—
341C2×C42—3161no—
342C2×C42—3161no—
343C2×C42—791no—
344Q32—794no—
345Q32—794no—
346SD32—794no—
347SD32C4×D7914yescomplemented
348M32—794no—
349M32—1582no—
350C2×C16—791no—
351C2×C16—791no—
352C2×C16C22×D7911yescharacteristic
353C2×C16C22×D7911yescharacteristic
354#107—791no—
355#107—791no—
356#107—1581no—
357#107C22×D7911yescharacteristic
358#107C22×D7911yescharacteristic
359#106—1581no—
360#103—21no—
361#103—1581no—
362#103—1581no—
363#103—1581no—
364#103—1581no—
365#103—1581no—
366#102—21no—
367#102—1581no—
368C4×C8—791no—
369C4×C8—791no—
370C4×C8—791no—
371C4×C8C22×D7911yescharacteristic
372#41—3162no—
373#41—1584no—
374C2×Q8—21no—
375C2×Q8—3161no—
376C2×Q8—3162no—
377C2×Q8—1584no—
378C2×Q8—1581no—
379C2×Q8—1581no—
380C2×D4—21no—
381C2×D4—3161no—
382C22×C4—41no—
383C22×C4—3161no—
384C22×C4—3161no—
385C22×C4—3161no—
386C22×C4—3161no—
387C22×C4—791no—
388C22×C4—791no—
389C22×C4—791no—
390Q16—21no—
391Q16—792no—
392Q16—792no—
393Q16—1584no—
394Q16—1581no—
395Q16—1581no—
396Q16C2×C4×D7912yes—
397SD16—1584no—
398D8—21no—
399D8C2×C4×D7912yes—
400C2×C8—791no—
401C2×C8—792no—
402C2×C8—792no—
403C2×C8—791no—
404C2×C8—1582no—
405C2×C8D4×D7911yescharacteristic
406C2×C8C23×D7911yescharacteristic
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—41no—
410C4⋊C4—3161no—
411C4⋊C4—3161no—
412C4⋊C4—3161no—
413C4⋊C4—3161no—
414C4⋊C4—3161no—
415C4⋊C4—3161no—
416C4⋊C4—1581no—
417C4⋊C4—1581no—
418C22⋊C4—41no—
419C22⋊C4—3161no—
420C22⋊C4—3161no—
421C42—41no—
422C42—3161no—
423C42—3161no—
424C42—3161no—
425C42—3161no—
426C42—3161no—
427C42—3161no—
428C42—791no—
429C42—791no—
430C42—791no—
431C42D4×D7911yescharacteristic
432C16—21no—
433C16—792no—
434C16—1581no—
435C16C2×C4×D7912yes—
436C23—41no—
437C23—791no—
438Q8—22no—
439Q8—41no—
440Q8—3164no—
441Q8—3161no—
442Q8—3161no—
443Q8—1582no—
444Q8—1582no—
445D4—22no—
446D4—41no—
447D4—3164no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—3161no—
453C2×C4—3164no—
454C2×C4—3161no—
455C2×C4—3161no—
456C2×C4—3164no—
457C2×C4—3161no—
458C2×C4—3161no—
459C2×C4—791no—
460C2×C4—791no—
461C2×C4—792no—
462C2×C4—792no—
463C2×C4—791no—
464C2×C4—6322no—
465C2×C4—6322no—
466C2×C4—1582no—
467C2×C4D8×D7911yescharacteristic
468C2×C4D8×D7911yescharacteristic
469C2×C4C2×D4×D7911yescharacteristic
470C8—21no—
471C8—792no—
472C8—1581no—
473C8C22×C4×D7911yescharacteristic
474C8#41×D7911yescharacteristic
475C22—42no—
476C22—81no—
477C22—81no—
478C22—792no—
479C22—1581no—
480C22—1581no—
481C22C2×D8×D7911yescharacteristic
482C4—21no—
483C4—21no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—3162no—
488C4—3162no—
489C4—792no—
490C4—792no—
491C4—6321no—
492C4—6321no—
493C4—1581no—
494C4C4×D4×D7911yescharacteristic
495C4#135×D7911yescharacteristic
496C2—82no—
497C2—1582no—
498C2C4×D8×D7911yescharacteristic
499C2#510×D7911yescharacteristic
500C2#2377653211yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2175C2
socleC22×C79C22×D8
prime radicalC2×#14579180C22
cosocleC2×C8×C79C24
hypercenter#2175D79
hypocenterC79C4×#510
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
3. C4×Q16×D79C2semidirect
4. #23765272C2semidirect
5. #23765269C2semidirect
6. #23765264C2semidirect
7. #23765258C2semidirect
8. #23765256C2semidirect
9. #23765255C2semidirect
10. SD32×Dic79C2semidirect
11. #23765185C2semidirect
12. #23765118C2semidirect
14. #23764993C2semidirect
15. C79×#2175C2semidirect
20. #19099002C4semidirect
68. Q16×D79C2×C4semidirect
89. Dic632C2×C4semidirect
90. #14579107C2×C4semidirect
91. #14579105C2×C4semidirect
93. #14579100C2×C4semidirect
113. SD32×C79C2×C4semidirect
244. Dic79#510semidirect
257. #2175D79semidirect
266. C79C4×#510semidirect
347. SD32C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190990022
C4×#145791802
C4×Q16×D792
#237652722
#237652692
#237652642
#237652582
#237652562
#237652552
SD32×Dic792
#237651852
#237651182
#237651092
#237649932
C79×#21752
C4×#51079