Finite Group Databasebeta

← #34441619

Subgroups — #34441619

Order 20224.

660 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099002C211yesmaximal, characteristic
2#23772270C211yesmaximal, characteristic
3#23772263C211yesmaximal, characteristic, complemented
4#23765272C211yesmaximal, characteristic, complemented
5#23765268C211yesmaximal, characteristic, complemented
6#23765261C211yesmaximal, characteristic, complemented
7#23765258C211yesmaximal, characteristic, complemented
8C4×#14579107C211yesmaximal, characteristic, complemented
9SD32×Dic79C211yesmaximal, characteristic, complemented
10C4×#14579105C211yesmaximal, characteristic, complemented
11#23765185C211yesmaximal, characteristic, complemented
12#23765104C211yesmaximal, characteristic
13#358×D79C211yesmaximal, characteristic, complemented
14#23764998C211yesmaximal, characteristic, complemented
15C79×#2175C211yesmaximal, characteristic, complemented
16C2×#14579180C2211yescharacteristic
17C2×Q16×D79C2211yescharacteristic
18#19099042—21no—
19#19099039—21no—
20#19099002C418yescomplemented
21C2×Dic632C2211yescharacteristic
22C2×#14579107C2211yescharacteristic
23C2×#14579105C2211yescharacteristic
24C2×#14579100C2211yescharacteristic
25C2×C16×D79C2211yescharacteristic
26#19098668—21no—
27#19098667—21no—
28#19098663—21no—
29#19098660—21no—
30C4×Dic316C2211yescharacteristic
31C4×Dic316C2211yescharacteristic
32Q16×Dic79C2211yescharacteristic
33#19098657C2211yescharacteristic
34#19098657C2211yescharacteristic
35D8×Dic79C2211yescharacteristic
36#19098600C2211yescharacteristic
37#107×D79C2211yescharacteristic
38#19098570—21no—
39#19098561—21no—
40#19098517C2211yescharacteristic
41#19098391C2211yescharacteristic
42#19098390C2211yescharacteristic
43#19098388C2211yescharacteristic
44#19098368C2211yescharacteristic
45#19098368C2211yescharacteristic
46#19098367C2211yescharacteristic
47#19098366C2211yescharacteristic
48#19098366C2211yescharacteristic
49#19098364C2211yescharacteristic
50#19098338C2211yescharacteristic
51C4×C79⋊C16C2211yescharacteristic
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×#11929607D411yescharacteristic
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#14579119—41no—
84#14579118—41no—
85#14579118D411yescharacteristic
86#14579117—41no—
87C2×Dic79⋊C4—41no—
88C4⋊C4×D79—21no—
89Dic632C2×C414yescomplemented
90#14579107C2×C414yescomplemented
91#14579105C2×C414yescomplemented
92#14579100C2×C414yescomplemented
93C2×C79⋊C16C2311yescharacteristic
94C2×C79⋊C16C2311yescharacteristic
95C16×D79—22no—
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—
106#14579072D411yescharacteristic
107#14579071D411yescharacteristic
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×D79C2×D411yescharacteristic
134Dic316—21no—
135Dic316—21no—
136Dic316C22×C412yes—
137Dic316C22×C412yes—
138#11929614—24no—
139#11929612—24no—
140#11929607—22no—
141C2×C79⋊C8C2×D411yescharacteristic
142C2×C79⋊C8C2×D411yescharacteristic
143C8×D79C22×C412yes—
144C8×D79#4112yes—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—41no—
148Dic79⋊C4—41no—
149Dic79⋊C4—41no—
150Dic79⋊C4—41no—
151Dic79⋊C4—41no—
152Dic79⋊C4—41no—
153Dic79⋊C4D811yescharacteristic
154Dic79⋊C4D811yescharacteristic
155C22⋊Dic79—41no—
156(C2×D79)⋊C4—41no—
157(C2×D79)⋊C4D811yescharacteristic
158(C2×D79)⋊C4D811yescharacteristic
159C4×Dic79—41no—
160C4×Dic79—41no—
161C4×Dic79—41no—
162C4×Dic79—41no—
163C4×Dic79—41no—
164C4×Dic79—41no—
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—161no—
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—161no—
205C2×Dic79C2×D811yescharacteristic
206C2×Dic79C2×D811yescharacteristic
207C2×Dic79C2×D811yescharacteristic
208C4×D79—44no—
209C4×D79—161no—
210C4×D79C4×D412yes—
211C4×D79#13512yes—
212Dic158—22no—
213Dic158—22no—
214Dic158—41no—
215Dic158—41no—
216Dic158—44no—
217C8×C79—21no—
218C8×C79C23×C411yescharacteristic
219C8×C79C2×#4111yescharacteristic
220C79⋊C8—21no—
221C79⋊C8C4×D412yes—
222C22×C79—42no—
223C22×C79—81no—
224C22×C79—81no—
225C22×C79C22×D811yescharacteristic
226C2×D79—21no—
227C2×D79—21no—
228C2×D79C4×D812yes—
229C4×C79—21no—
230C4×C79—21no—
231C4×C79—42no—
232C4×C79—81no—
233C4×C79—81no—
234C4×C79C2×C4×D411yescharacteristic
235C4×C79C2×#13511yescharacteristic
236Dic79—21no—
237Dic79—21no—
238Dic79—42no—
239Dic79—42no—
240Dic79—81no—
241Dic79—81no—
242Dic79C4×D812yes—
243#33455—791nomaximal, Sylow
244C2×C79—82no—
245C2×C79C2×C4×D811yescharacteristic
246C2×C79#341611yescharacteristic
247C2×C79#341711yescharacteristic
248D79—22no—
249C2×#510—791no—
250#2951—791no—
251#2950—791no—
252#2176—791no—
253#2176—791no—
254#2175—791no—
255#2175D7911yescharacteristic, complemented
256C4×Q32—791no—
257C4×Q32—791no—
258C4×SD32—791no—
259C4×SD32—791no—
260#2159—791no—
261#2138—791no—
262C2×#358—791no—
263#2108—791no—
264C79#3345511yescharacteristic, complemented, Sylow
265C2×#135—791no—
266C22×Q16—791no—
267#520—1581no—
268#519—1581no—
269#510—798no—
270C2×Q32—791no—
271C2×Q32—791no—
272C2×SD32—791no—
273C2×SD32C2×D7911yescharacteristic
274C2×M32—791no—
275C22×C16—791no—
276#441—1581no—
277#441—1581no—
278#440—1581no—
279#440—1581no—
280C4×Q16—791no—
281C4×Q16—791no—
282C4×Q16—791no—
283C4×Q16—791no—
284C4×Q16—791no—
285C4×Q16C2×D7911yescharacteristic
286C4×D8—791no—
287C4×D8C2×D7911yescharacteristic
288#427—791no—
289C2×#107—791no—
290#419—1581no—
291#417—1581no—
292#405—791no—
293#367—791no—
294#367—791no—
295#366—791no—
296#366C2×D7911yescharacteristic
297#358—791no—
298#358—791no—
299#358—791no—
300#358—791no—
301#358—791no—
302#358C2×D7911yescharacteristic
303#357—791no—
304#357C2×D7911yescharacteristic
305#346—791no—
306#346C2×D7911yescharacteristic
307C4×C16—791no—
308C4×C16—791no—
309C2×#41—1581no—
310C22×Q8—1581no—
311#135—794no—
312#135—1582no—
313C2×Q16—794no—
314C2×Q16—791no—
315C2×Q16—791no—
316C2×Q16—1582no—
317C2×Q16—1581no—
318C2×Q16C22×D7911yescharacteristic
319C2×SD16—1581no—
320C2×D8C22×D7911yescharacteristic
321C2×M16—791no—
322C22×C8—791no—
323C4×Q8—21no—
324C4×Q8—3161no—
325C4×Q8—3161no—
326C4×Q8—1581no—
327C4×Q8—1581no—
328C4×Q8—1581no—
329C4×Q8—1581no—
330C4×Q8—1581no—
331C4×D4—21no—
332C4×D4—3161no—
333C4×D4—3161no—
334C4×D4—1581no—
335#117—3161no—
336#117—3161no—
337#117—3161no—
338#117—791no—
339#117—1581no—
340C2×C4⋊C4—3161no—
341C2×C4⋊C4—1581no—
342Q32—794no—
343Q32—794no—
344SD32—794no—
345SD32C4×D7914yescomplemented
346M32—794no—
347C2×C16—791no—
348C2×C16—791no—
349C2×C16—1582no—
350C2×C16C22×D7911yescharacteristic
351C2×C16C22×D7911yescharacteristic
352#107—791no—
353#107—791no—
354#107—1581no—
355#107C22×D7911yescharacteristic
356#107C22×D7911yescharacteristic
357#106—1581no—
358#103—21no—
359#103—1581no—
360#103—1581no—
361#103—1581no—
362#103—1581no—
363#103—1581no—
364#102—21no—
365#102—1581no—
366#97—791no—
367#97—791no—
368C4×C8—791no—
369C4×C8C22×D7911yescharacteristic
370#41—3162no—
371#41—1584no—
372C2×Q8—21no—
373C2×Q8—3161no—
374C2×Q8—3162no—
375C2×Q8—1584no—
376C2×Q8—1581no—
377C2×Q8—1581no—
378C2×D4—21no—
379C2×D4—3161no—
380C22×C4—41no—
381C22×C4—3161no—
382C22×C4—3161no—
383C22×C4—3161no—
384C22×C4—3161no—
385C22×C4—791no—
386Q16—21no—
387Q16—792no—
388Q16—792no—
389Q16—1584no—
390Q16—1581no—
391Q16—1581no—
392Q16C2×C4×D7912yes—
393SD16—1584no—
394D8—21no—
395D8C2×C4×D7912yes—
396M16—1582no—
397C2×C8—791no—
398C2×C8—792no—
399C2×C8—792no—
400C2×C8—791no—
401C2×C8D4×D7911yescharacteristic
402C2×C8C23×D7911yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—3161no—
407C4⋊C4—3161no—
408C4⋊C4—3161no—
409C4⋊C4—3161no—
410C4⋊C4—3161no—
411C4⋊C4—3161no—
412C4⋊C4—791no—
413C4⋊C4—791no—
414C4⋊C4—1581no—
415C4⋊C4—1581no—
416C22⋊C4—41no—
417C22⋊C4—3161no—
418C22⋊C4—3161no—
419C22⋊C4—791no—
420C22⋊C4—791no—
421C42—41no—
422C42—3161no—
423C42—3161no—
424C42—3161no—
425C42—3161no—
426C42—3161no—
427C42—3161no—
428C42—791no—
429C42D4×D7911yescharacteristic
430C16—21no—
431C16—792no—
432C16—1581no—
433C16C2×C4×D7912yes—
434C23—41no—
435C23—791no—
436Q8—22no—
437Q8—41no—
438Q8—3164no—
439Q8—3161no—
440Q8—3161no—
441Q8—1582no—
442Q8—1582no—
443D4—22no—
444D4—41no—
445D4—3164no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—161no—
450C2×C4—3164no—
451C2×C4—3161no—
452C2×C4—3161no—
453C2×C4—3164no—
454C2×C4—3161no—
455C2×C4—3161no—
456C2×C4—791no—
457C2×C4—792no—
458C2×C4—792no—
459C2×C4—791no—
460C2×C4—791no—
461C2×C4—12641no—
462C2×C4—12641no—
463C2×C4D8×D7911yescharacteristic
464C2×C4D8×D7911yescharacteristic
465C2×C4C2×D4×D7911yescharacteristic
466C8—21no—
467C8—792no—
468C8—1581no—
469C8C22×C4×D7911yescharacteristic
470C8#41×D7911yescharacteristic
471C22—42no—
472C22—81no—
473C22—81no—
474C22—792no—
475C22—1581no—
476C22—1581no—
477C22C2×D8×D7911yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—3162no—
484C4—3162no—
485C4—792no—
486C4—6321no—
487C4—6321no—
488C4—1581no—
489C4—1581no—
490C4C4×D4×D7911yescharacteristic
491C4#135×D7911yescharacteristic
492C2—82no—
493C2—1582no—
494C2C4×D8×D7911yescharacteristic
495C2#2377654611yescharacteristic
496C2#2377655211yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2175C2
socleC22×C79C22×D8
prime radicalC2×#14579180C22
cosocleC2×C8×C79C24
hypercenter#2175D79
hypocenterC79#33455
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. #23772263C2semidirect
4. #23765272C2semidirect
5. #23765268C2semidirect
6. #23765261C2semidirect
7. #23765258C2semidirect
8. C4×#14579107C2semidirect
9. SD32×Dic79C2semidirect
10. C4×#14579105C2semidirect
11. #23765185C2semidirect
13. #358×D79C2semidirect
14. #23764998C2semidirect
15. C79×#2175C2semidirect
20. #19099002C4semidirect
68. Q16×D79C2×C4semidirect
89. Dic632C2×C4semidirect
90. #14579107C2×C4semidirect
91. #14579105C2×C4semidirect
92. #14579100C2×C4semidirect
113. SD32×C79C2×C4semidirect
255. #2175D79semidirect
264. C79#33455semidirect
345. SD32C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190990022
#237722702
#237722632
#237652722
#237652682
#237652612
#237652582
C4×#145791072
SD32×Dic792
C4×#145791052
#237651852
#237651042
#358×D792
#237649982
C79×#21752
#3345579