Finite Group Databasebeta

← #34183130

Subgroups — #34183130

Order 20224.

578 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
1D4×#11929608C211yesmaximal, characteristic, complemented
2#23764320C211yesmaximal, characteristic, complemented
3C4×#14579082C211yesmaximal, characteristic, complemented
4#23761851C211yesmaximal, characteristic, complemented
5#23761302C211yesmaximal, characteristic, complemented
6#23761236C211yesmaximal, characteristic, complemented
7C79×#3100C211yesmaximal, characteristic
8C22×#11929608—21no—
9#19098641—21no—
10#19098635—22no—
11D4×C79⋊C8—21no—
12#19098587C2211yescharacteristic
13C2×#14579088—21no—
14C2×#14579088—21no—
15C2×#14579082—21no—
16#19098527—21no—
17C4×#11929608C2211yescharacteristic
18C2×C4×C79⋊C8C2211yescharacteristic
19C4×C4⋊Dic79C2211yescharacteristic
20#19098325—21no—
21D4⋊(C79⋊C8)—22no—
22#532×C79C2211yescharacteristic
23C2×C4×D4×C79C2211yescharacteristic
24#480×C79—21no—
25#474×C79C411yescharacteristic, complemented
26#474×C79C411yescharacteristic, complemented
27#459×C79C411yescharacteristic, complemented
28#459×C79C411yescharacteristic, complemented
29#447×C79—21no—
30C4×D8×C79—21no—
31C4×D8×C79—21no—
32#431×C79C2211yescharacteristic
33C2×#11929608—22no—
34C2×#11929608—21no—
35C2×#11929608—42no—
36C2×#11929608—42no—
37C2×#11929608D411yescharacteristic
38C2×#11929608D411yescharacteristic
39C22×C79⋊C8—22no—
40C22×C79⋊C8—21no—
41C2×C4⋊Dic79—21no—
42C2×C4×Dic79—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)D412yes—
45#14579088—42no—
46#14579088—42no—
47#14579082—24no—
48C22⋊(C79⋊C8)—22no—
49C22⋊(C79⋊C8)—22no—
50#14579072—21no—
51C4×C79⋊C8—21no—
52C4×C79⋊C8—21no—
53C4×C79⋊C8—21no—
54C79⋊#95—21no—
55C2×#41×C79—21no—
56C22×D4×C79D411yescharacteristic
57C23×C4×C79—21no—
58C2×D8×C79—21no—
59C2×D8×C79—21no—
60C4⋊Q8×C79C2×C411yescharacteristic
61#127×C79C2×C411yescharacteristic
62#124×C79—21no—
63#121×C79—21no—
64#121×C79—21no—
65C4×D4×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79—21no—
68C4×D4×C79—21no—
69C4×D4×C79C2×C411yescharacteristic
70C4×D4×C79C2×C411yescharacteristic
71C4×D4×C79D411yescharacteristic
72C4×D4×C79D411yescharacteristic
73C2×C4⋊C4×C79D411yescharacteristic
74C2×C22⋊C4×C79—21no—
75C2×C42×C79C2311yescharacteristic
76#107×C79—21no—
77#107×C79—21no—
78C4⋊C8×C79C2×C411yescharacteristic
79C4⋊C8×C79C2×C411yescharacteristic
80#102×C79—21no—
81#102×C79—21no—
82#102×C79—21no—
83#102×C79—21no—
84C22⋊C8×C79—21no—
85C4×C8×C79—21no—
86C22×Dic79—41no—
87#11929608—21no—
88#11929608—21no—
89#11929608—21no—
90#11929608—44no—
91#11929608—41no—
92#11929608—41no—
93#11929608D812yescomplemented
94C2×C79⋊C8—22no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—21no—
97C2×C79⋊C8—22no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—44no—
100C2×C79⋊C8—41no—
101C4⋊Dic79—22no—
102C4⋊Dic79—41no—
103C4×Dic79—41no—
104C4×Dic79—41no—
105C24×C79—21no—
106#41×C79—41no—
107#41×C79—41no—
108#41×C79—41no—
109#41×C79—41no—
110C2×Q8×C79—21no—
111C2×D4×C79—22no—
112C2×D4×C79—21no—
113C2×D4×C79—21no—
114C2×D4×C79—21no—
115C2×D4×C79—21no—
116C2×D4×C79—41no—
117C2×D4×C79—41no—
118C2×D4×C79C22⋊C411yescharacteristic
119C2×D4×C79C22⋊C411yescharacteristic
120C22×C4×C79—21no—
121C22×C4×C79—21no—
122C22×C4×C79—21no—
123C22×C4×C79—21no—
124C22×C4×C79—21no—
125C22×C4×C79—41no—
126C22×C4×C79—41no—
127C22×C4×C79—41no—
128C22×C4×C79—41no—
129C22×C4×C79—41no—
130C22×C4×C79C2×D411yescharacteristic
131C22×C4×C79C2×D411yescharacteristic
132C22×C4×C79#4111yescharacteristic
133D8×C79—42no—
134D8×C79—42no—
135C2×C8×C79—21no—
136C2×C8×C79—21no—
137C4⋊C4×C79—21no—
138C4⋊C4×C79—21no—
139C4⋊C4×C79—21no—
140C4⋊C4×C79C22⋊C411yescharacteristic
141C4⋊C4×C79C22⋊C411yescharacteristic
142C22⋊C4×C79—21no—
143C22⋊C4×C79—21no—
144C22⋊C4×C79—21no—
145C22⋊C4×C79—21no—
146C22⋊C4×C79—41no—
147C22⋊C4×C79—41no—
148C42×C79C22×C411yescharacteristic
149C42×C79C2×D411yescharacteristic
150C42×C79C2×D411yescharacteristic
151C42×C79#4111yescharacteristic
152C23×C79—21no—
153C23×C79—21no—
154C23×C79—42no—
155C23×C79—41no—
156C23×C79—41no—
157C23×C79—41no—
158C23×C79—81no—
159C23×C79#12111yescharacteristic
160Q8×C79—41no—
161Q8×C79—41no—
162D4×C79—21no—
163D4×C79—21no—
164D4×C79—21no—
165D4×C79—42no—
166D4×C79—42no—
167D4×C79—41no—
168D4×C79—41no—
169D4×C79—41no—
170D4×C79—41no—
171D4×C79—81no—
172D4×C79—81no—
173D4×C79#10412yescomplemented
174C2×C4×C79—21no—
175C2×C4×C79—21no—
176C2×C4×C79—21no—
177C2×C4×C79—21no—
178C2×C4×C79—41no—
179C2×C4×C79—41no—
180C2×C4×C79—41no—
181C2×C4×C79—41no—
182C2×C4×C79—41no—
183C2×C4×C79—41no—
184C2×C4×C79—41no—
185C2×C4×C79—41no—
186C2×C4×C79—41no—
187C2×C4×C79—41no—
188C2×C4×C79—41no—
189C2×C4×C79—41no—
190C2×C4×C79—81no—
191C2×C4×C79—81no—
192C2×C4×C79C2×C22⋊C411yescharacteristic
193C2×C4×C79C4×D411yescharacteristic
194C2×C4×C79C4×D411yescharacteristic
195C2×C4×C79#12011yescharacteristic
196C2×C4×C79#12111yescharacteristic
197C2×C4×C79#12311yescharacteristic
198C2×C4×C79C2×D811yescharacteristic
199C2×C4×C79#13511yescharacteristic
200C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
201C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
202C2×Dic79—42no—
203C2×Dic79—81no—
204C2×Dic79—81no—
205C8×C79—41no—
206C8×C79—41no—
207C79⋊C8—22no—
208C79⋊C8—42no—
209C79⋊C8—42no—
210C79⋊C8—41no—
211C22×C79—21no—
212C22×C79—21no—
213C22×C79—22no—
214C22×C79—41no—
215C22×C79—41no—
216C22×C79—41no—
217C22×C79—41no—
218C22×C79—41no—
219C22×C79—82no—
220C22×C79—82no—
221C22×C79—81no—
222C22×C79—81no—
223C22×C79—81no—
224C22×C79#38611yescharacteristic
225C22×C79#45911yescharacteristic
226C22×C79#46311yescharacteristic
227C4×C79—21no—
228C4×C79—21no—
229C4×C79—21no—
230C4×C79—41no—
231C4×C79—41no—
232C4×C79—81no—
233C4×C79C2×#10411yescharacteristic
234C4×C79#42111yescharacteristic
235C4×C79C4×D811yescharacteristic
236C4×C79#44211yescharacteristic
237C4×C79#44711yescharacteristic
238C4×C79#44911yescharacteristic
239Dic79—82no—
240#8991—791nomaximal, Sylow
241C2×C79—21no—
242C2×C79—21no—
243C2×C79—42no—
244C2×C79—81no—
245C2×C79—161no—
246C2×C79#187111yescharacteristic
247C2×C79#187911yescharacteristic
248C2×C79#188111yescharacteristic
249#3100D7911yescharacteristic
250D4×M16—791no—
251#1980—791no—
252C4×#102—791no—
253#1492—791no—
254#1303—791no—
255#1278—791no—
256C79#899111yescharacteristic, complemented, Sylow
257C22×M16—1581no—
258#532C2×D7911yescharacteristic
259C2×C4×D4C2×D7911yescharacteristic
260#480—21no—
261#474Dic7911yescharacteristic, complemented
262#474Dic7911yescharacteristic, complemented
263#459Dic7911yescharacteristic, complemented
264#459Dic7911yescharacteristic, complemented
265#447—21no—
266C4×D8—21no—
267C4×D8—21no—
268#436—1581no—
269#435—1582no—
270C8×D4—1581no—
271#431C2×D7911yescharacteristic
272#423—791no—
273C2×#104—1581no—
274C2×#104—1581no—
275C2×#102—1581no—
276#407—1581no—
277C4×M16—791no—
278C2×C4×C8—791no—
279C4×C4⋊C4—791no—
280#340—1581no—
281D4⋊C8—1582no—
282C2×#41—21no—
283C22×D4C22⋊D7911yescharacteristic
284C23×C4—21no—
285C2×D8—21no—
286C2×D8—21no—
287C2×M16—3162no—
288C2×M16—3162no—
289C2×M16—791no—
290C2×M16—791no—
291C2×M16—1582no—
292C2×M16—1581no—
293C22×C8—1582no—
294C22×C8—1581no—
295C4⋊Q8C2×Dic7911yescharacteristic
296#127C2×Dic7911yescharacteristic
297#124—21no—
298#121—21no—
299#121—21no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4C2×Dic7911yescharacteristic
305C4×D4C2×Dic7911yescharacteristic
306C4×D4C22⋊D7911yescharacteristic
307C4×D4C22⋊D7911yescharacteristic
308C2×C4⋊C4—1581no—
309C2×C4⋊C4C22⋊D7911yescharacteristic
310C2×C22⋊C4—21no—
311C2×C42—1581no—
312C2×C42C22×D7911yescharacteristic
313#107—21no—
314#107—21no—
315C4⋊C8—792no—
316C4⋊C8—1581no—
317C4⋊C8C2×Dic7911yescharacteristic
318C4⋊C8C2×Dic7911yescharacteristic
319#104—3162no—
320#104—3162no—
321#102—21no—
322#102—21no—
323#102—21no—
324#102—21no—
325#102—1584no—
326C22⋊C8—21no—
327C22⋊C8—1582no—
328C22⋊C8—1582no—
329#97—1581no—
330C4×C8—21no—
331C4×C8—1581no—
332C4×C8—1581no—
333C4×C8—1581no—
334#95—1581no—
335C24—21no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340C2×Q8—21no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4C22⋊Dic7911yescharacteristic
349C2×D4C22⋊Dic7911yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—3161no—
361C22×C4D4×D7911yescharacteristic
362C22×C4C2×C22⋊D7911yescharacteristic
363C22×C4#1192962311yescharacteristic
364D8—42no—
365D8—42no—
366M16—3161no—
367M16—3161no—
368M16—3164no—
369M16—792no—
370M16—1581no—
371M16—1581no—
372M16—1581no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—3161no—
376C2×C8—3161no—
377C2×C8—3164no—
378C2×C8—1582no—
379C2×C8—1581no—
380C2×C8—1582no—
381C2×C8—1581no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—3161no—
386C4⋊C4—1582no—
387C4⋊C4C22⋊Dic7911yescharacteristic
388C4⋊C4C22⋊Dic7911yescharacteristic
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42—3161no—
396C42—3161no—
397C42C22×Dic7911yescharacteristic
398C42D4×D7911yescharacteristic
399C42C2×C22⋊D7911yescharacteristic
400C42#1192962311yescharacteristic
401C23—21no—
402C23—21no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—81no—
408C23#1457913311yescharacteristic
409Q8—41no—
410Q8—41no—
411D4—21no—
412D4—21no—
413D4—21no—
414D4—42no—
415D4—42no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—81no—
421D4—81no—
422D4#1457908812yescomplemented
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—3162no—
442C2×C4—6321no—
443C2×C4—6321no—
444C2×C4C2×C22⋊Dic7911yescharacteristic
445C2×C4D4×Dic7911yescharacteristic
446C2×C4D4×Dic7911yescharacteristic
447C2×C4#1457912911yescharacteristic
448C2×C4#1457913311yescharacteristic
449C2×C4#1457914211yescharacteristic
450C2×C4D8×D7911yescharacteristic
451C2×C4#1457918011yescharacteristic
452C2×C4#1457918511yescharacteristic
453C2×C4#1457918511yescharacteristic
454C8—41no—
455C8—41no—
456C8—3162no—
457C8—3162no—
458C8—3161no—
459C8—1582no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—82no—
469C22—82no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22#1909844311yescharacteristic
474C22#1909875611yescharacteristic
475C22#1909877611yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—41no—
481C4—81no—
482C4—6322no—
483C4C2×#1457908811yescharacteristic
484C4#1909858011yescharacteristic
485C4D8×Dic7911yescharacteristic
486C4#1909867511yescharacteristic
487C4#1909869511yescharacteristic
488C4#1909870511yescharacteristic
489C2—21no—
490C2—21no—
491C2—42no—
492C2—81no—
493C2—161no—
494C2#2376370311yescharacteristic
495C2#2376375011yescharacteristic
496C2#2376376411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3100C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3100D79
hypocenterC79#8991
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. D4×#11929608C2semidirect
2. #23764320C2semidirect
3. C4×#14579082C2semidirect
4. #23761851C2semidirect
5. #23761302C2semidirect
6. #23761236C2semidirect
25. #474×C79C4semidirect
26. #474×C79C4semidirect
27. #459×C79C4semidirect
28. #459×C79C4semidirect
93. #11929608D8semidirect
173. D4×C79#104semidirect
256. C79#8991semidirect
261. #474Dic79semidirect
262. #474Dic79semidirect
263. #459Dic79semidirect
264. #459Dic79semidirect
422. D4#14579088semidirect

Maximal subgroups

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

subgroupindex
D4×#119296082
#237643202
C4×#145790822
#237618512
#237613022
#237612362
C79×#31002
#899179