Finite Group Databasebeta

← #34181013

Subgroups — #34181013

Order 20224.

698 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14579135C211yesmaximal, characteristic
2C2×#19098298C212yesmaximal
3#23761627C211yesmaximal, characteristic, complemented
4#23761280C212yesmaximal, complemented
5C79×#1797C211yesmaximal, characteristic, complemented
6C2×#14579135C2211yescharacteristic
7C2×C4×D316C411yescharacteristic, complemented
8C2×C4×C22⋊D79—21no—
9#19098465C411yescharacteristic, complemented
10#19098461C411yescharacteristic, complemented
11#19098454—21no—
12#19098442—21no—
13#19098423C2211yescharacteristic
14C4×C4⋊Dic79C411yescharacteristic, complemented
15C4×(C2×D79)⋊C4—21no—
16C2×C79⋊#95—21no—
17C2×C79⋊#95—21no—
18C2×C79⋊#95C2212yes—
19#19098298—24no—
20C22×C42×C79C2211yescharacteristic
21C2×C4⋊C8×C79—21no—
22C2×C22⋊C8×C79C2212yes—
23#336×C79—21no—
24C22×D316C2×C411yescharacteristic
25C22×C22⋊D79—21no—
26C23×Dic79—22no—
27C22×C4×D79—21no—
28#14579135—21no—
29#14579135—21no—
30#14579135D412yes—
31#14579135Q812yes—
32C4×D316—42no—
33C4×C22⋊D79—44no—
34C2×C4⋊Dic79C2×C411yescharacteristic
35C2×C4⋊Dic79C2×C411yescharacteristic
36C2×C4⋊Dic79C2×C411yescharacteristic
37C2×Dic79⋊C4—21no—
38C2×C22⋊Dic79—21no—
39C2×(C2×D79)⋊C4—21no—
40C2×(C2×D79)⋊C4—21no—
41C2×(C2×D79)⋊C4—21no—
42C2×(C2×D79)⋊C4—21no—
43C2×C4×Dic79—21no—
44C79⋊#95—24no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—22no—
49C79⋊#95—22no—
50C79⋊#95—42no—
51C23×C4×C79D411yescharacteristic
52C23×C4×C79D411yescharacteristic
53C23×C4×C79C2311yescharacteristic
54C22×C8×C79—22no—
55C2×C42×C79—42no—
56C2×C42×C79C2×C411yescharacteristic
57C2×C42×C79C2×C411yescharacteristic
58C2×C42×C79D411yescharacteristic
59C2×C42×C79D411yescharacteristic
60C4⋊C8×C79—42no—
61C22⋊C8×C79—22no—
62C22⋊C8×C79—22no—
63C23×D79—21no—
64C2×D316—21no—
65C2×D316—41no—
66C2×D316—41no—
67C2×D316C4⋊C412yes—
68C2×C22⋊D79—22no—
69C2×C22⋊D79—22no—
70C2×C22⋊D79—42no—
71C2×C22⋊D79—42no—
72C22×Dic79—24no—
73C22×Dic79—21no—
74C22×Dic79—21no—
75C22×Dic79—21no—
76C22×Dic79—42no—
77C22×Dic79—42no—
78C22×Dic79—42no—
79C22×Dic79—42no—
80C22×Dic79—42no—
81C22×Dic79—42no—
82C2×C4×D79—42no—
83C2×C4×D79—42no—
84C2×C4×D79—42no—
85C4⋊Dic79—21no—
86C4⋊Dic79—41no—
87C4⋊Dic79—41no—
88C4⋊Dic79C4⋊C412yes—
89Dic79⋊C4—42no—
90C22⋊Dic79—42no—
91(C2×D79)⋊C4—22no—
92(C2×D79)⋊C4—22no—
93(C2×D79)⋊C4—42no—
94(C2×D79)⋊C4—42no—
95(C2×D79)⋊C4—42no—
96C4×Dic79—42no—
97C24×C79C2×D411yescharacteristic
98C22×C4×C79—22no—
99C22×C4×C79—22no—
100C22×C4×C79—22no—
101C22×C4×C79—22no—
102C22×C4×C79—41no—
103C22×C4×C79—41no—
104C22×C4×C79—41no—
105C22×C4×C79—41no—
106C22×C4×C79C22⋊C411yescharacteristic
107C22×C4×C79C22⋊C411yescharacteristic
108C22×C4×C79C22⋊C411yescharacteristic
109C22×C4×C79C22⋊C411yescharacteristic
110C22×C4×C79C22×C411yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×Q811yescharacteristic
114C22×C4×C79#4111yescharacteristic
115C22×C4×C79#4111yescharacteristic
116C2×C8×C79—42no—
117C2×C8×C79—42no—
118C2×C8×C79—42no—
119C42×C79—22no—
120C42×C79—22no—
121C42×C79—41no—
122C42×C79—41no—
123C23×C79—21no—
124C23×C79—21no—
125C23×C79—21no—
126C23×C79—21no—
127C23×C79C2×C22⋊C411yescharacteristic
128C23×C79#12111yescharacteristic
129C23×C79C22⋊Q811yescharacteristic
130C23×C79#13512yes—
131C23×C79AGL(1,ℤ/8)12yes—
132C22×D79—22no—
133C22×D79—41no—
134C22×D79—41no—
135C22×D79—41no—
136C22×D79—41no—
137C22×D79—41no—
138C22×D79—41no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—44no—
158C2×C4×C79—44no—
159C2×C4×C79#9912yes—
160C2×C4×C79#10412yes—
161C2×C4×C79C2×C4⋊C411yescharacteristic
162C2×C4×C79#11711yescharacteristic
163C2×C4×C79#12111yescharacteristic
164C2×C4×C79C22⋊Q811yescharacteristic
165C22⋊D79—44no—
166C22⋊D79—44no—
167C2×Dic79—22no—
168C2×Dic79—41no—
169C2×Dic79—41no—
170C2×Dic79—44no—
171C2×Dic79—44no—
172C2×Dic79—44no—
173C2×Dic79—41no—
174C2×Dic79—41no—
175C2×Dic79—41no—
176C2×Dic79—41no—
177C2×Dic79—41no—
178C2×Dic79—41no—
179C2×Dic79—82no—
180C2×Dic79—82no—
181C2×Dic79—82no—
182C2×Dic79—82no—
183D316—42no—
184D316—42no—
185C4×D79—84no—
186C8×C79—82no—
187C22×C79—21no—
188C22×C79—21no—
189C22×C79—21no—
190C22×C79—21no—
191C22×C79—22no—
192C22×C79—22no—
193C22×C79—22no—
194C22×C79—22no—
195C22×C79#38011yescharacteristic
196C22×C79C2×#9911yescharacteristic
197C22×C79#41011yescharacteristic
198C22×C79#41611yescharacteristic
199C22×C79#41811yescharacteristic
200C22×C79C2×#10411yescharacteristic
201C22×C79#42111yescharacteristic
202C22×C79#46312yes—
203C22×C79#47812yes—
204C2×D79—42no—
205C2×D79—42no—
206C2×D79—42no—
207C2×D79—81no—
208C2×D79—81no—
209C2×D79—81no—
210C2×D79—81no—
211C4×C79—21no—
212C4×C79—21no—
213C4×C79—21no—
214C4×C79—21no—
215C4×C79—22no—
216C4×C79—22no—
217C4×C79—41no—
218C4×C79—41no—
219Dic79—42no—
220Dic79—84no—
221Dic79—81no—
222Dic79—81no—
223#8575—791nomaximal, Sylow
224C2×C79—22no—
225C2×C79—22no—
226C2×C79#150111yescharacteristic
227C2×C79#150511yescharacteristic
228C2×C79#151011yescharacteristic
229C2×C79#152111yescharacteristic
230C2×C79#179911yescharacteristic
231C2×C79#180411yescharacteristic
232C2×C79#180611yescharacteristic
233D79—82no—
234C4×#121—791no—
235#1797D7911yescharacteristic, complemented
236C2×#327—792no—
237#1434—791no—
238#1294—792no—
239C79#857511yescharacteristic, complemented, Sylow
240C2×#121—791no—
241C2×C4×D4—791no—
242C2×C4×D4—1581no—
243C22×C42C2×D7911yescharacteristic
244C2×C4⋊C8—21no—
245C2×C22⋊C8C2×D7912yes—
246#390—791no—
247#389—791no—
248#388—1581no—
249#386—1581no—
250#380—791no—
251C4×C4⋊C4—791no—
252C4×C22⋊C4—1581no—
253C2×#95—792no—
254C2×#95—1581no—
255C2×#95—1581no—
256#336—21no—
257#327—1584no—
258C22×D4—791no—
259C22×D4—1581no—
260C23×C4—1582no—
261C23×C4—1581no—
262C23×C4D31611yescharacteristic
263C23×C4D31611yescharacteristic
264C23×C4C22×D7911yescharacteristic
265C22×C8—22no—
266#121—792no—
267#121—792no—
268#121—1581no—
269#121—1581no—
270C4×D4—3162no—
271C4×D4—3164no—
272C2×C4⋊C4—791no—
273C2×C4⋊C4—791no—
274C2×C4⋊C4—791no—
275C2×C4⋊C4—1581no—
276C2×C22⋊C4—1581no—
277C2×C22⋊C4—1581no—
278C2×C22⋊C4—1581no—
279C2×C22⋊C4—1581no—
280C2×C22⋊C4—1581no—
281C2×C42—42no—
282C2×C42—1581no—
283C2×C42C4×D7911yescharacteristic
284C2×C42C4×D7911yescharacteristic
285C2×C42C22⋊D7911yescharacteristic
286C2×C42C22⋊D7911yescharacteristic
287C4⋊C8—42no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290#95—3162no—
291#95—1584no—
292#95—1582no—
293#95—1582no—
294#95—1581no—
295#95—1581no—
296#95—1581no—
297C24—1581no—
298C24C2×D31611yescharacteristic
299C2×D4—3161no—
300C2×D4—3161no—
301C2×D4—3162no—
302C2×D4—3162no—
303C2×D4—792no—
304C2×D4—1582no—
305C2×D4—1582no—
306C2×D4—1581no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—3162no—
316C22×C4—3162no—
317C22×C4—3162no—
318C22×C4—3162no—
319C22×C4—3162no—
320C22×C4—3162no—
321C22×C4—3162no—
322C22×C4—3162no—
323C22×C4—3162no—
324C22×C4—1584no—
325C22×C4—1581no—
326C22×C4—1581no—
327C22×C4—1581no—
328C22×C4(C2×D79)⋊C411yescharacteristic
329C22×C4(C2×D79)⋊C411yescharacteristic
330C22×C4(C2×D79)⋊C411yescharacteristic
331C22×C4(C2×D79)⋊C411yescharacteristic
332C22×C4C2×C4×D7911yescharacteristic
333C22×C4D4×D7911yescharacteristic
334C22×C4C2×C22⋊D7911yescharacteristic
335C22×C4Q8×D7911yescharacteristic
336C22×C4#1192962311yescharacteristic
337C22×C4#1192962511yescharacteristic
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—3161no—
342C4⋊C4—3161no—
343C4⋊C4—3162no—
344C4⋊C4—792no—
345C4⋊C4—1581no—
346C22⋊C4—3162no—
347C22⋊C4—3162no—
348C22⋊C4—3162no—
349C22⋊C4—3162no—
350C22⋊C4—1582no—
351C22⋊C4—1582no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—3162no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—3161no—
362C23—3161no—
363C23—3161no—
364C23—3161no—
365C23—3161no—
366C23—3161no—
367C23—1582no—
368C23C2×(C2×D79)⋊C411yescharacteristic
369C23#1457913611yescharacteristic
370C23#1457914111yescharacteristic
371C23#1457918312yes—
372C23#1457918912yes—
373D4—3162no—
374D4—3162no—
375D4—3164no—
376D4—3164no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—3161no—
398C2×C4—3161no—
399C2×C4—3164no—
400C2×C4—3164no—
401C2×C4—3164no—
402C2×C4—3161no—
403C2×C4—3161no—
404C2×C4—3161no—
405C2×C4—3161no—
406C2×C4—3161no—
407C2×C4—3161no—
408C2×C4—6324no—
409C2×C4—6322no—
410C2×C4—6322no—
411C2×C4—6322no—
412C2×C4—6322no—
413C2×C4—1582no—
414C2×C4#1457907512yes—
415C2×C4#1457908712yes—
416C2×C4C4⋊C4×D7911yescharacteristic
417C2×C4#1457911911yescharacteristic
418C2×C4#1457913211yescharacteristic
419C2×C4#1457913711yescharacteristic
420C8—82no—
421C22—22no—
422C22—22no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—22no—
428C22—22no—
429C22—3162no—
430C22—3162no—
431C22—3162no—
432C22—6321no—
433C22—6321no—
434C22—6321no—
435C22—6321no—
436C22#1909842011yescharacteristic
437C22C2×#1457907511yescharacteristic
438C22#1909853611yescharacteristic
439C22#1909856011yescharacteristic
440C22#1909856911yescharacteristic
441C22C2×#1457908711yescharacteristic
442C22#1909857911yescharacteristic
443C22#1909877912yes—
444C22#1909886412yes—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—3162no—
454C4—6324no—
455C4—6321no—
456C4—6321no—
457C2—22no—
458C2—22no—
459C2—6322no—
460C2#2376189211yescharacteristic
461C2#2376190911yescharacteristic
462C2#2376193011yescharacteristic
463C2#2376197811yescharacteristic
464C2#2376338511yescharacteristic
465C2#2376341111yescharacteristic
466C2#2376341711yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1797C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenter#1797D79
hypocenterC79#8575
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #23761627C2semidirect
4. #23761280C2semidirect
5. C79×#1797C2semidirect
7. C2×C4×D316C4semidirect
9. #19098465C4semidirect
10. #19098461C4semidirect
14. C4×C4⋊Dic79C4semidirect
235. #1797D79semidirect
239. C79#8575semidirect

Maximal subgroups

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

subgroupindex
C4×#145791352
C2×#190982982
#237616272
#237612802
C79×#17972
#857579