Finite Group Databasebeta

← #34198736

Subgroups — #34198736

Order 20224.

586 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767674C211yesmaximal, characteristic
2C2×#19098330C211yesmaximal, characteristic
3#23761425C212yesmaximal, complemented
4#23761401C212yesmaximal, complemented
5C79×#1462C211yesmaximal, characteristic
6C2×#14579120C411yescharacteristic, complemented
7C22×C22⋊Dic79—21no—
8#19098499C411yescharacteristic, complemented
9#19098499C2211yescharacteristic
10#19098488C411yescharacteristic, complemented
11#19098488C2211yescharacteristic
12#19098451—21no—
13#19098451—21no—
14#19098436—21no—
15#19098426C411yescharacteristic, complemented
16#19098419—21no—
17#19098330—21no—
18#19098330C2212yes—
19C22×C22⋊C4×C79C2211yescharacteristic
20#407×C79—21no—
21#323×C79—21no—
22#323×C79C2212yes—
23C23×Dic79—41no—
24#14579120—41no—
25#14579120—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic, complemented
27C2×C4⋊Dic79C2×C411yescharacteristic, complemented
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—24no—
31C2×C22⋊Dic79—22no—
32C2×C22⋊Dic79—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—21no—
36C2×C22⋊Dic79—21no—
37C2×C22⋊Dic79—21no—
38C2×C22⋊Dic79—41no—
39C2×C22⋊Dic79—42no—
40C2×C4×Dic79—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95C2×C411yescharacteristic, complemented
45C79⋊#95C2×C411yescharacteristic, complemented
46C25×C79D411yescharacteristic
47C23×C4×C79C2×C411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C2×M16×C79—41no—
50C2×C22⋊C4×C79—41no—
51C2×C22⋊C4×C79—41no—
52C2×C22⋊C4×C79C2×C411yescharacteristic
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79D411yescharacteristic
55C2×C22⋊C4×C79C2311yescharacteristic
56C22⋊C8×C79—22no—
57C22⋊C8×C79—41no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—44no—
63C22×Dic79—42no—
64C22×Dic79—82no—
65C22×Dic79—82no—
66C4⋊Dic79—41no—
67C4⋊Dic79—41no—
68Dic79⋊C4—81no—
69Dic79⋊C4—81no—
70C22⋊Dic79—24no—
71C22⋊Dic79—42no—
72C22⋊Dic79—44no—
73C22⋊Dic79—41no—
74C22⋊Dic79—41no—
75C22⋊Dic79—42no—
76C22⋊Dic79—41no—
77C22⋊Dic79—42no—
78C22⋊Dic79—41no—
79C22⋊Dic79—81no—
80C22⋊Dic79—81no—
81C4×Dic79—42no—
82C24×C79—21no—
83C24×C79—21no—
84C24×C79—22no—
85C24×C79—21no—
86C24×C79—21no—
87C24×C79—42no—
88C24×C79—81no—
89C24×C79C22⋊C411yescharacteristic
90C24×C79C22⋊C411yescharacteristic
91C24×C79C2×D411yescharacteristic
92C22×C4×C79—21no—
93C22×C4×C79—21no—
94C22×C4×C79—21no—
95C22×C4×C79—21no—
96C22×C4×C79—42no—
97C22×C4×C79—81no—
98C22×C4×C79C22⋊C411yescharacteristic
99C22×C4×C79C22⋊C411yescharacteristic
100C22×C4×C79C22×C411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102M16×C79—82no—
103C2×C8×C79—42no—
104C22⋊C4×C79—41no—
105C22⋊C4×C79—41no—
106C22⋊C4×C79—41no—
107C22⋊C4×C79—41no—
108C23×C79—22no—
109C23×C79—22no—
110C23×C79—22no—
111C23×C79—22no—
112C23×C79—22no—
113C23×C79—21no—
114C23×C79—21no—
115C23×C79—21no—
116C23×C79—21no—
117C23×C79—21no—
118C23×C79—21no—
119C23×C79—41no—
120C23×C79—42no—
121C23×C79—41no—
122C23×C79—41no—
123C23×C79—41no—
124C23×C79—41no—
125C23×C79—42no—
126C23×C79—81no—
127C23×C79—81no—
128C23×C79—84no—
129C23×C79—82no—
130C23×C79—82no—
131C23×C79#9911yescharacteristic
132C23×C79#9911yescharacteristic
133C23×C79#10412yes—
134C23×C79C2×C22⋊C411yescharacteristic
135C23×C79AGL(1,ℤ/8)11yescharacteristic
136C23×C79#13711yescharacteristic
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—42no—
142C2×C4×C79—42no—
143C2×C4×C79—42no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—81no—
149C2×Dic79—44no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—41no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—41no—
161C2×Dic79—41no—
162C2×Dic79—88no—
163C2×Dic79—82no—
164C2×Dic79—82no—
165C8×C79—82no—
166C22×C79—21no—
167C22×C79—21no—
168C22×C79—21no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—22no—
174C22×C79—21no—
175C22×C79—21no—
176C22×C79—42no—
177C22×C79—42no—
178C22×C79—42no—
179C22×C79—41no—
180C22×C79—41no—
181C22×C79—44no—
182C22×C79—44no—
183C22×C79—41no—
184C22×C79—41no—
185C22×C79—41no—
186C22×C79—41no—
187C22×C79—81no—
188C22×C79—81no—
189C22×C79—82no—
190C22×C79—82no—
191C22×C79#35212yes—
192C22×C79C2×#9911yescharacteristic
193C22×C79#41711yescharacteristic
194C22×C79C2×#10411yescharacteristic
195C4×C79—41no—
196C4×C79—41no—
197C4×C79—41no—
198C4×C79—41no—
199Dic79—81no—
200Dic79—81no—
201Dic79—84no—
202Dic79—81no—
203Dic79—81no—
204#11755—791nomaximal, Sylow
205C2×C79—21no—
206C2×C79—21no—
207C2×C79—22no—
208C2×C79—41no—
209C2×C79—41no—
210C2×C79—42no—
211C2×C79—41no—
212C2×C79#150411yescharacteristic
213C2×C79C2×#35211yescharacteristic
214C2×C79#212711yescharacteristic
215#2482—791no—
216C2×#342—791no—
217#1462D7911yescharacteristic
218#1352—792no—
219#1344—792no—
220C79#1175511yescharacteristic, complemented, Sylow
221C2×#117—791no—
222C22×C22⋊C4—1581no—
223C22×C22⋊C4C2×D7911yescharacteristic
224#407—21no—
225#399—791no—
226#399—791no—
227#397—791no—
228#397—791no—
229#388—1581no—
230#388—1581no—
231#385—1581no—
232#381—791no—
233#379—1581no—
234#342—792no—
235#342—1581no—
236#323—21no—
237#323C2×D7912yes—
238C25D31611yescharacteristic
239C23×C4—3161no—
240C23×C4C4×D7911yescharacteristic
241C23×C4C22⋊D7911yescharacteristic
242C2×M16—41no—
243#117—3161no—
244#117—3161no—
245C2×C4⋊C4—791no—
246C2×C4⋊C4—791no—
247C2×C4⋊C4—1581no—
248C2×C4⋊C4—1581no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—3161no—
252C2×C22⋊C4—3162no—
253C2×C22⋊C4—1581no—
254C2×C22⋊C4—1581no—
255C2×C22⋊C4—1581no—
256C2×C22⋊C4—1581no—
257C2×C22⋊C4—1584no—
258C2×C22⋊C4—1582no—
259C2×C22⋊C4—1581no—
260C2×C22⋊C4—1581no—
261C2×C22⋊C4C4×D7911yescharacteristic
262C2×C22⋊C4D31611yescharacteristic
263C2×C22⋊C4C22⋊D7911yescharacteristic
264C2×C22⋊C4C22×D7911yescharacteristic
265C2×C42—1581no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268#95—791no—
269#95—791no—
270#95—1581no—
271#95—1581no—
272#95—1581no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—42no—
279C24—81no—
280C24(C2×D79)⋊C411yescharacteristic
281C24(C2×D79)⋊C411yescharacteristic
282C24C2×D31611yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4—3164no—
290C22×C4—3162no—
291C22×C4—6322no—
292C22×C4—6322no—
293C22×C4—1581no—
294C22×C4—1581no—
295C22×C4—1581no—
296C22×C4—1581no—
297C22×C4(C2×D79)⋊C411yescharacteristic
298C22×C4(C2×D79)⋊C411yescharacteristic
299C22×C4C2×C4×D7911yescharacteristic
300C22×C4C2×C22⋊D7911yescharacteristic
301M16—82no—
302C2×C8—42no—
303C4⋊C4—3161no—
304C4⋊C4—3161no—
305C4⋊C4—6321no—
306C4⋊C4—6321no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—3164no—
312C22⋊C4—3162no—
313C22⋊C4—3161no—
314C22⋊C4—3161no—
315C22⋊C4—3162no—
316C22⋊C4—3162no—
317C22⋊C4—3161no—
318C22⋊C4—3161no—
319C22⋊C4—6321no—
320C22⋊C4—6321no—
321C22⋊C4—1584no—
322C42—3162no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—84no—
344C23—82no—
345C23—82no—
346C23#1457907511yescharacteristic
347C23#1457907511yescharacteristic
348C23#1457908712yes—
349C23C2×(C2×D79)⋊C411yescharacteristic
350C23#1457918911yescharacteristic
351C23#1457919411yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—3164no—
365C2×C4—3161no—
366C2×C4—3161no—
367C2×C4—3161no—
368C2×C4—3161no—
369C2×C4—3161no—
370C2×C4—3161no—
371C2×C4—3161no—
372C2×C4—3161no—
373C2×C4—3161no—
374C2×C4—3161no—
375C2×C4—3161no—
376C2×C4—3161no—
377C2×C4—6328no—
378C2×C4—6322no—
379C2×C4—6322no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—44no—
397C22—44no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—82no—
405C22—82no—
406C22#1909835412yes—
407C22C2×#1457907511yescharacteristic
408C22#1909856511yescharacteristic
409C22C2×#1457908711yescharacteristic
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—6321no—
415C4—6321no—
416C4—6324no—
417C4—6321no—
418C4—6321no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—41no—
423C2—41no—
424C2—42no—
425C2—41no—
426C2#2376190511yescharacteristic
427C2C2×#1909835411yescharacteristic
428C2#2376506711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1462C2
socleC22×C79C2×#99
prime radicalC25×C79D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1462D79
hypocenterC79#11755
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #23761425C2semidirect
4. #23761401C2semidirect
6. C2×#14579120C4semidirect
8. #19098499C4semidirect
10. #19098488C4semidirect
15. #19098426C4semidirect
26. C2×C4⋊Dic79C2×C4semidirect
27. C2×C4⋊Dic79C2×C4semidirect
44. C79⋊#95C2×C4semidirect
45. C79⋊#95C2×C4semidirect
220. C79#11755semidirect

Maximal subgroups

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

subgroupindex
#237676742
C2×#190983302
#237614252
#237614012
C79×#14622
#1175579