Finite Group Databasebeta

← #34178819

Subgroups — #34178819

Order 20224.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098631C211yesmaximal, characteristic
2C4×#14579075C211yesmaximal, characteristic
3#23763179C211yesmaximal, characteristic, complemented
4C22⋊C4×C79⋊C8C211yesmaximal, characteristic, complemented
5#23761716C211yesmaximal, characteristic
6#23761247C211yesmaximal, characteristic, complemented
7C79×#1282C211yesmaximal, characteristic, complemented
8C22×#11929607C411yescharacteristic, complemented
9C23×C79⋊C8C2211yescharacteristic
10C2×C4×C22⋊D79C2211yescharacteristic
11#19098631—21no—
12#19098631—21no—
13#19098631—21no—
14#19098631—21no—
15C2×Dic79⋊C8C411yescharacteristic, complemented
16C2×#14579075C412yes—
17C2×C22⋊(C79⋊C8)C2211yescharacteristic
18C2×(C2×D79)⋊C8C411yescharacteristic, complemented
19C2×#14579071—21no—
20C2×#14579071C411yescharacteristic, complemented
21C2×C4×C79⋊C8—21no—
22C4×C22⋊Dic79C2211yescharacteristic
23#19098329C412yes—
24#19098318—21no—
25#19098317—21no—
26#19098288C412yescomplemented
27#19098286C412yescomplemented
28C2×C22⋊C8×C79C2211yescharacteristic
29C2×C22⋊C8×C79C2211yescharacteristic
30C4×C22⋊C4×C79C2211yescharacteristic
31C22×C22⋊D79C2×C411yescharacteristic
32C22×C4×D79C2×C411yescharacteristic
33C2×#11929607—21no—
34C2×#11929607—21no—
35C2×#11929607—42no—
36C22×C79⋊C8—21no—
37C22×C79⋊C8—42no—
38C22×C79⋊C8C2×C411yescharacteristic
39C22×C79⋊C8C2×C411yescharacteristic
40C22×C79⋊C8D411yescharacteristic
41C22×C79⋊C8D411yescharacteristic
42C22×C79⋊C8D411yescharacteristic
43C22×C79⋊C8D411yescharacteristic
44C4×C22⋊D79—21no—
45C4×C22⋊D79—21no—
46C4×C22⋊D79—21no—
47C4×C22⋊D79—21no—
48C2×Dic79⋊C4C2×C411yescharacteristic
49C2×C22⋊Dic79C2×C412yes—
50C2×C22⋊Dic79C2×C411yescharacteristic
51C2×(C2×D79)⋊C4C2×C411yescharacteristic
52C2×C4×Dic79—21no—
53C2×C4×Dic79C2×C411yescharacteristic
54Dic79⋊C8—21no—
55Dic79⋊C8—21no—
56#14579075—24no—
57C22⋊(C79⋊C8)—21no—
58C22⋊(C79⋊C8)C2×C412yes—
59(C2×D79)⋊C8—21no—
60(C2×D79)⋊C8—21no—
61#14579071—21no—
62#14579071—21no—
63#14579071—42no—
64C4×C79⋊C8—42no—
65C79⋊#95—21no—
66C23×C4×C79C2311yescharacteristic
67C22×C8×C79—21no—
68C22×C8×C79C2×C411yescharacteristic
69C22×C8×C79C2×C411yescharacteristic
70C2×C22⋊C4×C79C2×C412yes—
71C2×C42×C79—21no—
72C22⋊C8×C79—21no—
73C22⋊C8×C79—21no—
74C22⋊C8×C79—21no—
75C22⋊C8×C79C2×C412yes—
76#95×C79—21no—
77C23×D79C4211yescharacteristic
78C2×C22⋊D79—21no—
79C2×C22⋊D79—21no—
80C2×C22⋊D79—22no—
81C2×C22⋊D79—22no—
82C22×Dic79—22no—
83C22×Dic79C4211yescharacteristic
84C22×Dic79C4211yescharacteristic
85C22×Dic79C4211yescharacteristic
86C2×C4×D79—21no—
87C2×C4×D79—21no—
88C2×C4×D79—41no—
89C2×C4×D79—41no—
90#11929607—42no—
91#11929607—42no—
92C2×C79⋊C8—22no—
93C2×C79⋊C8—22no—
94C2×C79⋊C8—22no—
95C2×C79⋊C8—22no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—41no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8C22⋊C411yescharacteristic
102C2×C79⋊C8C22⋊C411yescharacteristic
103C2×C79⋊C8C22⋊C411yescharacteristic
104C2×C79⋊C8C22⋊C411yescharacteristic
105Dic79⋊C4—21no—
106Dic79⋊C4—21no—
107C22⋊Dic79—21no—
108C22⋊Dic79—21no—
109C22⋊Dic79—22no—
110C22⋊Dic79—22no—
111(C2×D79)⋊C4—21no—
112(C2×D79)⋊C4—21no—
113C4×Dic79—21no—
114C4×Dic79—21no—
115C4×Dic79—41no—
116C4×Dic79—41no—
117C24×C79C22×C411yescharacteristic
118C22×C4×C79—22no—
119C22×C4×C79—41no—
120C22×C4×C79—41no—
121C22×C4×C79C22×C411yescharacteristic
122C22×C4×C79C22×C411yescharacteristic
123C22×C4×C79C2×D411yescharacteristic
124C22×C4×C79C2×D411yescharacteristic
125C22×C4×C79#4111yescharacteristic
126C22×C4×C79#4111yescharacteristic
127C2×C8×C79—21no—
128C2×C8×C79—21no—
129C2×C8×C79—22no—
130C2×C8×C79—41no—
131C2×C8×C79—41no—
132C2×C8×C79—41no—
133C2×C8×C79—41no—
134C22⋊C4×C79—22no—
135C22⋊C4×C79M1614yescomplemented
136C42×C79—41no—
137C42×C79—41no—
138C23×C79—41no—
139C23×C79—41no—
140C23×C79C2×C4211yescharacteristic
141C23×C79C4×D411yescharacteristic
142C23×C79C4×D411yescharacteristic
143C23×C79C2×M1612yes—
144C23×C79#13112yes—
145C22×D79—21no—
146C22×D79—21no—
147C22×D79—21no—
148C22×D79—41no—
149C22×D79—41no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—21no—
155C2×C4×C79—21no—
156C2×C4×C79—24no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—42no—
161C2×C4×C79—42no—
162C2×C4×C79C2×C22⋊C411yescharacteristic
163C2×C4×C79#11711yescharacteristic
164C2×C4×C79C4×D411yescharacteristic
165C2×C4×C79C4×D411yescharacteristic
166C22⋊D79—42no—
167C22⋊D79—42no—
168C2×Dic79—21no—
169C2×Dic79—21no—
170C2×Dic79—21no—
171C2×Dic79—21no—
172C2×Dic79—21no—
173C2×Dic79—41no—
174C2×Dic79—41no—
175C2×Dic79—42no—
176C2×Dic79—42no—
177C2×Dic79—42no—
178C4×D79—42no—
179C4×D79—41no—
180C4×D79—41no—
181C8×C79—42no—
182C8×C79—41no—
183C8×C79—41no—
184C79⋊C8—21no—
185C79⋊C8—21no—
186C79⋊C8—21no—
187C79⋊C8—42no—
188C79⋊C8#9912yescomplemented
189C22×C79—21no—
190C22×C79—21no—
191C22×C79—21no—
192C22×C79—21no—
193C22×C79—21no—
194C22×C79—21no—
195C22×C79—42no—
196C22×C79—42no—
197C22×C79C4×C22⋊C411yescharacteristic
198C22×C79C4×M1611yescharacteristic
199C22×C79#40511yescharacteristic
200C22×C79#43512yes—
201C22×C79#43612yes—
202C2×D79—41no—
203C2×D79—41no—
204C2×D79—41no—
205C2×D79—81no—
206C2×D79—81no—
207C4×C79—21no—
208C4×C79—21no—
209C4×C79—21no—
210C4×C79—44no—
211C4×C79C2×#9911yescharacteristic
212C4×C79#41011yescharacteristic
213Dic79—41no—
214Dic79—41no—
215Dic79—41no—
216Dic79—82no—
217#8161—791nomaximal, Sylow
218C2×C79—22no—
219C2×C79—22no—
220C2×C79—21no—
221C2×C79—21no—
222C2×C79#175311yescharacteristic
223C2×C79C4×#9911yescharacteristic
224C2×C79#175711yescharacteristic
225D79—81no—
226C2×#435—791no—
227C4×#99—791no—
228#1752—791no—
229C8×C22⋊C4—791no—
230#1459—791no—
231#1282—791no—
232#1282D7911yescharacteristic, complemented
233C79#816111yescharacteristic, complemented, Sylow
234C22×M16—791no—
235C23×C8—791no—
236C2×C4×D4—791no—
237#435—1581no—
238#435—1581no—
239#435—1581no—
240#435—1581no—
241C2×C4⋊C8—791no—
242C2×#99—792no—
243C2×C22⋊C8—791no—
244C2×C22⋊C8—791no—
245C2×C22⋊C8C2×D7911yescharacteristic
246C2×C22⋊C8C2×D7911yescharacteristic
247C2×#97—791no—
248C2×#97—1581no—
249C2×C4×C8—1581no—
250C4×C22⋊C4—791no—
251C4×C22⋊C4C2×D7911yescharacteristic
252#342—792no—
253#336—1581no—
254#336—1581no—
255#324—792no—
256#323—792no—
257C22×D4—791no—
258C23×C4—791no—
259C23×C4C22×D7911yescharacteristic
260C2×M16—3162no—
261C2×M16—1581no—
262C2×M16—1581no—
263C22×C8—21no—
264C22×C8—3162no—
265C22×C8—791no—
266C22×C8—791no—
267C22×C8—791no—
268C22×C8—791no—
269C22×C8—791no—
270C22×C8—791no—
271C22×C8—1581no—
272C22×C8C4×D7911yescharacteristic
273C22×C8C4×D7911yescharacteristic
274C4×D4—1581no—
275C4×D4—1581no—
276C4×D4—1581no—
277C4×D4—1581no—
278C2×C4⋊C4—791no—
279C2×C22⋊C4—791no—
280C2×C22⋊C4—791no—
281C2×C22⋊C4—792no—
282C2×C22⋊C4C4×D7912yes—
283C2×C42—21no—
284C2×C42—791no—
285C2×C42—1581no—
286C4⋊C8—1581no—
287C4⋊C8—1581no—
288#99—1584no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—792no—
293C22⋊C8—1581no—
294C22⋊C8—1581no—
295C22⋊C8—1581no—
296C22⋊C8C4×D7912yes—
297#97—3162no—
298#97—1581no—
299#97—1581no—
300C4×C8—3162no—
301#95—21no—
302#95—1581no—
303C24—791no—
304C24C2×C4×D7911yescharacteristic
305C2×D4—1581no—
306C2×D4—1581no—
307C2×D4—1582no—
308C2×D4—1582no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—3161no—
313C22×C4—3161no—
314C22×C4—791no—
315C22×C4—791no—
316C22×C4—791no—
317C22×C4—1582no—
318C22×C4—1581no—
319C22×C4—1581no—
320C22×C4C2×C4×D7911yescharacteristic
321C22×C4C2×C4×D7911yescharacteristic
322C22×C4D4×D7911yescharacteristic
323C22×C4D4×D7911yescharacteristic
324C22×C4#1192962311yescharacteristic
325C22×C4#1192962511yescharacteristic
326M16—3162no—
327M16—3162no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—3161no—
336C2×C8—3161no—
337C2×C8—3161no—
338C2×C8—3161no—
339C2×C8—3161no—
340C2×C8—791no—
341C2×C8—791no—
342C2×C8—791no—
343C2×C8—791no—
344C2×C8—1582no—
345C2×C8—1582no—
346C2×C8—1582no—
347C2×C8—1582no—
348C4⋊C4—1581no—
349C4⋊C4—1581no—
350C22⋊C4—22no—
351C22⋊C4—1582no—
352C22⋊C4—1582no—
353C22⋊C4—1581no—
354C22⋊C4—1581no—
355C22⋊C4—1581no—
356C22⋊C4—1581no—
357C22⋊C4#1192960714yescomplemented
358C42—41no—
359C42—41no—
360C42—3161no—
361C42—3161no—
362C42—1581no—
363C42—1581no—
364C23—41no—
365C23—41no—
366C23—3161no—
367C23—3161no—
368C23—1581no—
369C23—1581no—
370C23—1581no—
371C23C42×D7911yescharacteristic
372C23(C22⋊D79)⋊C411yescharacteristic
373C23#1457912311yescharacteristic
374C23C2×#1192960712yes—
375C23#1457916912yes—
376D4—3162no—
377D4—3162no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—24no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—3162no—
391C2×C4—3161no—
392C2×C4—3161no—
393C2×C4—3161no—
394C2×C4—3161no—
395C2×C4—3162no—
396C2×C4—3162no—
397C2×C4—3162no—
398C2×C4—1581no—
399C2×C4—1581no—
400C2×C4—1581no—
401C2×C4—1581no—
402C2×C4—1581no—
403C2×C4C22⋊C4×D7911yescharacteristic
404C2×C4#1457911911yescharacteristic
405C2×C4(C22⋊D79)⋊C411yescharacteristic
406C2×C4#1457912311yescharacteristic
407C8—42no—
408C8—41no—
409C8—41no—
410C8—3162no—
411C8—792no—
412C8—1581no—
413C8—1581no—
414C8—1581no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—3161no—
424C22—3161no—
425C22—3161no—
426C22—6321no—
427C22—6321no—
428C22#1909841011yescharacteristic
429C22C4×#1192960711yescharacteristic
430C22#1909852011yescharacteristic
431C22#1909863112yes—
432C22#1909864012yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—44no—
437C4—3161no—
438C4—3161no—
439C4—3161no—
440C4—6322no—
441C4#99×D7911yescharacteristic
442C4#1909853511yescharacteristic
443C2—22no—
444C2—22no—
445C2—21no—
446C2—21no—
447C2—6321no—
448C2#2376318011yescharacteristic
449C2#2376318711yescharacteristic
450C2#2376319911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D79
commutatorC23×C79C2×C42
FrattiniC23×C4C22×D79
FittingC79×#1282C2
socleC22×C79C4×C22⋊C4
prime radicalC22×C22⋊D79C2×C4
cosocleC23×C4×C79C23
hypercenter#1282D79
hypocenterC79#8161
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
3. #23763179C2semidirect
4. C22⋊C4×C79⋊C8C2semidirect
6. #23761247C2semidirect
7. C79×#1282C2semidirect
8. C22×#11929607C4semidirect
15. C2×Dic79⋊C8C4semidirect
18. C2×(C2×D79)⋊C8C4semidirect
20. C2×#14579071C4semidirect
26. #19098288C4semidirect
27. #19098286C4semidirect
135. C22⋊C4×C79M16semidirect
188. C79⋊C8#99semidirect
232. #1282D79semidirect
233. C79#8161semidirect
357. C22⋊C4#11929607semidirect

Maximal subgroups

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

subgroupindex
C2×#190986312
C4×#145790752
#237631792
C22⋊C4×C79⋊C82
#237617162
#237612472
C79×#12822
#816179