Finite Group Databasebeta

← #34178824

Subgroups — #34178824

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×#19098634C211yesmaximal, characteristic
2C4×#14579075C211yesmaximal, characteristic
3C8×C22⋊Dic79C211yesmaximal, characteristic, complemented
4#23761716C211yesmaximal, characteristic
5#23761248C211yesmaximal, characteristic, complemented
6#23761246C211yesmaximal, characteristic, complemented
7C79×#1752C211yesmaximal, characteristic, complemented
8C22×#11929607C411yescharacteristic, complemented
9C2×C4×C22⋊D79C2211yescharacteristic
10#19098634—21no—
11#19098634—21no—
12#19098634—21no—
13#19098634—21no—
14C2×Dic79⋊C8C411yescharacteristic, complemented
15C2×#14579075C412yes—
16C2×C22⋊(C79⋊C8)C2211yescharacteristic
17C2×C22⋊(C79⋊C8)C2211yescharacteristic
18C2×(C2×D79)⋊C8C411yescharacteristic, complemented
19C2×#14579071C411yescharacteristic, complemented
20C2×C8×Dic79—21no—
21C4×C22⋊Dic79C2211yescharacteristic
22#19098329C412yes—
23#19098317—21no—
24#19098288C412yescomplemented
25#19098286C412yescomplemented
26C23×C8×C79C2211yescharacteristic
27C2×C22⋊C8×C79C2211yescharacteristic
28C2×#97×C79—21no—
29C4×C22⋊C4×C79C2211yescharacteristic
30#336×C79—21no—
31C22×C22⋊D79C2×C411yescharacteristic
32C22×C4×D79C2×C411yescharacteristic
33C2×#11929607—21no—
34C2×#11929607—21no—
35C2×#11929607—42no—
36C22×C79⋊C8—21no—
37C22×C79⋊C8C2×C411yescharacteristic
38C22×C79⋊C8C2×C411yescharacteristic
39C4×C22⋊D79—21no—
40C4×C22⋊D79—21no—
41C4×C22⋊D79—21no—
42C4×C22⋊D79—21no—
43C2×Dic79⋊C4C2×C411yescharacteristic
44C2×C22⋊Dic79C2×C411yescharacteristic
45C2×C22⋊Dic79C2×C412yes—
46C2×(C2×D79)⋊C4C2×C411yescharacteristic
47C2×C4×Dic79—21no—
48C2×C4×Dic79C2×C411yescharacteristic
49Dic79⋊C8—21no—
50Dic79⋊C8—21no—
51#14579075—24no—
52C22⋊(C79⋊C8)—21no—
53C22⋊(C79⋊C8)—21no—
54C22⋊(C79⋊C8)—21no—
55C22⋊(C79⋊C8)C2×C412yes—
56(C2×D79)⋊C8—21no—
57(C2×D79)⋊C8—21no—
58#14579071—21no—
59#14579071—21no—
60C8×Dic79—42no—
61C79⋊#95—21no—
62C23×C4×C79C2311yescharacteristic
63C22×C8×C79—21no—
64C22×C8×C79—42no—
65C22×C8×C79C2×C411yescharacteristic
66C22×C8×C79C2×C411yescharacteristic
67C22×C8×C79D411yescharacteristic
68C22×C8×C79D411yescharacteristic
69C22×C8×C79D411yescharacteristic
70C22×C8×C79D411yescharacteristic
71C2×C22⋊C4×C79C2×C412yes—
72C2×C42×C79—21no—
73C22⋊C8×C79—21no—
74C22⋊C8×C79C2×C412yes—
75#97×C79—42no—
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—21no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—22no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99Dic79⋊C4—21no—
100Dic79⋊C4—21no—
101C22⋊Dic79—21no—
102C22⋊Dic79—21no—
103C22⋊Dic79—22no—
104C22⋊Dic79M1614yescomplemented
105(C2×D79)⋊C4—21no—
106(C2×D79)⋊C4—21no—
107C4×Dic79—21no—
108C4×Dic79—21no—
109C4×Dic79—41no—
110C4×Dic79—41no—
111C24×C79C22×C411yescharacteristic
112C22×C4×C79—22no—
113C22×C4×C79—41no—
114C22×C4×C79—41no—
115C22×C4×C79C22×C411yescharacteristic
116C22×C4×C79C22×C411yescharacteristic
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79#4111yescharacteristic
120C22×C4×C79#4111yescharacteristic
121C2×C8×C79—22no—
122C2×C8×C79—22no—
123C2×C8×C79—22no—
124C2×C8×C79—22no—
125C2×C8×C79—41no—
126C2×C8×C79—41no—
127C2×C8×C79—41no—
128C2×C8×C79—41no—
129C2×C8×C79—41no—
130C2×C8×C79C22⋊C411yescharacteristic
131C2×C8×C79C22⋊C411yescharacteristic
132C2×C8×C79C22⋊C411yescharacteristic
133C2×C8×C79C22⋊C411yescharacteristic
134C22⋊C4×C79—22no—
135C22⋊C4×C79—22no—
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—41no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—42no—
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—24no—
174C2×Dic79—42no—
175C2×Dic79—42no—
176C2×Dic79—41no—
177C2×Dic79—41no—
178C4×D79—42no—
179C4×D79—41no—
180C4×D79—41no—
181C8×C79—21no—
182C8×C79—21no—
183C8×C79—21no—
184C8×C79—42no—
185C8×C79#9912yescomplemented
186C79⋊C8—42no—
187C79⋊C8—41no—
188C79⋊C8—41no—
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—82no—
211C4×C79C2×#9911yescharacteristic
212C4×C79#41011yescharacteristic
213Dic79—44no—
214Dic79—41no—
215Dic79—41no—
216Dic79—41no—
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#1752D7911yescharacteristic, complemented
229C8×C22⋊C4—791no—
230#1459—791no—
231#1282—791no—
232#1282—791no—
233C79#816111yescharacteristic, complemented, Sylow
234C22×M16—791no—
235C23×C8C2×D7911yescharacteristic
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⋊C8—791no—
246C2×C22⋊C8C2×D7911yescharacteristic
247C2×#97—21no—
248C2×#97—791no—
249C2×C4×C8—1581no—
250C4×C22⋊C4—791no—
251C4×C22⋊C4C2×D7911yescharacteristic
252#342—792no—
253#336—21no—
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—42no—
265C22×C8—791no—
266C22×C8—791no—
267C22×C8—1581no—
268C22×C8C4×D7911yescharacteristic
269C22×C8C4×D7911yescharacteristic
270C22×C8D31611yescharacteristic
271C22×C8D31611yescharacteristic
272C22×C8C22⋊D7911yescharacteristic
273C22×C8C22⋊D7911yescharacteristic
274C4×D4—1581no—
275C4×D4—1581no—
276C4×D4—1581no—
277C4×D4—1581no—
278C2×C4⋊C4—791no—
279C2×C22⋊C4—792no—
280C2×C22⋊C4—791no—
281C2×C22⋊C4—791no—
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—792no—
291C22⋊C8—1581no—
292C22⋊C8—1581no—
293C22⋊C8—1581no—
294C22⋊C8—1581no—
295C22⋊C8—1581no—
296C22⋊C8C4×D7912yes—
297#97—42no—
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×C4C2×C22⋊D7911yescharacteristic
323C22×C4C2×D31611yescharacteristic
324C22×C4#1192962411yescharacteristic
325C22×C4#1192962411yescharacteristic
326M16—3162no—
327M16—3162no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—3161no—
338C2×C8—3161no—
339C2×C8—3161no—
340C2×C8—3161no—
341C2×C8—1581no—
342C2×C8—1581no—
343C2×C8—1582no—
344C2×C8(C2×D79)⋊C411yescharacteristic
345C2×C8(C2×D79)⋊C411yescharacteristic
346C2×C8(C2×D79)⋊C411yescharacteristic
347C2×C8(C2×D79)⋊C411yescharacteristic
348C4⋊C4—1581no—
349C4⋊C4—1581no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—794no—
353C22⋊C4—1581no—
354C22⋊C4—1581no—
355C22⋊C4—1582no—
356C22⋊C4—1581no—
357C22⋊C4—1581no—
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
372C23C4×C22⋊D7911yescharacteristic
373C23C4×D31611yescharacteristic
374C23M16×D7912yes—
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—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—3162no—
391C2×C4—3162no—
392C2×C4—3162no—
393C2×C4—3161no—
394C2×C4—3161no—
395C2×C4—3161no—
396C2×C4—3161no—
397C2×C4—1584no—
398C2×C4—1581no—
399C2×C4—1581no—
400C2×C4—1581no—
401C2×C4—1581no—
402C2×C4—1581no—
403C2×C4C2×(C2×D79)⋊C411yescharacteristic
404C2×C4#1457911811yescharacteristic
405C2×C4C4×C22⋊D7911yescharacteristic
406C2×C4C4×D31611yescharacteristic
407C8—21no—
408C8—21no—
409C8—21no—
410C8—42no—
411C8—3162no—
412C8—3161no—
413C8—3161no—
414C8#1457907512yescomplemented
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—
428C22C4×(C2×D79)⋊C411yescharacteristic
429C22#1909851511yescharacteristic
430C22#1909852011yescharacteristic
431C22#1909863612yes—
432C22#1909863912yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—82no—
437C4—3164no—
438C4—3161no—
439C4—3161no—
440C4—3161no—
441C4C2×#1457907511yescharacteristic
442C4#1909853611yescharacteristic
443C2—22no—
444C2—22no—
445C2—21no—
446C2—21no—
447C2—6321no—
448C2#2376318511yescharacteristic
449C2C4×#1457907511yescharacteristic
450C2#2376320111yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D79)⋊C4
commutatorC23×C79C2×C42
FrattiniC23×C4C22×D79
FittingC79×#1752C2
socleC22×C79C4×C22⋊C4
prime radicalC22×C22⋊D79C2×C4
cosocleC23×C4×C79C23
hypercenter#1752D79
hypocenterC79#8161
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #1752

Complements

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

NH ≅ G/Ntype
3. C8×C22⋊Dic79C2semidirect
5. #23761248C2semidirect
6. #23761246C2semidirect
7. C79×#1752C2semidirect
8. C22×#11929607C4semidirect
14. C2×Dic79⋊C8C4semidirect
18. C2×(C2×D79)⋊C8C4semidirect
19. C2×#14579071C4semidirect
24. #19098288C4semidirect
25. #19098286C4semidirect
104. C22⋊Dic79M16semidirect
185. C8×C79#99semidirect
228. #1752D79semidirect
233. C79#8161semidirect
414. C8#14579075semidirect

Maximal subgroups

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

subgroupindex
C2×#190986342
C4×#145790752
C8×C22⋊Dic792
#237617162
#237612482
#237612462
C79×#17522
#816179