Finite Group Databasebeta

← #34191709

Subgroups — #34191709

Order 20224.

556 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23768354C211yesmaximal, characteristic
2#23764161C211yesmaximal, characteristic, complemented
3#23763397C211yesmaximal, characteristic
4C2×#19098309C211yesmaximal, characteristic, complemented
5C2×#19098292C211yesmaximal, characteristic
6#23761240C211yesmaximal, characteristic, complemented
7C79×#1990C211yesmaximal, characteristic, complemented
8C2×#14579137—21no—
9C2×#14579137—21no—
10C2×C4×D316C2211yescharacteristic
11C2×#14579092—21no—
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579081—21no—
15#19098498—22no—
16#19098464—21no—
17#19098432—21no—
18C4×C4⋊Dic79C2211yescharacteristic
19#19098325—21no—
20#19098325—21no—
21#19098309—22no—
22#19098292—22no—
23C2×C4⋊Q8×C79C2211yescharacteristic
24C2×C4×C8×C79C2211yescharacteristic
25#384×C79C2211yescharacteristic
26#340×C79—22no—
27C22×D316D411yescharacteristic
28C22×C4×D79—21no—
29C22×C79⋊C8—21no—
30C22×C79⋊C8—21no—
31#14579137—42no—
32#14579137—42no—
33#14579137—42no—
34#14579137—42no—
35C4×D316—21no—
36C4×D316—21no—
37C4×D316—21no—
38C4×D316—21no—
39C2×C4⋊Dic79D411yescharacteristic
40C2×C4⋊Dic79D411yescharacteristic
41C2×C4⋊Dic79D411yescharacteristic
42C2×Dic79⋊C4—22no—
43C2×Dic79⋊C4—22no—
44C2×(C2×D79)⋊C4—22no—
45C2×(C2×D79)⋊C4—22no—
46C2×(C2×D79)⋊C4—21no—
47C2×C4×Dic79—21no—
48#14579092—42no—
49C4⋊(C79⋊C8)—21no—
50C4⋊(C79⋊C8)—21no—
51C4⋊(C79⋊C8)—21no—
52C4⋊(C79⋊C8)D412yes—
53#14579081—42no—
54C79⋊#95—21no—
55C22×Q8×C79—21no—
56C22×C8×C79—21no—
57C4⋊Q8×C79—21no—
58C4⋊Q8×C79—21no—
59C4⋊Q8×C79—21no—
60C4⋊Q8×C79—21no—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C4⋊C4×C79—21no—
65C2×C4⋊C4×C79D412yes—
66C2×C42×C79C2311yescharacteristic
67C4×C8×C79—21no—
68C4×C8×C79—21no—
69C23×D79—21no—
70C2×D316—21no—
71C2×D316—21no—
72C2×D316—21no—
73C2×D316—21no—
74C2×D316—21no—
75C2×D316SD1612yes—
76C22×Dic79—21no—
77C22×Dic79—21no—
78C22×Dic79—21no—
79C2×C4×D79—41no—
80C2×C4×D79—41no—
81C2×C4×D79—41no—
82C2×C4×D79—41no—
83C2×C4×D79—41no—
84C2×C4×D79—41no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C4⋊Dic79—21no—
92C4⋊Dic79—21no—
93C4⋊Dic79—21no—
94C4⋊Dic79—21no—
95C4⋊Dic79D812yes—
96C4⋊Dic79SD1612yes—
97Dic79⋊C4—44no—
98Dic79⋊C4—44no—
99(C2×D79)⋊C4—44no—
100(C2×D79)⋊C4—44no—
101(C2×D79)⋊C4—41no—
102(C2×D79)⋊C4—41no—
103C4×Dic79—41no—
104C4×Dic79—41no—
105C2×Q8×C79—41no—
106C2×Q8×C79—41no—
107C2×Q8×C79—41no—
108C2×Q8×C79—41no—
109C2×Q8×C79—41no—
110C2×Q8×C79—41no—
111C22×C4×C79—22no—
112C22×C4×C79—41no—
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C2×C8×C79—22no—
117C2×C8×C79—41no—
118C2×C8×C79—41no—
119C4⋊C4×C79—22no—
120C4⋊C4×C79—22no—
121C4⋊C4×C79—42no—
122C4⋊C4×C79—41no—
123C4⋊C4×C79—41no—
124C4⋊C4×C79—41no—
125C4⋊C4×C79—41no—
126C4⋊C4×C79—41no—
127C4⋊C4×C79—41no—
128C42×C79C2×D411yescharacteristic
129C42×C79#4111yescharacteristic
130C42×C79#4111yescharacteristic
131C42×C79#4111yescharacteristic
132C23×C79#12011yescharacteristic
133C22×D79—22no—
134C22×D79—41no—
135C22×D79—41no—
136C22×D79—41no—
137C22×D79—41no—
138C22×D79—41no—
139C22×D79—41no—
140Q8×C79—81no—
141Q8×C79—81no—
142Q8×C79—81no—
143Q8×C79—81no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—42no—
147C2×C4×C79—42no—
148C2×C4×C79—42no—
149C2×C4×C79—81no—
150C2×C4×C79—81no—
151C2×C4×C79—81no—
152C2×C4×C79#12111yescharacteristic
153C2×C4×C79#12111yescharacteristic
154C2×C4×C79#12111yescharacteristic
155C2×C4×C79#12311yescharacteristic
156C2×C4×C79#12311yescharacteristic
157C2×C4×C79#12411yescharacteristic
158C2×C4×C79C2×D811yescharacteristic
159C2×C4×C79C2×SD1611yescharacteristic
160C2×C4×C79C2×SD1611yescharacteristic
161C2×C4×C79#13511yescharacteristic
162C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C79#13711yescharacteristic
165C2×C4×C79#13711yescharacteristic
166C2×Dic79—22no—
167C2×Dic79—22no—
168C2×Dic79—41no—
169C2×Dic79—41no—
170C2×Dic79—41no—
171C2×Dic79—41no—
172C2×Dic79—41no—
173C2×Dic79—41no—
174C2×Dic79—41no—
175D316—22no—
176D316—22no—
177D316—41no—
178D316—41no—
179C4×D79—81no—
180C4×D79—81no—
181C4×D79—81no—
182C4×D79—81no—
183C8×C79—42no—
184C79⋊C8—42no—
185C79⋊C8—81no—
186C22×C79#39411yescharacteristic
187C22×C79#44711yescharacteristic
188C22×C79#44811yescharacteristic
189C22×C79#45011yescharacteristic
190C22×C79#45211yescharacteristic
191C22×C79#45411yescharacteristic
192C22×C79#45611yescharacteristic
193C2×D79—42no—
194C2×D79—42no—
195C2×D79—42no—
196C2×D79—81no—
197C2×D79—81no—
198C2×D79—81no—
199C2×D79—81no—
200C4×C79—21no—
201C4×C79—21no—
202C4×C79—82no—
203C4×C79—161no—
204C4×C79#45911yescharacteristic
205C4×C79#46011yescharacteristic
206C4×C79#46511yescharacteristic
207C4×C79#47011yescharacteristic
208C4×C79#48311yescharacteristic
209C4×C79#48511yescharacteristic
210C4×C79#48611yescharacteristic
211C4×C79#49011yescharacteristic
212Dic79—42no—
213Dic79—42no—
214Dic79—81no—
215#10420—791nomaximal, Sylow
216C2×C79#162711yescharacteristic
217C2×C79#163511yescharacteristic
218C2×C79#164711yescharacteristic
219C2×C79#165111yescharacteristic
220C2×C79#203511yescharacteristic
221C2×C79#203811yescharacteristic
222C2×C79#204111yescharacteristic
223D79—82no—
224#2542—791no—
225#1990D7911yescharacteristic, complemented
226#1948—791no—
227#1802—791no—
228C2×#332—791no—
229C2×D4⋊C8—791no—
230#1279—791no—
231C79#1042011yescharacteristic, complemented, Sylow
232C2×C4⋊Q8C2×D7911yescharacteristic
233C2×C22⋊Q8—1581no—
234C2×C22⋊Q8—1581no—
235C2×C4×D4—791no—
236C2×#106—1581no—
237C2×C4⋊C8—791no—
238C2×C4⋊C8—791no—
239C2×#102—1581no—
240C2×C4×C8C2×D7911yescharacteristic
241#399—1582no—
242#390—1581no—
243#384—1581no—
244#384C2×D7911yescharacteristic
245C4×C4⋊C4—791no—
246#340—22no—
247#340—1581no—
248#340—1581no—
249#332—1582no—
250D4⋊C8—1582no—
251C22×Q8—21no—
252C22×D4—791no—
253C23×C4—1581no—
254C22×C8—21no—
255C22×C8—1581no—
256C22×C8—1581no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C22⋊Q8—3162no—
262C22⋊Q8—3162no—
263C22⋊Q8—3162no—
264C22⋊Q8—3162no—
265C4×D4—1581no—
266C4×D4—1581no—
267C4×D4—1581no—
268C4×D4—1581no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—791no—
274C2×C4⋊C4—791no—
275C2×C4⋊C4—791no—
276C2×C4⋊C4—1582no—
277C2×C4⋊C4—1582no—
278C2×C4⋊C4C22⋊D7912yes—
279C2×C22⋊C4—1582no—
280C2×C22⋊C4—1582no—
281C2×C22⋊C4—1581no—
282C2×C42—1581no—
283C2×C42C22×D7911yescharacteristic
284#106—3162no—
285C4⋊C8—792no—
286C4⋊C8—1581no—
287C4⋊C8—1581no—
288C4⋊C8—1581no—
289#102—3162no—
290C4×C8—21no—
291C4×C8—21no—
292#95—1581no—
293C24—1581no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—792no—
301C2×D4—1581no—
302C2×D4—1581no—
303C2×D4—1581no—
304C2×D4—1581no—
305C2×D4—1581no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—3161no—
309C22×C4—3161no—
310C22×C4—3161no—
311C22×C4—3161no—
312C22×C4—3161no—
313C22×C4—3161no—
314C22×C4—1581no—
315C22×C4—1581no—
316C22×C4—1581no—
317C22×C4D4×D7911yescharacteristic
318C22×C4D4×D7911yescharacteristic
319C22×C4C2×C22⋊D7911yescharacteristic
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—3161no—
324C2×C8—3161no—
325C2×C8—3161no—
326C2×C8—3161no—
327C2×C8—3161no—
328C2×C8—1582no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—3164no—
339C4⋊C4—3164no—
340C4⋊C4—792no—
341C4⋊C4—792no—
342C4⋊C4—1581no—
343C4⋊C4—1581no—
344C4⋊C4—1581no—
345C4⋊C4—1581no—
346C22⋊C4—3164no—
347C22⋊C4—3164no—
348C22⋊C4—3161no—
349C22⋊C4—3161no—
350C42—3161no—
351C42—3161no—
352C42D4×D7911yescharacteristic
353C42#1192962311yescharacteristic
354C42#1192962411yescharacteristic
355C42#1192962411yescharacteristic
356C23—3161no—
357C23—3161no—
358C23—3161no—
359C23—3161no—
360C23—3161no—
361C23—3161no—
362C23—1582no—
363C23#1457912911yescharacteristic
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368D4—3161no—
369D4—3161no—
370D4—1582no—
371D4—1582no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—3161no—
381C2×C4—3161no—
382C2×C4—3161no—
383C2×C4—3161no—
384C2×C4—3161no—
385C2×C4—3161no—
386C2×C4—3161no—
387C2×C4—6321no—
388C2×C4—6321no—
389C2×C4—6321no—
390C2×C4—6321no—
391C2×C4—1582no—
392C2×C4—1582no—
393C2×C4#1457913311yescharacteristic
394C2×C4#1457913411yescharacteristic
395C2×C4#1457913411yescharacteristic
396C2×C4#1457914311yescharacteristic
397C2×C4#1457914411yescharacteristic
398C2×C4#1457914711yescharacteristic
399C2×C4D8×D7911yescharacteristic
400C2×C4SD16×D7911yescharacteristic
401C2×C4SD16×D7911yescharacteristic
402C2×C4#1457918411yescharacteristic
403C2×C4#1457918511yescharacteristic
404C2×C4#1457918811yescharacteristic
405C2×C4#1457919011yescharacteristic
406C2×C4#1457919311yescharacteristic
407C8—42no—
408C8—3162no—
409C8—6321no—
410C22—3162no—
411C22—3162no—
412C22—3162no—
413C22—6321no—
414C22—6321no—
415C22—6321no—
416C22—6321no—
417C22#1909847811yescharacteristic
418C22#1909869511yescharacteristic
419C22#1909870011yescharacteristic
420C22#1909871211yescharacteristic
421C22#1909872411yescharacteristic
422C22#1909873511yescharacteristic
423C22#1909874511yescharacteristic
424C4—21no—
425C4—21no—
426C4—82no—
427C4—161no—
428C4—3162no—
429C4—3162no—
430C4—6321no—
431C4#1909875711yescharacteristic
432C4#1909876211yescharacteristic
433C4#1909879511yescharacteristic
434C4#1909882011yescharacteristic
435C4#1909889211yescharacteristic
436C4#1909890311yescharacteristic
437C4#1909890811yescharacteristic
438C4#1909892611yescharacteristic
439C2—6322no—
440C2#2376250011yescharacteristic
441C2#2376254811yescharacteristic
442C2#2376263211yescharacteristic
443C2#2376266011yescharacteristic
444C2#2376459411yescharacteristic
445C2#2376461711yescharacteristic
446C2#2376463211yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1990C2
socleC23×C79#120
prime radicalC22×D316D4
hypercenter#1990D79
hypocenterC79#10420
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. #23764161C2semidirect
4. C2×#19098309C2semidirect
6. #23761240C2semidirect
7. C79×#1990C2semidirect
225. #1990D79semidirect
231. C79#10420semidirect

Maximal subgroups

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

subgroupindex
#237683542
#237641612
#237633972
C2×#190983092
C2×#190982922
#237612402
C79×#19902
#1042079