Finite Group Databasebeta

← #34184375

Subgroups — #34184375

Order 20224.

510 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772367C211yesmaximal, characteristic, complemented
2#23763420C211yesmaximal, characteristic, complemented
3#23761895C211yesmaximal, characteristic, complemented
4#23761851C211yesmaximal, characteristic, complemented
5#23761297C211yesmaximal, characteristic, complemented
6#23761239C211yesmaximal, characteristic, complemented
7C79×#3035C211yesmaximal, characteristic
8#19098694—21no—
9#19098641—21no—
10#19098635—22no—
11#19098616—21no—
12#19098616C2211yescharacteristic
13#19098587C2211yescharacteristic
14C2×#14579088—21no—
15C4×#11929608—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17C4×C4⋊Dic79C2211yescharacteristic
18#19098322—21no—
19#19098297—22no—
20D4⋊(C79⋊C8)—21no—
21D4⋊(C79⋊C8)—21no—
22C2×#135×C79—21no—
23#532×C79C2211yescharacteristic
24C4×#41×C79C2211yescharacteristic
25#464×C79—21no—
26#463×C79—21no—
27#462×C79C411yescharacteristic, complemented
28#461×C79C411yescharacteristic, complemented
29#460×C79C411yescharacteristic, complemented
30#459×C79C411yescharacteristic, complemented
31C2×C4⋊C8×C79C2211yescharacteristic
32#416×C79—21no—
33C2×#11929608—22no—
34C2×#11929608—21no—
35C22×C79⋊C8—21no—
36C2×C4⋊Dic79—21no—
37C2×C4×Dic79—21no—
38C4⋊(C79⋊C8)—22no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42#14579088—42no—
43C22⋊(C79⋊C8)—22no—
44C22⋊(C79⋊C8)D412yes—
45#14579072—22no—
46C4×C79⋊C8—21no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8D411yescharacteristic
49C4×C79⋊C8D411yescharacteristic
50C79⋊#95—21no—
51C2×#41×C79—21no—
52C2×#41×C79D411yescharacteristic
53#135×C79—42no—
54#135×C79—42no—
55C2×Q16×C79—21no—
56C2×SD16×C79—21no—
57C2×SD16×C79—21no—
58C2×D8×C79—21no—
59C22×C8×C79—21no—
60C4⋊Q8×C79C2×C411yescharacteristic
61#127×C79C2×C411yescharacteristic
62#124×C79—21no—
63#121×C79—21no—
64#121×C79—21no—
65C4×Q8×C79C2×C411yescharacteristic
66C4×Q8×C79D411yescharacteristic
67C4×D4×C79—21no—
68C4×D4×C79—21no—
69C4×D4×C79—21no—
70C4×D4×C79—21no—
71C4×D4×C79C2×C411yescharacteristic
72C4×D4×C79D411yescharacteristic
73#117×C79—21no—
74#117×C79D411yescharacteristic
75C2×C42×C79—21no—
76C2×C42×C79C2311yescharacteristic
77C4⋊C8×C79—21no—
78C4⋊C8×C79C2×C411yescharacteristic
79C4⋊C8×C79C2×C411yescharacteristic
80#103×C79—21no—
81#103×C79—21no—
82#102×C79—21no—
83#102×C79—21no—
84C22×Dic79—41no—
85#11929608—44no—
86#11929608—41no—
87#11929608—41no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—42no—
92C2×C79⋊C8—41no—
93C4⋊Dic79—22no—
94C4⋊Dic79—41no—
95C4×Dic79—41no—
96C4×Dic79—41no—
97#41×C79—21no—
98#41×C79—21no—
99#41×C79—21no—
100#41×C79—21no—
101#41×C79—41no—
102#41×C79—41no—
103#41×C79—41no—
104#41×C79—41no—
105C2×Q8×C79—21no—
106C2×Q8×C79C22⋊C411yescharacteristic
107C2×D4×C79—21no—
108C2×D4×C79—21no—
109C2×D4×C79—21no—
110C2×D4×C79—21no—
111C2×D4×C79—41no—
112C2×D4×C79C22⋊C411yescharacteristic
113C22×C4×C79—21no—
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79—41no—
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79#4111yescharacteristic
120Q16×C79—42no—
121SD16×C79—42no—
122SD16×C79—42no—
123D8×C79—42no—
124C2×C8×C79—21no—
125C2×C8×C79—21no—
126C2×C8×C79—41no—
127C2×C8×C79—41no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79—21no—
131C4⋊C4×C79—21no—
132C4⋊C4×C79C22⋊C411yescharacteristic
133C4⋊C4×C79C22⋊C411yescharacteristic
134C22⋊C4×C79—21no—
135C22⋊C4×C79—21no—
136C22⋊C4×C79—21no—
137C22⋊C4×C79—41no—
138C22⋊C4×C79—41no—
139C42×C79—21no—
140C42×C79—21no—
141C42×C79—21no—
142C42×C79C22×C411yescharacteristic
143C42×C79C2×D411yescharacteristic
144C42×C79C2×D411yescharacteristic
145C42×C79#4111yescharacteristic
146C23×C79—21no—
147C23×C79—41no—
148C23×C79#12111yescharacteristic
149Q8×C79—21no—
150Q8×C79—21no—
151Q8×C79—41no—
152Q8×C79—41no—
153D4×C79—21no—
154D4×C79—21no—
155D4×C79—41no—
156D4×C79—41no—
157D4×C79—41no—
158D4×C79—41no—
159D4×C79—41no—
160D4×C79—41no—
161D4×C79—81no—
162D4×C79—81no—
163D4×C79—81no—
164C2×C4×C79—21no—
165C2×C4×C79—21no—
166C2×C4×C79—21no—
167C2×C4×C79—21no—
168C2×C4×C79—21no—
169C2×C4×C79—41no—
170C2×C4×C79—42no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79—41no—
174C2×C4×C79—41no—
175C2×C4×C79—41no—
176C2×C4×C79—81no—
177C2×C4×C79—81no—
178C2×C4×C79C2×C22⋊C411yescharacteristic
179C2×C4×C79C4×D411yescharacteristic
180C2×C4×C79C4×D411yescharacteristic
181C2×C4×C79#12011yescharacteristic
182C2×C4×C79#12111yescharacteristic
183C2×C4×C79#12311yescharacteristic
184C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C79#13711yescharacteristic
188C2×Dic79—42no—
189C2×Dic79—81no—
190C2×Dic79—81no—
191C8×C79—41no—
192C8×C79—41no—
193C79⋊C8—22no—
194C79⋊C8—41no—
195C79⋊C8—42no—
196C79⋊C8—81no—
197C22×C79—21no—
198C22×C79—21no—
199C22×C79—41no—
200C22×C79—41no—
201C22×C79—41no—
202C22×C79—81no—
203C22×C79—81no—
204C22×C79—81no—
205C22×C79#38611yescharacteristic
206C22×C79#46811yescharacteristic
207C22×C79#46911yescharacteristic
208C4×C79—21no—
209C4×C79—21no—
210C4×C79—21no—
211C4×C79—41no—
212C4×C79—41no—
213C4×C79—41no—
214C4×C79—81no—
215C4×C79#41711yescharacteristic
216C4×C79#41811yescharacteristic
217C4×C79#44311yescharacteristic
218C4×C79#44411yescharacteristic
219C4×C79#46811yescharacteristic
220C4×C79#46911yescharacteristic
221Dic79—82no—
222#9210—791nomaximal, Sylow
223C2×C79—21no—
224C2×C79—21no—
225C2×C79—81no—
226C2×C79—161no—
227C2×C79#187611yescharacteristic
228C2×C79#189511yescharacteristic
229C2×C79#190611yescharacteristic
230#3035D7911yescharacteristic
231#2965—791no—
232#1806—791no—
233#1502—791no—
234#1492—791no—
235#1301—791no—
236#1279—791no—
237C79#921011yescharacteristic, complemented, Sylow
238C2×#135—21no—
239#532C2×D7911yescharacteristic
240C4×#41C2×D7911yescharacteristic
241#464—21no—
242#463—21no—
243#462Dic7911yescharacteristic, complemented
244#461Dic7911yescharacteristic, complemented
245#460Dic7911yescharacteristic, complemented
246#459Dic7911yescharacteristic, complemented
247#446—1581no—
248#436—1581no—
249#435—1582no—
250#431—791no—
251#431—1581no—
252#423—791no—
253C2×C4⋊C8C2×D7911yescharacteristic
254C2×#104—1581no—
255#416—21no—
256C4×M16—1581no—
257C2×C4×C8—791no—
258C4×C4⋊C4—791no—
259#339—1581no—
260#327—1582no—
261D4⋊C8—1581no—
262D4⋊C8—1581no—
263C2×#41—21no—
264C2×#41C22⋊D7911yescharacteristic
265#135—42no—
266#135—42no—
267C2×Q16—21no—
268C2×SD16—21no—
269C2×SD16—21no—
270C2×D8—21no—
271C2×M16—1581no—
272C2×M16—1582no—
273C22×C8—21no—
274C22×C8—1581no—
275C4⋊Q8C2×Dic7911yescharacteristic
276#127C2×Dic7911yescharacteristic
277#124—21no—
278#121—21no—
279#121—21no—
280C4×Q8C2×Dic7911yescharacteristic
281C4×Q8C22⋊D7911yescharacteristic
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C2×Dic7911yescharacteristic
287C4×D4C22⋊D7911yescharacteristic
288#117—21no—
289#117C22⋊D7911yescharacteristic
290C2×C4⋊C4—1581no—
291C2×C42—21no—
292C2×C42—1581no—
293C2×C42C22×D7911yescharacteristic
294C4⋊C8—21no—
295C4⋊C8—1582no—
296C4⋊C8—1581no—
297C4⋊C8—1581no—
298C4⋊C8—1581no—
299C4⋊C8C2×Dic7911yescharacteristic
300C4⋊C8C2×Dic7911yescharacteristic
301#104—3162no—
302#103—21no—
303#103—21no—
304#102—21no—
305#102—21no—
306C22⋊C8—792no—
307C22⋊C8—1582no—
308#97—1582no—
309C4×C8—791no—
310C4×C8—791no—
311C4×C8—1581no—
312C4×C8—1581no—
313#95—1581no—
314#41—21no—
315#41—21no—
316#41—21no—
317#41—21no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322C2×Q8—21no—
323C2×Q8C22⋊Dic7911yescharacteristic
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4C22⋊Dic7911yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—3161no—
335C22×C4D4×D7911yescharacteristic
336C22×C4C2×C22⋊D7911yescharacteristic
337C22×C4#1192962311yescharacteristic
338Q16—42no—
339SD16—42no—
340SD16—42no—
341D8—42no—
342M16—3164no—
343M16—3161no—
344M16—3161no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—3162no—
350C2×C8—3161no—
351C2×C8—1581no—
352C2×C8—1582no—
353C2×C8—1581no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—3161no—
359C4⋊C4—1582no—
360C4⋊C4C22⋊Dic7911yescharacteristic
361C4⋊C4C22⋊Dic7911yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—3161no—
371C42—3161no—
372C42C22×Dic7911yescharacteristic
373C42D4×D7911yescharacteristic
374C42C2×C22⋊D7911yescharacteristic
375C42#1192962311yescharacteristic
376C23—21no—
377C23—41no—
378C23#1457913311yescharacteristic
379Q8—21no—
380Q8—21no—
381Q8—41no—
382Q8—41no—
383D4—21no—
384D4—21no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—81no—
392D4—81no—
393D4—81no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—3162no—
409C2×C4—6321no—
410C2×C4—6321no—
411C2×C4C2×C22⋊Dic7911yescharacteristic
412C2×C4D4×Dic7911yescharacteristic
413C2×C4D4×Dic7911yescharacteristic
414C2×C4#1457912911yescharacteristic
415C2×C4#1457913311yescharacteristic
416C2×C4#1457914211yescharacteristic
417C2×C4#1457918611yescharacteristic
418C2×C4#1457918711yescharacteristic
419C2×C4#1457918711yescharacteristic
420C2×C4#1457919111yescharacteristic
421C8—41no—
422C8—41no—
423C8—3161no—
424C8—3162no—
425C8—6321no—
426C8—1582no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22#1909844311yescharacteristic
436C22#1909880711yescharacteristic
437C22#1909881411yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—81no—
445C4—6322no—
446C4#1909856411yescharacteristic
447C4#1909856811yescharacteristic
448C4#1909868011yescharacteristic
449C4#1909868511yescharacteristic
450C4#1909880711yescharacteristic
451C4#1909881411yescharacteristic
452C2—21no—
453C2—21no—
454C2—81no—
455C2—161no—
456C2#2376372911yescharacteristic
457C2#2376383911yescharacteristic
458C2#2376390311yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3035C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3035D79
hypocenterC79#9210
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #23772367C2semidirect
2. #23763420C2semidirect
3. #23761895C2semidirect
4. #23761851C2semidirect
5. #23761297C2semidirect
6. #23761239C2semidirect
27. #462×C79C4semidirect
28. #461×C79C4semidirect
29. #460×C79C4semidirect
30. #459×C79C4semidirect
237. C79#9210semidirect
243. #462Dic79semidirect
244. #461Dic79semidirect
245. #460Dic79semidirect
246. #459Dic79semidirect

Maximal subgroups

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

subgroupindex
#237723672
#237634202
#237618952
#237618512
#237612972
#237612392
C79×#30352
#921079