Finite Group Databasebeta

← #34184606

Subgroups — #34184606

Order 20224.

514 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772435C211yesmaximal, characteristic, complemented
2#23761998C211yesmaximal, characteristic, complemented
3#23761843C211yesmaximal, characteristic, complemented
4#23761752C211yesmaximal, characteristic, complemented
5#23761270C211yesmaximal, characteristic, complemented
6#23761250C211yesmaximal, characteristic, complemented
7C2×#463×C79C211yesmaximal, characteristic
8#19098641—21no—
9#19098635—22no—
10D4×C79⋊C8—21no—
11#19098620C2211yescharacteristic
12#19098620C2211yescharacteristic
13#19098616C2211yescharacteristic
14#19098527—21no—
15C2×C22⋊(C79⋊C8)—21no—
16C4×#11929608C2211yescharacteristic
17#19098303—22no—
18D4⋊(C79⋊C8)—21no—
19D4⋊(C79⋊C8)—21no—
20#19098287—21no—
21#19098287—21no—
22C22×SD16×C79C411yescharacteristic
23C22×D8×C79C411yescharacteristic
24C2×#124×C79C2211yescharacteristic
25C2×C4×D4×C79C2211yescharacteristic
26#463×C79—21no—
27#463×C79—21no—
28#463×C79—21no—
29#463×C79—21no—
30C2×C4⋊C8×C79C2211yescharacteristic
31C2×#103×C79C411yescharacteristic
32C2×#102×C79C411yescharacteristic
33C2×#11929608—22no—
34C2×#11929608—21no—
35C22×C79⋊C8—22no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C22⋊(C79⋊C8)—22no—
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)—21no—
42C22⋊(C79⋊C8)—21no—
43C22⋊(C79⋊C8)D411yescharacteristic
44C22⋊(C79⋊C8)D411yescharacteristic
45#14579072—21no—
46#14579072—21no—
47#14579072D412yes—
48C4×C79⋊C8—21no—
49C4×C79⋊C8—21no—
50C22×Q8×C79C2×C411yescharacteristic
51C22×D4×C79C2×C411yescharacteristic
52C22×D4×C79C2×C411yescharacteristic
53C23×C4×C79—21no—
54C2×SD16×C79—21no—
55C2×SD16×C79—21no—
56C2×SD16×C79—42no—
57C2×D8×C79—21no—
58C2×D8×C79—21no—
59C2×D8×C79—42no—
60C22×C8×C79C2×C411yescharacteristic
61C22×C8×C79C2×C411yescharacteristic
62#124×C79—21no—
63#124×C79—21no—
64#124×C79—21no—
65#124×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79—21no—
68C4×D4×C79D411yescharacteristic
69C4×D4×C79D411yescharacteristic
70C4×D4×C79D411yescharacteristic
71C4×D4×C79D411yescharacteristic
72C2×C4⋊C4×C79C2×C411yescharacteristic
73C2×C22⋊C4×C79—21no—
74C2×C22⋊C4×C79—21no—
75C2×C22⋊C4×C79—21no—
76C2×C42×C79C2311yescharacteristic
77C4⋊C8×C79—21no—
78C4⋊C8×C79—21no—
79#103×C79—21no—
80#103×C79—21no—
81#102×C79—21no—
82#102×C79—21no—
83#11929608—22no—
84#11929608—41no—
85#11929608—82no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—82no—
92C24×C79—21no—
93C24×C79—21no—
94C2×Q8×C79—21no—
95C2×Q8×C79—21no—
96C2×Q8×C79—41no—
97C2×Q8×C79—41no—
98C2×D4×C79—22no—
99C2×D4×C79—21no—
100C2×D4×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79—41no—
103C2×D4×C79—41no—
104C2×D4×C79—41no—
105C2×D4×C79C22⋊C411yescharacteristic
106C2×D4×C79C22⋊C411yescharacteristic
107C22×C4×C79—21no—
108C22×C4×C79—21no—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—21no—
112C22×C4×C79—21no—
113C22×C4×C79—82no—
114C22×C4×C79C22×C411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79#4111yescharacteristic
117SD16×C79—42no—
118SD16×C79—42no—
119D8×C79—42no—
120D8×C79—42no—
121C2×C8×C79—21no—
122C2×C8×C79—21no—
123C2×C8×C79—41no—
124C2×C8×C79—41no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79C22⋊C411yescharacteristic
127C4⋊C4×C79C22⋊C411yescharacteristic
128C22⋊C4×C79—21no—
129C22⋊C4×C79—21no—
130C22⋊C4×C79—21no—
131C22⋊C4×C79—21no—
132C22⋊C4×C79—41no—
133C22⋊C4×C79—41no—
134C22⋊C4×C79—41no—
135C22⋊C4×C79—41no—
136C42×C79C2×D411yescharacteristic
137C42×C79C2×D411yescharacteristic
138C42×C79C2×D411yescharacteristic
139C42×C79#4111yescharacteristic
140C23×C79—21no—
141C23×C79—21no—
142C23×C79—42no—
143C23×C79—41no—
144C23×C79—41no—
145C23×C79—41no—
146C23×C79—41no—
147C23×C79—41no—
148C23×C79—81no—
149C23×C79—81no—
150C23×C79—81no—
151C23×C79C4×D411yescharacteristic
152Q8×C79—41no—
153Q8×C79—41no—
154Q8×C79—81no—
155D4×C79—21no—
156D4×C79—21no—
157D4×C79—21no—
158D4×C79—42no—
159D4×C79—41no—
160D4×C79—41no—
161D4×C79—81no—
162D4×C79#10012yes—
163C2×C4×C79—21no—
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—42no—
172C2×C4×C79—41no—
173C2×C4×C79—81no—
174C2×C4×C79—81no—
175C2×C4×C79—81no—
176C2×C4×C79—81no—
177C2×C4×C79C2×C22⋊C411yescharacteristic
178C2×C4×C79C4×D411yescharacteristic
179C2×C4×C79#12011yescharacteristic
180C2×C4×C79#12111yescharacteristic
181C2×C4×C79#12111yescharacteristic
182C2×C4×C79#12311yescharacteristic
183C8×C79—41no—
184C8×C79—41no—
185C79⋊C8—41no—
186C79⋊C8—42no—
187C79⋊C8—41no—
188C79⋊C8—42no—
189C79⋊C8—81no—
190C22×C79—21no—
191C22×C79—21no—
192C22×C79—22no—
193C22×C79—41no—
194C22×C79—41no—
195C22×C79—41no—
196C22×C79—41no—
197C22×C79—41no—
198C22×C79—82no—
199C22×C79—82no—
200C22×C79—81no—
201C22×C79—81no—
202C22×C79—81no—
203C22×C79—161no—
204C22×C79—161no—
205C22×C79#38611yescharacteristic
206C22×C79#44311yescharacteristic
207C22×C79#44411yescharacteristic
208C4×C79—21no—
209C4×C79—21no—
210C4×C79—21no—
211C4×C79—21no—
212C4×C79—21no—
213C4×C79—41no—
214C4×C79—41no—
215C4×C79—81no—
216C4×C79C2×#10011yescharacteristic
217C4×C79#41311yescharacteristic
218#9243—791nomaximal, Sylow
219C2×C79—21no—
220C2×C79—21no—
221C2×C79—42no—
222C2×C79—81no—
223C2×C79—161no—
224C2×C79#187611yescharacteristic
225C2×C79#190011yescharacteristic
226C2×C79#190411yescharacteristic
227C2×#463D7911yescharacteristic
228#2974—791no—
229#1527—791no—
230#1490—791no—
231#1469—791no—
232#1290—791no—
233#1283—791no—
234C79#924311yescharacteristic, complemented, Sylow
235C22×SD16Dic7911yescharacteristic
236C22×D8Dic7911yescharacteristic
237C2×#124C2×D7911yescharacteristic
238C2×C4×D4C2×D7911yescharacteristic
239#463—21no—
240#463—21no—
241#463—21no—
242#463—21no—
243#436—1581no—
244#435—1582no—
245C8×D4—1581no—
246#432—791no—
247#432—791no—
248#431—791no—
249C2×C4⋊C8C2×D7911yescharacteristic
250C2×#103Dic7911yescharacteristic
251C2×#102Dic7911yescharacteristic
252#407—1581no—
253C2×C22⋊C8—1581no—
254C4×M16—791no—
255#329—1582no—
256D4⋊C8—1581no—
257D4⋊C8—1581no—
258#323—1581no—
259#323—1581no—
260C22×Q8C2×Dic7911yescharacteristic
261C22×D4C2×Dic7911yescharacteristic
262C22×D4C2×Dic7911yescharacteristic
263C23×C4—21no—
264C2×SD16—21no—
265C2×SD16—21no—
266C2×SD16—42no—
267C2×D8—21no—
268C2×D8—21no—
269C2×D8—42no—
270C2×M16—1581no—
271C2×M16—1582no—
272C22×C8—1582no—
273C22×C8C2×Dic7911yescharacteristic
274C22×C8C2×Dic7911yescharacteristic
275#124—21no—
276#124—21no—
277#124—21no—
278#124—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4C22⋊D7911yescharacteristic
282C4×D4C22⋊D7911yescharacteristic
283C4×D4C22⋊D7911yescharacteristic
284C4×D4C22⋊D7911yescharacteristic
285C2×C4⋊C4C2×Dic7911yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C42C22×D7911yescharacteristic
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—1581no—
293C4⋊C8—1581no—
294C4⋊C8—1581no—
295#103—21no—
296#103—21no—
297#102—21no—
298#102—21no—
299C22⋊C8—791no—
300C22⋊C8—791no—
301C22⋊C8—1581no—
302C22⋊C8—1582no—
303C22⋊C8—1582no—
304C22⋊C8—1581no—
305#97—792no—
306#97—1581no—
307#97—1581no—
308C4×C8—1581no—
309C4×C8—1581no—
310C24—21no—
311C24—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×D4—22no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4C22⋊Dic7911yescharacteristic
324C2×D4C22⋊Dic7911yescharacteristic
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—82no—
332C22×C4C22×Dic7911yescharacteristic
333C22×C4D4×D7911yescharacteristic
334C22×C4#1192962311yescharacteristic
335SD16—42no—
336SD16—42no—
337D8—42no—
338D8—42no—
339M16—3161no—
340M16—6322no—
341M16—1582no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—3161no—
347C2×C8—3161no—
348C2×C8—6322no—
349C2×C8—1581no—
350C2×C8—1581no—
351C2×C8—1582no—
352C4⋊C4—21no—
353C4⋊C4C22⋊Dic7911yescharacteristic
354C4⋊C4C22⋊Dic7911yescharacteristic
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C42D4×D7911yescharacteristic
364C42C2×C22⋊D7911yescharacteristic
365C42C2×C22⋊D7911yescharacteristic
366C42#1192962311yescharacteristic
367C23—21no—
368C23—21no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23D4×Dic7911yescharacteristic
379Q8—41no—
380Q8—41no—
381Q8—81no—
382D4—21no—
383D4—21no—
384D4—21no—
385D4—42no—
386D4—41no—
387D4—41no—
388D4—81no—
389D4#1457907812yes—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4C2×C22⋊Dic7911yescharacteristic
405C2×C4D4×Dic7911yescharacteristic
406C2×C4#1457912911yescharacteristic
407C2×C4#1457913311yescharacteristic
408C2×C4#1457913311yescharacteristic
409C2×C4#1457914211yescharacteristic
410C8—41no—
411C8—41no—
412C8—3162no—
413C8—3161no—
414C8—3162no—
415C8—3161no—
416C8—6321no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—82no—
426C22—82no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—161no—
431C22—161no—
432C22#1909844311yescharacteristic
433C22#1909868011yescharacteristic
434C22#1909868511yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4C2×#1457907811yescharacteristic
444C4#1909854711yescharacteristic
445C2—21no—
446C2—21no—
447C2—42no—
448C2—81no—
449C2—161no—
450C2#2376372911yescharacteristic
451C2#2376386411yescharacteristic
452C2#2376388911yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC2×#463×C79C2
socleC22×C79#386
prime radicalC22×D8×C79C4
cosocleC2×C42×C79C23
hypercenterC2×#463D79
hypocenterC79#9243
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
1. #23772435C2semidirect
2. #23761998C2semidirect
3. #23761843C2semidirect
4. #23761752C2semidirect
5. #23761270C2semidirect
6. #23761250C2semidirect
234. C79#9243semidirect

Maximal subgroups

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

subgroupindex
#237724352
#237619982
#237618432
#237617522
#237612702
#237612502
C2×#463×C792
#924379