Finite Group Databasebeta

← #34181239

Subgroups — #34181239

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772095C211yesmaximal, characteristic
2#23763436C211yesmaximal, characteristic, complemented
3#23763423C211yesmaximal, characteristic, complemented
4#23761943C211yesmaximal, characteristic
5#23761307C211yesmaximal, characteristic, complemented
6#23761306C211yesmaximal, characteristic, complemented
7C79×#1513C211yesmaximal, characteristic
8C2×#14579168C411yescharacteristic, complemented
9C23×C79⋊C8C2211yescharacteristic
10#19098635—21no—
11#19098635—21no—
12D4×C79⋊C8—21no—
13D4×C79⋊C8—21no—
14C2×#14579096—21no—
15C2×#14579092—21no—
16#19098590C411yescharacteristic, complemented
17#19098530C411yescharacteristic, complemented
18#19098527C2211yescharacteristic
19#19098527C2211yescharacteristic
20#19098519C411yescharacteristic, complemented
21#19098423C2211yescharacteristic
22#19098327—21no—
23#19098325—21no—
24#19098300—21no—
25#19098297—21no—
26#525×C79C2211yescharacteristic
27#407×C79C2211yescharacteristic
28#380×C79C2211yescharacteristic
29#328×C79—21no—
30#327×C79—21no—
31#14579168—42no—
32C2×#11929608—21no—
33C2×#11929608—21no—
34C2×#11929608C2×C411yescharacteristic
35C2×#11929608C2×C411yescharacteristic
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
44C2×C4⋊Dic79—21no—
45C2×C22⋊Dic79—21no—
46#14579096—42no—
47#14579092—42no—
48C4⋊(C79⋊C8)—21no—
49C4⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)—21no—
51C22⋊(C79⋊C8)—21no—
52C22⋊(C79⋊C8)—21no—
53C22⋊(C79⋊C8)—21no—
54C22⋊(C79⋊C8)—21no—
55C22⋊(C79⋊C8)—21no—
56#14579072—21no—
57C4×C79⋊C8—21no—
58C79⋊#95—21no—
59C2×#41×C79C2×C411yescharacteristic
60C23×C4×C79C2311yescharacteristic
61C2×M16×C79—21no—
62#123×C79—21no—
63C22⋊Q8×C79D411yescharacteristic
64C22⋊Q8×C79Q811yescharacteristic
65#121×C79D411yescharacteristic
66#121×C79Q811yescharacteristic
67#120×C79—21no—
68C4×D4×C79—21no—
69C4×D4×C79—21no—
70#117×C79C2×C411yescharacteristic
71C2×C4⋊C4×C79—21no—
72C2×C22⋊C4×C79—21no—
73C22⋊C8×C79—21no—
74C22⋊C8×C79—21no—
75#95×C79—21no—
76C22×Dic79—41no—
77#11929608—41no—
78#11929608—41no—
79#11929608—41no—
80#11929608—41no—
81#11929608—42no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—22no—
84C2×C79⋊C8—22no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—42no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8C22⋊C411yescharacteristic
93C2×C79⋊C8C22⋊C411yescharacteristic
94C2×C79⋊C8C22⋊C411yescharacteristic
95C2×C79⋊C8C22⋊C411yescharacteristic
96C4⋊Dic79—41no—
97C4⋊Dic79—41no—
98C22⋊Dic79—81no—
99C24×C79#4111yescharacteristic
100#41×C79—41no—
101#41×C79—41no—
102C2×Q8×C79C4⋊C411yescharacteristic
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79C4⋊C411yescharacteristic
107C22×C4×C79—21no—
108C22×C4×C79—41no—
109C22×C4×C79—41no—
110C22×C4×C79—41no—
111C22×C4×C79C22×C411yescharacteristic
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79C2×Q811yescharacteristic
116C22×C4×C79#4111yescharacteristic
117M16×C79—41no—
118M16×C79—41no—
119C2×C8×C79—41no—
120C4⋊C4×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—41no—
123C4⋊C4×C79—41no—
124C4⋊C4×C79C4⋊C411yescharacteristic
125C4⋊C4×C79C4⋊C411yescharacteristic
126C22⋊C4×C79—21no—
127C22⋊C4×C79—21no—
128C22⋊C4×C79—21no—
129C22⋊C4×C79—21no—
130C22⋊C4×C79—21no—
131C22⋊C4×C79—81no—
132C42×C79—21no—
133C23×C79—21no—
134C23×C79—21no—
135C23×C79—21no—
136C23×C79—41no—
137C23×C79—41no—
138C23×C79#11711yescharacteristic
139C23×C79#12111yescharacteristic
140C23×C79C22⋊Q811yescharacteristic
141Q8×C79—41no—
142D4×C79—41no—
143D4×C79—42no—
144D4×C79—42no—
145D4×C79—41no—
146D4×C79—41no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
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—41no—
160C2×C4×C79—41no—
161C2×C4×C79—81no—
162C2×C4×C79—81no—
163C2×C4×C79—81no—
164C2×C4×C79C2×C22⋊C411yescharacteristic
165C2×C4×C79C2×C4⋊C411yescharacteristic
166C2×C4×C79#12111yescharacteristic
167C2×C4×C79C22⋊Q811yescharacteristic
168C2×Dic79—81no—
169C2×Dic79—81no—
170C2×Dic79—81no—
171C8×C79—81no—
172C79⋊C8—21no—
173C79⋊C8—21no—
174C79⋊C8—21no—
175C79⋊C8—41no—
176C79⋊C8—41no—
177C79⋊C8—81no—
178C79⋊C8#9912yes—
179C22×C79—21no—
180C22×C79—21no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—21no—
187C22×C79—42no—
188C22×C79—42no—
189C22×C79—41no—
190C22×C79—41no—
191C22×C79—41no—
192C22×C79#38011yescharacteristic
193C22×C79#47111yescharacteristic
194C22×C79#47111yescharacteristic
195C4×C79—21no—
196C4×C79—21no—
197C4×C79—21no—
198C4×C79—41no—
199C4×C79—41no—
200C4×C79—41no—
201C4×C79—161no—
202C4×C79C2×#9911yescharacteristic
203C4×C79#41011yescharacteristic
204Dic79—161no—
205#8626—791nomaximal, Sylow
206C2×C79—21no—
207C2×C79—21no—
208C2×C79—41no—
209C2×C79—41no—
210C2×C79—81no—
211C2×C79#179911yescharacteristic
212C2×C79#181411yescharacteristic
213C2×C79#181511yescharacteristic
214#2929—791no—
215#1810—791no—
216#1807—791no—
217#1513—791no—
218#1513D7911yescharacteristic
219#1305—791no—
220#1305—791no—
221C79#862611yescharacteristic, complemented, Sylow
222C2×#131—791no—
223C23×C8—791no—
224#525C2×D7911yescharacteristic
225#435—1581no—
226#435—1581no—
227C8×D4—1581no—
228C8×D4—1581no—
229C2×#108—1581no—
230C2×#106—1581no—
231#424—791no—
232#408—791no—
233#407—791no—
234#407—791no—
235#407C2×D7911yescharacteristic
236#405—791no—
237#380—791no—
238#380C2×D7911yescharacteristic
239#341—1581no—
240#340—1581no—
241#328—21no—
242#328—1581no—
243#327—21no—
244#327—1581no—
245C2×#41C4×D7911yescharacteristic
246C23×C4C22×D7911yescharacteristic
247#131—3162no—
248C2×M16—21no—
249C2×M16—791no—
250C2×M16—791no—
251C2×M16—1581no—
252C2×M16—1581no—
253C22×C8—3162no—
254C22×C8—791no—
255C22×C8—791no—
256C22×C8—791no—
257C22×C8—791no—
258C22×C8—791no—
259C22×C8—791no—
260C22×C8—1581no—
261#123—21no—
262C22⋊Q8Dic15811yescharacteristic
263C22⋊Q8C22⋊D7911yescharacteristic
264#121Dic15811yescharacteristic
265#121C22⋊D7911yescharacteristic
266#120—21no—
267C4×D4—21no—
268C4×D4—21no—
269#117C4×D7911yescharacteristic
270C2×C4⋊C4—21no—
271C2×C4⋊C4—1581no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—1581no—
274#108—3162no—
275#106—3162no—
276C4⋊C8—1581no—
277C4⋊C8—1581no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—1581no—
281C22⋊C8—1581no—
282C22⋊C8—1581no—
283C22⋊C8—1581no—
284C22⋊C8—1581no—
285C22⋊C8—1581no—
286#97—1581no—
287C4×C8—1581no—
288#95—21no—
289#95—1581no—
290C24#1192962311yescharacteristic
291#41—41no—
292#41—41no—
293C2×Q8Dic79⋊C411yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4Dic79⋊C411yescharacteristic
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—3161no—
303C22×C4C2×C4×D7911yescharacteristic
304C22×C4D4×D7911yescharacteristic
305C22×C4D4×D7911yescharacteristic
306C22×C4C2×C22⋊D7911yescharacteristic
307C22×C4C2×Dic15811yescharacteristic
308C22×C4#1192962311yescharacteristic
309M16—41no—
310M16—41no—
311M16—3162no—
312M16—3161no—
313M16—3161no—
314M16—3161no—
315M16—3161no—
316C2×C8—41no—
317C2×C8—3162no—
318C2×C8—3161no—
319C2×C8—3161no—
320C2×C8—3161no—
321C2×C8—3161no—
322C2×C8—791no—
323C2×C8—791no—
324C2×C8—791no—
325C2×C8—791no—
326C2×C8—1581no—
327C2×C8—1581no—
328C2×C8—1582no—
329C2×C8—1582no—
330C2×C8—1582no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—3161no—
336C4⋊C4—3161no—
337C4⋊C4Dic79⋊C411yescharacteristic
338C4⋊C4Dic79⋊C411yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—81no—
345C22⋊C4—6321no—
346C42—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23#1457911711yescharacteristic
353C23#1457913311yescharacteristic
354C23#1457913811yescharacteristic
355Q8—41no—
356D4—41no—
357D4—42no—
358D4—42no—
359D4—41no—
360D4—41no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—6321no—
379C2×C4—6321no—
380C2×C4—6321no—
381C2×C4C22⋊C4×D7911yescharacteristic
382C2×C4C2×Dic79⋊C411yescharacteristic
383C2×C4#1457913311yescharacteristic
384C2×C4#1457913811yescharacteristic
385C8—81no—
386C8—3161no—
387C8—3161no—
388C8—792no—
389C8—6321no—
390C8—1581no—
391C8—1581no—
392C8—1581no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22#1909842111yescharacteristic
407C22#1909882411yescharacteristic
408C22#1909882811yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—161no—
416C4—12641no—
417C4#99×D7911yescharacteristic
418C4#1909853511yescharacteristic
419C2—21no—
420C2—21no—
421C2—41no—
422C2—41no—
423C2—81no—
424C2#2376338611yescharacteristic
425C2#2376345211yescharacteristic
426C2#2376345711yescharacteristic
subgroupG/H
centerC22#19098421
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1513C2
socleC22×C79#380
prime radical#525×C79C22
cosocleC23×C4×C79C23
hypercenter#1513D79
hypocenterC79#8626
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1513

Complements

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

NH ≅ G/Ntype
2. #23763436C2semidirect
3. #23763423C2semidirect
5. #23761307C2semidirect
6. #23761306C2semidirect
8. C2×#14579168C4semidirect
16. #19098590C4semidirect
17. #19098530C4semidirect
20. #19098519C4semidirect
221. C79#8626semidirect

Maximal subgroups

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

subgroupindex
#237720952
#237634362
#237634232
#237619432
#237613072
#237613062
C79×#15132
#862679