Finite Group Databasebeta

← #34182380

Subgroups — #34182380

Order 20224.

464 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23763703C211yesmaximal, characteristic
2#23763604C211yesmaximal, characteristic, complemented
3#23763582C211yesmaximal, characteristic, complemented
4#23761891C211yesmaximal, characteristic
5#23761285C211yesmaximal, characteristic, complemented
6#23761247C211yesmaximal, characteristic, complemented
7C79×#2976C211yesmaximal, characteristic
8C22×#11929608C2211yescharacteristic
9C2×#14579096—21no—
10C2×Dic79⋊C8—21no—
11C2×#14579088—21no—
12C2×C22⋊(C79⋊C8)C2211yescharacteristic
13C4×C22⋊Dic79C2211yescharacteristic
14C4×C22⋊Dic79C2211yescharacteristic
15#19098327—21no—
16#19098322—21no—
17#19098317—21no—
18#19098300C411yescharacteristic, complemented
19#19098300C411yescharacteristic, complemented
20#19098297C411yescharacteristic, complemented
21#19098297C411yescharacteristic, complemented
22#525×C79C2211yescharacteristic
23#436×C79—21no—
24#435×C79—21no—
25#435×C79—21no—
26C8×D4×C79—21no—
27#433×C79—21no—
28#408×C79—21no—
29#407×C79C2211yescharacteristic
30C2×C22⋊C8×C79C2211yescharacteristic
31C2×#11929608—21no—
32C2×#11929608—21no—
33C2×#11929608—42no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36C2×C22⋊Dic79—21no—
37C2×C22⋊Dic79—21no—
38C2×C4×Dic79—21no—
39C2×C4×Dic79—21no—
40#14579096—42no—
41Dic79⋊C8—42no—
42#14579088—42no—
43C22⋊(C79⋊C8)—21no—
44C22⋊(C79⋊C8)C2×C411yescharacteristic
45C22⋊(C79⋊C8)C2×C411yescharacteristic
46C79⋊#95—21no—
47C79⋊#95C2×C411yescharacteristic
48C79⋊#95C2×C411yescharacteristic
49C2×#41×C79D411yescharacteristic
50C23×C4×C79C2311yescharacteristic
51C2×M16×C79—21no—
52C2×M16×C79D411yescharacteristic
53C2×M16×C79D411yescharacteristic
54C22×C8×C79—21no—
55C22×C8×C79D411yescharacteristic
56C22×C8×C79D411yescharacteristic
57#123×C79—21no—
58C22⋊Q8×C79C2×C411yescharacteristic
59C22⋊Q8×C79Q811yescharacteristic
60#121×C79C2×C411yescharacteristic
61#121×C79Q811yescharacteristic
62#120×C79—21no—
63C4×D4×C79—21no—
64C4×D4×C79—21no—
65#117×C79D411yescharacteristic
66C4⋊C8×C79—21no—
67C4⋊C8×C79—21no—
68C22⋊C8×C79—21no—
69C22⋊C8×C79—21no—
70C22⋊C8×C79—21no—
71C22⋊C8×C79—21no—
72C22⋊C8×C79—21no—
73C22⋊C8×C79—21no—
74#97×C79—21no—
75C4×C8×C79—21no—
76C22×Dic79—21no—
77C22×Dic79—21no—
78C22×Dic79—41no—
79#11929608—22no—
80#11929608—22no—
81#11929608—41no—
82#11929608—41no—
83C2×C79⋊C8—22no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C22⋊Dic79—22no—
91C22⋊Dic79—41no—
92C22⋊Dic79—42no—
93C4×Dic79—41no—
94C4×Dic79—41no—
95C4×Dic79—41no—
96C4×Dic79—41no—
97C24×C79#4111yescharacteristic
98#41×C79—41no—
99#41×C79—41no—
100C2×Q8×C79C4⋊C411yescharacteristic
101C2×D4×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79C4⋊C411yescharacteristic
105C22×C4×C79—21no—
106C22×C4×C79—41no—
107C22×C4×C79—41no—
108C22×C4×C79C22×C411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79C2×Q811yescharacteristic
113C22×C4×C79#4111yescharacteristic
114M16×C79—41no—
115M16×C79—41no—
116M16×C79—81no—
117C2×C8×C79—21no—
118C2×C8×C79—21no—
119C2×C8×C79—21no—
120C2×C8×C79—21no—
121C2×C8×C79—41no—
122C2×C8×C79—41no—
123C2×C8×C79—81no—
124C4⋊C4×C79—21no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79C4⋊C411yescharacteristic
127C4⋊C4×C79C4⋊C411yescharacteristic
128C22⋊C4×C79—21no—
129C22⋊C4×C79—21no—
130C22⋊C4×C79—21no—
131C22⋊C4×C79—21no—
132C22⋊C4×C79—21no—
133C42×C79—21no—
134C23×C79—21no—
135C23×C79—21no—
136C23×C79—21no—
137C23×C79—41no—
138C23×C79—41no—
139C23×C79C4×D411yescharacteristic
140C23×C79C22⋊Q811yescharacteristic
141C23×C79#12311yescharacteristic
142Q8×C79—41no—
143D4×C79—41no—
144D4×C79—41no—
145D4×C79—41no—
146D4×C79—81no—
147D4×C79—81no—
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—21no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79C2×C4⋊C411yescharacteristic
163C2×C4×C79C4×D411yescharacteristic
164C2×C4×C79#12011yescharacteristic
165C2×C4×C79C22⋊Q811yescharacteristic
166C2×Dic79—22no—
167C2×Dic79—41no—
168C2×Dic79—41no—
169C2×Dic79—41no—
170C2×Dic79—42no—
171C2×Dic79—41no—
172C2×Dic79—41no—
173C2×Dic79—81no—
174C2×Dic79—81no—
175C8×C79—41no—
176C8×C79—41no—
177C8×C79—41no—
178C8×C79—41no—
179C79⋊C8—41no—
180C79⋊C8—41no—
181C79⋊C8—42no—
182C22×C79—21no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—21no—
187C22×C79—21no—
188C22×C79—21no—
189C22×C79—21no—
190C22×C79—41no—
191C22×C79—41no—
192C22×C79—41no—
193C22×C79—81no—
194C22×C79—81no—
195C22×C79#38511yescharacteristic
196C22×C79#44311yescharacteristic
197C22×C79#44411yescharacteristic
198C4×C79—21no—
199C4×C79—21no—
200C4×C79—21no—
201C4×C79—41no—
202C4×C79—41no—
203C4×C79—41no—
204C4×C79B(4,𝔽2)11yescharacteristic
205C4×C79#45811yescharacteristic
206Dic79—42no—
207Dic79—81no—
208Dic79—82no—
209#8875—791nomaximal, Sylow
210C2×C79—21no—
211C2×C79—21no—
212C2×C79—41no—
213C2×C79—41no—
214C2×C79—81no—
215C2×C79#186111yescharacteristic
216C2×C79#186611yescharacteristic
217C2×C79#186711yescharacteristic
218#2976D7911yescharacteristic
219#1871—791no—
220#1855—791no—
221#1851—791no—
222#1501—791no—
223#1297—791no—
224#1282—791no—
225C79#887511yescharacteristic, complemented, Sylow
226C22×M16—791no—
227#525C2×D7911yescharacteristic
228#436—21no—
229#435—21no—
230#435—21no—
231C8×D4—21no—
232#433—21no—
233C2×#108—1581no—
234C2×C4⋊C8—1581no—
235C2×#104—1581no—
236#408—21no—
237#407C2×D7911yescharacteristic
238C2×C22⋊C8—791no—
239C2×C22⋊C8C2×D7911yescharacteristic
240C4×C22⋊C4—791no—
241C4×C22⋊C4—791no—
242#341—1581no—
243#339—1581no—
244#336—1581no—
245#328—791no—
246#328—791no—
247#327—791no—
248#327—791no—
249C2×#41C22⋊D7911yescharacteristic
250C23×C4C22×D7911yescharacteristic
251C2×M16—21no—
252C2×M16—3162no—
253C2×M16—1581no—
254C2×M16—1581no—
255C2×M16C22⋊D7911yescharacteristic
256C2×M16C22⋊D7911yescharacteristic
257C22×C8—21no—
258C22×C8—1581no—
259C22×C8—1581no—
260C22×C8C22⋊D7911yescharacteristic
261C22×C8C22⋊D7911yescharacteristic
262#123—21no—
263C22⋊Q8Dic15811yescharacteristic
264C22⋊Q8C4×D7911yescharacteristic
265#121Dic15811yescharacteristic
266#121C4×D7911yescharacteristic
267#120—21no—
268C4×D4—21no—
269C4×D4—21no—
270#117C22⋊D7911yescharacteristic
271C2×C22⋊C4—1581no—
272C2×C22⋊C4—1581no—
273C2×C42—1581no—
274C2×C42—1581no—
275#108—3162no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—3162no—
279#104—3162no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—791no—
287C22⋊C8—791no—
288C22⋊C8—1581no—
289#97—21no—
290C4×C8—21no—
291#95—791no—
292#95—791no—
293#95—1581no—
294C24#1192962411yescharacteristic
295#41—41no—
296#41—41no—
297C2×Q8Dic79⋊C411yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4Dic79⋊C411yescharacteristic
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—3161no—
306C22×C4—1581no—
307C22×C4—1581no—
308C22×C4C2×C4×D7911yescharacteristic
309C22×C4C2×C22⋊D7911yescharacteristic
310C22×C4C2×C22⋊D7911yescharacteristic
311C22×C4C2×C22⋊D7911yescharacteristic
312C22×C4C2×Dic15811yescharacteristic
313C22×C4#1192962411yescharacteristic
314M16—41no—
315M16—41no—
316M16—81no—
317M16—3161no—
318M16—3161no—
319M16—1582no—
320M16—1582no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—81no—
328C2×C8—3161no—
329C2×C8—3161no—
330C2×C8—3161no—
331C2×C8—3161no—
332C2×C8—1582no—
333C2×C8—1581no—
334C2×C8—1581no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4Dic79⋊C411yescharacteristic
338C4⋊C4Dic79⋊C411yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—3161no—
345C22⋊C4—3162no—
346C22⋊C4—1582no—
347C42—21no—
348C42—3161no—
349C42—3161no—
350C42—3161no—
351C42—3161no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23C4×C22⋊D7911yescharacteristic
358C23#1457914011yescharacteristic
359C23#1457914411yescharacteristic
360Q8—41no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—81no—
365D4—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—3161no—
381C2×C4—3161no—
382C2×C4—3161no—
383C2×C4—3161no—
384C2×C4—3162no—
385C2×C4—3161no—
386C2×C4—6321no—
387C2×C4—6321no—
388C2×C4—1582no—
389C2×C4C2×Dic79⋊C411yescharacteristic
390C2×C4C4×C22⋊D7911yescharacteristic
391C2×C4#1457913011yescharacteristic
392C2×C4#1457914011yescharacteristic
393C8—41no—
394C8—41no—
395C8—41no—
396C8—41no—
397C8—3161no—
398C8—3161no—
399C8—3162no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—81no—
412C22—81no—
413C22#1909843611yescharacteristic
414C22#1909867811yescharacteristic
415C22#1909868311yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—3162no—
423C4—6321no—
424C4—6322no—
425C4#1909875011yescharacteristic
426C4#1909875311yescharacteristic
427C2—21no—
428C2—21no—
429C2—41no—
430C2—41no—
431C2—81no—
432C2#2376364011yescharacteristic
433C2#2376367511yescharacteristic
434C2#2376368111yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2976C2
socleC22×C79#385
prime radical#525×C79C22
cosocleC23×C4×C79C23
hypercenter#2976D79
hypocenterC79#8875
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2976

Complements

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

NH ≅ G/Ntype
2. #23763604C2semidirect
3. #23763582C2semidirect
5. #23761285C2semidirect
6. #23761247C2semidirect
18. #19098300C4semidirect
19. #19098300C4semidirect
20. #19098297C4semidirect
21. #19098297C4semidirect
225. C79#8875semidirect

Maximal subgroups

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

subgroupindex
#237637032
#237636042
#237635822
#237618912
#237612852
#237612472
C79×#29762
#887579