Finite Group Databasebeta

← #34184087

Subgroups — #34184087

Order 20224.

472 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772420C211yesmaximal, characteristic, complemented
2#23763210C211yesmaximal, characteristic, complemented
3#23761992C211yesmaximal, characteristic, complemented
4#23761908C211yesmaximal, characteristic, complemented
5#23761292C211yesmaximal, characteristic, complemented
6#23761271C211yesmaximal, characteristic, complemented
7C79×#3115C211yesmaximal, characteristic
8Q8×C79⋊C8—21no—
9#19098641—22no—
10#19098641—21no—
11#19098620—21no—
12#19098620C2211yescharacteristic
13C2×#14579088—21no—
14C4×#11929608—21no—
15C4×#11929608C2211yescharacteristic
16C4×#11929608C2211yescharacteristic
17#19098409C2211yescharacteristic
18#19098322—21no—
19#19098303—21no—
20#19098303—21no—
21#19098297—22no—
22#532×C79C2211yescharacteristic
23C4×#41×C79C2211yescharacteristic
24#482×C79—21no—
25#475×C79C411yescharacteristic, complemented
26#474×C79C411yescharacteristic, complemented
27#460×C79C411yescharacteristic, complemented
28#459×C79C411yescharacteristic, complemented
29#449×C79—21no—
30#442×C79—21no—
31#440×C79—21no—
32#432×C79C2211yescharacteristic
33C2×#11929608—22no—
34C2×#11929608—21no—
35C2×#11929608—21no—
36C2×C4×Dic79—21no—
37C4⋊(C79⋊C8)—22no—
38C4⋊(C79⋊C8)—21no—
39#14579088—24no—
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)D412yes—
42#14579072—21no—
43#14579072—21no—
44#14579072D411yescharacteristic
45#14579072D411yescharacteristic
46C4×C79⋊C8—22no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8—21no—
49C79⋊#95—21no—
50C79⋊#95—21no—
51C2×#41×C79—21no—
52C2×#41×C79D411yescharacteristic
53C2×SD16×C79—21no—
54C2×D8×C79—21no—
55C4⋊Q8×C79C2×C411yescharacteristic
56#127×C79C2×C411yescharacteristic
57#124×C79—21no—
58#121×C79—21no—
59#121×C79—21no—
60C4×Q8×C79C2×C411yescharacteristic
61C4×Q8×C79D411yescharacteristic
62C4×D4×C79—21no—
63C4×D4×C79—21no—
64C4×D4×C79—21no—
65C4×D4×C79—21no—
66C4×D4×C79C2×C411yescharacteristic
67C4×D4×C79D411yescharacteristic
68#117×C79—21no—
69#117×C79D411yescharacteristic
70C2×C42×C79—21no—
71C2×C42×C79C2311yescharacteristic
72#107×C79—21no—
73#106×C79—21no—
74C4⋊C8×C79C2×C411yescharacteristic
75C4⋊C8×C79C2×C411yescharacteristic
76#103×C79—21no—
77#102×C79—21no—
78#102×C79—21no—
79#102×C79—21no—
80C22⋊C8×C79—21no—
81#97×C79—21no—
82C22×Dic79—41no—
83#11929608—22no—
84#11929608—41no—
85#11929608—42no—
86#11929608—44no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—41no—
91C4×Dic79—22no—
92C4×Dic79—41no—
93#41×C79—22no—
94#41×C79—41no—
95#41×C79—82no—
96C2×Q8×C79—21no—
97C2×Q8×C79C22⋊C411yescharacteristic
98C2×D4×C79—21no—
99C2×D4×C79—21no—
100C2×D4×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79—41no—
103C2×D4×C79C22⋊C411yescharacteristic
104C22×C4×C79—21no—
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79—41no—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79#4111yescharacteristic
111SD16×C79—42no—
112D8×C79—42no—
113C2×C8×C79—21no—
114C2×C8×C79—21no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79C22⋊C411yescharacteristic
120C4⋊C4×C79C22⋊C411yescharacteristic
121C22⋊C4×C79—21no—
122C22⋊C4×C79—21no—
123C22⋊C4×C79—21no—
124C22⋊C4×C79—41no—
125C22⋊C4×C79—41no—
126C42×C79—21no—
127C42×C79—21no—
128C42×C79—21no—
129C42×C79C22×C411yescharacteristic
130C42×C79C2×D411yescharacteristic
131C42×C79C2×D411yescharacteristic
132C42×C79#4111yescharacteristic
133C23×C79—21no—
134C23×C79—41no—
135C23×C79#12111yescharacteristic
136Q8×C79—21no—
137Q8×C79—81no—
138Q8×C79#10412yes—
139D4×C79—21no—
140D4×C79—21no—
141D4×C79—42no—
142D4×C79—41no—
143D4×C79—41no—
144D4×C79—81no—
145D4×C79—81no—
146D4×C79—82no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—42no—
155C2×C4×C79—42no—
156C2×C4×C79—41no—
157C2×C4×C79—81no—
158C2×C4×C79—82no—
159C2×C4×C79C2×C22⋊C411yescharacteristic
160C2×C4×C79C4×D411yescharacteristic
161C2×C4×C79C4×D411yescharacteristic
162C2×C4×C79#12011yescharacteristic
163C2×C4×C79#12111yescharacteristic
164C2×C4×C79#12311yescharacteristic
165C2×Dic79—42no—
166C2×Dic79—81no—
167C2×Dic79—81no—
168C8×C79—41no—
169C8×C79—41no—
170C79⋊C8—22no—
171C79⋊C8—42no—
172C79⋊C8—41no—
173C79⋊C8—42no—
174C22×C79—21no—
175C22×C79—21no—
176C22×C79—21no—
177C22×C79—41no—
178C22×C79—41no—
179C22×C79—41no—
180C22×C79—81no—
181C22×C79—81no—
182C22×C79—81no—
183C22×C79#38611yescharacteristic
184C4×C79—21no—
185C4×C79—22no—
186C4×C79—21no—
187C4×C79—21no—
188C4×C79—21no—
189C4×C79—21no—
190C4×C79—41no—
191C4×C79—41no—
192C4×C79—81no—
193C4×C79C2×#10411yescharacteristic
194C4×C79#42111yescharacteristic
195Dic79—82no—
196#9162—791nomaximal, Sylow
197C2×C79—41no—
198C2×C79—81no—
199C2×C79—161no—
200C2×C79#187111yescharacteristic
201C2×C79#190311yescharacteristic
202C2×C79#190611yescharacteristic
203#3115D7911yescharacteristic
204#2971—791no—
205#1759—791no—
206#1525—791no—
207#1505—791no—
208#1300—791no—
209#1291—791no—
210C79#916211yescharacteristic, complemented, Sylow
211#532C2×D7911yescharacteristic
212C4×#41C2×D7911yescharacteristic
213#482—21no—
214#475Dic7911yescharacteristic, complemented
215#474Dic7911yescharacteristic, complemented
216#460Dic7911yescharacteristic, complemented
217#459Dic7911yescharacteristic, complemented
218#449—21no—
219C8×Q8—1581no—
220#442—21no—
221#440—21no—
222#436—1582no—
223#436—1581no—
224#432—791no—
225#432—1581no—
226#432C2×D7911yescharacteristic
227C2×#104—1581no—
228C4×M16—791no—
229C4×M16—791no—
230C4×M16—1581no—
231#376—791no—
232#339—1581no—
233#329—1581no—
234#329—1581no—
235#327—1582no—
236C2×#41—21no—
237C2×#41C22⋊D7911yescharacteristic
238C2×SD16—21no—
239C2×D8—21no—
240C2×M16—1581no—
241C2×M16—1582no—
242C2×M16—1581no—
243C4⋊Q8C2×Dic7911yescharacteristic
244#127C2×Dic7911yescharacteristic
245#124—21no—
246#121—21no—
247#121—21no—
248C4×Q8C2×Dic7911yescharacteristic
249C4×Q8C22⋊D7911yescharacteristic
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C2×Dic7911yescharacteristic
255C4×D4C22⋊D7911yescharacteristic
256#117—21no—
257#117C22⋊D7911yescharacteristic
258C2×C42—21no—
259C2×C42—1581no—
260C2×C42C22×D7911yescharacteristic
261#107—21no—
262#106—21no—
263C4⋊C8—1582no—
264C4⋊C8—1581no—
265C4⋊C8C2×Dic7911yescharacteristic
266C4⋊C8C2×Dic7911yescharacteristic
267#104—1584no—
268#103—21no—
269#102—21no—
270#102—21no—
271#102—21no—
272C22⋊C8—21no—
273C22⋊C8—792no—
274C22⋊C8—1582no—
275#97—21no—
276#97—791no—
277#97—791no—
278#97—1581no—
279#97—1581no—
280C4×C8—1582no—
281C4×C8—1581no—
282C4×C8—1581no—
283#95—1581no—
284#95—1581no—
285#41—22no—
286#41—41no—
287#41—82no—
288C2×Q8—21no—
289C2×Q8C22⋊Dic7911yescharacteristic
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4C22⋊Dic7911yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—3161no—
301C22×C4D4×D7911yescharacteristic
302C22×C4C2×C22⋊D7911yescharacteristic
303C22×C4#1192962311yescharacteristic
304SD16—42no—
305D8—42no—
306M16—3161no—
307M16—3162no—
308M16—3164no—
309M16—1582no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—3161no—
313C2×C8—1581no—
314C2×C8—1582no—
315C2×C8—1581no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4C22⋊Dic7911yescharacteristic
321C4⋊C4C22⋊Dic7911yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—3161no—
331C42—1582no—
332C42C22×Dic7911yescharacteristic
333C42D4×D7911yescharacteristic
334C42C2×C22⋊D7911yescharacteristic
335C42#1192962311yescharacteristic
336C23—21no—
337C23—41no—
338C23#1457913311yescharacteristic
339Q8—21no—
340Q8—81no—
341Q8#1457908812yes—
342D4—21no—
343D4—21no—
344D4—42no—
345D4—41no—
346D4—41no—
347D4—81no—
348D4—81no—
349D4—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—3162no—
363C2×C4—6321no—
364C2×C4—6321no—
365C2×C4C2×C22⋊Dic7911yescharacteristic
366C2×C4D4×Dic7911yescharacteristic
367C2×C4D4×Dic7911yescharacteristic
368C2×C4#1457912911yescharacteristic
369C2×C4#1457913311yescharacteristic
370C2×C4#1457914211yescharacteristic
371C8—41no—
372C8—41no—
373C8—3162no—
374C8—3161no—
375C8—3162no—
376C8—1582no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1909844311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—6322no—
397C4C2×#1457908811yescharacteristic
398C4#1909858011yescharacteristic
399C2—41no—
400C2—81no—
401C2—161no—
402C2#2376370311yescharacteristic
403C2#2376388211yescharacteristic
404C2#2376390311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3115C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3115D79
hypocenterC79#9162
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #23772420C2semidirect
2. #23763210C2semidirect
3. #23761992C2semidirect
4. #23761908C2semidirect
5. #23761292C2semidirect
6. #23761271C2semidirect
25. #475×C79C4semidirect
26. #474×C79C4semidirect
27. #460×C79C4semidirect
28. #459×C79C4semidirect
210. C79#9162semidirect
214. #475Dic79semidirect
215. #474Dic79semidirect
216. #460Dic79semidirect
217. #459Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724202
#237632102
#237619922
#237619082
#237612922
#237612712
C79×#31152
#916279