Finite Group Databasebeta

← #34184396

Subgroups — #34184396

Order 20224.

450 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772044C211yesmaximal, characteristic, complemented
2#23761992C211yesmaximal, characteristic, complemented
3#23761851C211yesmaximal, characteristic, complemented
4#23761839C211yesmaximal, characteristic, complemented
5#23761291C211yesmaximal, characteristic, complemented
6#23761236C211yesmaximal, characteristic, complemented
7C79×#3106C211yesmaximal, characteristic
8#19098624—22no—
9#19098616—22no—
10#19098590—21no—
11#19098587C2211yescharacteristic
12#19098530—21no—
13C4×#11929608C2211yescharacteristic
14C4×#11929608C2211yescharacteristic
15C2×C4×C79⋊C8C2211yescharacteristic
16#19098303—22no—
17Q8⋊(C79⋊C8)—21no—
18D4⋊(C79⋊C8)—22no—
19D4⋊(C79⋊C8)—21no—
20#532×C79C2211yescharacteristic
21C4×#41×C79C2211yescharacteristic
22#482×C79—21no—
23#477×C79C411yescharacteristic
24#476×C79C411yescharacteristic
25#460×C79C411yescharacteristic
26#459×C79C411yescharacteristic
27#449×C79—21no—
28C4×SD16×C79—21no—
29C4×D8×C79—21no—
30#431×C79C2211yescharacteristic
31C2×#11929608—21no—
32C2×#11929608—21no—
33C2×#11929608—21no—
34C22×C79⋊C8—21no—
35C4⋊(C79⋊C8)—24no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38C22⋊(C79⋊C8)—24no—
39#14579072—21no—
40#14579072D412yes—
41C4×C79⋊C8—21no—
42C4×C79⋊C8—21no—
43C4×C79⋊C8—21no—
44C4×C79⋊C8D412yes—
45C2×#41×C79—21no—
46C2×#41×C79D411yescharacteristic
47C2×SD16×C79—21no—
48C2×D8×C79—21no—
49C4⋊Q8×C79C2×C411yescharacteristic
50#127×C79C2×C411yescharacteristic
51#124×C79—21no—
52#121×C79—21no—
53#121×C79—21no—
54C4×Q8×C79C2×C411yescharacteristic
55C4×Q8×C79D411yescharacteristic
56C4×D4×C79—21no—
57C4×D4×C79—21no—
58C4×D4×C79—21no—
59C4×D4×C79—21no—
60C4×D4×C79C2×C411yescharacteristic
61C4×D4×C79D411yescharacteristic
62#117×C79—21no—
63#117×C79D411yescharacteristic
64C2×C42×C79—21no—
65C2×C42×C79C2311yescharacteristic
66#107×C79—21no—
67#106×C79—21no—
68C4⋊C8×C79C2×C411yescharacteristic
69C4⋊C8×C79C2×C411yescharacteristic
70#103×C79—21no—
71#102×C79—21no—
72#102×C79—21no—
73#102×C79—21no—
74C22⋊C8×C79—21no—
75C4×C8×C79—21no—
76#11929608—22no—
77#11929608—41no—
78#11929608—41no—
79#11929608—41no—
80#11929608—81no—
81C2×C79⋊C8—22no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87#41×C79—41no—
88#41×C79—41no—
89#41×C79—81no—
90#41×C79—81no—
91C2×Q8×C79—21no—
92C2×Q8×C79C22⋊C411yescharacteristic
93C2×D4×C79—21no—
94C2×D4×C79—21no—
95C2×D4×C79—21no—
96C2×D4×C79—21no—
97C2×D4×C79—41no—
98C2×D4×C79C22⋊C411yescharacteristic
99C22×C4×C79—21no—
100C22×C4×C79—21no—
101C22×C4×C79—21no—
102C22×C4×C79—41no—
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79#4111yescharacteristic
105C22×C4×C79#4111yescharacteristic
106SD16×C79—42no—
107D8×C79—42no—
108C2×C8×C79—21no—
109C2×C8×C79—21no—
110C4⋊C4×C79—21no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79C22⋊C411yescharacteristic
115C4⋊C4×C79C22⋊C411yescharacteristic
116C22⋊C4×C79—21no—
117C22⋊C4×C79—21no—
118C22⋊C4×C79—21no—
119C22⋊C4×C79—41no—
120C22⋊C4×C79—41no—
121C42×C79—21no—
122C42×C79—21no—
123C42×C79—21no—
124C42×C79C22×C411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79C2×D411yescharacteristic
127C42×C79C2×D411yescharacteristic
128C23×C79—21no—
129C23×C79—41no—
130C23×C79#12311yescharacteristic
131Q8×C79—21no—
132Q8×C79—21no—
133Q8×C79—81no—
134D4×C79—21no—
135D4×C79—21no—
136D4×C79—42no—
137D4×C79—41no—
138D4×C79—41no—
139D4×C79—41no—
140D4×C79—41no—
141D4×C79—81no—
142D4×C79—82no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—42no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—81no—
159C2×C4×C79—81no—
160C2×C4×C79C2×C22⋊C411yescharacteristic
161C2×C4×C79C4×D411yescharacteristic
162C2×C4×C79C4×D411yescharacteristic
163C2×C4×C79#12011yescharacteristic
164C2×C4×C79#12111yescharacteristic
165C2×C4×C79#12111yescharacteristic
166C8×C79—41no—
167C8×C79—41no—
168C79⋊C8—42no—
169C79⋊C8—42no—
170C79⋊C8—42no—
171C79⋊C8—81no—
172C22×C79—21no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79—41no—
176C22×C79—41no—
177C22×C79—41no—
178C22×C79—81no—
179C22×C79—81no—
180C22×C79—81no—
181C22×C79#38611yescharacteristic
182C4×C79—21no—
183C4×C79—21no—
184C4×C79—21no—
185C4×C79—41no—
186C4×C79—41no—
187C4×C79—41no—
188C4×C79—81no—
189C4×C79#42111yescharacteristic
190C4×C79#42111yescharacteristic
191C4×C79#44311yescharacteristic
192C4×C79#44411yescharacteristic
193C4×C79#47111yescharacteristic
194C4×C79#47211yescharacteristic
195#9213—791nomaximal, Sylow
196C2×C79—41no—
197C2×C79—81no—
198C2×C79—161no—
199C2×C79#187611yescharacteristic
200C2×C79#189711yescharacteristic
201C2×C79#190711yescharacteristic
202#3106D7911yescharacteristic
203#2922—791no—
204#1525—791no—
205#1492—791no—
206#1489—791no—
207#1299—791no—
208#1278—791no—
209C79#921311yescharacteristic, complemented, Sylow
210#532C2×D7911yescharacteristic
211C4×#41C2×D7911yescharacteristic
212#482—21no—
213#477Dic7911yescharacteristic
214#476Dic7911yescharacteristic
215#460Dic7911yescharacteristic
216#459Dic7911yescharacteristic
217#449—21no—
218C4×SD16—21no—
219C4×D8—21no—
220#433—1582no—
221#431—1582no—
222#431C2×D7911yescharacteristic
223#424—1581no—
224#423—791no—
225#408—1581no—
226C4×M16—791no—
227C4×M16—791no—
228C2×C4×C8—791no—
229#329—1582no—
230Q8⋊C8—1581no—
231D4⋊C8—1582no—
232D4⋊C8—1581no—
233C2×#41—21no—
234C2×#41C22⋊D7911yescharacteristic
235C2×SD16—21no—
236C2×D8—21no—
237C2×M16—1581no—
238C2×M16—1581no—
239C2×M16—1581no—
240C22×C8—1581no—
241C4⋊Q8C2×Dic7911yescharacteristic
242#127C2×Dic7911yescharacteristic
243#124—21no—
244#121—21no—
245#121—21no—
246C4×Q8C2×Dic7911yescharacteristic
247C4×Q8C22⋊D7911yescharacteristic
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4C2×Dic7911yescharacteristic
253C4×D4C22⋊D7911yescharacteristic
254#117—21no—
255#117C22⋊D7911yescharacteristic
256C2×C42—21no—
257C2×C42C22×D7911yescharacteristic
258#107—21no—
259#106—21no—
260C4⋊C8—1584no—
261C4⋊C8—1581no—
262C4⋊C8—1581no—
263C4⋊C8C2×Dic7911yescharacteristic
264C4⋊C8C2×Dic7911yescharacteristic
265#103—21no—
266#102—21no—
267#102—21no—
268#102—21no—
269C22⋊C8—21no—
270C22⋊C8—1584no—
271#97—792no—
272#97—1581no—
273C4×C8—21no—
274C4×C8—792no—
275C4×C8—1581no—
276C4×C8—1581no—
277C4×C8—1581no—
278#41—41no—
279#41—41no—
280#41—81no—
281#41—81no—
282C2×Q8—21no—
283C2×Q8C22⋊Dic7911yescharacteristic
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4C22⋊Dic7911yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4C2×C22⋊D7911yescharacteristic
295C22×C4#1192962311yescharacteristic
296C22×C4#1192962311yescharacteristic
297SD16—42no—
298D8—42no—
299M16—3161no—
300M16—3161no—
301M16—3161no—
302M16—6321no—
303M16—1582no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—3161no—
307C2×C8—3161no—
308C2×C8—3161no—
309C2×C8—3161no—
310C2×C8—1582no—
311C2×C8—1582no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4C22⋊Dic7911yescharacteristic
317C4⋊C4C22⋊Dic7911yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42C22×Dic7911yescharacteristic
327C42D4×D7911yescharacteristic
328C42D4×D7911yescharacteristic
329C42C2×C22⋊D7911yescharacteristic
330C23—21no—
331C23—41no—
332C23#1457914211yescharacteristic
333Q8—21no—
334Q8—21no—
335Q8—81no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—81no—
344D4—82no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4C2×C22⋊Dic7911yescharacteristic
363C2×C4D4×Dic7911yescharacteristic
364C2×C4D4×Dic7911yescharacteristic
365C2×C4#1457912911yescharacteristic
366C2×C4#1457913311yescharacteristic
367C2×C4#1457913311yescharacteristic
368C8—41no—
369C8—41no—
370C8—3162no—
371C8—3162no—
372C8—3162no—
373C8—6321no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22#1909844311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—81no—
391C4#1909858011yescharacteristic
392C4#1909858011yescharacteristic
393C4#1909868011yescharacteristic
394C4#1909868511yescharacteristic
395C4#1909882411yescharacteristic
396C4#1909883111yescharacteristic
397C2—41no—
398C2—81no—
399C2—161no—
400C2#2376372911yescharacteristic
401C2#2376384911yescharacteristic
402C2#2376391011yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3106C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3106D79
hypocenterC79#9213
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #23772044C2semidirect
2. #23761992C2semidirect
3. #23761851C2semidirect
4. #23761839C2semidirect
5. #23761291C2semidirect
6. #23761236C2semidirect
209. C79#9213semidirect

Maximal subgroups

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

subgroupindex
#237720442
#237619922
#237618512
#237618392
#237612912
#237612362
C79×#31062
#921379