Finite Group Databasebeta

← #34185068

Subgroups — #34185068

Order 20224.

518 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772441C211yesmaximal, characteristic, complemented
2#23763410C211yesmaximal, characteristic, complemented
3#23761960C211yesmaximal, characteristic, complemented
4#23761799C211yesmaximal, characteristic, complemented
5#23761295C211yesmaximal, characteristic, complemented
6#23761260C211yesmaximal, characteristic, complemented
7C79×#3036C211yesmaximal, characteristic
8#19098641—22no—
9#19098641—22no—
10#19098616C2211yescharacteristic
11#19098616C2211yescharacteristic
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×#14579082—21no—
14#19098527—22no—
15#19098429C2211yescharacteristic
16#19098325—21no—
17#19098297—22no—
18D4⋊(C79⋊C8)—22no—
19C2×AGL(1,ℤ/8)×C79—21no—
20#532×C79C2211yescharacteristic
21C2×C4×D4×C79C2211yescharacteristic
22#463×C79—22no—
23#461×C79C412yescomplemented
24#459×C79C412yescomplemented
25#423×C79C2211yescharacteristic
26#417×C79—21no—
27C2×#11929608—24no—
28C22×C79⋊C8—21no—
29C2×C4⋊Dic79—21no—
30C4⋊(C79⋊C8)—21no—
31C4⋊(C79⋊C8)—21no—
32C4⋊(C79⋊C8)—21no—
33C4⋊(C79⋊C8)—21no—
34#14579082—42no—
35C22⋊(C79⋊C8)—24no—
36C22⋊(C79⋊C8)—21no—
37C22⋊(C79⋊C8)D412yes—
38C4×C79⋊C8—21no—
39C4×C79⋊C8D412yes—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C2×#41×C79—21no—
43C22×D4×C79D411yescharacteristic
44C23×C4×C79—21no—
45AGL(1,ℤ/8)×C79—42no—
46AGL(1,ℤ/8)×C79—42no—
47C2×SD16×C79—22no—
48C2×D8×C79—22no—
49C2×M16×C79—21no—
50C4⋊Q8×C79C2×C411yescharacteristic
51#127×C79C2×C411yescharacteristic
52#124×C79—21no—
53#121×C79—21no—
54#121×C79—21no—
55C4×D4×C79—21no—
56C4×D4×C79—21no—
57C4×D4×C79—21no—
58C4×D4×C79—21no—
59C4×D4×C79C2×C412yes—
60C4×D4×C79D412yes—
61C2×C4⋊C4×C79D411yescharacteristic
62C2×C22⋊C4×C79—21no—
63C2×C42×C79C2311yescharacteristic
64C4⋊C8×C79—21no—
65C4⋊C8×C79C2×C412yes—
66#103×C79—22no—
67#102×C79—22no—
68C22×Dic79—41no—
69#11929608—84no—
70C2×C79⋊C8—22no—
71C2×C79⋊C8—22no—
72C2×C79⋊C8—41no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8—41no—
75C4⋊Dic79—41no—
76C4⋊Dic79—41no—
77C24×C79—21no—
78#41×C79—41no—
79#41×C79—41no—
80#41×C79—41no—
81#41×C79—41no—
82C2×Q8×C79—21no—
83C2×D4×C79—21no—
84C2×D4×C79—21no—
85C2×D4×C79—21no—
86C2×D4×C79—21no—
87C2×D4×C79—21no—
88C2×D4×C79—21no—
89C2×D4×C79—21no—
90C2×D4×C79—21no—
91C2×D4×C79—41no—
92C2×D4×C79C22⋊C412yes—
93C22×C4×C79—21no—
94C22×C4×C79—22no—
95C22×C4×C79—22no—
96C22×C4×C79—41no—
97C22×C4×C79—82no—
98C22×C4×C79C2×D411yescharacteristic
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79#4111yescharacteristic
101SD16×C79—44no—
102D8×C79—44no—
103M16×C79—41no—
104M16×C79—41no—
105C2×C8×C79—22no—
106C4⋊C4×C79—21no—
107C4⋊C4×C79—21no—
108C4⋊C4×C79—21no—
109C4⋊C4×C79C22⋊C412yes—
110C22⋊C4×C79—22no—
111C22⋊C4×C79—22no—
112C22⋊C4×C79—41no—
113C22⋊C4×C79—41no—
114C42×C79C22×C411yescharacteristic
115C42×C79C2×D411yescharacteristic
116C42×C79C2×D411yescharacteristic
117C42×C79#4111yescharacteristic
118C23×C79—22no—
119C23×C79—41no—
120C23×C79—41no—
121C23×C79—41no—
122C23×C79—81no—
123C23×C79—81no—
124C23×C79#12111yescharacteristic
125Q8×C79—41no—
126Q8×C79—41no—
127D4×C79—22no—
128D4×C79—22no—
129D4×C79—42no—
130D4×C79—41no—
131D4×C79—41no—
132D4×C79—41no—
133D4×C79—41no—
134D4×C79—81no—
135D4×C79—81no—
136D4×C79—81no—
137C2×C4×C79—22no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—44no—
141C2×C4×C79—41no—
142C2×C4×C79—82no—
143C2×C4×C79—81no—
144C2×C4×C79—81no—
145C2×C4×C79—81no—
146C2×C4×C79C2×C22⋊C411yescharacteristic
147C2×C4×C79C4×D411yescharacteristic
148C2×C4×C79C4×D411yescharacteristic
149C2×C4×C79#12011yescharacteristic
150C2×C4×C79#12111yescharacteristic
151C2×C4×C79#12311yescharacteristic
152C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C79#13711yescharacteristic
156C2×Dic79—81no—
157C2×Dic79—81no—
158C2×Dic79—81no—
159C8×C79—42no—
160C79⋊C8—42no—
161C79⋊C8—42no—
162C79⋊C8—81no—
163C22×C79—21no—
164C22×C79—21no—
165C22×C79—42no—
166C22×C79—42no—
167C22×C79—42no—
168C22×C79—82no—
169C22×C79—82no—
170C22×C79—81no—
171C22×C79—81no—
172C22×C79—81no—
173C22×C79#38611yescharacteristic
174C22×C79#46811yescharacteristic
175C22×C79#46911yescharacteristic
176C4×C79—21no—
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—42no—
181C4×C79—81no—
182C4×C79#41711yescharacteristic
183C4×C79#41811yescharacteristic
184C4×C79#46811yescharacteristic
185C4×C79#46911yescharacteristic
186Dic79—161no—
187#9309—791nomaximal, Sylow
188C2×C79—21no—
189C2×C79—21no—
190C2×C79—82no—
191C2×C79—161no—
192C2×C79#188111yescharacteristic
193C2×C79#189511yescharacteristic
194C2×C79#190411yescharacteristic
195#3036D7911yescharacteristic
196#2975—791no—
197#1804—791no—
198#1517—791no—
199#1480—791no—
200#1301—791no—
201#1286—791no—
202C79#930911yescharacteristic, complemented, Sylow
203C2×AGL(1,ℤ/8)—21no—
204#532C2×D7911yescharacteristic
205C2×C4×D4C2×D7911yescharacteristic
206#463—22no—
207#461Dic7912yescomplemented
208#459Dic7912yescomplemented
209#436—1582no—
210#436—1582no—
211#431—791no—
212#431—791no—
213#423C2×D7911yescharacteristic
214C2×C4⋊C8—791no—
215#417—21no—
216C2×#102—1581no—
217#407—1582no—
218#382—791no—
219#340—1581no—
220#327—1582no—
221D4⋊C8—1582no—
222C2×#41—21no—
223C22×D4C22⋊D7911yescharacteristic
224C23×C4—21no—
225AGL(1,ℤ/8)—42no—
226AGL(1,ℤ/8)—42no—
227C2×SD16—22no—
228C2×D8—22no—
229C2×M16—21no—
230C2×M16—1584no—
231C22×C8—1581no—
232C4⋊Q8C2×Dic7911yescharacteristic
233#127C2×Dic7911yescharacteristic
234#124—21no—
235#121—21no—
236#121—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4C2×Dic7912yes—
242C4×D4C22⋊D7912yes—
243C2×C4⋊C4—1581no—
244C2×C4⋊C4C22⋊D7911yescharacteristic
245C2×C22⋊C4—21no—
246C2×C42C22×D7911yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—1581no—
249C4⋊C8—1581no—
250C4⋊C8—1581no—
251C4⋊C8—1581no—
252C4⋊C8C2×Dic7912yes—
253#103—22no—
254#102—22no—
255#102—3162no—
256C22⋊C8—792no—
257C22⋊C8—1584no—
258C22⋊C8—1581no—
259C4×C8—792no—
260C4×C8—1581no—
261#95—1581no—
262#95—1581no—
263C24—21no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4C22⋊Dic7912yes—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—3161no—
285C22×C4D4×D7911yescharacteristic
286C22×C4C2×C22⋊D7911yescharacteristic
287C22×C4#1192962311yescharacteristic
288SD16—44no—
289D8—44no—
290M16—41no—
291M16—41no—
292M16—6324no—
293C2×C8—22no—
294C2×C8—3161no—
295C2×C8—3161no—
296C2×C8—3161no—
297C2×C8—1582no—
298C2×C8—1582no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—3161no—
303C4⋊C4—3161no—
304C4⋊C4C22⋊Dic7912yes—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42C22×Dic7911yescharacteristic
310C42D4×D7911yescharacteristic
311C42C2×C22⋊D7911yescharacteristic
312C42#1192962311yescharacteristic
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—81no—
318C23—81no—
319C23#1457913311yescharacteristic
320Q8—41no—
321Q8—41no—
322D4—22no—
323D4—22no—
324D4—42no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—81no—
331D4—81no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—44no—
336C2×C4—41no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—6321no—
342C2×C4—6321no—
343C2×C4—6321no—
344C2×C4C2×C22⋊Dic7911yescharacteristic
345C2×C4D4×Dic7911yescharacteristic
346C2×C4D4×Dic7911yescharacteristic
347C2×C4#1457912911yescharacteristic
348C2×C4#1457913311yescharacteristic
349C2×C4#1457914211yescharacteristic
350C2×C4#1457918611yescharacteristic
351C2×C4#1457918711yescharacteristic
352C2×C4#1457918711yescharacteristic
353C2×C4#1457919111yescharacteristic
354C8—42no—
355C8—3162no—
356C8—3162no—
357C8—6321no—
358C22—21no—
359C22—21no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—82no—
364C22—82no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22#1909844311yescharacteristic
369C22#1909880711yescharacteristic
370C22#1909881411yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—12641no—
378C4#1909856411yescharacteristic
379C4#1909856811yescharacteristic
380C4#1909880711yescharacteristic
381C4#1909881411yescharacteristic
382C2—21no—
383C2—21no—
384C2—82no—
385C2—161no—
386C2#2376376411yescharacteristic
387C2#2376383911yescharacteristic
388C2#2376388911yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3036C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3036D79
hypocenterC79#9309
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #23772441C2semidirect
2. #23763410C2semidirect
3. #23761960C2semidirect
4. #23761799C2semidirect
5. #23761295C2semidirect
6. #23761260C2semidirect
23. #461×C79C4semidirect
24. #459×C79C4semidirect
202. C79#9309semidirect
207. #461Dic79semidirect
208. #459Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724412
#237634102
#237619602
#237617992
#237612952
#237612602
C79×#30362
#930979