Finite Group Databasebeta

← #34187619

Subgroups — #34187619

Order 20224.

428 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773722C211yesmaximal, characteristic
2#23763265C211yesmaximal, characteristic, complemented
3#23762146C211yesmaximal, characteristic, complemented
4#23761900C211yesmaximal, characteristic, complemented
5#23761895C211yesmaximal, characteristic
6#23761802C211yesmaximal, characteristic, complemented
7C79×#1480C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098917—21no—
10#19098917—21no—
11#19098913C411yescharacteristic, complemented
12#19098913C411yescharacteristic, complemented
13#19098909C411yescharacteristic, complemented
14#19098909C411yescharacteristic, complemented
15#19098616C2211yescharacteristic
16#19098559—21no—
17#19098559—21no—
18C2×C4×C79⋊C8C2211yescharacteristic
19C2×C8×Dic79—21no—
20C4×C4⋊Dic79C2211yescharacteristic
21#19098317—21no—
22#19098315—21no—
23#19098313—21no—
24#19098301—22no—
25#19098297—22no—
26#19098293—22no—
27#532×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29C2×C4⋊C8×C79C2211yescharacteristic
30D4⋊C8×C79—22no—
31C22×C79⋊C8D411yescharacteristic
32C22×C79⋊C8D411yescharacteristic
33#14579161C2×C411yescharacteristic
34#14579161C2×C411yescharacteristic
35#14579153—21no—
36#14579140—21no—
37#14579140—21no—
38C4×Dic158—21no—
39C4×Dic158—21no—
40#14579120—21no—
41C2×C4⋊Dic79—21no—
42C2×C4×Dic79—21no—
43C4⋊(C79⋊C8)—21no—
44#14579085—21no—
45#14579085—21no—
46#14579085—21no—
47#14579085—21no—
48#14579082—21no—
49#14579082—21no—
50#14579082—21no—
51#14579082—21no—
52C22⋊(C79⋊C8)—21no—
53C4×C79⋊C8—21no—
54C4×C79⋊C8C2×C411yescharacteristic
55C4×C79⋊C8C2×C411yescharacteristic
56C4×C79⋊C8D411yescharacteristic
57C4×C79⋊C8D411yescharacteristic
58C8×Dic79—42no—
59C79⋊#95D412yes—
60C2×#41×C79—21no—
61C22×C8×C79—21no—
62C4⋊Q8×C79C2×C411yescharacteristic
63#127×C79C2×C411yescharacteristic
64#124×C79—21no—
65#121×C79—21no—
66#121×C79—21no—
67C4×D4×C79—21no—
68C4×D4×C79—21no—
69C2×C42×C79C2311yescharacteristic
70C4⋊C8×C79—21no—
71C4⋊C8×C79—21no—
72C4⋊C8×C79D412yes—
73C22⋊C8×C79—21no—
74C4×C8×C79—21no—
75C2×Dic158—21no—
76C2×Dic158—21no—
77C22×Dic79—22no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8C22⋊C411yescharacteristic
84C2×C79⋊C8C22⋊C411yescharacteristic
85C2×C79⋊C8C22⋊C411yescharacteristic
86C2×C79⋊C8C22⋊C411yescharacteristic
87C4⋊Dic79—24no—
88C4⋊Dic79—21no—
89C4⋊Dic79—21no—
90C4⋊Dic79—21no—
91C4⋊Dic79—21no—
92Dic79⋊C4—41no—
93Dic79⋊C4—41no—
94C22⋊Dic79—41no—
95C4×Dic79—41no—
96C4×Dic79—41no—
97C4×Dic79—41no—
98#41×C79—41no—
99#41×C79—41no—
100#41×C79—41no—
101#41×C79—41no—
102C2×Q8×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79—41no—
107C22×C4×C79—41no—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79#4111yescharacteristic
111C2×C8×C79—22no—
112C2×C8×C79—41no—
113C2×C8×C79—41no—
114C2×C8×C79—41no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—21no—
117C22⋊C4×C79—41no—
118C22⋊C4×C79—41no—
119C42×C79C22×C411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79C2×D411yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79—41no—
124C23×C79#12111yescharacteristic
125Q8×C79—41no—
126Q8×C79—41no—
127D4×C79—41no—
128D4×C79—41no—
129D4×C79—42no—
130D4×C79—81no—
131D4×C79—81no—
132D4×C79—81no—
133C2×C4×C79—22no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79C2×C22⋊C411yescharacteristic
139C2×C4×C79C4×D411yescharacteristic
140C2×C4×C79C4×D411yescharacteristic
141C2×C4×C79#12111yescharacteristic
142C2×C4×C79#12411yescharacteristic
143C2×C4×C79#12711yescharacteristic
144C2×C4×C79#13511yescharacteristic
145C2×C4×C79#13511yescharacteristic
146C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C79#13711yescharacteristic
148C2×Dic79—24no—
149C2×Dic79—42no—
150C2×Dic79—42no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153Dic158—42no—
154Dic158—81no—
155C8×C79—42no—
156C8×C79—81no—
157C79⋊C8—21no—
158C79⋊C8—21no—
159C79⋊C8—21no—
160C79⋊C8—21no—
161C79⋊C8—81no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79—81no—
165C22×C79—81no—
166C22×C79—81no—
167C22×C79#39011yescharacteristic
168C22×C79#46311yescharacteristic
169C22×C79#46411yescharacteristic
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—81no—
174C4×C79#41611yescharacteristic
175C4×C79#41711yescharacteristic
176C4×C79#44312yes—
177C4×C79#46311yescharacteristic
178C4×C79#46411yescharacteristic
179Dic79—44no—
180Dic79—81no—
181Dic79—81no—
182#9751—791nomaximal, Sylow
183C2×C79—21no—
184C2×C79—21no—
185C2×C79—161no—
186C2×C79#197411yescharacteristic
187C2×C79#197611yescharacteristic
188C2×C79#198311yescharacteristic
189#3131—791no—
190C8×C4⋊C4—791no—
191#1564—791no—
192#1503—791no—
193#1502—791no—
194#1481—791no—
195#1480D7911yescharacteristic
196C79#975111yescharacteristic, complemented, Sylow
197#533—791no—
198#532C2×D7911yescharacteristic
199#488—1581no—
200#488—1581no—
201#487—791no—
202#487—791no—
203#486—791no—
204#486—791no—
205#431—791no—
206#431C2×D7911yescharacteristic
207C2×C4⋊C8C2×D7911yescharacteristic
208#416—1581no—
209#416—1581no—
210C2×C4×C8—791no—
211C2×C4×C8—1581no—
212C4×C4⋊C4—791no—
213#336—1581no—
214#335—1581no—
215#334—1581no—
216#328—1582no—
217#327—1582no—
218Q8⋊C8—1582no—
219D4⋊C8—22no—
220C2×#41—21no—
221C22×C8—21no—
222C22×C8—791no—
223C22×C8—791no—
224C4⋊Q8—791no—
225C4⋊Q8—791no—
226C4⋊Q8C4×D7911yescharacteristic
227#127C4×D7911yescharacteristic
228#125—1581no—
229#124—21no—
230C22⋊Q8—1581no—
231C22⋊Q8—1581no—
232#121—21no—
233#121—21no—
234C4×Q8—1581no—
235C4×Q8—1581no—
236C4×D4—21no—
237C4×D4—21no—
238#117—1581no—
239C2×C4⋊C4—1581no—
240C2×C42—1581no—
241C2×C42C22×D7911yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—1581no—
245C4⋊C8C22⋊D7912yes—
246#103—1581no—
247#103—1581no—
248#103—1581no—
249#103—1581no—
250#102—1581no—
251#102—1581no—
252#102—1581no—
253#102—1581no—
254C22⋊C8—21no—
255C22⋊C8—1581no—
256C4×C8—21no—
257C4×C8—3162no—
258C4×C8—791no—
259C4×C8—791no—
260C4×C8—791no—
261C4×C8—791no—
262C4×C8—1581no—
263#95—792no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—1581no—
270C2×Q8—1581no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—41no—
276C22×C4—1582no—
277C22×C4D4×D7911yescharacteristic
278C22×C4D4×D7911yescharacteristic
279C22×C4#1192962411yescharacteristic
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—3161no—
285C2×C8—791no—
286C2×C8—791no—
287C2×C8—791no—
288C2×C8—791no—
289C2×C8—1581no—
290C2×C8—1581no—
291C2×C8—1581no—
292C2×C8—1581no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—3161no—
296C4⋊C4—3161no—
297C4⋊C4—1584no—
298C4⋊C4—1581no—
299C4⋊C4—1581no—
300C4⋊C4—1581no—
301C4⋊C4—1581no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—3161no—
305C42—3161no—
306C42—3161no—
307C42—3161no—
308C42C2×C4×D7911yescharacteristic
309C42D4×D7911yescharacteristic
310C42C2×C22⋊D7911yescharacteristic
311C42#1192962311yescharacteristic
312C23—41no—
313C23#1457913411yescharacteristic
314Q8—41no—
315Q8—41no—
316Q8—3162no—
317Q8—6321no—
318D4—42no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—3162no—
330C2×C4—3162no—
331C2×C4—3161no—
332C2×C4—3161no—
333C2×C4—1584no—
334C2×C4C22⋊C4×D7911yescharacteristic
335C2×C4(C22⋊D79)⋊C411yescharacteristic
336C2×C4C4×C22⋊D7911yescharacteristic
337C2×C4#1457913311yescharacteristic
338C2×C4#1457914711yescharacteristic
339C2×C4#1457915711yescharacteristic
340C2×C4#1457918011yescharacteristic
341C2×C4#1457918211yescharacteristic
342C2×C4#1457918511yescharacteristic
343C2×C4#1457919011yescharacteristic
344C8—42no—
345C8—81no—
346C8—6321no—
347C8—1581no—
348C8—1581no—
349C8—1581no—
350C8—1581no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1909846311yescharacteristic
357C22#1909877711yescharacteristic
358C22#1909878411yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—3164no—
364C4—6321no—
365C4—6321no—
366C4#1909855611yescharacteristic
367C4#1909856111yescharacteristic
368C4#1909867812yes—
369C4#1909877611yescharacteristic
370C4#1909878311yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2#2376428711yescharacteristic
375C2#2376429711yescharacteristic
376C2#2376433111yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1480C2
socleC22×C79#390
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#1480D79
hypocenterC79#9751
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #23763265C2semidirect
3. #23762146C2semidirect
4. #23761900C2semidirect
6. #23761802C2semidirect
11. #19098913C4semidirect
12. #19098913C4semidirect
13. #19098909C4semidirect
14. #19098909C4semidirect
196. C79#9751semidirect

Maximal subgroups

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

subgroupindex
#237737222
#237632652
#237621462
#237619002
#237618952
#237618022
C79×#14802
#975179