Finite Group Databasebeta

← #34187621

Subgroups — #34187621

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
2C4⋊C4×C79⋊C8C211yesmaximal, characteristic, complemented
3#23762146C211yesmaximal, characteristic, complemented
4#23761898C211yesmaximal, characteristic, complemented
5#23761804C211yesmaximal, characteristic, complemented
6#23761799C211yesmaximal, characteristic
7C79×#1502C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098917—21no—
10#19098917—21no—
11#19098913C411yescharacteristic, complemented
12#19098913C411yescharacteristic, complemented
13#19098909C411yescharacteristic, complemented
14#19098909C411yescharacteristic, complemented
15#19098616C2211yescharacteristic
16C2×C4⋊(C79⋊C8)C2211yescharacteristic
17#19098559—21no—
18#19098559—21no—
19C2×C4×C79⋊C8—21no—
20C2×C4×C79⋊C8C2211yescharacteristic
21#19098318—21no—
22#19098315—21no—
23#19098313—21no—
24#19098299—22no—
25#19098295—22no—
26D4⋊(C79⋊C8)—22no—
27#532×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29C4×C4⋊C4×C79C2211yescharacteristic
30#327×C79—22no—
31C22×C79⋊C8—21no—
32C22×C79⋊C8D411yescharacteristic
33C22×C79⋊C8D411yescharacteristic
34#14579161C2×C411yescharacteristic
35#14579161C2×C411yescharacteristic
36#14579153—21no—
37#14579140—21no—
38#14579140—21no—
39C4×Dic158—21no—
40C4×Dic158—21no—
41#14579120—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)D412yes—
45#14579085—21no—
46#14579085—21no—
47#14579085—21no—
48#14579085—21no—
49#14579082—21no—
50#14579082—21no—
51#14579082—21no—
52#14579082—21no—
53C22⋊(C79⋊C8)—21no—
54C4×C79⋊C8—21no—
55C4×C79⋊C8—42no—
56C4×C79⋊C8C2×C411yescharacteristic
57C4×C79⋊C8C2×C411yescharacteristic
58C4×C79⋊C8D411yescharacteristic
59C4×C79⋊C8D411yescharacteristic
60C2×#41×C79—21no—
61C4⋊Q8×C79C2×C411yescharacteristic
62#127×C79C2×C411yescharacteristic
63#124×C79—21no—
64#121×C79—21no—
65#121×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79—21no—
68C2×C4⋊C4×C79—21no—
69C2×C42×C79—21no—
70C2×C42×C79C2311yescharacteristic
71C4⋊C8×C79—21no—
72C22⋊C8×C79—21no—
73C4×C8×C79—21no—
74#95×C79D412yes—
75C2×Dic158—21no—
76C2×Dic158—21no—
77C2×C79⋊C8—22no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8C22⋊C411yescharacteristic
86C2×C79⋊C8C22⋊C411yescharacteristic
87C2×C79⋊C8C22⋊C411yescharacteristic
88C2×C79⋊C8C22⋊C411yescharacteristic
89C4⋊Dic79—21no—
90C4⋊Dic79—21no—
91C4⋊Dic79—21no—
92C4⋊Dic79—21no—
93Dic79⋊C4—41no—
94Dic79⋊C4—41no—
95C22⋊Dic79—41no—
96C4×Dic79—41no—
97#41×C79—41no—
98#41×C79—41no—
99#41×C79—41no—
100#41×C79—41no—
101C2×Q8×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—41no—
106C22×C4×C79—22no—
107C22×C4×C79—41no—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79#4111yescharacteristic
111C2×C8×C79—41no—
112C4⋊C4×C79—24no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—21no—
115C22⋊C4×C79—41no—
116C22⋊C4×C79—41no—
117C42×C79—41no—
118C42×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—24no—
135C2×C4×C79—41no—
136C2×C4×C79—42no—
137C2×C4×C79—42no—
138C2×C4×C79—41no—
139C2×C4×C79—81no—
140C2×C4×C79—81no—
141C2×C4×C79C2×C22⋊C411yescharacteristic
142C2×C4×C79C4×D411yescharacteristic
143C2×C4×C79C4×D411yescharacteristic
144C2×C4×C79#12111yescharacteristic
145C2×C4×C79#12411yescharacteristic
146C2×C4×C79#12711yescharacteristic
147C2×C4×C79#13511yescharacteristic
148C2×C4×C79#13511yescharacteristic
149C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C79#13711yescharacteristic
151C2×Dic79—41no—
152C2×Dic79—41no—
153Dic158—42no—
154Dic158—81no—
155C8×C79—81no—
156C79⋊C8—21no—
157C79⋊C8—21no—
158C79⋊C8—21no—
159C79⋊C8—21no—
160C79⋊C8—42no—
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—44no—
174C4×C79—81no—
175C4×C79#41611yescharacteristic
176C4×C79#41711yescharacteristic
177C4×C79#44312yes—
178C4×C79#46311yescharacteristic
179C4×C79#46411yescharacteristic
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#1502D7911yescharacteristic
194#1481—791no—
195#1480—791no—
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⋊C8—791no—
208#416—1581no—
209#416—1581no—
210C2×C4×C8—791no—
211C2×C4×C8—1581no—
212C4×C4⋊C4C2×D7911yescharacteristic
213#336—1581no—
214#335—1581no—
215#334—1581no—
216#328—1582no—
217#327—22no—
218Q8⋊C8—1582no—
219D4⋊C8—1582no—
220C2×#41—21no—
221C22×C8—791no—
222C22×C8—791no—
223C22×C8—1581no—
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—21no—
240C2×C42—21no—
241C2×C42C22×D7911yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—792no—
244C4⋊C8—1581no—
245C4⋊C8—1581no—
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#95C22⋊D7912yes—
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—22no—
276C22×C4—41no—
277C22×C4D4×D7911yescharacteristic
278C22×C4C2×C22⋊D7911yescharacteristic
279C22×C4#1192962311yescharacteristic
280C2×C8—41no—
281C2×C8—3161no—
282C2×C8—3161no—
283C2×C8—3161no—
284C2×C8—791no—
285C2×C8—791no—
286C2×C8—791no—
287C2×C8—791no—
288C2×C8—1582no—
289C2×C8—1581no—
290C2×C8—1581no—
291C2×C8—1581no—
292C2×C8—1581no—
293C4⋊C4—24no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—3161no—
297C4⋊C4—3161no—
298C4⋊C4—1581no—
299C4⋊C4—1581no—
300C4⋊C4—1581no—
301C4⋊C4—1581no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—3161no—
305C42—41no—
306C42—41no—
307C42—3161no—
308C42C2×C4×D7911yescharacteristic
309C42D4×D7911yescharacteristic
310C42D4×D7911yescharacteristic
311C42#1192962411yescharacteristic
312C23—41no—
313C23#1457913311yescharacteristic
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—24no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—3161no—
333C2×C4—3161no—
334C2×C4C22⋊C4×D7911yescharacteristic
335C2×C4(C22⋊D79)⋊C411yescharacteristic
336C2×C4C4×C22⋊D7911yescharacteristic
337C2×C4#1457913411yescharacteristic
338C2×C4#1457914711yescharacteristic
339C2×C4#1457915711yescharacteristic
340C2×C4#1457918011yescharacteristic
341C2×C4#1457918211yescharacteristic
342C2×C4#1457918511yescharacteristic
343C2×C4#1457919011yescharacteristic
344C8—81no—
345C8—3162no—
346C8—6321no—
347C8—1581no—
348C8—1581no—
349C8—1581no—
350C8—1581no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1909846311yescharacteristic
357C22#1909877611yescharacteristic
358C22#1909878311yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—44no—
363C4—81no—
364C4—6321no—
365C4—6321no—
366C4#1909855611yescharacteristic
367C4#1909856111yescharacteristic
368C4#1909867712yes—
369C4#1909877711yescharacteristic
370C4#1909878411yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2#2376428511yescharacteristic
375C2#2376429511yescharacteristic
376C2#2376433111yescharacteristic
subgroupG/H
centerC2×C4#14579134
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1502C2
socleC22×C79#390
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#1502D79
hypocenterC79#9751
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C79⋊C8C2semidirect
3. #23762146C2semidirect
4. #23761898C2semidirect
5. #23761804C2semidirect
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
C4⋊C4×C79⋊C82
#237621462
#237618982
#237618042
#237617992
C79×#15022
#975179