Finite Group Databasebeta

← #34187620

Subgroups — #34187620

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#23773723C211yesmaximal, characteristic
2#23762148C211yesmaximal, characteristic, complemented
3#23761898C211yesmaximal, characteristic, complemented
4#23761894C211yesmaximal, characteristic
5#23761802C211yesmaximal, characteristic, complemented
6#23761798C211yesmaximal, characteristic
7C8×C4⋊C4×C79C211yesmaximal, characteristic, complemented
8#19099120C2211yescharacteristic
9#19099113C2211yescharacteristic
10#19098918—21no—
11#19098918—21no—
12#19098914C411yescharacteristic, complemented
13#19098914C411yescharacteristic, complemented
14#19098910C411yescharacteristic, complemented
15#19098910C411yescharacteristic, complemented
16#19098616C2211yescharacteristic
17#19098616C2211yescharacteristic
18#19098560—21no—
19#19098560—21no—
20#19098316—21no—
21#19098314—21no—
22#19098299—22no—
23#19098296—22no—
24#19098293—22no—
25#19098290—22no—
26C2×C4⋊C8×C79C2211yescharacteristic
27C2×C4×C8×C79—21no—
28C2×C4×C8×C79C2211yescharacteristic
29C4×C4⋊C4×C79C2211yescharacteristic
30#336×C79—21no—
31C2×#11929624—21no—
32#14579161C2×C411yescharacteristic
33#14579161C2×C411yescharacteristic
34#14579161C2×C411yescharacteristic
35#14579158C2×C411yescharacteristic
36#14579153—21no—
37#14579149—21no—
38#14579140—21no—
39#14579140—21no—
40#14579135—21no—
41#14579135—21no—
42C4×Dic158—21no—
43C4×Dic158—21no—
44C4×D316—21no—
45C4×D316—21no—
46#14579120—21no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)—21no—
49#14579086—21no—
50#14579086—21no—
51#14579086—21no—
52#14579086—21no—
53#14579083—21no—
54#14579083—21no—
55#14579083—21no—
56#14579083—21no—
57C22⋊(C79⋊C8)—21no—
58C22⋊(C79⋊C8)—21no—
59C4×C79⋊C8—21no—
60C4×C79⋊C8—21no—
61C22×C8×C79—21no—
62C22×C8×C79D411yescharacteristic
63C22×C8×C79D411yescharacteristic
64C2×C4⋊C4×C79—21no—
65C2×C42×C79—21no—
66C2×C42×C79C2311yescharacteristic
67C4⋊C8×C79—21no—
68C4⋊C8×C79D412yes—
69C4×C8×C79—42no—
70C4×C8×C79C2×C411yescharacteristic
71C4×C8×C79C2×C411yescharacteristic
72C4×C8×C79D411yescharacteristic
73C4×C8×C79D411yescharacteristic
74#95×C79D412yes—
75#11929624—41no—
76#11929624—41no—
77#11929624—41no—
78#11929624—41no—
79C2×Dic158—21no—
80C2×Dic158—21no—
81C2×Dic158—21no—
82C2×D316—21no—
83C2×D316—21no—
84C2×D316—21no—
85C2×C22⋊D79—41no—
86C2×C4×D79—41no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C4⋊Dic79—21no—
90C4⋊Dic79—21no—
91C4⋊Dic79—21no—
92C4⋊Dic79—21no—
93C4⋊Dic79—21no—
94C4⋊Dic79—21no—
95Dic79⋊C4—41no—
96Dic79⋊C4—41no—
97C22⋊Dic79—41no—
98(C2×D79)⋊C4—41no—
99(C2×D79)⋊C4—41no—
100C4×Dic79—41no—
101C22×C4×C79—22no—
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79#4111yescharacteristic
105C2×C8×C79—22no—
106C2×C8×C79—21no—
107C2×C8×C79—21no—
108C2×C8×C79—21no—
109C2×C8×C79—21no—
110C2×C8×C79—41no—
111C2×C8×C79—41no—
112C2×C8×C79C22⋊C411yescharacteristic
113C2×C8×C79C22⋊C411yescharacteristic
114C2×C8×C79C22⋊C411yescharacteristic
115C2×C8×C79C22⋊C411yescharacteristic
116C4⋊C4×C79—24no—
117C42×C79—41no—
118C42×C79—41no—
119C42×C79C22×C411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79C2×D411yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79#12111yescharacteristic
124C22×D79—41no—
125C2×C4×C79—22no—
126C2×C4×C79—24no—
127C2×C4×C79—41no—
128C2×C4×C79—42no—
129C2×C4×C79—42no—
130C2×C4×C79C2×C22⋊C411yescharacteristic
131C2×C4×C79C4×D411yescharacteristic
132C2×C4×C79C4×D411yescharacteristic
133C2×C4×C79#12111yescharacteristic
134C2×C4×C79#12411yescharacteristic
135C2×C4×C79#12711yescharacteristic
136C2×C4×C79#13511yescharacteristic
137C2×C4×C79#13511yescharacteristic
138C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C79#13711yescharacteristic
140C22⋊D79—81no—
141C22⋊D79—81no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145D316—41no—
146D316—41no—
147D316—42no—
148D316—81no—
149C4×D79—81no—
150C4×D79—81no—
151Dic158—41no—
152Dic158—41no—
153Dic158—42no—
154Dic158—81no—
155C8×C79—21no—
156C8×C79—21no—
157C8×C79—21no—
158C8×C79—21no—
159C8×C79—42no—
160C79⋊C8—81no—
161C79⋊C8—81no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79#39011yescharacteristic
165C22×C79#46311yescharacteristic
166C22×C79#46411yescharacteristic
167C2×D79—81no—
168C2×D79—81no—
169C2×D79—81no—
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—44no—
174C4×C79#41611yescharacteristic
175C4×C79#41711yescharacteristic
176C4×C79#44312yes—
177C4×C79#46311yescharacteristic
178C4×C79#46411yescharacteristic
179Dic79—81no—
180Dic79—81no—
181Dic79—81no—
182#9751—791nomaximal, Sylow
183C2×C79—21no—
184C2×C79—21no—
185C2×C79#197411yescharacteristic
186C2×C79#197611yescharacteristic
187C2×C79#198311yescharacteristic
188D79—161no—
189#3131—791no—
190C8×C4⋊C4D7911yescharacteristic, complemented
191#1564—791no—
192#1503—791no—
193#1502—791no—
194#1481—791no—
195#1480—791no—
196C79#975111yescharacteristic, complemented, Sylow
197#533—791no—
198#532—791no—
199#488—1581no—
200#488—1581no—
201#487—791no—
202#487—791no—
203#486—791no—
204#486—791no—
205#431—791no—
206#431—791no—
207C2×C4⋊C8C2×D7911yescharacteristic
208#416—1581no—
209#416—1581no—
210C2×C4×C8—21no—
211C2×C4×C8C2×D7911yescharacteristic
212C4×C4⋊C4C2×D7911yescharacteristic
213#336—21no—
214#335—1581no—
215#334—1581no—
216#328—1582no—
217#327—1582no—
218Q8⋊C8—1582no—
219D4⋊C8—1582no—
220C2×#41—1581no—
221C22×C8—21no—
222C22×C8C22⋊D7911yescharacteristic
223C22×C8C22⋊D7911yescharacteristic
224C4⋊Q8—791no—
225C4⋊Q8—791no—
226C4⋊Q8—791no—
227#127—791no—
228#125—1581no—
229#124—1581no—
230C22⋊Q8—1581no—
231C22⋊Q8—1581no—
232#121—1581no—
233#121—1581no—
234C4×Q8—1581no—
235C4×Q8—1581no—
236C4×D4—1581no—
237C4×D4—1581no—
238#117—1581no—
239C2×C4⋊C4—21no—
240C2×C42—21no—
241C2×C42C22×D7911yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—1581no—
244C4⋊C8—1581no—
245C4⋊C8D31612yes—
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—1581no—
255C22⋊C8—1581no—
256C4×C8—42no—
257C4×C8—1581no—
258C4×C8—1581no—
259C4×C8C4×D7911yescharacteristic
260C4×C8C4×D7911yescharacteristic
261C4×C8D31611yescharacteristic
262C4×C8D31611yescharacteristic
263#95D31612yes—
264#41—3161no—
265#41—3161no—
266#41—3161no—
267#41—3161no—
268C2×Q8—1581no—
269C2×Q8—1581no—
270C2×Q8—1581no—
271C2×D4—3161no—
272C2×D4—1581no—
273C2×D4—1581no—
274C2×D4—1581no—
275C22×C4—22no—
276C22×C4—3161no—
277C22×C4C2×C22⋊D7911yescharacteristic
278C22×C4C2×D31611yescharacteristic
279C22×C4#1192962411yescharacteristic
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—3161no—
288C2×C8—3161no—
289C2×C8(C2×D79)⋊C411yescharacteristic
290C2×C8(C2×D79)⋊C411yescharacteristic
291C2×C8(C2×D79)⋊C411yescharacteristic
292C2×C8(C2×D79)⋊C411yescharacteristic
293C4⋊C4—24no—
294C4⋊C4—3161no—
295C4⋊C4—3161no—
296C4⋊C4—1581no—
297C4⋊C4—1581no—
298C4⋊C4—1581no—
299C4⋊C4—1581no—
300C4⋊C4—1581no—
301C4⋊C4—1581no—
302C22⋊C4—3161no—
303C22⋊C4—3161no—
304C22⋊C4—3161no—
305C42—41no—
306C42—41no—
307C42—3161no—
308C42C2×C4×D7911yescharacteristic
309C42C2×D31611yescharacteristic
310C42C2×D31611yescharacteristic
311C42#1192962411yescharacteristic
312C23—3161no—
313C23#1457913511yescharacteristic
314Q8—3162no—
315Q8—3161no—
316Q8—3161no—
317Q8—6321no—
318D4—3162no—
319D4—3161no—
320D4—3161no—
321D4—6321no—
322D4—6321no—
323D4—6321no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—3161no—
330C2×C4—3161no—
331C2×C4—3161no—
332C2×C4—6321no—
333C2×C4—6321no—
334C2×C4C2×(C2×D79)⋊C411yescharacteristic
335C2×C4C4×D31611yescharacteristic
336C2×C4C4×D31611yescharacteristic
337C2×C4#1457913511yescharacteristic
338C2×C4#1457914911yescharacteristic
339C2×C4#1457915811yescharacteristic
340C2×C4#1457918111yescharacteristic
341C2×C4#1457918111yescharacteristic
342C2×C4#1457918711yescharacteristic
343C2×C4#1457919211yescharacteristic
344C8—21no—
345C8—21no—
346C8—21no—
347C8—21no—
348C8—42no—
349C8—6321no—
350C8—6321no—
351C22—21no—
352C22—21no—
353C22—6321no—
354C22—6321no—
355C22—6321no—
356C22#1909846511yescharacteristic
357C22#1909877811yescharacteristic
358C22#1909878511yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—44no—
363C4—6321no—
364C4—6321no—
365C4—6321no—
366C4#1909855711yescharacteristic
367C4#1909856211yescharacteristic
368C4#1909868112yes—
369C4#1909877811yescharacteristic
370C4#1909878511yescharacteristic
371C2—21no—
372C2—21no—
373C2—12641no—
374C2#2376428811yescharacteristic
375C2#2376429811yescharacteristic
376C2#2376433211yescharacteristic
subgroupG/H
centerC2×C4#14579135
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC8×C4⋊C4×C79C2
socleC22×C79#390
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenterC8×C4⋊C4D79
hypocenterC79#9751
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #23762148C2semidirect
3. #23761898C2semidirect
5. #23761802C2semidirect
7. C8×C4⋊C4×C79C2semidirect
12. #19098914C4semidirect
13. #19098914C4semidirect
14. #19098910C4semidirect
15. #19098910C4semidirect
190. C8×C4⋊C4D79semidirect
196. C79#9751semidirect

Maximal subgroups

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

subgroupindex
#237737232
#237621482
#237618982
#237618942
#237618022
#237617982
C8×C4⋊C4×C792
#975179