Finite Group Databasebeta

← #34182584

Subgroups — #34182584

Order 20224.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771969C211yesmaximal, characteristic
2#23763421C211yesmaximal, characteristic
3#23762171C211yesmaximal, characteristic, complemented
4#23762155C211yesmaximal, characteristic, complemented
5#23761780C211yesmaximal, characteristic
6#23761241C211yesmaximal, characteristic, complemented
7C79×#1303C211yesmaximal, characteristic, complemented
8C2×#14579169—21no—
9C4×#11929624C2211yescharacteristic
10#19098591—21no—
11#19098588—22no—
12#19098587C2211yescharacteristic
13#19098585—21no—
14C2×C4⋊(C79⋊C8)C2211yescharacteristic
15C4⋊C8×D79—21no—
16C2×#14579089—21no—
17C2×C4×C79⋊C8C2211yescharacteristic
18#19098323—21no—
19#19098314—21no—
20#19098314—21no—
21#19098313—22no—
22#19098293C411yescharacteristic, complemented
23#19098293C411yescharacteristic, complemented
24#19098290C411yescharacteristic, complemented
25#19098290C411yescharacteristic, complemented
26#431×C79C2211yescharacteristic
27#423×C79C2211yescharacteristic
28C4×C4⋊C4×C79C2211yescharacteristic
29C2×#11929624D411yescharacteristic
30#14579169—42no—
31#14579169—42no—
32C2×#11929608—21no—
33C2×#11929607—22no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36C2×C8×D79—22no—
37C4×Dic158C2×C411yescharacteristic
38C4×Dic158Q811yescharacteristic
39C4×D316C2×C411yescharacteristic
40C4×D316Q811yescharacteristic
41C4×C22⋊D79—21no—
42C4×C22⋊D79—21no—
43#14579120D411yescharacteristic
44#14579118—21no—
45C42×D79—21no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)—21no—
49C4⋊(C79⋊C8)D411yescharacteristic
50C4⋊(C79⋊C8)D411yescharacteristic
51Dic79⋊C8—22no—
52Dic79⋊C8—22no—
53#14579089—42no—
54C4×C79⋊C8—21no—
55C4×C79⋊C8C2×C411yescharacteristic
56C4×C79⋊C8C2×C411yescharacteristic
57C2×M16×C79—21no—
58C2×C4⋊C4×C79—21no—
59C2×C42×C79—21no—
60C2×C42×C79C2311yescharacteristic
61C4⋊C8×C79—21no—
62C4⋊C8×C79C2×C411yescharacteristic
63C4⋊C8×C79C2×C411yescharacteristic
64C4⋊C8×C79D412yes—
65C22⋊C8×C79—21no—
66C4×C8×C79—21no—
67#95×C79—21no—
68#11929624—21no—
69#11929624—21no—
70#11929624SD1612yes—
71#11929624SD1612yes—
72C2×Dic158C4⋊C411yescharacteristic
73C2×D316C4⋊C411yescharacteristic
74C2×C22⋊D79—21no—
75C2×C4×D79—21no—
76C2×C4×D79—21no—
77C2×C4×D79—21no—
78#11929608—41no—
79#11929608—41no—
80#11929607—44no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—21no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—42no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C8×D79—44no—
90C4⋊Dic79C4⋊C411yescharacteristic
91C4⋊Dic79C4⋊C411yescharacteristic
92Dic79⋊C4—21no—
93Dic79⋊C4—21no—
94C22⋊Dic79—21no—
95(C2×D79)⋊C4—21no—
96(C2×D79)⋊C4—21no—
97C4×Dic79—21no—
98C4×Dic79—21no—
99C4×Dic79—21no—
100C22×C4×C79—41no—
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79#4111yescharacteristic
103C22×C4×C79#4111yescharacteristic
104M16×C79—41no—
105M16×C79—41no—
106C2×C8×C79—21no—
107C2×C8×C79—21no—
108C2×C8×C79—22no—
109C4⋊C4×C79—22no—
110C4⋊C4×C79—41no—
111C42×C79—41no—
112C42×C79—41no—
113C42×C79C22×C411yescharacteristic
114C42×C79C2×D411yescharacteristic
115C42×C79C2×D411yescharacteristic
116C42×C79C2×Q811yescharacteristic
117C23×C79#12311yescharacteristic
118C22×D79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—41no—
122C2×C4×C79—42no—
123C2×C4×C79—81no—
124C2×C4×C79—81no—
125C2×C4×C79C2×C4⋊C411yescharacteristic
126C2×C4×C79C4×D411yescharacteristic
127C2×C4×C79C4×D411yescharacteristic
128C2×C4×C79#12011yescharacteristic
129C2×C4×C79C22⋊Q811yescharacteristic
130C2×C4×C79C22⋊Q811yescharacteristic
131C2×C4×C79C2×SD1611yescharacteristic
132C2×C4×C79C2×SD1611yescharacteristic
133C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C79#13711yescharacteristic
135C22⋊D79—22no—
136C22⋊D79—22no—
137C2×Dic79—21no—
138C2×Dic79—21no—
139C2×Dic79—21no—
140D316—21no—
141D316#10612yes—
142C4×D79—22no—
143C4×D79—22no—
144C4×D79—41no—
145C4×D79—41no—
146C4×D79—81no—
147Dic158—21no—
148Dic158#10612yes—
149C8×C79—41no—
150C8×C79—41no—
151C8×C79—42no—
152C79⋊C8—22no—
153C79⋊C8—41no—
154C79⋊C8—41no—
155C79⋊C8—41no—
156C79⋊C8—81no—
157C22×C79—21no—
158C22×C79—21no—
159C22×C79#38511yescharacteristic
160C22×C79#48111yescharacteristic
161C22×C79#48311yescharacteristic
162C2×D79—22no—
163C2×D79—41no—
164C2×D79—41no—
165C4×C79—21no—
166C4×C79—21no—
167C4×C79—21no—
168C4×C79—82no—
169C4×C79C2×#10611yescharacteristic
170C4×C79#42811yescharacteristic
171C4×C79#44311yescharacteristic
172C4×C79#44411yescharacteristic
173C4×C79#44811yescharacteristic
174C4×C79#45011yescharacteristic
175Dic79—22no—
176Dic79—41no—
177Dic79—41no—
178#8904—791nomaximal, Sylow
179C2×C79—21no—
180C2×C79—21no—
181C2×C79#185311yescharacteristic
182C2×C79#186111yescharacteristic
183C2×C79#186611yescharacteristic
184D79—42no—
185#2906—791no—
186#1806—791no—
187#1570—791no—
188#1566—791no—
189#1476—791no—
190#1303D7911yescharacteristic, complemented
191#1279—791no—
192C79#890411yescharacteristic, complemented, Sylow
193C2×#131—1581no—
194C4×#41—791no—
195#431C2×D7911yescharacteristic
196#424—1581no—
197#424—1582no—
198#423—791no—
199#423—1581no—
200#423C2×D7911yescharacteristic
201C2×C4⋊C8—791no—
202C2×C4⋊C8—1581no—
203C2×#104—1581no—
204C2×C4×C8—791no—
205C4×C4⋊C4C2×D7911yescharacteristic
206#339—1581no—
207#334—1581no—
208#334—1581no—
209#334—1582no—
210Q8⋊C8—791no—
211Q8⋊C8—791no—
212D4⋊C8—791no—
213D4⋊C8—791no—
214C2×#41—791no—
215#131—3162no—
216#131—3162no—
217C2×M16—21no—
218C2×M16—1582no—
219C2×M16—1581no—
220C22×C8—1582no—
221C22×C8—1581no—
222C22×C8—1581no—
223C4×Q8—791no—
224C4×Q8—791no—
225C4×D4—791no—
226C4×D4—791no—
227C4×D4—1581no—
228C4×D4—1581no—
229#117—791no—
230#117—1581no—
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—1581no—
234C2×C42C22×D7911yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—791no—
237C4⋊C8—791no—
238C4⋊C8—1582no—
239C4⋊C8—1582no—
240C4⋊C8—1581no—
241C4⋊C8—1581no—
242C4⋊C8—1581no—
243C4⋊C8C4×D7911yescharacteristic
244C4⋊C8C4×D7911yescharacteristic
245C4⋊C8D31612yes—
246#104—3162no—
247C22⋊C8—21no—
248C4×C8—21no—
249C4×C8—791no—
250C4×C8—791no—
251C4×C8—1581no—
252#95—21no—
253#41—792no—
254#41—792no—
255#41—1581no—
256#41—1581no—
257C2×Q8—791no—
258C2×D4—791no—
259C2×D4—1581no—
260C22×C4—41no—
261C22×C4—1581no—
262C22×C4—1581no—
263C22×C4—1581no—
264C22×C4D4×D7911yescharacteristic
265C22×C4#1192962311yescharacteristic
266C22×C4#1192962411yescharacteristic
267M16—41no—
268M16—41no—
269M16—3164no—
270M16—3161no—
271M16—3161no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—22no—
275C2×C8—3164no—
276C2×C8—3161no—
277C2×C8—3161no—
278C2×C8—3162no—
279C2×C8—3161no—
280C2×C8—1581no—
281C2×C8—1581no—
282C2×C8—1581no—
283C2×C8—1581no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—791no—
287C4⋊C4—791no—
288C4⋊C4—1581no—
289C4⋊C4—1581no—
290C22⋊C4—1581no—
291C22⋊C4—1581no—
292C22⋊C4—1581no—
293C42—41no—
294C42—41no—
295C42—1581no—
296C42—1581no—
297C42—1581no—
298C42C2×C4×D7911yescharacteristic
299C42D4×D7911yescharacteristic
300C42C2×D31611yescharacteristic
301C42Q8×D7911yescharacteristic
302C23—1581no—
303C23#1457914311yescharacteristic
304Q8—792no—
305Q8—1581no—
306D4—792no—
307D4—1582no—
308D4—1582no—
309D4—1581no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—3161no—
317C2×C4—3161no—
318C2×C4—6321no—
319C2×C4—1581no—
320C2×C4—1581no—
321C2×C4—1581no—
322C2×C4—1582no—
323C2×C4—1582no—
324C2×C4C4⋊C4×D7911yescharacteristic
325C2×C4(C22⋊D79)⋊C411yescharacteristic
326C2×C4C4×D31611yescharacteristic
327C2×C4#1457913111yescharacteristic
328C2×C4#1457913911yescharacteristic
329C2×C4#1457914111yescharacteristic
330C2×C4SD16×D7911yescharacteristic
331C2×C4SD16×D7911yescharacteristic
332C2×C4#1457918511yescharacteristic
333C2×C4#1457919311yescharacteristic
334C8—41no—
335C8—41no—
336C8—42no—
337C8—3161no—
338C8—3161no—
339C8—3161no—
340C8—6321no—
341C8—1582no—
342C22—21no—
343C22—21no—
344C22—3161no—
345C22—3161no—
346C22—1582no—
347C22#1909843711yescharacteristic
348C22#1909888011yescharacteristic
349C22#1909889211yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—82no—
354C4—3161no—
355C4—3161no—
356C4—1582no—
357C4#106×D7911yescharacteristic
358C4#1909860411yescharacteristic
359C4#1909867711yescharacteristic
360C4#1909868211yescharacteristic
361C4#1909870311yescharacteristic
362C4#1909871511yescharacteristic
363C2—21no—
364C2—21no—
365C2—3162no—
366C2#2376359411yescharacteristic
367C2#2376364311yescharacteristic
368C2#2376367611yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1303C2
socleC22×C79#385
prime radicalC2×#11929624D4
cosocleC2×C42×C79C23
hypercenter#1303D79
hypocenterC79#8904
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #23762171C2semidirect
4. #23762155C2semidirect
6. #23761241C2semidirect
7. C79×#1303C2semidirect
22. #19098293C4semidirect
23. #19098293C4semidirect
24. #19098290C4semidirect
25. #19098290C4semidirect
190. #1303D79semidirect
192. C79#8904semidirect

Maximal subgroups

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

subgroupindex
#237719692
#237634212
#237621712
#237621552
#237617802
#237612412
C79×#13032
#890479