Finite Group Databasebeta

← #34182575

Subgroups — #34182575

Order 20224.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772065C211yesmaximal, characteristic
2#23763417C211yesmaximal, characteristic
3#23762158C211yesmaximal, characteristic, complemented
4#23761780C211yesmaximal, characteristic
5#23761302C211yesmaximal, characteristic, complemented
6#23761239C211yesmaximal, characteristic, complemented
7C79×#1570C211yesmaximal, characteristic, complemented
8C4×#11929624C2211yescharacteristic
9#19098623—22no—
10#19098619—21no—
11#19098615—21no—
12#19098587C2211yescharacteristic
13C2×#14579087—21no—
14#19098529—21no—
15#19098518—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17C4×C4⋊Dic79C2211yescharacteristic
18#19098324—21no—
19#19098315—22no—
20#19098293C411yescharacteristic, complemented
21#19098293C411yescharacteristic, complemented
22#19098290C411yescharacteristic, complemented
23#19098290C411yescharacteristic, complemented
24#431×C79C2211yescharacteristic
25#423×C79C2211yescharacteristic
26C2×C4⋊C8×C79C2211yescharacteristic
27#334×C79—21no—
28#334×C79—21no—
29C2×#11929624D411yescharacteristic
30C2×#11929608—21no—
31C22×C79⋊C8—21no—
32C4×Dic158C2×C411yescharacteristic
33C4×Dic158Q811yescharacteristic
34C4×D316C2×C411yescharacteristic
35C4×D316Q811yescharacteristic
36C4×C22⋊D79—21no—
37C4×C22⋊D79—21no—
38#14579120D411yescharacteristic
39#14579118—21no—
40C2×C4⋊Dic79—21no—
41C2×C4×Dic79—21no—
42C42×D79—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)D412yes—
45#14579087—42no—
46(C2×D79)⋊C8—22no—
47(C2×D79)⋊C8—22no—
48#14579071—22no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8C2×C411yescharacteristic
51C4×C79⋊C8C2×C411yescharacteristic
52C8×Dic79—22no—
53C79⋊#95—21no—
54C2×M16×C79—21no—
55C22×C8×C79—21no—
56C2×C42×C79C2311yescharacteristic
57C4⋊C8×C79—21no—
58C4⋊C8×C79—21no—
59C4⋊C8×C79—21no—
60C4⋊C8×C79C2×C411yescharacteristic
61C4⋊C8×C79C2×C411yescharacteristic
62C4⋊C8×C79D411yescharacteristic
63C4⋊C8×C79D411yescharacteristic
64C22⋊C8×C79—21no—
65C4×C8×C79—21no—
66#11929624—21no—
67#11929624—21no—
68#11929624—21no—
69#11929624—21no—
70C2×Dic158C4⋊C411yescharacteristic
71C2×D316C4⋊C411yescharacteristic
72C2×C22⋊D79—21no—
73C22×Dic79—41no—
74C2×C4×D79—21no—
75C2×C4×D79—21no—
76C2×C4×D79—21no—
77#11929608—41no—
78#11929608—41no—
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—22no—
82C2×C79⋊C8—42no—
83C4⋊Dic79—22no—
84C4⋊Dic79—41no—
85C4⋊Dic79C4⋊C411yescharacteristic
86C4⋊Dic79C4⋊C411yescharacteristic
87Dic79⋊C4—21no—
88Dic79⋊C4—21no—
89C22⋊Dic79—21no—
90(C2×D79)⋊C4—21no—
91(C2×D79)⋊C4—21no—
92C4×Dic79—21no—
93C4×Dic79—21no—
94C4×Dic79—21no—
95C4×Dic79—41no—
96C4×Dic79—41no—
97C22×C4×C79C2×D411yescharacteristic
98C22×C4×C79#4111yescharacteristic
99C22×C4×C79#4111yescharacteristic
100M16×C79—41no—
101M16×C79—41no—
102C2×C8×C79—21no—
103C2×C8×C79—21no—
104C2×C8×C79—21no—
105C2×C8×C79—21no—
106C2×C8×C79—41no—
107C2×C8×C79—41no—
108C2×C8×C79—41no—
109C42×C79C22×C411yescharacteristic
110C42×C79C2×D411yescharacteristic
111C42×C79C2×D411yescharacteristic
112C42×C79C2×Q811yescharacteristic
113C23×C79#12311yescharacteristic
114C22×D79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—41no—
118C2×C4×C79C2×C4⋊C411yescharacteristic
119C2×C4×C79C4×D411yescharacteristic
120C2×C4×C79C4×D411yescharacteristic
121C2×C4×C79#12011yescharacteristic
122C2×C4×C79C22⋊Q811yescharacteristic
123C2×C4×C79C22⋊Q811yescharacteristic
124C2×C4×C79#13511yescharacteristic
125C2×C4×C79#13511yescharacteristic
126C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C79#13711yescharacteristic
128C22⋊D79—41no—
129C22⋊D79—41no—
130C2×Dic79—21no—
131C2×Dic79—21no—
132C2×Dic79—21no—
133C2×Dic79—42no—
134C2×Dic79—81no—
135C2×Dic79—81no—
136D316—21no—
137D316—21no—
138C4×D79—41no—
139C4×D79—41no—
140C4×D79—41no—
141C4×D79—41no—
142C4×D79—81no—
143Dic158—21no—
144Dic158—21no—
145C8×C79—41no—
146C8×C79—41no—
147C8×C79—41no—
148C8×C79—41no—
149C8×C79—81no—
150C79⋊C8—22no—
151C79⋊C8—41no—
152C79⋊C8—42no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79#38511yescharacteristic
156C22×C79#48211yescharacteristic
157C22×C79#48511yescharacteristic
158C2×D79—41no—
159C2×D79—41no—
160C2×D79—41no—
161C4×C79—21no—
162C4×C79—21no—
163C4×C79—21no—
164C4×C79#42711yescharacteristic
165C4×C79#42811yescharacteristic
166C4×C79#44311yescharacteristic
167C4×C79#44411yescharacteristic
168C4×C79#44911yescharacteristic
169C4×C79#45211yescharacteristic
170Dic79—41no—
171Dic79—41no—
172Dic79—41no—
173Dic79—82no—
174#8903—791nomaximal, Sylow
175C2×C79—21no—
176C2×C79—21no—
177C2×C79#185411yescharacteristic
178C2×C79#186111yescharacteristic
179C2×C79#186611yescharacteristic
180D79—81no—
181#2925—791no—
182#1806—791no—
183#1570D7911yescharacteristic, complemented
184#1567—791no—
185#1476—791no—
186#1303—791no—
187#1279—791no—
188C79#890311yescharacteristic, complemented, Sylow
189C4×#41—791no—
190#433—1582no—
191#432—1581no—
192#431—1581no—
193#431C2×D7911yescharacteristic
194#423—791no—
195#423C2×D7911yescharacteristic
196C2×C4⋊C8C2×D7911yescharacteristic
197C2×#104—1581no—
198#408—1581no—
199#405—1581no—
200C2×C4×C8—791no—
201C4×C4⋊C4—791no—
202#339—1581no—
203#335—1582no—
204#334—21no—
205#334—21no—
206Q8⋊C8—791no—
207Q8⋊C8—791no—
208D4⋊C8—791no—
209D4⋊C8—791no—
210C2×#41—791no—
211C2×M16—21no—
212C2×M16—1581no—
213C22×C8—21no—
214C22×C8—1581no—
215C4×Q8—791no—
216C4×Q8—791no—
217C4×D4—791no—
218C4×D4—791no—
219C4×D4—1581no—
220C4×D4—1581no—
221#117—791no—
222#117—1581no—
223C2×C4⋊C4—1581no—
224C2×C42—1581no—
225C2×C42—1581no—
226C2×C42C22×D7911yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—792no—
231C4⋊C8—1581no—
232C4⋊C8C4×D7911yescharacteristic
233C4⋊C8C4×D7911yescharacteristic
234C4⋊C8C22⋊D7911yescharacteristic
235C4⋊C8C22⋊D7911yescharacteristic
236#104—3162no—
237C22⋊C8—21no—
238C22⋊C8—1582no—
239C22⋊C8—1582no—
240#97—1582no—
241C4×C8—21no—
242C4×C8—791no—
243C4×C8—791no—
244C4×C8—1582no—
245C4×C8—1581no—
246#95—1581no—
247#41—1581no—
248#41—1581no—
249#41—1581no—
250#41—1581no—
251C2×Q8—791no—
252C2×D4—791no—
253C2×D4—1581no—
254C22×C4—3161no—
255C22×C4—1581no—
256C22×C4—1581no—
257C22×C4—1581no—
258C22×C4D4×D7911yescharacteristic
259C22×C4#1192962411yescharacteristic
260C22×C4#1192962511yescharacteristic
261M16—41no—
262M16—41no—
263M16—3161no—
264M16—3161no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—3162no—
273C2×C8—1581no—
274C2×C8—1582no—
275C2×C8—1581no—
276C4⋊C4—3161no—
277C4⋊C4—791no—
278C4⋊C4—791no—
279C4⋊C4—1582no—
280C4⋊C4—1581no—
281C4⋊C4—1581no—
282C22⋊C4—1581no—
283C22⋊C4—1581no—
284C22⋊C4—1581no—
285C42—3161no—
286C42—3161no—
287C42—1581no—
288C42—1581no—
289C42—1581no—
290C42C2×C4×D7911yescharacteristic
291C42D4×D7911yescharacteristic
292C42C2×C22⋊D7911yescharacteristic
293C42Q8×D7911yescharacteristic
294C23—1581no—
295C23#1457914611yescharacteristic
296Q8—1581no—
297Q8—1581no—
298D4—3161no—
299D4—3161no—
300D4—1581no—
301D4—1581no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—3161no—
306C2×C4—3161no—
307C2×C4—3161no—
308C2×C4—3161no—
309C2×C4—3162no—
310C2×C4—6321no—
311C2×C4—6321no—
312C2×C4—6321no—
313C2×C4—1581no—
314C2×C4—1581no—
315C2×C4—1581no—
316C2×C4C4⋊C4×D7911yescharacteristic
317C2×C4C4×C22⋊D7911yescharacteristic
318C2×C4#1457912311yescharacteristic
319C2×C4#1457912911yescharacteristic
320C2×C4#1457913711yescharacteristic
321C2×C4#1457913911yescharacteristic
322C2×C4#1457918211yescharacteristic
323C2×C4#1457918211yescharacteristic
324C2×C4#1457918811yescharacteristic
325C2×C4#1457919011yescharacteristic
326C8—41no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—81no—
331C8—3161no—
332C8—3162no—
333C8—1582no—
334C22—21no—
335C22—21no—
336C22—3161no—
337C22—3161no—
338C22—3161no—
339C22#1909843411yescharacteristic
340C22#1909888911yescharacteristic
341C22#1909890711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—3161no—
346C4—3161no—
347C4—3161no—
348C4—6322no—
349C4#1909859811yescharacteristic
350C4#1909860211yescharacteristic
351C4#1909867811yescharacteristic
352C4#1909868311yescharacteristic
353C4#1909870711yescharacteristic
354C4#1909872211yescharacteristic
355C2—21no—
356C2—21no—
357C2—6321no—
358C2#2376359811yescharacteristic
359C2#2376363711yescharacteristic
360C2#2376367411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1570C2
socleC22×C79#385
prime radicalC2×#11929624D4
cosocleC2×C42×C79C23
hypercenter#1570D79
hypocenterC79#8903
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
3. #23762158C2semidirect
5. #23761302C2semidirect
6. #23761239C2semidirect
7. C79×#1570C2semidirect
20. #19098293C4semidirect
21. #19098293C4semidirect
22. #19098290C4semidirect
23. #19098290C4semidirect
183. #1570D79semidirect
188. C79#8903semidirect

Maximal subgroups

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

subgroupindex
#237720652
#237634172
#237621582
#237617802
#237613022
#237612392
C79×#15702
#890379