Finite Group Databasebeta

← #34199227

Subgroups — #34199227

Order 20224.

336 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773971C211yesmaximal, characteristic
2#23765200C211yesmaximal, characteristic, complemented
3#23763526C211yesmaximal, characteristic, complemented
4#23761412C211yesmaximal, characteristic, complemented
5#23761410C211yesmaximal, characteristic
6#23761408C211yesmaximal, characteristic
7C79×#1348C211yesmaximal, characteristic, complemented
8#19099099—21no—
9#19098964C411yescharacteristic, complemented
10#19098948C411yescharacteristic, complemented
11#19098919—21no—
12#19098799C2211yescharacteristic
13#19098794—21no—
14Q16×Dic79C411yescharacteristic, complemented
15#19098648—21no—
16#19098645C411yescharacteristic, complemented
17C2×C8×Dic79C2211yescharacteristic
18#19098432—21no—
19Dic79⋊C16—21no—
20C22⋊(C79⋊C16)C2211yescharacteristic
21#19098338—21no—
22#19098326C2211yescharacteristic
23#19098325—21no—
24#467×C79C2211yescharacteristic
25C22⋊C16×C79C2211yescharacteristic
26#340×C79C2211yescharacteristic
27C2×#11929611—21no—
28C2×D632C2×C411yescharacteristic
29C22×C79⋊C8D411yescharacteristic
30#14579156—41no—
31#14579152—21no—
32#14579150—21no—
33#14579145—41no—
34#14579135—21no—
35Q8×Dic79—21no—
36#14579123—21no—
37(C22⋊D79)⋊C4—41no—
38#14579119—41no—
39C2×C4⋊Dic79—21no—
40C2×Dic79⋊C4—41no—
41C2×Dic79⋊C4—41no—
42C2×C4×Dic79D411yescharacteristic
43C2×C79⋊C16—21no—
44#14579095C2×C411yescharacteristic
45#14579092—21no—
46#14579085—21no—
47#14579083—21no—
48#14579081—21no—
49C8×Dic79C2×C411yescharacteristic
50C8×Dic79C2×C411yescharacteristic
51C8×Dic79D411yescharacteristic
52C8×Dic79D411yescharacteristic
53C2×Q16×C79C2×C411yescharacteristic
54C22×C8×C79C2311yescharacteristic
55C22⋊Q8×C79—21no—
56C2×C4⋊C4×C79—21no—
57C2×C16×C79—21no—
58#107×C79C2×C411yescharacteristic
59#103×C79—21no—
60C2×D316—21no—
61C2×C22⋊D79—41no—
62C22×Dic79—41no—
63C22×Dic79D811yescharacteristic
64C22×Dic79D811yescharacteristic
65C2×C4×D79—41no—
66#11929611—42no—
67D632—41no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8C22⋊C411yescharacteristic
70C2×C79⋊C8C22⋊C411yescharacteristic
71C4⋊Dic79—21no—
72C4⋊Dic79—41no—
73C4⋊Dic79—41no—
74C4⋊Dic79—41no—
75C4⋊Dic79—41no—
76Dic79⋊C4—41no—
77Dic79⋊C4—82no—
78Dic79⋊C4—82no—
79(C2×D79)⋊C4—41no—
80(C2×D79)⋊C4—41no—
81(C2×D79)⋊C4—41no—
82(C2×D79)⋊C4—41no—
83C4×Dic79—41no—
84C4×Dic79C22⋊C411yescharacteristic
85C4×Dic79C22⋊C411yescharacteristic
86C4×Dic79SD1611yescharacteristic
87C4×Dic79SD1611yescharacteristic
88C79⋊C16—41no—
89C2×Q8×C79—21no—
90C22×C4×C79—41no—
91C22×C4×C79C2×D411yescharacteristic
92Q16×C79—41no—
93C2×C8×C79—41no—
94C2×C8×C79C22×C411yescharacteristic
95C2×C8×C79C2×D411yescharacteristic
96C4⋊C4×C79—21no—
97C4⋊C4×C79—41no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C22⋊C4×C79—41no—
101C16×C79—41no—
102C23×C79C2×D811yescharacteristic
103C22×D79—41no—
104Q8×C79—41no—
105Q8×C79—41no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—81no—
111C2×C4×C79—81no—
112C2×C4×C79—81no—
113C2×C4×C79C2×C22⋊C411yescharacteristic
114C2×C4×C79C2×SD1611yescharacteristic
115C22⋊D79—81no—
116C22⋊D79—81no—
117C2×Dic79—21no—
118C2×Dic79—21no—
119C2×Dic79—22no—
120C2×Dic79—41no—
121C2×Dic79—81no—
122C2×Dic79—81no—
123C2×Dic79—81no—
124C2×Dic79#10211yescharacteristic
125C2×Dic79#10211yescharacteristic
126C2×Dic79#10211yescharacteristic
127C2×Dic79#10211yescharacteristic
128D316—41no—
129D316—41no—
130C4×D79—82no—
131C8×C79—21no—
132C8×C79—21no—
133C79⋊C8—21no—
134C79⋊C8—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79C2×#10211yescharacteristic
138C22×C79#50911yescharacteristic
139C22×C79#51011yescharacteristic
140C2×D79—81no—
141C2×D79—81no—
142C2×D79—81no—
143C4×C79—21no—
144C4×C79—81no—
145C4×C79—81no—
146C4×C79—161no—
147C4×C79#41011yescharacteristic
148C4×C79#42111yescharacteristic
149Dic79—21no—
150Dic79—21no—
151Dic79—21no—
152Dic79—81no—
153Dic79—161no—
154Dic79#36012yes—
155#11860—791nomaximal, Sylow
156C2×C79—21no—
157C2×C79—21no—
158C2×C79#151711yescharacteristic
159C2×C79#213911yescharacteristic
160C2×C79C2×#36011yescharacteristic
161D79—161no—
162#3168—791no—
163#2163—791no—
164#1834—791no—
165#1348—791no—
166#1348D7911yescharacteristic, complemented
167#1347—791no—
168#1347—791no—
169C79#1186011yescharacteristic, complemented, Sylow
170#529—1581no—
171#499—791no—
172#495—791no—
173#488—1581no—
174#467C2×D7911yescharacteristic
175#466—791no—
176#465—1581no—
177C4×Q16—791no—
178C4×SD16—1581no—
179C4×D8—791no—
180C2×C4×C8—791no—
181#384—1581no—
182C4⋊C16—1581no—
183C22⋊C16—791no—
184C22⋊C16C2×D7911yescharacteristic
185#346—1581no—
186#340—791no—
187#340—1581no—
188#340C2×D7911yescharacteristic
189C2×Q16C4×D7911yescharacteristic
190C2×SD16—1581no—
191C2×D8—791no—
192C22×C8—791no—
193C22×C8C22×D7911yescharacteristic
194#126—3161no—
195#125—1581no—
196#124—1581no—
197#123—3161no—
198C22⋊Q8—21no—
199#121—1581no—
200C4×Q8—1581no—
201C4×D4—3161no—
202C4×D4—1581no—
203#117—3161no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—3161no—
206C2×C4⋊C4—3161no—
207C2×C4⋊C4—1581no—
208C2×C42—791no—
209C2×C16—21no—
210C2×C16—1581no—
211#107—791no—
212#107C4×D7911yescharacteristic
213#106—1581no—
214#103—21no—
215#103—1581no—
216#102—1581no—
217#102—1581no—
218C4×C8—791no—
219C4×C8—791no—
220C4×C8—791no—
221C4×C8—791no—
222C2×Q8—21no—
223C2×D4—3161no—
224C2×D4—1581no—
225C22×C4—41no—
226C22×C4—3161no—
227C22×C4—3161no—
228C22×C4—791no—
229C22×C4—791no—
230C22×C4D4×D7911yescharacteristic
231Q16—41no—
232SD16—3162no—
233D8—3161no—
234C2×C8—41no—
235C2×C8—3161no—
236C2×C8—791no—
237C2×C8—791no—
238C2×C8C2×C4×D7911yescharacteristic
239C2×C8D4×D7911yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—3161no—
245C4⋊C4—3161no—
246C4⋊C4—3161no—
247C4⋊C4—3161no—
248C4⋊C4—3161no—
249C4⋊C4—6322no—
250C4⋊C4—6322no—
251C4⋊C4—1581no—
252C22⋊C4—41no—
253C22⋊C4—3161no—
254C22⋊C4—3161no—
255C22⋊C4—3161no—
256C22⋊C4—3161no—
257C42—3161no—
258C42—791no—
259C42—791no—
260C42—791no—
261C42—791no—
262C16—41no—
263C16—3161no—
264C23—3161no—
265C23D8×D7911yescharacteristic
266Q8—41no—
267Q8—41no—
268D4—3161no—
269D4—3161no—
270D4—6321no—
271D4—6321no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—3161no—
280C2×C4—791no—
281C2×C4—791no—
282C2×C4—791no—
283C2×C4—791no—
284C2×C4—6322no—
285C2×C4—6321no—
286C2×C4—6321no—
287C2×C4—6321no—
288C2×C4—1582no—
289C2×C4—1581no—
290C2×C4—1581no—
291C2×C4C22⋊C4×D7911yescharacteristic
292C2×C4SD16×D7911yescharacteristic
293C8—21no—
294C8—21no—
295C8—1581no—
296C8—1581no—
297C22—21no—
298C22—21no—
299C22—6321no—
300C22—6321no—
301C22—6321no—
302C22#102×D7911yescharacteristic
303C22#1909900011yescharacteristic
304C22#1909900511yescharacteristic
305C4—21no—
306C4—81no—
307C4—81no—
308C4—161no—
309C4—792no—
310C4—6321no—
311C4—1581no—
312C4—1581no—
313C4—1581no—
314C4—12641no—
315C4#1909853511yescharacteristic
316C4#1909857811yescharacteristic
317C2—21no—
318C2—21no—
319C2—12641no—
320C2#2376195811yescharacteristic
321C2#2376511111yescharacteristic
322C2#360×D7911yescharacteristic
subgroupG/H
centerC22#102×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC79×#1348C2
socleC22×C79C2×#102
prime radical#19098799C22
cosocleC22×C8×C79C23
hypercenter#1348D79
hypocenterC79#11860
LCSG ▷ C2×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #23765200C2semidirect
3. #23763526C2semidirect
4. #23761412C2semidirect
7. C79×#1348C2semidirect
9. #19098964C4semidirect
10. #19098948C4semidirect
14. Q16×Dic79C4semidirect
16. #19098645C4semidirect
166. #1348D79semidirect
169. C79#11860semidirect

Maximal subgroups

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

subgroupindex
#237739712
#237652002
#237635262
#237614122
#237614102
#237614082
C79×#13482
#1186079