Finite Group Databasebeta

← #34199206

Subgroups — #34199206

Order 20224.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 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#23765196C211yesmaximal, characteristic, complemented
3#23763529C211yesmaximal, 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#19098428—21no—
19Dic79⋊C16—21no—
20C22⋊(C79⋊C16)C2211yescharacteristic
21C16×Dic79—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×C4×Dic79D411yescharacteristic
41C2×C79⋊C16—21no—
42#14579095C2×C411yescharacteristic
43#14579092—21no—
44#14579085—21no—
45#14579083—21no—
46#14579081—21no—
47C8×Dic79C2×C411yescharacteristic
48C8×Dic79C2×C411yescharacteristic
49C8×Dic79D411yescharacteristic
50C8×Dic79D411yescharacteristic
51C79⋊#95—41no—
52C79⋊#95—41no—
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—
77(C2×D79)⋊C4—41no—
78(C2×D79)⋊C4—41no—
79(C2×D79)⋊C4—41no—
80(C2×D79)⋊C4—41no—
81C4×Dic79—41no—
82C4×Dic79C22⋊C411yescharacteristic
83C4×Dic79C22⋊C411yescharacteristic
84C4×Dic79SD1611yescharacteristic
85C4×Dic79SD1611yescharacteristic
86C79⋊C16—41no—
87C2×Q8×C79—21no—
88C22×C4×C79—41no—
89C22×C4×C79C2×D411yescharacteristic
90Q16×C79—41no—
91C2×C8×C79—41no—
92C2×C8×C79C22×C411yescharacteristic
93C2×C8×C79C2×D411yescharacteristic
94C4⋊C4×C79—21no—
95C4⋊C4×C79—41no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C22⋊C4×C79—41no—
99C16×C79—41no—
100C23×C79C2×D811yescharacteristic
101C22×D79—41no—
102Q8×C79—41no—
103Q8×C79—41no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—81no—
109C2×C4×C79—81no—
110C2×C4×C79—81no—
111C2×C4×C79C2×C22⋊C411yescharacteristic
112C2×C4×C79C2×SD1611yescharacteristic
113C22⋊D79—81no—
114C22⋊D79—81no—
115C2×Dic79—21no—
116C2×Dic79—21no—
117C2×Dic79—41no—
118C2×Dic79—41no—
119C2×Dic79—81no—
120C2×Dic79—81no—
121C2×Dic79—81no—
122C2×Dic79#10211yescharacteristic
123C2×Dic79#10211yescharacteristic
124C2×Dic79#10211yescharacteristic
125C2×Dic79#10211yescharacteristic
126D316—41no—
127D316—41no—
128C4×D79—82no—
129C8×C79—21no—
130C8×C79—21no—
131C79⋊C8—21no—
132C79⋊C8—21no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79C2×#10211yescharacteristic
136C22×C79#50811yescharacteristic
137C22×C79#50811yescharacteristic
138C2×D79—81no—
139C2×D79—81no—
140C2×D79—81no—
141C4×C79—21no—
142C4×C79—81no—
143C4×C79—81no—
144C4×C79—161no—
145C4×C79#41011yescharacteristic
146C4×C79#42111yescharacteristic
147Dic79—21no—
148Dic79—21no—
149Dic79—21no—
150Dic79—21no—
151Dic79—81no—
152Dic79—161no—
153#11855—791nomaximal, Sylow
154C2×C79—21no—
155C2×C79—21no—
156C2×C79#151711yescharacteristic
157C2×C79#213811yescharacteristic
158C2×C79#214511yescharacteristic
159D79—161no—
160#3168—791no—
161#2162—791no—
162#1835—791no—
163#1348—791no—
164#1348D7911yescharacteristic, complemented
165#1347—791no—
166#1347—791no—
167C79#1185511yescharacteristic, complemented, Sylow
168#529—1581no—
169#499—791no—
170#495—791no—
171#488—1581no—
172#467C2×D7911yescharacteristic
173#466—791no—
174#465—1581no—
175C4×Q16—791no—
176C4×SD16—1581no—
177C4×D8—791no—
178C2×C4×C8—791no—
179#382—1581no—
180C4⋊C16—1581no—
181C22⋊C16—791no—
182C22⋊C16C2×D7911yescharacteristic
183C4×C16—1581no—
184#340—791no—
185#340—1581no—
186#340C2×D7911yescharacteristic
187C2×Q16C4×D7911yescharacteristic
188C2×SD16—1581no—
189C2×D8—791no—
190C22×C8—791no—
191C22×C8C22×D7911yescharacteristic
192#126—3161no—
193#125—1581no—
194#124—1581no—
195#123—3161no—
196C22⋊Q8—21no—
197#121—1581no—
198C4×Q8—1581no—
199C4×D4—3161no—
200C4×D4—1581no—
201#117—3161no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1581no—
204C2×C42—791no—
205C2×C16—21no—
206C2×C16—1581no—
207#107—791no—
208#107C4×D7911yescharacteristic
209#106—1581no—
210#103—21no—
211#103—1581no—
212#102—1581no—
213#102—1581no—
214C4×C8—791no—
215C4×C8—791no—
216C4×C8—791no—
217C4×C8—791no—
218#95—3161no—
219#95—3161no—
220C2×Q8—21no—
221C2×D4—3161no—
222C2×D4—1581no—
223C22×C4—41no—
224C22×C4—3161no—
225C22×C4—3161no—
226C22×C4—791no—
227C22×C4—791no—
228C22×C4D4×D7911yescharacteristic
229Q16—41no—
230SD16—3162no—
231D8—3161no—
232C2×C8—41no—
233C2×C8—3161no—
234C2×C8—791no—
235C2×C8—791no—
236C2×C8C2×C4×D7911yescharacteristic
237C2×C8D4×D7911yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—3161no—
243C4⋊C4—3161no—
244C4⋊C4—3161no—
245C4⋊C4—3161no—
246C4⋊C4—3161no—
247C4⋊C4—1581no—
248C22⋊C4—41no—
249C22⋊C4—3161no—
250C22⋊C4—3161no—
251C22⋊C4—3161no—
252C22⋊C4—3161no—
253C42—3161no—
254C42—791no—
255C42—791no—
256C42—791no—
257C42—791no—
258C16—41no—
259C16—3161no—
260C23—3161no—
261C23D8×D7911yescharacteristic
262Q8—41no—
263Q8—41no—
264D4—3161no—
265D4—3161no—
266D4—6321no—
267D4—6321no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—3161no—
276C2×C4—3161no—
277C2×C4—791no—
278C2×C4—791no—
279C2×C4—791no—
280C2×C4—791no—
281C2×C4—6322no—
282C2×C4—6321no—
283C2×C4—6321no—
284C2×C4—6321no—
285C2×C4—1581no—
286C2×C4—1581no—
287C2×C4C22⋊C4×D7911yescharacteristic
288C2×C4SD16×D7911yescharacteristic
289C8—21no—
290C8—21no—
291C8—1581no—
292C8—1581no—
293C22—21no—
294C22—21no—
295C22—6321no—
296C22—6321no—
297C22—6321no—
298C22#102×D7911yescharacteristic
299C22#1909899411yescharacteristic
300C22#1909899611yescharacteristic
301C4—21no—
302C4—81no—
303C4—81no—
304C4—161no—
305C4—6321no—
306C4—1581no—
307C4—1581no—
308C4—1581no—
309C4—1581no—
310C4—12641no—
311C4#1909853511yescharacteristic
312C4#1909857811yescharacteristic
313C2—21no—
314C2—21no—
315C2—12641no—
316C2#2376195811yescharacteristic
317C2#2376510611yescharacteristic
318C2#2376513411yescharacteristic
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#11855
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. #23765196C2semidirect
3. #23763529C2semidirect
4. #23761412C2semidirect
7. C79×#1348C2semidirect
9. #19098964C4semidirect
10. #19098948C4semidirect
14. Q16×Dic79C4semidirect
16. #19098645C4semidirect
164. #1348D79semidirect
167. C79#11855semidirect

Maximal subgroups

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

subgroupindex
#237739712
#237651962
#237635292
#237614122
#237614102
#237614082
C79×#13482
#1185579