Finite Group Databasebeta

← #34186078

Subgroups — #34186078

Order 20224.

734 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 111 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#23767674C211yesmaximal, characteristic
2#23763500C212yesmaximal, complemented
3#23761727C211yesmaximal, characteristic
4#23761253C212yesmaximal, complemented
5C79×#2273C211yesmaximal, characteristic
6C2×#14579120C2211yescharacteristic
7#19098527C2212yes—
8#19098499C412yescomplemented
9#19098488C412yescomplemented
10#19098451—22no—
11#19098436—21no—
12#19098426C2211yescharacteristic
13#19098419—21no—
14#19098323—24no—
15#19098287—22no—
16C22×C22⋊C4×C79—21no—
17C22×C22⋊C4×C79C2211yescharacteristic
18C4×C22⋊C4×C79—21no—
19C4×C22⋊C4×C79—24no—
20C4×C22⋊C4×C79C2212yes—
21C2×#11929608—22no—
22#14579120—22no—
23#14579120—22no—
24C2×C4⋊Dic79C2×C411yescharacteristic, complemented
25C2×C4⋊Dic79C2×C411yescharacteristic, complemented
26C2×Dic79⋊C4—22no—
27C2×C22⋊Dic79—22no—
28C2×C22⋊Dic79—21no—
29C2×C22⋊Dic79—21no—
30C2×C22⋊Dic79—22no—
31C2×C4×Dic79—21no—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)—22no—
34C79⋊#95—22no—
35C79⋊#95—21no—
36C79⋊#95C2×C412yescomplemented
37C25×C79D411yescharacteristic
38C23×C4×C79—22no—
39C23×C4×C79D411yescharacteristic
40C23×C4×C79C2311yescharacteristic
41C2×C22⋊C4×C79—28no—
42C2×C22⋊C4×C79—22no—
43C2×C22⋊C4×C79—22no—
44C2×C22⋊C4×C79—22no—
45C2×C22⋊C4×C79—22no—
46C2×C22⋊C4×C79C2×C412yes—
47C2×C22⋊C4×C79D412yes—
48C2×C42×C79—22no—
49C2×C42×C79D414yes—
50#95×C79—22no—
51#95×C79—22no—
52C22×Dic79—22no—
53C22×Dic79—21no—
54C22×Dic79—21no—
55#11929608—44no—
56C2×C79⋊C8—42no—
57C4⋊Dic79—22no—
58C4⋊Dic79—22no—
59Dic79⋊C4—82no—
60C22⋊Dic79—42no—
61C22⋊Dic79—44no—
62C22⋊Dic79—41no—
63C22⋊Dic79—41no—
64C22⋊Dic79—82no—
65C4×Dic79—42no—
66C24×C79—21no—
67C24×C79—24no—
68C24×C79—21no—
69C24×C79—42no—
70C24×C79—81no—
71C24×C79C22⋊C412yes—
72C24×C79C2×D411yescharacteristic
73C22×C4×C79—21no—
74C22×C4×C79—24no—
75C22×C4×C79—28no—
76C22×C4×C79—21no—
77C22×C4×C79—42no—
78C22×C4×C79—81no—
79C22×C4×C79—82no—
80C22×C4×C79—82no—
81C22×C4×C79C22⋊C412yes—
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D412yes—
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79#4112yes—
86C22⋊C4×C79—28no—
87C22⋊C4×C79—28no—
88C22⋊C4×C79—24no—
89C22⋊C4×C79—24no—
90C42×C79—44no—
91C42×C79—82no—
92C23×C79—22no—
93C23×C79—22no—
94C23×C79—24no—
95C23×C79—22no—
96C23×C79—22no—
97C23×C79—22no—
98C23×C79—22no—
99C23×C79—41no—
100C23×C79—42no—
101C23×C79—42no—
102C23×C79—42no—
103C23×C79—42no—
104C23×C79—81no—
105C23×C79—81no—
106C23×C79—88no—
107C23×C79#10414yes—
108C23×C79C2×C22⋊C411yescharacteristic
109C23×C79#12011yescharacteristic
110C23×C79#12311yescharacteristic
111C2×C4×C79—24no—
112C2×C4×C79—28no—
113C2×C4×C79—22no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—42no—
117C2×C4×C79—44no—
118C2×C4×C79—42no—
119C2×C4×C79—44no—
120C2×C4×C79—42no—
121C2×C4×C79—81no—
122C2×C4×C79—88no—
123C2×C4×C79C4×D412yes—
124C2×C4×C79#12112yes—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—42no—
131C2×Dic79—42no—
132C79⋊C8—82no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—24no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—44no—
142C22×C79—44no—
143C22×C79—44no—
144C22×C79—44no—
145C22×C79—42no—
146C22×C79—42no—
147C22×C79—82no—
148C22×C79—84no—
149C22×C79#38611yescharacteristic
150C22×C79C2×#10412yes—
151C22×C79#45812yes—
152C4×C79—22no—
153C4×C79—22no—
154C4×C79—48no—
155C4×C79—42no—
156Dic79—82no—
157Dic79—81no—
158Dic79—81no—
159#9477—791nomaximal, Sylow
160C2×C79—22no—
161C2×C79—22no—
162C2×C79—41no—
163C2×C79—44no—
164C2×C79#189011yescharacteristic
165C2×C79#189611yescharacteristic
166C2×C79#191111yescharacteristic
167#2482—791no—
168#2273D7911yescharacteristic
169#1826—792no—
170#1462—791no—
171#1284—792no—
172C79#947711yescharacteristic, complemented, Sylow
173C2×#117—791no—
174C22×C22⋊C4—21no—
175C22×C22⋊C4C2×D7911yescharacteristic
176#407—792no—
177#399—792no—
178#397—792no—
179#388—1582no—
180#385—1581no—
181#381—791no—
182#379—1581no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—24no—
185C4×C22⋊C4C2×D7912yes—
186#339—1584no—
187#323—1582no—
188C25C22⋊D7911yescharacteristic
189C23×C4—22no—
190C23×C4C22⋊D7911yescharacteristic
191C23×C4C22×D7911yescharacteristic
192C2×M16—1582no—
193#117—1582no—
194#117—1582no—
195C2×C4⋊C4—791no—
196C2×C4⋊C4—791no—
197C2×C4⋊C4—1582no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—1582no—
204C2×C22⋊C4—1581no—
205C2×C22⋊C4—1581no—
206C2×C22⋊C4—1582no—
207C2×C22⋊C4C4×D7912yes—
208C2×C22⋊C4D31612yes—
209C2×C42—22no—
210C2×C42—1581no—
211C2×C42C22⋊D7914yes—
212C22⋊C8—1582no—
213C22⋊C8—1582no—
214#95—22no—
215#95—22no—
216#95—792no—
217#95—1582no—
218#95—1581no—
219C24—21no—
220C24—24no—
221C24—21no—
222C24—42no—
223C24—81no—
224C24(C2×D79)⋊C412yes—
225C24C2×C22⋊D7911yescharacteristic
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—81no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—1582no—
235C22×C4—1581no—
236C22×C4—1581no—
237C22×C4(C2×D79)⋊C412yes—
238C22×C4C2×C4×D7911yescharacteristic
239C22×C4C2×C22⋊D7912yes—
240C22×C4C2×D31611yescharacteristic
241C22×C4#1192962412yes—
242M16—3164no—
243C2×C8—3162no—
244C4⋊C4—6322no—
245C4⋊C4—1582no—
246C4⋊C4—1582no—
247C22⋊C4—28no—
248C22⋊C4—28no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—3162no—
252C22⋊C4—3164no—
253C22⋊C4—3161no—
254C22⋊C4—3161no—
255C22⋊C4—6322no—
256C42—44no—
257C42—82no—
258C42—3162no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—81no—
272C23—81no—
273C23—88no—
274C23#1457908914yes—
275C23C2×(C2×D79)⋊C411yescharacteristic
276C23#1457913011yescharacteristic
277C23#1457914411yescharacteristic
278C2×C4—24no—
279C2×C4—28no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—88no—
290C2×C4—3162no—
291C2×C4—3162no—
292C2×C4—3162no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—3162no—
296C2×C4—3162no—
297C2×C4C4×C22⋊D7912yes—
298C2×C4#1457913512yes—
299C8—6322no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22#1909844211yescharacteristic
317C22C2×#1457908912yes—
318C22#1909875312yes—
319C4—22no—
320C4—22no—
321C4—48no—
322C4—42no—
323C4—6322no—
324C4—6321no—
325C4—6321no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2#2376381311yescharacteristic
331C2#2376384311yescharacteristic
332C2#2376393211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2273C2
socleC22×C79#386
prime radicalC25×C79D4
cosocleC23×C4×C79C23
hypercenter#2273D79
hypocenterC79#9477
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2273

Complements

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

NH ≅ G/Ntype
2. #23763500C2semidirect
4. #23761253C2semidirect
8. #19098499C4semidirect
9. #19098488C4semidirect
24. C2×C4⋊Dic79C2×C4semidirect
25. C2×C4⋊Dic79C2×C4semidirect
36. C79⋊#95C2×C4semidirect
172. C79#9477semidirect

Maximal subgroups

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

subgroupindex
#237676742
#237635002
#237617272
#237612532
C79×#22732
#947779