Finite Group Databasebeta

← #34185725

Subgroups — #34185725

Order 20224.

606 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 107 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#23765998C211yesmaximal, characteristic
2#23761920C212yesmaximal, complemented
3#23761727C211yesmaximal, characteristic
4#23761286C212yesmaximal, complemented
5C79×#2561C211yesmaximal, characteristic
6#19098527C2212yes—
7#19098456—24no—
8#19098423—21no—
9#19098419—21no—
10C4×C22⋊Dic79C2212yes—
11#19098300—24no—
12#19098287—22no—
13C2×C22⋊Q8×C79C2211yescharacteristic
14C22×C22⋊C4×C79C2211yescharacteristic
15#397×C79C412yescomplemented
16#393×C79C412yescomplemented
17#388×C79—22no—
18#381×C79—21no—
19#379×C79—21no—
20C4×C22⋊C4×C79C2211yescharacteristic
21C2×#11929608—22no—
22C2×C4⋊Dic79—22no—
23C2×C22⋊Dic79—22no—
24C2×C22⋊Dic79—22no—
25C2×C22⋊Dic79—22no—
26C2×C4×Dic79—22no—
27C22⋊(C79⋊C8)—22no—
28C22⋊(C79⋊C8)—22no—
29C79⋊#95—22no—
30C79⋊#95D414yes—
31C25×C79D411yescharacteristic
32C22×Q8×C79C2×C411yescharacteristic
33C23×C4×C79D411yescharacteristic
34C23×C4×C79C2311yescharacteristic
35C22⋊Q8×C79—22no—
36C22⋊Q8×C79—22no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79C2×C411yescharacteristic
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79—22no—
42C2×C22⋊C4×C79—21no—
43C2×C22⋊C4×C79—21no—
44C2×C22⋊C4×C79—22no—
45C2×C22⋊C4×C79C2×C412yes—
46C2×C22⋊C4×C79D412yes—
47C2×C42×C79—21no—
48#95×C79—22no—
49#95×C79—22no—
50#95×C79C2×C412yes—
51C22×Dic79—24no—
52#11929608—82no—
53C2×C79⋊C8—42no—
54C4⋊Dic79—82no—
55C22⋊Dic79—44no—
56C22⋊Dic79—82no—
57C22⋊Dic79—82no—
58C4×Dic79—82no—
59C24×C79—21no—
60C24×C79—21no—
61C24×C79—42no—
62C24×C79—42no—
63C24×C79—81no—
64C24×C79C22⋊C412yes—
65C24×C79C2×D411yescharacteristic
66C2×Q8×C79—22no—
67C2×Q8×C79—81no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—22no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—42no—
74C22×C4×C79—81no—
75C22×C4×C79C22⋊C412yes—
76C22×C4×C79C22×C411yescharacteristic
77C22×C4×C79C2×D412yes—
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79#4112yes—
80C4⋊C4×C79—22no—
81C4⋊C4×C79—42no—
82C22⋊C4×C79—24no—
83C22⋊C4×C79—24no—
84C22⋊C4×C79—42no—
85C22⋊C4×C79—44no—
86C22⋊C4×C79—41no—
87C22⋊C4×C79—41no—
88C22⋊C4×C79—82no—
89C42×C79—81no—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79—22no—
93C23×C79—22no—
94C23×C79—22no—
95C23×C79—41no—
96C23×C79—42no—
97C23×C79—42no—
98C23×C79—42no—
99C23×C79—42no—
100C23×C79—42no—
101C23×C79—42no—
102C23×C79—81no—
103C23×C79—81no—
104C23×C79—88no—
105C23×C79C2×C22⋊C411yescharacteristic
106C23×C79#12011yescharacteristic
107C23×C79#12311yescharacteristic
108Q8×C79—82no—
109C2×C4×C79—22no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—42no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—42no—
122C2×C4×C79—42no—
123C2×C4×C79—81no—
124C2×C4×C79C4×D412yes—
125C2×C4×C79#12112yes—
126C2×Dic79—44no—
127C2×Dic79—44no—
128C2×Dic79—44no—
129C79⋊C8—82no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—44no—
138C22×C79—44no—
139C22×C79—42no—
140C22×C79—42no—
141C22×C79—42no—
142C22×C79—82no—
143C22×C79—84no—
144C22×C79—84no—
145C22×C79#38611yescharacteristic
146C22×C79#42112yes—
147C22×C79#45512yes—
148C4×C79—22no—
149C4×C79—22no—
150C4×C79—42no—
151C4×C79—82no—
152C4×C79—81no—
153C4×C79—81no—
154Dic79—84no—
155#9411—791nomaximal, Sylow
156C2×C79—22no—
157C2×C79—22no—
158C2×C79—41no—
159C2×C79—82no—
160C2×C79#188511yescharacteristic
161C2×C79#189711yescharacteristic
162C2×C79#191311yescharacteristic
163#2561D7911yescharacteristic
164#2312—791no—
165#1508—792no—
166#1462—791no—
167#1297—792no—
168C79#941111yescharacteristic, complemented, Sylow
169C2×C22⋊Q8C2×D7911yescharacteristic
170C22×C22⋊C4C2×D7911yescharacteristic
171#407—792no—
172#397Dic7912yescomplemented
173#393Dic7912yescomplemented
174#388—22no—
175#388—1584no—
176#381—21no—
177#380—1581no—
178#379—21no—
179#379—1581no—
180C4×C22⋊C4—792no—
181C4×C22⋊C4C2×D7911yescharacteristic
182#328—1584no—
183#323—1582no—
184C25C22⋊D7911yescharacteristic
185C22×Q8C2×Dic7911yescharacteristic
186C23×C4C22⋊D7911yescharacteristic
187C23×C4C22×D7911yescharacteristic
188C2×M16—1582no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—1582no—
193C2×C4⋊C4C2×Dic7911yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1582no—
201C2×C22⋊C4—1582no—
202C2×C22⋊C4—1582no—
203C2×C22⋊C4C2×Dic7912yes—
204C2×C22⋊C4C22⋊D7912yes—
205C2×C42—21no—
206C2×C42—1582no—
207C22⋊C8—1582no—
208C22⋊C8—1582no—
209#95—22no—
210#95—22no—
211#95—794no—
212#95—1582no—
213#95C2×Dic7912yes—
214C24—21no—
215C24—21no—
216C24—42no—
217C24—42no—
218C24—81no—
219C24C22⋊Dic7912yes—
220C24C2×C22⋊D7911yescharacteristic
221C2×Q8—22no—
222C2×Q8—81no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—1584no—
231C22×C4C22⋊Dic7912yes—
232C22×C4C22×Dic7911yescharacteristic
233C22×C4D4×D7912yes—
234C22×C4C2×C22⋊D7911yescharacteristic
235C22×C4#1192962312yes—
236M16—6322no—
237C2×C8—3162no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—6322no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—42no—
244C22⋊C4—44no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—82no—
248C22⋊C4—3164no—
249C22⋊C4—6322no—
250C22⋊C4—6322no—
251C42—81no—
252C42—6322no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—88no—
268C23C2×C22⋊Dic7911yescharacteristic
269C23#1457912911yescharacteristic
270C23#1457914211yescharacteristic
271Q8—82no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—3164no—
288C2×C4—3164no—
289C2×C4—3164no—
290C2×C4D4×Dic7912yes—
291C2×C4#1457913312yes—
292C8—6322no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—84no—
307C22—84no—
308C22#1909844311yescharacteristic
309C22#1909858012yes—
310C22#1909874012yes—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—82no—
315C4—81no—
316C4—81no—
317C4—6324no—
318C2—22no—
319C2—22no—
320C2—41no—
321C2—82no—
322C2#2376378911yescharacteristic
323C2#2376384911yescharacteristic
324C2#2376394311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2561C2
socleC22×C79#386
prime radicalC25×C79D4
cosocleC23×C4×C79C23
hypercenter#2561D79
hypocenterC79#9411
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
2. #23761920C2semidirect
4. #23761286C2semidirect
15. #397×C79C4semidirect
16. #393×C79C4semidirect
168. C79#9411semidirect
172. #397Dic79semidirect
173. #393Dic79semidirect

Maximal subgroups

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

subgroupindex
#237659982
#237619202
#237617272
#237612862
C79×#25612
#941179