Finite Group Databasebeta

← #34186014

Subgroups — #34186014

Order 20224.

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

normal subgroups only

This group has 610 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#23766003C211yesmaximal, characteristic, complemented
2#23763480C211yesmaximal, characteristic, complemented
3#23761929C212yesmaximal
4#23761253C212yesmaximal, complemented
5C79×#2501C211yesmaximal, characteristic
6#19098527C2212yes—
7#19098449—24no—
8#19098426C2212yes—
9#19098423—21no—
10C4×C22⋊Dic79—21no—
11#19098332—22no—
12#19098297—24no—
13C2×#123×C79—21no—
14C2×#123×C79—21no—
15C2×#121×C79C2211yescharacteristic
16C22×C4⋊C4×C79C2211yescharacteristic
17#399×C79—22no—
18#390×C79C412yescomplemented
19#384×C79C412yescomplemented
20C4×C22⋊C4×C79C2211yescharacteristic
21C2×#11929608—22no—
22C2×C4⋊Dic79—22no—
23C2×C22⋊Dic79—22no—
24C2×C4×Dic79—22no—
25C22⋊(C79⋊C8)—22no—
26C22⋊(C79⋊C8)—22no—
27C79⋊#95—22no—
28C79⋊#95—22no—
29C79⋊#95—22no—
30C79⋊#95D414yes—
31C22×D4×C79—21no—
32C22×D4×C79C2×C411yescharacteristic
33C23×C4×C79D411yescharacteristic
34C23×C4×C79D411yescharacteristic
35C23×C4×C79C2311yescharacteristic
36#123×C79—44no—
37#123×C79—44no—
38#121×C79—22no—
39#121×C79—22no—
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79C2×C411yescharacteristic
45C2×C4⋊C4×C79C2×C412yes—
46C2×C4⋊C4×C79D412yes—
47C2×C22⋊C4×C79—21no—
48C2×C22⋊C4×C79—22no—
49C2×C22⋊C4×C79—22no—
50C2×C22⋊C4×C79—21no—
51C2×C42×C79C2×C412yes—
52#95×C79—21no—
53C22×Dic79—24no—
54#11929608—44no—
55C2×C79⋊C8—42no—
56C4⋊Dic79—82no—
57C22⋊Dic79—44no—
58C4×Dic79—82no—
59C24×C79—21no—
60C24×C79C2×D411yescharacteristic
61C2×D4×C79—22no—
62C2×D4×C79—42no—
63C2×D4×C79—42no—
64C2×D4×C79—42no—
65C2×D4×C79—41no—
66C2×D4×C79—41no—
67C22×C4×C79—22no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—42no—
74C22×C4×C79—42no—
75C22×C4×C79—42no—
76C22×C4×C79C22⋊C412yes—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C22×C4×C79C2×D412yes—
81C22×C4×C79#4112yes—
82C4⋊C4×C79—24no—
83C4⋊C4×C79—24no—
84C4⋊C4×C79—22no—
85C4⋊C4×C79—44no—
86C4⋊C4×C79—44no—
87C22⋊C4×C79—24no—
88C22⋊C4×C79—44no—
89C22⋊C4×C79—44no—
90C22⋊C4×C79—42no—
91C42×C79—42no—
92C23×C79—22no—
93C23×C79—22no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79—42no—
98C23×C79—41no—
99C23×C79—42no—
100C23×C79C2×C22⋊C411yescharacteristic
101C23×C79#12011yescharacteristic
102C23×C79#12311yescharacteristic
103D4×C79—82no—
104D4×C79—82no—
105D4×C79—82no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—24no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
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—41no—
122C2×C4×C79—81no—
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—82no—
139C22×C79—82no—
140C22×C79—81no—
141C22×C79—81no—
142C22×C79—81no—
143C22×C79#38611yescharacteristic
144C22×C79#41012yes—
145C22×C79#45312yes—
146C4×C79—22no—
147C4×C79—22no—
148C4×C79—44no—
149C4×C79—42no—
150C4×C79—42no—
151C4×C79—81no—
152Dic79—84no—
153#9464—791nomaximal, Sylow
154C2×C79—22no—
155C2×C79—22no—
156C2×C79—41no—
157C2×C79—161no—
158C2×C79#188711yescharacteristic
159C2×C79#189911yescharacteristic
160C2×C79#191211yescharacteristic
161#2501D7911yescharacteristic
162#2313—791no—
163#1821—791no—
164#1510—792no—
165#1284—792no—
166C79#946411yescharacteristic, complemented, Sylow
167C2×#123—21no—
168C2×#123—21no—
169C2×#121C2×D7911yescharacteristic
170C22×C4⋊C4C2×D7911yescharacteristic
171#407—792no—
172#399—22no—
173#390Dic7912yescomplemented
174#387—1584no—
175#384Dic7912yescomplemented
176#381—792no—
177#380—1581no—
178C4×C22⋊C4—1581no—
179C4×C22⋊C4C2×D7911yescharacteristic
180#343—1582no—
181#327—1584no—
182C22×D4—21no—
183C22×D4C2×Dic7911yescharacteristic
184C23×C4C22⋊D7911yescharacteristic
185C23×C4C22⋊D7911yescharacteristic
186C23×C4C22×D7911yescharacteristic
187C2×M16—1582no—
188#123—44no—
189#123—44no—
190#121—22no—
191#121—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—1582no—
197C2×C4⋊C4C2×Dic7911yescharacteristic
198C2×C4⋊C4C2×Dic7912yes—
199C2×C4⋊C4C22⋊D7912yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—1582no—
205C2×C42—1582no—
206C2×C42C2×Dic7912yes—
207C22⋊C8—1582no—
208C22⋊C8—1582no—
209#95—21no—
210#95—794no—
211#95—1582no—
212#95—1582no—
213#95—1582no—
214C24—21no—
215C24C2×C22⋊D7911yescharacteristic
216C2×D4—22no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—1584no—
232C22×C4C22⋊Dic7912yes—
233C22×C4C22⋊Dic7912yes—
234C22×C4C22×Dic7911yescharacteristic
235C22×C4D4×D7912yes—
236C22×C4C2×C22⋊D7911yescharacteristic
237C22×C4#1192962312yes—
238M16—3164no—
239C2×C8—3162no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—6322no—
246C22⋊C4—24no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—42no—
250C22⋊C4—3164no—
251C42—42no—
252C42—6322no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23C2×C22⋊Dic7911yescharacteristic
262C23#1457912911yescharacteristic
263C23#1457914211yescharacteristic
264D4—82no—
265D4—82no—
266D4—82no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—3164no—
286C2×C4—3164no—
287C2×C4—3164no—
288C2×C4D4×Dic7912yes—
289C2×C4#1457913312yes—
290C8—6322no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—82no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22#1909844311yescharacteristic
305C22#1909853712yes—
306C22#1909873012yes—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—42no—
312C4—81no—
313C4—6324no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—161no—
318C2#2376379911yescharacteristic
319C2#2376385911yescharacteristic
320C2#2376393811yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2501C2
socleC22×C79#386
prime radicalC2×#121×C79C22
cosocleC23×C4×C79C23
hypercenter#2501D79
hypocenterC79#9464
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
1. #23766003C2semidirect
2. #23763480C2semidirect
4. #23761253C2semidirect
18. #390×C79C4semidirect
19. #384×C79C4semidirect
166. C79#9464semidirect
173. #390Dic79semidirect
175. #384Dic79semidirect

Maximal subgroups

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

subgroupindex
#237660032
#237634802
#237619292
#237612532
C79×#25012
#946479