Finite Group Databasebeta

← #34186019

Subgroups — #34186019

Order 20224.

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

normal subgroups only

This group has 554 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×#2504C211yesmaximal, characteristic
6#19098527C2212yes—
7#19098449—24no—
8#19098426C2212yes—
9#19098423—21no—
10C4×C22⋊Dic79—21no—
11#19098332—22no—
12#19098297—24no—
13C2×#121×C79C2211yescharacteristic
14C22×C4⋊C4×C79C2211yescharacteristic
15#397×C79—22no—
16#390×C79C412yescomplemented
17#386×C79—21no—
18#386×C79—21no—
19#382×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#121×C79—22no—
37#121×C79—22no—
38C2×C4⋊C4×C79—22no—
39C2×C4⋊C4×C79—22no—
40C2×C4⋊C4×C79C2×C411yescharacteristic
41C2×C4⋊C4×C79C2×C412yes—
42C2×C4⋊C4×C79D412yes—
43C2×C22⋊C4×C79—21no—
44C2×C22⋊C4×C79—21no—
45C2×C22⋊C4×C79—22no—
46C2×C22⋊C4×C79—22no—
47C2×C42×C79C2×C412yes—
48#95×C79—22no—
49#95×C79—22no—
50#95×C79—21no—
51C22×Dic79—24no—
52#11929608—44no—
53C2×C79⋊C8—42no—
54C4⋊Dic79—82no—
55C22⋊Dic79—44no—
56C4×Dic79—82no—
57C24×C79—21no—
58C24×C79C2×D411yescharacteristic
59C2×D4×C79—22no—
60C2×D4×C79—42no—
61C2×D4×C79—44no—
62C2×D4×C79—41no—
63C2×D4×C79—41no—
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—22no—
70C22×C4×C79—42no—
71C22×C4×C79—81no—
72C22×C4×C79—81no—
73C22×C4×C79C22⋊C412yes—
74C22×C4×C79C22⋊C412yes—
75C22×C4×C79C22×C411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77C22×C4×C79C2×D412yes—
78C22×C4×C79#4112yes—
79C4⋊C4×C79—24no—
80C4⋊C4×C79—24no—
81C4⋊C4×C79—22no—
82C22⋊C4×C79—42no—
83C22⋊C4×C79—42no—
84C22⋊C4×C79—82no—
85C22⋊C4×C79—82no—
86C42×C79—42no—
87C23×C79—22no—
88C23×C79—22no—
89C23×C79—41no—
90C23×C79—42no—
91C23×C79—42no—
92C23×C79—41no—
93C23×C79—42no—
94C23×C79C2×C22⋊C411yescharacteristic
95C23×C79#12011yescharacteristic
96C23×C79#12311yescharacteristic
97D4×C79—82no—
98D4×C79—82no—
99D4×C79—82no—
100C2×C4×C79—22no—
101C2×C4×C79—22no—
102C2×C4×C79—22no—
103C2×C4×C79—22no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—41no—
114C2×C4×C79—42no—
115C2×C4×C79—81no—
116C2×C4×C79—81no—
117C2×C4×C79C4×D412yes—
118C2×C4×C79#12112yes—
119C2×Dic79—44no—
120C2×Dic79—44no—
121C2×Dic79—44no—
122C79⋊C8—82no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—21no—
127C22×C79—21no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—44no—
131C22×C79—84no—
132C22×C79—81no—
133C22×C79—81no—
134C22×C79—81no—
135C22×C79#38611yescharacteristic
136C22×C79#41012yes—
137C22×C79#45412yes—
138C4×C79—22no—
139C4×C79—22no—
140C4×C79—42no—
141C4×C79—42no—
142C4×C79—82no—
143C4×C79—81no—
144Dic79—84no—
145#9465—791nomaximal, Sylow
146C2×C79—22no—
147C2×C79—22no—
148C2×C79—41no—
149C2×C79—161no—
150C2×C79#188811yescharacteristic
151C2×C79#189911yescharacteristic
152C2×C79#191111yescharacteristic
153#2504D7911yescharacteristic
154#2313—791no—
155#1821—791no—
156#1510—792no—
157#1284—792no—
158C79#946511yescharacteristic, complemented, Sylow
159C2×#121C2×D7911yescharacteristic
160C22×C4⋊C4C2×D7911yescharacteristic
161#407—792no—
162#397—22no—
163#390Dic7912yescomplemented
164#387—1584no—
165#386—21no—
166#386—21no—
167#382Dic7912yescomplemented
168#381—792no—
169#380—1581no—
170C4×C22⋊C4—1581no—
171C4×C22⋊C4C2×D7911yescharacteristic
172#343—1582no—
173#327—1584no—
174C22×D4—21no—
175C22×D4C2×Dic7911yescharacteristic
176C23×C4C22⋊D7911yescharacteristic
177C23×C4C22⋊D7911yescharacteristic
178C23×C4C22×D7911yescharacteristic
179C2×M16—1582no—
180#121—22no—
181#121—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—1582no—
185C2×C4⋊C4C2×Dic7911yescharacteristic
186C2×C4⋊C4C2×Dic7912yes—
187C2×C4⋊C4C22⋊D7912yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—1582no—
193C2×C42—1582no—
194C2×C42C2×Dic7912yes—
195C22⋊C8—1582no—
196C22⋊C8—1582no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—794no—
201#95—1582no—
202#95—1582no—
203#95—1582no—
204C24—21no—
205C24C2×C22⋊D7911yescharacteristic
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—44no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—1584no—
221C22×C4C22⋊Dic7912yes—
222C22×C4C22⋊Dic7912yes—
223C22×C4C22×Dic7911yescharacteristic
224C22×C4D4×D7912yes—
225C22×C4C2×C22⋊D7911yescharacteristic
226C22×C4#1192962312yes—
227M16—3164no—
228C2×C8—3162no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—6322no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—3164no—
238C42—42no—
239C42—6322no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23C2×C22⋊Dic7911yescharacteristic
248C23#1457912911yescharacteristic
249C23#1457914211yescharacteristic
250D4—82no—
251D4—82no—
252D4—82no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—3164no—
271C2×C4—3164no—
272C2×C4—3164no—
273C2×C4D4×Dic7912yes—
274C2×C4#1457913312yes—
275C8—6322no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#1909844311yescharacteristic
289C22#1909853712yes—
290C22#1909873512yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—81no—
297C4—6324no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2#2376380411yescharacteristic
303C2#2376385911yescharacteristic
304C2#2376393311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2504C2
socleC22×C79#386
prime radicalC2×#121×C79C22
cosocleC23×C4×C79C23
hypercenter#2504D79
hypocenterC79#9465
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. #23766003C2semidirect
2. #23763480C2semidirect
4. #23761253C2semidirect
16. #390×C79C4semidirect
19. #382×C79C4semidirect
158. C79#9465semidirect
163. #390Dic79semidirect
167. #382Dic79semidirect

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×#25042
#946579