Finite Group Databasebeta

← #34203439

Subgroups — #34203439

Order 20224.

564 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#19098358C212yesmaximal
2C4×#14579076C212yesmaximal
3#23761525C212yesmaximal
4C79×#2326C211yesmaximal, characteristic
5C2×#14579076C2214yes—
6C4×C22⋊Dic79—22no—
7#19098358C418yescomplemented
8#19098330—22no—
9C2×#124×C79C2211yescharacteristic
10C2×C4×Q8×C79C411yescharacteristic, complemented
11C2×C4×D4×C79C2211yescharacteristic
12#391×C79C411yescharacteristic, complemented
13#390×C79C2211yescharacteristic
14#388×C79—22no—
15#385×C79—22no—
16#384×C79C411yescharacteristic, complemented
17#376×C79C411yescharacteristic, complemented
18C2×C22⋊Dic79—24no—
19C2×C4×Dic79—42no—
20#14579076—24no—
21#14579076C2×C418yes—
22C79⋊#95—42no—
23C22×Q8×C79C2×C411yescharacteristic
24C22×D4×C79C2311yescharacteristic
25C23×C4×C79D412yes—
26#124×C79—22no—
27#124×C79C2×C414yes—
28C4×Q8×C79—42no—
29C4×D4×C79—42no—
30C2×C4⋊C4×C79—21no—
31C2×C4⋊C4×C79—22no—
32C2×C4⋊C4×C79C2×C411yescharacteristic
33C2×C4⋊C4×C79C2×C411yescharacteristic
34C2×C22⋊C4×C79—22no—
35C2×C22⋊C4×C79—22no—
36C2×C22⋊C4×C79D412yes—
37C2×C42×C79—21no—
38C2×C42×C79C2×C411yescharacteristic
39C2×C42×C79C2×C411yescharacteristic
40C2×C42×C79C2×C411yescharacteristic
41#95×C79—21no—
42#95×C79—21no—
43#95×C79—22no—
44C22×Dic79—44no—
45C22⋊Dic79—28no—
46C22⋊Dic79—44no—
47C4×Dic79—84no—
48C24×C79C2×D412yes—
49C2×Q8×C79—21no—
50C2×Q8×C79—41no—
51C2×Q8×C79—41no—
52C2×Q8×C79C4212yescomplemented
53C2×D4×C79—41no—
54C2×D4×C79—41no—
55C2×D4×C79C22×C412yes—
56C2×D4×C79#4112yes—
57C22×C4×C79—21no—
58C22×C4×C79—21no—
59C22×C4×C79—22no—
60C22×C4×C79—22no—
61C22×C4×C79—21no—
62C22×C4×C79—21no—
63C22×C4×C79—82no—
64C22×C4×C79C22⋊C411yescharacteristic
65C22×C4×C79C22⋊C412yes—
66C22×C4×C79C22⋊C411yescharacteristic
67C22×C4×C79C22×C411yescharacteristic
68C4⋊C4×C79—41no—
69C4⋊C4×C79—41no—
70C4⋊C4×C79—42no—
71C4⋊C4×C79—82no—
72C22⋊C4×C79—28no—
73C22⋊C4×C79—42no—
74C22⋊C4×C79—82no—
75C42×C79—22no—
76C42×C79—21no—
77C42×C79—42no—
78C42×C79—41no—
79C42×C79C4212yescomplemented
80C23×C79—22no—
81C23×C79—22no—
82C23×C79—42no—
83C23×C79—42no—
84C23×C79C2×C22⋊C411yescharacteristic
85C23×C79C4×D414yes—
86Q8×C79—42no—
87Q8×C79—42no—
88D4×C79—42no—
89D4×C79—42no—
90C2×C4×C79—21no—
91C2×C4×C79—21no—
92C2×C4×C79—22no—
93C2×C4×C79—22no—
94C2×C4×C79—21no—
95C2×C4×C79—22no—
96C2×C4×C79—41no—
97C2×C4×C79—44no—
98C2×C4×C79—41no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79—41no—
102C2×C4×C79—42no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79—42no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—82no—
110C2×C4×C79#9912yes—
111C2×C4×C79C2×C4211yescharacteristic
112C2×C4×C79#11711yescharacteristic
113C2×Dic79—48no—
114C2×Dic79—84no—
115C2×Dic79—84no—
116C22×C79—21no—
117C22×C79—21no—
118C22×C79—24no—
119C22×C79—44no—
120C22×C79—42no—
121C22×C79—42no—
122C22×C79—82no—
123C22×C79C4×C22⋊C411yescharacteristic
124C22×C79C2×#9911yescharacteristic
125C22×C79#41011yescharacteristic
126C4×C79—22no—
127C4×C79—21no—
128C4×C79—21no—
129C4×C79—42no—
130C4×C79—42no—
131C4×C79—41no—
132C4×C79—42no—
133C4×C79—81no—
134C4×C79—81no—
135Dic79—88no—
136#12627—791nomaximal, Sylow
137C2×C79—21no—
138C2×C79—21no—
139C2×C79—44no—
140C2×C79C4×#9911yescharacteristic
141C2×C79#212711yescharacteristic
142C2×C79#212911yescharacteristic
143#2326D7911yescharacteristic
144C2×#354—792no—
145C4×#99—792no—
146#1393—792no—
147C79#1262711yescharacteristic, complemented, Sylow
148C2×#124C2×D7911yescharacteristic
149C2×C4×Q8Dic7911yescharacteristic, complemented
150C2×C4×D4C2×D7911yescharacteristic
151C2×#99—794no—
152#391Dic7911yescharacteristic, complemented
153#390C2×D7911yescharacteristic
154#388—22no—
155#385—22no—
156#384Dic7911yescharacteristic, complemented
157C4×C22⋊C4—1582no—
158#376Dic7911yescharacteristic, complemented
159#354—798no—
160#342—1582no—
161C22×Q8C2×Dic7911yescharacteristic
162C22×D4C22×D7911yescharacteristic
163C23×C4C22⋊D7912yes—
164#124—22no—
165#124C4×D7914yes—
166C4×Q8—42no—
167C4×D4—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×Dic7911yescharacteristic
171C2×C4⋊C4C2×Dic7911yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—1584no—
175C2×C22⋊C4C22⋊D7912yes—
176C2×C42—21no—
177C2×C42—3162no—
178C2×C42C2×Dic7911yescharacteristic
179C2×C42C2×Dic7911yescharacteristic
180C2×C42C2×Dic7911yescharacteristic
181#99—798no—
182#99—1584no—
183#95—21no—
184#95—22no—
185#95—21no—
186#95—3162no—
187C24C2×C22⋊D7912yes—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C4×Dic7912yescomplemented
192C2×D4—41no—
193C2×D4—41no—
194C2×D4C2×C4×D7912yes—
195C2×D4#1192962412yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—82no—
203C22×C4—3164no—
204C22×C4C22⋊Dic7911yescharacteristic
205C22×C4C22⋊Dic7912yes—
206C22×C4C22⋊Dic7911yescharacteristic
207C22×C4C22×Dic7911yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C22⋊C4—28no—
213C22⋊C4—42no—
214C22⋊C4—82no—
215C22⋊C4—3164no—
216C22⋊C4—1588no—
217C42—22no—
218C42—21no—
219C42—41no—
220C42—42no—
221C42—6324no—
222C42C4×Dic7912yescomplemented
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23C2×C22⋊Dic7911yescharacteristic
228C23C4×C22⋊D7914yes—
229Q8—42no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—3168no—
254C2×C4—6324no—
255C2×C4—6324no—
256C2×C4#1457907612yes—
257C2×C4C2×C4×Dic7911yescharacteristic
258C2×C4#1457912011yescharacteristic
259C22—21no—
260C22—21no—
261C22—24no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22—82no—
266C22C4×C22⋊Dic7911yescharacteristic
267C22C2×#1457907611yescharacteristic
268C22#1909853711yescharacteristic
269C4—22no—
270C4—21no—
271C4—21no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—81no—
277C4—81no—
278C4—6328no—
279C2—21no—
280C2—21no—
281C2—44no—
282C2C4×#1457907611yescharacteristic
283C2#2376506911yescharacteristic
284C2#2376507511yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC2×C4×C79C2×C42
FrattiniC22×D4C22×D79
FittingC79×#2326C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenter#2326D79
hypocenterC79#12627
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
7. #19098358C4semidirect
10. C2×C4×Q8×C79C4semidirect
12. #391×C79C4semidirect
16. #384×C79C4semidirect
17. #376×C79C4semidirect
52. C2×Q8×C79C42semidirect
79. C42×C79C42semidirect
147. C79#12627semidirect
149. C2×C4×Q8Dic79semidirect
152. #391Dic79semidirect
156. #384Dic79semidirect
158. #376Dic79semidirect
191. C2×Q8C4×Dic79semidirect
222. C42C4×Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190983582
C4×#145790762
#237615252
C79×#23262
#1262779