Finite Group Databasebeta

← #34198668

Subgroups — #34198668

Order 20224.

358 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#14579080C211yesmaximal, characteristic
2#23761724C211yesmaximal, characteristic
3#23761427C212yesmaximal
4#23761406C212yesmaximal
5C79×#2634C211yesmaximal, characteristic
6#19098616—21no—
7C2×#14579080—21no—
8C4×#11929608—21no—
9#19098332C2212yes—
10#19098289—21no—
11#19098289C2212yes—
12C2×#125×C79C411yescharacteristic
13C2×C4×Q8×C79C2211yescharacteristic
14#400×C79—21no—
15#400×C79C411yescharacteristic
16#400×C79C2211yescharacteristic
17#398×C79C411yescharacteristic
18#398×C79C2211yescharacteristic
19#395×C79—21no—
20#387×C79—21no—
21#387×C79—21no—
22#383×C79C411yescharacteristic
23C2×#11929608—22no—
24C4⋊(C79⋊C8)—41no—
25#14579080—24no—
26C22⋊(C79⋊C8)—22no—
27#14579072—41no—
28C4×C79⋊C8—41no—
29C4×C79⋊C8—41no—
30C22×Q8×C79D411yescharacteristic
31#125×C79—41no—
32#125×C79—41no—
33C4×Q8×C79—41no—
34C4×Q8×C79—41no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79C2×C411yescharacteristic
40C2×C4⋊C4×C79C2×C411yescharacteristic
41C2×C4⋊C4×C79C2×C411yescharacteristic
42C2×C4⋊C4×C79D411yescharacteristic
43C2×C4⋊C4×C79C2311yescharacteristic
44C2×C42×C79C2×C411yescharacteristic
45C2×C42×C79D411yescharacteristic
46C2×C42×C79D411yescharacteristic
47#95×C79—21no—
48#95×C79—21no—
49#95×C79—21no—
50#95×C79—21no—
51#95×C79—21no—
52#95×C79—21no—
53#95×C79—21no—
54#95×C79—21no—
55#95×C79C2×C411yescharacteristic
56#95×C79C2×C411yescharacteristic
57#11929608—24no—
58#11929608—42no—
59C2×C79⋊C8—42no—
60C2×C79⋊C8—42no—
61C2×Q8×C79—22no—
62C2×Q8×C79—22no—
63C2×Q8×C79—41no—
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79C22⋊C411yescharacteristic
69C22×C4×C79C22⋊C411yescharacteristic
70C22×C4×C79C22⋊C411yescharacteristic
71C22×C4×C79C22⋊C411yescharacteristic
72C22×C4×C79C22×C411yescharacteristic
73C22×C4×C79C2×D411yescharacteristic
74C22×C4×C79C2×D411yescharacteristic
75C4⋊C4×C79—41no—
76C4⋊C4×C79—41no—
77C4⋊C4×C79—41no—
78C4⋊C4×C79—42no—
79C4⋊C4×C79—41no—
80C4⋊C4×C79—41no—
81C4⋊C4×C79—41no—
82C4⋊C4×C79—41no—
83C4⋊C4×C79—42no—
84C4⋊C4×C79—42no—
85C42×C79—21no—
86C42×C79—21no—
87C42×C79—21no—
88C42×C79—21no—
89C42×C79—21no—
90C42×C79—21no—
91C23×C79C2×C22⋊C411yescharacteristic
92Q8×C79—44no—
93C2×C4×C79—21no—
94C2×C4×C79—21no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—42no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79#9911yescharacteristic
119C2×C4×C79#9911yescharacteristic
120C2×C4×C79#10412yes—
121C2×C4×C79#13512yes—
122C79⋊C8—44no—
123C79⋊C8—82no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79C2×#9911yescharacteristic
127C22×C79#41611yescharacteristic
128C22×C79C2×#10411yescharacteristic
129C4×C79—21no—
130C4×C79—21no—
131C4×C79—22no—
132C4×C79—22no—
133C4×C79—41no—
134C4×C79—41no—
135C4×C79—41no—
136C4×C79—41no—
137C4×C79—81no—
138C4×C79—81no—
139C4×C79—81no—
140C4×C79—81no—
141#11740—791nomaximal, Sylow
142C2×C79—21no—
143C2×C79—21no—
144C2×C79#150111yescharacteristic
145C2×C79#212311yescharacteristic
146C2×C79#213311yescharacteristic
147#2634D7911yescharacteristic
148C4×#101—791no—
149#1461—791no—
150#1353—792no—
151#1346—792no—
152C79#1174011yescharacteristic, complemented, Sylow
153C2×#125Dic7911yescharacteristic
154C2×C4×Q8C2×D7911yescharacteristic
155#431—1581no—
156C2×#101—1581no—
157C4×M16—1581no—
158#400—21no—
159#400Dic7911yescharacteristic
160#400C2×D7911yescharacteristic
161#398Dic7911yescharacteristic
162#398C2×D7911yescharacteristic
163#395—21no—
164#387—21no—
165#387—21no—
166#383Dic7911yescharacteristic
167#343—792no—
168#324—792no—
169#324—1581no—
170C22×Q8C22⋊D7911yescharacteristic
171C2×M16—1582no—
172#125—41no—
173#125—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C2×Dic7911yescharacteristic
181C2×C4⋊C4C2×Dic7911yescharacteristic
182C2×C4⋊C4C2×Dic7911yescharacteristic
183C2×C4⋊C4C22⋊D7911yescharacteristic
184C2×C4⋊C4C22×D7911yescharacteristic
185C2×C42C2×Dic7911yescharacteristic
186C2×C42C22⋊D7911yescharacteristic
187C2×C42C22⋊D7911yescharacteristic
188C4⋊C8—3161no—
189#101—1584no—
190C22⋊C8—1582no—
191#97—3161no—
192C4×C8—3161no—
193C4×C8—3161no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95C2×Dic7911yescharacteristic
203#95C2×Dic7911yescharacteristic
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—41no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4C22⋊Dic7911yescharacteristic
212C22×C4C22⋊Dic7911yescharacteristic
213C22×C4C22⋊Dic7911yescharacteristic
214C22×C4C22⋊Dic7911yescharacteristic
215C22×C4C22×Dic7911yescharacteristic
216C22×C4C2×C22⋊D7911yescharacteristic
217C22×C4C2×C22⋊D7911yescharacteristic
218M16—3162no—
219M16—1584no—
220C2×C8—3162no—
221C2×C8—3162no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C23C2×C22⋊Dic7911yescharacteristic
239Q8—44no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4#1457907611yescharacteristic
266C2×C4#1457907611yescharacteristic
267C2×C4#1457908812yes—
268C2×C4#1457918112yes—
269C8—3164no—
270C8—6322no—
271C22—21no—
272C22—21no—
273C22C2×#1457907611yescharacteristic
274C22#1909855911yescharacteristic
275C22C2×#1457908811yescharacteristic
276C4—21no—
277C4—21no—
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C2—21no—
289C2—21no—
290C2#2376189111yescharacteristic
291C2#2376505411yescharacteristic
292C2#2376508911yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2634C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#2634D79
hypocenterC79#11740
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2634

Complements

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

NH ≅ G/Ntype
152. C79#11740semidirect

Maximal subgroups

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

subgroupindex
C4×#145790802
#237617242
#237614272
#237614062
C79×#26342
#1174079