Finite Group Databasebeta

← #34198908

Subgroups — #34198908

Order 20224.

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

normal subgroups only

This group has 378 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#23761744C211yesmaximal, characteristic
3#23761430C212yesmaximal
4#23761427C212yesmaximal
5C79×#2581C211yesmaximal, characteristic
6#19098620—21no—
7C2×#14579080—21no—
8C4×#11929608—21no—
9#19098332C2212yes—
10#19098289—21no—
11#19098289C2212yes—
12C2×C4×Q8×C79C411yescharacteristic
13C2×C4×Q8×C79C2211yescharacteristic
14#400×C79C411yescharacteristic
15#400×C79C2211yescharacteristic
16#395×C79C411yescharacteristic
17#395×C79C2211yescharacteristic
18#391×C79—21no—
19#389×C79—21no—
20#387×C79—21no—
21#387×C79—21no—
22#382×C79C411yescharacteristic
23C2×#11929608—22no—
24C4⋊(C79⋊C8)—41no—
25#14579080—24no—
26C22⋊(C79⋊C8)—22no—
27#14579072—41no—
28#14579072—41no—
29C4×C79⋊C8—41no—
30C22×Q8×C79C2×C411yescharacteristic
31C22×Q8×C79D411yescharacteristic
32C4×Q8×C79—41no—
33C4×Q8×C79—41no—
34C4×Q8×C79—41no—
35C4×Q8×C79—41no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79C2×C411yescharacteristic
41C2×C4⋊C4×C79C2×C411yescharacteristic
42C2×C4⋊C4×C79D411yescharacteristic
43C2×C4⋊C4×C79C2311yescharacteristic
44C2×C42×C79—21no—
45C2×C42×C79C2×C411yescharacteristic
46C2×C42×C79D411yescharacteristic
47C2×C42×C79D411yescharacteristic
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—
64C2×Q8×C79—42no—
65C2×Q8×C79—41no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79C22⋊C411yescharacteristic
71C22×C4×C79C22⋊C411yescharacteristic
72C22×C4×C79C22⋊C411yescharacteristic
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22×C411yescharacteristic
75C22×C4×C79C2×D411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77C4⋊C4×C79—41no—
78C4⋊C4×C79—41no—
79C4⋊C4×C79—41no—
80C4⋊C4×C79—42no—
81C4⋊C4×C79—42no—
82C4⋊C4×C79—41no—
83C4⋊C4×C79—41no—
84C4⋊C4×C79—41no—
85C4⋊C4×C79—42no—
86C42×C79—21no—
87C42×C79—21no—
88C42×C79—21no—
89C42×C79—21no—
90C42×C79—21no—
91C42×C79—21no—
92C42×C79—42no—
93C23×C79C2×C22⋊C411yescharacteristic
94Q8×C79—44no—
95Q8×C79—42no—
96Q8×C79—42no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
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—42no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79#9911yescharacteristic
123C2×C4×C79#9911yescharacteristic
124C2×C4×C79#10412yes—
125C2×C4×C79#13712yes—
126C79⋊C8—44no—
127C79⋊C8—82no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79C2×#9911yescharacteristic
131C22×C79#41911yescharacteristic
132C22×C79C2×#10411yescharacteristic
133C4×C79—21no—
134C4×C79—21no—
135C4×C79—22no—
136C4×C79—22no—
137C4×C79—41no—
138C4×C79—42no—
139C4×C79—41no—
140C4×C79—41no—
141C4×C79—41no—
142C4×C79—81no—
143C4×C79—81no—
144C4×C79—81no—
145#11792—791nomaximal, Sylow
146C2×C79—21no—
147C2×C79—21no—
148C2×C79#150811yescharacteristic
149C2×C79#212011yescharacteristic
150C2×C79#213311yescharacteristic
151#2581D7911yescharacteristic
152C4×#101—791no—
153#1467—791no—
154#1354—792no—
155#1353—792no—
156C79#1179211yescharacteristic, complemented, Sylow
157C2×C4×Q8Dic7911yescharacteristic
158C2×C4×Q8C2×D7911yescharacteristic
159#432—1581no—
160C2×#101—1581no—
161C4×M16—1581no—
162#400Dic7911yescharacteristic
163#400C2×D7911yescharacteristic
164#395Dic7911yescharacteristic
165#395C2×D7911yescharacteristic
166#391—21no—
167#389—21no—
168#387—21no—
169#387—21no—
170#382Dic7911yescharacteristic
171#343—792no—
172#324—792no—
173#324—1581no—
174C22×Q8C2×Dic7911yescharacteristic
175C22×Q8C22⋊D7911yescharacteristic
176C2×M16—1582no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic7911yescharacteristic
186C2×C4⋊C4C2×Dic7911yescharacteristic
187C2×C4⋊C4C22⋊D7911yescharacteristic
188C2×C4⋊C4C22×D7911yescharacteristic
189C2×C42—21no—
190C2×C42C2×Dic7911yescharacteristic
191C2×C42C22⋊D7911yescharacteristic
192C2×C42C22⋊D7911yescharacteristic
193C4⋊C8—3161no—
194#101—1584no—
195C22⋊C8—1582no—
196#97—3161no—
197#97—3161no—
198C4×C8—3161no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95C2×Dic7911yescharacteristic
207#95C2×Dic7911yescharacteristic
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—41no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4C22⋊Dic7911yescharacteristic
218C22×C4C22⋊Dic7911yescharacteristic
219C22×C4C22⋊Dic7911yescharacteristic
220C22×C4C22⋊Dic7911yescharacteristic
221C22×C4C22×Dic7911yescharacteristic
222C22×C4C2×C22⋊D7911yescharacteristic
223C22×C4C2×C22⋊D7911yescharacteristic
224M16—3162no—
225M16—1584no—
226C2×C8—3162no—
227C2×C8—3162no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—42no—
244C23C2×C22⋊Dic7911yescharacteristic
245Q8—44no—
246Q8—42no—
247Q8—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4#1457907611yescharacteristic
274C2×C4#1457907611yescharacteristic
275C2×C4#1457908812yes—
276C2×C4#1457919212yes—
277C8—3164no—
278C8—6322no—
279C22—21no—
280C22—21no—
281C22C2×#1457907611yescharacteristic
282C22#1909857211yescharacteristic
283C22C2×#1457908811yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—42no—
291C4—41no—
292C4—41no—
293C4—81no—
294C4—81no—
295C4—81no—
296C2—21no—
297C2—21no—
298C2#2376192011yescharacteristic
299C2#2376504111yescharacteristic
300C2#2376508911yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2581C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#2581D79
hypocenterC79#11792
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
156. C79#11792semidirect

Maximal subgroups

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

subgroupindex
C4×#145790802
#237617442
#237614302
#237614272
C79×#25812
#1179279