Finite Group Databasebeta

← #34185719

Subgroups — #34185719

Order 20224.

644 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 111 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#23768880C211yesmaximal, characteristic
2#23761915C212yesmaximal
3#23761730C211yesmaximal, characteristic, complemented
4#23761286C212yesmaximal, complemented
5C79×#2274C211yesmaximal, characteristic, complemented
6C2×#14579135C2211yescharacteristic
7#19098527C2212yes—
8#19098502C412yescomplemented
9#19098480C412yescomplemented
10#19098453—22no—
11#19098442—21no—
12#19098442—21no—
13C4×C22⋊Dic79C2211yescharacteristic
14#19098296—24no—
15#19098289—22no—
16C2×#117×C79—21no—
17C22×C4⋊C4×C79C2211yescharacteristic
18C4×C4⋊C4×C79—24no—
19C4×C22⋊C4×C79—21no—
20C4×C22⋊C4×C79C2212yes—
21C22×D316C2×C411yescharacteristic
22C22×C22⋊D79—21no—
23C2×#11929608—22no—
24#14579135—22no—
25#14579135—22no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C22⋊Dic79—21no—
28C2×(C2×D79)⋊C4—22no—
29C2×(C2×D79)⋊C4—21no—
30C2×(C2×D79)⋊C4—22no—
31C2×C4×Dic79—21no—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)—22no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95C2×C412yes—
37C23×C4×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79C2311yescharacteristic
40#117×C79—28no—
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79C2×C412yes—
45C2×C4⋊C4×C79D412yes—
46C2×C22⋊C4×C79—22no—
47C2×C42×C79—22no—
48C2×C42×C79—22no—
49C2×C42×C79—22no—
50#95×C79—22no—
51#95×C79D414yes—
52C23×D79—21no—
53C2×D316—22no—
54C2×D316—81no—
55C2×C22⋊D79—44no—
56C2×C22⋊D79—41no—
57C2×C22⋊D79—41no—
58C22×Dic79—21no—
59C22×Dic79—22no—
60#11929608—82no—
61C2×C79⋊C8—42no—
62C4⋊Dic79—22no—
63C22⋊Dic79—42no—
64(C2×D79)⋊C4—44no—
65(C2×D79)⋊C4—42no—
66(C2×D79)⋊C4—82no—
67C4×Dic79—81no—
68C24×C79C2×D411yescharacteristic
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—24no—
72C22×C4×C79—21no—
73C22×C4×C79—24no—
74C22×C4×C79—21no—
75C22×C4×C79—81no—
76C22×C4×C79—81no—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79C22⋊C412yes—
79C22×C4×C79C22×C411yescharacteristic
80C22×C4×C79C2×D412yes—
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79#4112yes—
83C4⋊C4×C79—28no—
84C4⋊C4×C79—24no—
85C4⋊C4×C79—24no—
86C22⋊C4×C79—28no—
87C42×C79—28no—
88C42×C79—82no—
89C42×C79—82no—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79—41no—
93C23×C79C2×C22⋊C411yescharacteristic
94C23×C79#12011yescharacteristic
95C23×C79#12311yescharacteristic
96C22×D79—42no—
97C22×D79—42no—
98C22×D79—41no—
99C22×D79—42no—
100C2×C4×C79—22no—
101C2×C4×C79—22no—
102C2×C4×C79—24no—
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—28no—
106C2×C4×C79—22no—
107C2×C4×C79—41no—
108C2×C4×C79—42no—
109C2×C4×C79—41no—
110C2×C4×C79—42no—
111C2×C4×C79—44no—
112C2×C4×C79—42no—
113C2×C4×C79—44no—
114C2×C4×C79—42no—
115C2×C4×C79—81no—
116C2×C4×C79#10414yes—
117C2×C4×C79C4×D412yes—
118C2×C4×C79#12112yes—
119C22⋊D79—82no—
120C22⋊D79—82no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—42no—
124C2×Dic79—41no—
125C2×Dic79—42no—
126D316—82no—
127C79⋊C8—82no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—44no—
136C22×C79#38611yescharacteristic
137C22×C79C2×#10412yes—
138C22×C79#45412yes—
139C2×D79—84no—
140C2×D79—81no—
141C2×D79—81no—
142C2×D79—81no—
143C4×C79—24no—
144C4×C79—22no—
145C4×C79—22no—
146C4×C79—48no—
147C4×C79—42no—
148Dic79—82no—
149Dic79—81no—
150#9410—791nomaximal, Sylow
151C2×C79—22no—
152C2×C79—22no—
153C2×C79—41no—
154C2×C79#188811yescharacteristic
155C2×C79#189611yescharacteristic
156C2×C79#191311yescharacteristic
157D79—161no—
158#2591—791no—
159#2274D7911yescharacteristic, complemented
160#1507—792no—
161#1463—791no—
162#1297—792no—
163C79#941011yescharacteristic, complemented, Sylow
164C2×#121—791no—
165C2×#117—21no—
166C22×C4⋊C4C2×D7911yescharacteristic
167#407—792no—
168#400—792no—
169#394—792no—
170#388—1582no—
171#386—1581no—
172#386—1581no—
173C4×C4⋊C4—24no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—791no—
176C4×C22⋊C4C2×D7912yes—
177#327—1584no—
178#324—1582no—
179C22×D4—791no—
180C22×D4—1581no—
181C23×C4C22⋊D7911yescharacteristic
182C23×C4C22⋊D7911yescharacteristic
183C23×C4C22×D7911yescharacteristic
184C2×M16—1582no—
185#121—1582no—
186#121—1582no—
187#117—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—791no—
192C2×C4⋊C4C4×D7912yes—
193C2×C4⋊C4D31612yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1581no—
196C2×C22⋊C4—1582no—
197C2×C22⋊C4—1581no—
198C2×C22⋊C4—1582no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—1581no—
203C22⋊C8—1582no—
204C22⋊C8—1582no—
205#95—22no—
206#95—792no—
207#95—1582no—
208#95—1582no—
209#95C22⋊D7914yes—
210C24—1581no—
211C24C2×C22⋊D7911yescharacteristic
212C2×D4—3164no—
213C2×D4—3161no—
214C2×D4—3161no—
215C2×D4—6321no—
216C2×D4—1582no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—1582no—
226C22×C4—1581no—
227C22×C4(C2×D79)⋊C412yes—
228C22×C4(C2×D79)⋊C412yes—
229C22×C4C2×C4×D7911yescharacteristic
230C22×C4C2×C22⋊D7912yes—
231C22×C4C2×D31611yescharacteristic
232C22×C4#1192962412yes—
233M16—6322no—
234C2×C8—3162no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—1582no—
239C22⋊C4—28no—
240C22⋊C4—3162no—
241C22⋊C4—3164no—
242C22⋊C4—3162no—
243C22⋊C4—6322no—
244C42—28no—
245C42—82no—
246C42—82no—
247C42—6321no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—3162no—
252C23—3162no—
253C23—3161no—
254C23—3162no—
255C23C2×(C2×D79)⋊C411yescharacteristic
256C23#1457913011yescharacteristic
257C23#1457914411yescharacteristic
258D4—6322no—
259D4—6322no—
260D4—6322no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—3162no—
278C2×C4—3162no—
279C2×C4—3162no—
280C2×C4—3161no—
281C2×C4—3162no—
282C2×C4#1457908914yes—
283C2×C4C4×C22⋊D7912yes—
284C2×C4#1457913512yes—
285C8—6322no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—6324no—
295C22—6321no—
296C22—6321no—
297C22—6321no—
298C22#1909844211yescharacteristic
299C22C2×#1457908912yes—
300C22#1909873612yes—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—48no—
305C4—42no—
306C4—6322no—
307C4—6321no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—12641no—
312C2#2376380311yescharacteristic
313C2#2376384311yescharacteristic
314C2#2376394211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2274C2
socleC22×C79#386
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenter#2274D79
hypocenterC79#9410
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
3. #23761730C2semidirect
4. #23761286C2semidirect
5. C79×#2274C2semidirect
8. #19098502C4semidirect
9. #19098480C4semidirect
159. #2274D79semidirect
163. C79#9410semidirect

Maximal subgroups

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

subgroupindex
#237688802
#237619152
#237617302
#237612862
C79×#22742
#941079