Finite Group Databasebeta

← #34197754

Subgroups — #34197754

Order 20224.

424 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 75 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#23766110C211yesmaximal, characteristic
2#23761742C212yesmaximal, complemented
3#23761345C212yesmaximal, complemented
4#23761332C211yesmaximal, characteristic
5C79×#1318C211yesmaximal, characteristic
6C2×#14579148C411yescharacteristic
7C2×#14579126C2211yescharacteristic
8C2×D4×Dic79C411yescharacteristic
9#19098617—22no—
10#19098469C411yescharacteristic
11#19098457—22no—
12#19098435—22no—
13#19098432C2211yescharacteristic
14#19098409C411yescharacteristic
15#19098289C2212yes—
16#19098288—22no—
17#390×C79C2211yescharacteristic
18#324×C79C2212yes—
19C22×Dic158C2×C411yescharacteristic
20C23×Dic79—21no—
21#14579148—22no—
22#14579148—22no—
23#14579126—42no—
24D4×Dic79—42no—
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×Dic79⋊C4—22no—
27C2×Dic79⋊C4D412yes—
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—21no—
30C2×C4×Dic79C2×C411yescharacteristic
31C2×C4×Dic79C2×C411yescharacteristic
32C2×C4×Dic79D412yes—
33Dic79⋊C8—42no—
34C22⋊(C79⋊C8)—22no—
35#14579071—42no—
36C79⋊#95—21no—
37C79⋊#95—21no—
38C79⋊#95—22no—
39C22×D4×C79C2×C411yescharacteristic
40C2×C4⋊C4×C79C2311yescharacteristic
41C2×C22⋊C4×C79—22no—
42C2×C42×C79C2×C411yescharacteristic
43C22⋊C8×C79—22no—
44C2×Dic158—21no—
45C2×Dic158—21no—
46C2×Dic158—41no—
47C2×Dic158—41no—
48C22×Dic79—21no—
49C22×Dic79—21no—
50C22×Dic79—42no—
51C22×Dic79—82no—
52C22×Dic79C22⋊C412yes—
53C22×Dic79C22⋊C411yescharacteristic
54C22×Dic79C22⋊C411yescharacteristic
55C2×C79⋊C8—42no—
56C4⋊Dic79—41no—
57Dic79⋊C4—42no—
58Dic79⋊C4—82no—
59C22⋊Dic79—28no—
60C22⋊Dic79—42no—
61C4×Dic79—22no—
62C4×Dic79—22no—
63C4×Dic79—22no—
64C4×Dic79—21no—
65C4×Dic79M1612yes—
66C24×C79—21no—
67C2×D4×C79—21no—
68C2×D4×C79—41no—
69C2×D4×C79—41no—
70C2×D4×C79M1612yes—
71C22×C4×C79—21no—
72C22×C4×C79C22×C411yescharacteristic
73C22×C4×C79C2×D412yes—
74C2×C8×C79—42no—
75C4⋊C4×C79—41no—
76C22⋊C4×C79—82no—
77C42×C79—41no—
78C23×C79—22no—
79C23×C79—42no—
80C23×C79—42no—
81C23×C79—41no—
82C23×C79—41no—
83C23×C79C2×C22⋊C411yescharacteristic
84D4×C79—42no—
85D4×C79—42no—
86C2×C4×C79—21no—
87C2×C4×C79—21no—
88C2×C4×C79—22no—
89C2×C4×C79—42no—
90C2×C4×C79—41no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79C2×M1611yescharacteristic
94C2×C4×C79C2×M1611yescharacteristic
95C2×Dic79—22no—
96C2×Dic79—22no—
97C2×Dic79—21no—
98C2×Dic79—22no—
99C2×Dic79—22no—
100C2×Dic79—41no—
101C2×Dic79—41no—
102C2×Dic79—41no—
103C2×Dic79—41no—
104C2×Dic79—42no—
105C2×Dic79—41no—
106C2×Dic79—84no—
107C2×Dic79#9912yes—
108Dic158—42no—
109Dic158—42no—
110C8×C79—82no—
111C79⋊C8—82no—
112C22×C79—21no—
113C22×C79—21no—
114C22×C79—44no—
115C22×C79—42no—
116C22×C79—82no—
117C22×C79—81no—
118C22×C79—81no—
119C22×C79#40711yescharacteristic
120C22×C79C2×#9911yescharacteristic
121C22×C79#41311yescharacteristic
122C4×C79—21no—
123C4×C79—21no—
124C4×C79—42no—
125C4×C79—81no—
126Dic79—22no—
127Dic79—42no—
128Dic79—41no—
129Dic79—42no—
130Dic79—81no—
131#11534—791nomaximal, Sylow
132C2×C79—21no—
133C2×C79—21no—
134C2×C79—82no—
135C2×C79#146511yescharacteristic
136C2×C79#212811yescharacteristic
137C2×C79#213311yescharacteristic
138#2326—791no—
139#1467—792no—
140#1325—792no—
141#1318—791no—
142#1318D7911yescharacteristic
143C79#1153411yescharacteristic, complemented, Sylow
144C2×#124—791no—
145C2×C4×Q8—791no—
146C2×C4×D4—791no—
147#432—1582no—
148#391—791no—
149#390C2×D7911yescharacteristic
150#388—1582no—
151#385—1582no—
152#384—791no—
153#376—791no—
154#324—792no—
155#324—1582no—
156#324C2×D7912yes—
157C22×Q8—791no—
158C22×D4C4×D7911yescharacteristic
159C23×C4—1581no—
160#124—1582no—
161#124—1582no—
162C4×Q8—3162no—
163C4×D4—3162no—
164C2×C4⋊C4—792no—
165C2×C4⋊C4—791no—
166C2×C4⋊C4—1582no—
167C2×C4⋊C4C22×D7911yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—1581no—
170C2×C22⋊C4—1582no—
171C2×C42—791no—
172C2×C42—791no—
173C2×C42—792no—
174C2×C42C4×D7911yescharacteristic
175C4⋊C8—3162no—
176C22⋊C8—22no—
177C22⋊C8—1582no—
178#97—3162no—
179#95—1581no—
180#95—1581no—
181#95—1582no—
182C24—21no—
183C2×Q8—3161no—
184C2×Q8—3161no—
185C2×Q8—1581no—
186C2×Q8—1581no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4#1192960712yes—
191C22×C4—21no—
192C22×C4—3162no—
193C22×C4—791no—
194C22×C4—791no—
195C22×C4—792no—
196C22×C4—6322no—
197C22×C4—1581no—
198C22×C4—1581no—
199C22×C4C2×C4×D7911yescharacteristic
200C22×C4D4×D7912yes—
201C2×C8—42no—
202C2×C8—3162no—
203C4⋊C4—41no—
204C4⋊C4—3161no—
205C4⋊C4—3162no—
206C4⋊C4—6322no—
207C22⋊C4—82no—
208C22⋊C4—3162no—
209C22⋊C4—1588no—
210C42—41no—
211C42—792no—
212C42—1582no—
213C42—1582no—
214C42—1582no—
215C42—1581no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—41no—
220C23—41no—
221C23C22⋊C4×D7911yescharacteristic
222Q8—3162no—
223Q8—3162no—
224D4—42no—
225D4—42no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—3161no—
234C2×C4—3161no—
235C2×C4—3161no—
236C2×C4—3162no—
237C2×C4—3161no—
238C2×C4—3161no—
239C2×C4—792no—
240C2×C4—6324no—
241C2×C4—1582no—
242C2×C4—1582no—
243C2×C4—1581no—
244C2×C4—1582no—
245C2×C4—1582no—
246C2×C4M16×D7911yescharacteristic
247C2×C4C2×#1192960711yescharacteristic
248C8—82no—
249C8—6322no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22#1909852511yescharacteristic
258C22#99×D7911yescharacteristic
259C22#1909854411yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—81no—
264C4—3162no—
265C4—3162no—
266C4—3161no—
267C4—6321no—
268C4—1582no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2#2376173511yescharacteristic
273C2#2376507011yescharacteristic
274C2#2376508611yescharacteristic
subgroupG/H
centerC22#99×D79
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1318C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1318D79
hypocenterC79#11534
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
2. #23761742C2semidirect
3. #23761345C2semidirect
143. C79#11534semidirect

Maximal subgroups

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

subgroupindex
#237661102
#237617422
#237613452
#237613322
C79×#13182
#1153479