Finite Group Databasebeta

← #34184297

Subgroups — #34184297

Order 20224.

478 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 115 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#23769166C211yesmaximal, characteristic
2#23762073C212yesmaximal, complemented
3#23761714C211yesmaximal, characteristic
4#23761260C212yesmaximal, complemented
5C79×#2921C211yesmaximal, characteristic
6C2×#14579161C2211yescharacteristic
7C2×#14579140—21no—
8#19098616C2212yes—
9#19098493—21no—
10#19098493C412yes—
11#19098488—21no—
12#19098488C412yes—
13#19098469—21no—
14#19098429C2211yescharacteristic
15#19098310—24no—
16#19098289—21no—
17#19098287—21no—
18C2×#117×C79C2211yescharacteristic
19#433×C79—24no—
20#423×C79C2212yes—
21#407×C79—21no—
22C4×M16×C79—21no—
23C22×Dic158—21no—
24#14579161—22no—
25#14579161—22no—
26#14579140—44no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79C2×C412yes—
30C2×Dic79⋊C4—22no—
31C2×C22⋊Dic79—22no—
32C4⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)—22no—
34C4×C79⋊C8—22no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C79⋊#95—21no—
38C79⋊#95C2×C412yes—
39C23×C4×C79D411yescharacteristic
40C2×M16×C79—22no—
41#117×C79—22no—
42#117×C79—22no—
43C2×C4⋊C4×C79C2×C411yescharacteristic
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C22⋊C4×C79C2×C411yescharacteristic
46C2×C22⋊C4×C79D411yescharacteristic
47C2×C42×C79D411yescharacteristic
48C2×C42×C79C2311yescharacteristic
49C4⋊C8×C79—22no—
50C4⋊C8×C79D414yes—
51C22⋊C8×C79—24no—
52#97×C79—22no—
53C4×C8×C79—22no—
54C2×Dic158—41no—
55C2×Dic158—41no—
56C2×Dic158—42no—
57C2×Dic158—42no—
58C22×Dic79—22no—
59C22×Dic79—22no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—24no—
62C4⋊Dic79—42no—
63C4⋊Dic79—42no—
64Dic79⋊C4—44no—
65C22⋊Dic79—44no—
66C24×C79C22⋊C411yescharacteristic
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—22no—
70C22×C4×C79—22no—
71C22×C4×C79C22⋊C411yescharacteristic
72C22×C4×C79C22⋊C411yescharacteristic
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22×C411yescharacteristic
75C22×C4×C79C2×D411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77M16×C79—42no—
78M16×C79—42no—
79C2×C8×C79—24no—
80C4⋊C4×C79—22no—
81C4⋊C4×C79—22no—
82C22⋊C4×C79—22no—
83C22⋊C4×C79—22no—
84C42×C79—21no—
85C42×C79—21no—
86C42×C79C2×D412yes—
87C42×C79#4112yes—
88C23×C79—21no—
89C23×C79—21no—
90C23×C79—21no—
91C23×C79—41no—
92C23×C79—41no—
93C23×C79C2×C22⋊C411yescharacteristic
94C2×C4×C79—21no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79C4×D412yes—
106C2×C4×C79#12011yescharacteristic
107C2×C4×C79#12112yes—
108C2×C4×C79#12311yescharacteristic
109C2×C4×C79#13512yes—
110C2×C4×C79AGL(1,ℤ/8)12yes—
111C2×Dic79—42no—
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117Dic158—82no—
118Dic158—82no—
119C8×C79—44no—
120C79⋊C8—82no—
121C22×C79—21no—
122C22×C79—21no—
123C22×C79—42no—
124C22×C79—41no—
125C22×C79—41no—
126C22×C79—41no—
127C22×C79—81no—
128C22×C79#38611yescharacteristic
129C22×C79#41611yescharacteristic
130C22×C79#41811yescharacteristic
131C4×C79—22no—
132C4×C79—22no—
133C4×C79—41no—
134C4×C79—41no—
135C4×C79—41no—
136C4×C79#44912yes—
137C4×C79#48212yes—
138Dic79—82no—
139Dic79—82no—
140#9196—791nomaximal, Sylow
141C2×C79—21no—
142C2×C79—21no—
143C2×C79—81no—
144C2×C79#187511yescharacteristic
145C2×C79#190911yescharacteristic
146C2×C79#191511yescharacteristic
147#2921D7911yescharacteristic
148#2621—791no—
149#1546—792no—
150#1458—791no—
151#1286—792no—
152C79#919611yescharacteristic, complemented, Sylow
153C2×C4⋊Q8—791no—
154C2×C22⋊Q8—1581no—
155C2×#117C2×D7911yescharacteristic
156#433—24no—
157#431—792no—
158#423C2×D7912yes—
159#407—21no—
160C4×M16—21no—
161#398—792no—
162#398—1581no—
163#397—792no—
164#397—1581no—
165#391—1581no—
166#382—791no—
167#332—1584no—
168#324—1581no—
169#323—1581no—
170C22×Q8—1581no—
171C23×C4C22⋊D7911yescharacteristic
172C2×M16—22no—
173C4⋊Q8—1582no—
174C4⋊Q8—1582no—
175C22⋊Q8—3164no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—792no—
179C2×C4⋊C4—1582no—
180C2×C4⋊C4—1581no—
181C2×C4⋊C4—1581no—
182C2×C4⋊C4C4×D7911yescharacteristic
183C2×C4⋊C4D31611yescharacteristic
184C2×C22⋊C4—1582no—
185C2×C22⋊C4C4×D7911yescharacteristic
186C2×C22⋊C4D31611yescharacteristic
187C2×C42C22⋊D7911yescharacteristic
188C2×C42C22×D7911yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—1582no—
191C4⋊C8C22⋊D7914yes—
192C22⋊C8—24no—
193C22⋊C8—1582no—
194#97—22no—
195C4×C8—22no—
196C4×C8—1582no—
197#95—792no—
198#95—1582no—
199#95—1582no—
200#95—1581no—
201C24(C2×D79)⋊C411yescharacteristic
202C2×Q8—3161no—
203C2×Q8—3161no—
204C2×Q8—3162no—
205C2×Q8—3162no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—1582no—
211C22×C4—1582no—
212C22×C4(C2×D79)⋊C411yescharacteristic
213C22×C4(C2×D79)⋊C411yescharacteristic
214C22×C4(C2×D79)⋊C411yescharacteristic
215C22×C4C2×C4×D7911yescharacteristic
216C22×C4C2×C22⋊D7911yescharacteristic
217C22×C4C2×D31611yescharacteristic
218M16—42no—
219M16—42no—
220C2×C8—24no—
221C2×C8—3162no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—3164no—
225C4⋊C4—3162no—
226C4⋊C4—3162no—
227C4⋊C4—1584no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—3164no—
231C42—21no—
232C42—21no—
233C42C2×C22⋊D7912yes—
234C42#1192962412yes—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×(C2×D79)⋊C411yescharacteristic
241Q8—6322no—
242Q8—6322no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—3162no—
255C2×C4—3162no—
256C2×C4—3162no—
257C2×C4—3162no—
258C2×C4—3162no—
259C2×C4—3162no—
260C2×C4C4×C22⋊D7912yes—
261C2×C4#1457913011yescharacteristic
262C2×C4#1457913512yes—
263C2×C4#1457914411yescharacteristic
264C2×C4#1457918112yes—
265C2×C4#1457918612yes—
266C8—44no—
267C8—6322no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1909844211yescharacteristic
276C22#1909855711yescharacteristic
277C22#1909856711yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—6322no—
284C4—6322no—
285C4#1909870812yes—
286C4#1909888712yes—
287C2—21no—
288C2—21no—
289C2—81no—
290C2#2376372211yescharacteristic
291C2#2376392211yescharacteristic
292C2#2376395211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2921C2
socleC22×C79#386
prime radicalC24×C79C22⋊C4
cosocleC2×C42×C79C23
hypercenter#2921D79
hypocenterC79#9196
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #23762073C2semidirect
4. #23761260C2semidirect
152. C79#9196semidirect

Maximal subgroups

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

subgroupindex
#237691662
#237620732
#237617142
#237612602
C79×#29212
#919679