Finite Group Databasebeta

← #34185991

Subgroups — #34185991

Order 20224.

512 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 99 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#23767881C211yesmaximal, characteristic
2#23766003C211yesmaximal, characteristic
3#23761934C212yesmaximal
4#23761254C212yesmaximal
5C79×#1821C211yesmaximal, characteristic
6C2×#14579140C2211yescharacteristic
7#19098501—22no—
8#19098468C412yescomplemented
9#19098449—24no—
10#19098428C412yescomplemented
11#19098426—21no—
12#19098426—21no—
13#19098426C2212yes—
14#19098423—21no—
15C4×C22⋊Dic79—21no—
16C4×C22⋊Dic79C2211yescharacteristic
17#19098301—24no—
18C22×C4⋊C4×C79C2211yescharacteristic
19#407×C79C2212yes—
20#343×C79—22no—
21C22×Dic158C2×C411yescharacteristic
22#14579140—22no—
23#14579140—22no—
24C2×C4⋊Dic79—22no—
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×Dic79⋊C4—21no—
27C2×C22⋊Dic79—22no—
28C2×C22⋊Dic79—21no—
29C2×C22⋊Dic79—21no—
30C2×C4×Dic79—22no—
31C2×C4×Dic79C2×C412yes—
32C79⋊#95—22no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C79⋊#95—22no—
40C79⋊#95D414yes—
41C23×C4×C79D411yescharacteristic
42C23×C4×C79D411yescharacteristic
43C23×C4×C79C2311yescharacteristic
44C2×M16×C79—22no—
45C2×C4⋊C4×C79—22no—
46C2×C4⋊C4×C79—22no—
47C2×C4⋊C4×C79C2×C412yes—
48C2×C4⋊C4×C79D412yes—
49C22⋊C8×C79—22no—
50C22⋊C8×C79—22no—
51C2×Dic158—22no—
52C2×Dic158—42no—
53C22×Dic79—21no—
54C22×Dic79—22no—
55C22×Dic79—21no—
56C22×Dic79—24no—
57C4⋊Dic79—22no—
58C4⋊Dic79—82no—
59Dic79⋊C4—42no—
60C22⋊Dic79—44no—
61C22⋊Dic79—42no—
62C22⋊Dic79—41no—
63C22⋊Dic79—41no—
64C4×Dic79—42no—
65C4×Dic79—82no—
66C24×C79C2×D411yescharacteristic
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—42no—
72C22×C4×C79—42no—
73C22×C4×C79—42no—
74C22×C4×C79C22⋊C412yes—
75C22×C4×C79C22⋊C412yes—
76C22×C4×C79C22×C411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79C2×D412yes—
79C22×C4×C79#4112yes—
80M16×C79—44no—
81C2×C8×C79—42no—
82C4⋊C4×C79—24no—
83C4⋊C4×C79—24no—
84C23×C79—22no—
85C23×C79—22no—
86C23×C79—41no—
87C23×C79C2×C22⋊C411yescharacteristic
88C23×C79#12011yescharacteristic
89C23×C79#12311yescharacteristic
90C2×C4×C79—22no—
91C2×C4×C79—22no—
92C2×C4×C79—22no—
93C2×C4×C79—22no—
94C2×C4×C79—42no—
95C2×C4×C79—42no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—81no—
101C2×C4×C79—81no—
102C2×C4×C79C4×D412yes—
103C2×C4×C79#12112yes—
104C2×Dic79—42no—
105C2×Dic79—44no—
106C2×Dic79—42no—
107C2×Dic79—44no—
108C2×Dic79—42no—
109C2×Dic79—41no—
110C2×Dic79—41no—
111C2×Dic79—42no—
112C2×Dic79—42no—
113C2×Dic79—44no—
114Dic158—82no—
115C8×C79—82no—
116C22×C79—21no—
117C22×C79—21no—
118C22×C79—21no—
119C22×C79—21no—
120C22×C79—21no—
121C22×C79—21no—
122C22×C79—21no—
123C22×C79—44no—
124C22×C79#38611yescharacteristic
125C22×C79#41012yes—
126C22×C79#45512yes—
127C4×C79—22no—
128C4×C79—22no—
129C4×C79—42no—
130C4×C79—42no—
131Dic79—84no—
132Dic79—82no—
133Dic79—81no—
134Dic79—81no—
135#9460—791nomaximal, Sylow
136C2×C79—22no—
137C2×C79—22no—
138C2×C79—41no—
139C2×C79#188511yescharacteristic
140C2×C79#189911yescharacteristic
141C2×C79#191211yescharacteristic
142#2503—791no—
143#2313—791no—
144#1821D7911yescharacteristic
145#1511—792no—
146#1284—792no—
147C79#946011yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—791no—
149C22×C4⋊C4C2×D7911yescharacteristic
150#407C2×D7912yes—
151#400—1582no—
152#391—792no—
153#387—1584no—
154#382—792no—
155#381—792no—
156#381—1581no—
157#381—1581no—
158#380—1581no—
159C4×C22⋊C4—791no—
160C4×C22⋊C4—1581no—
161#343—22no—
162#328—1584no—
163C22×Q8—791no—
164C23×C4D31611yescharacteristic
165C23×C4D31611yescharacteristic
166C23×C4C22×D7911yescharacteristic
167C2×M16—22no—
168C22⋊Q8—1582no—
169C22⋊Q8—1582no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—791no—
173C2×C4⋊C4—1581no—
174C2×C4⋊C4—1582no—
175C2×C4⋊C4C4×D7912yes—
176C2×C4⋊C4C22⋊D7912yes—
177C2×C22⋊C4—1581no—
178C2×C22⋊C4—1581no—
179C2×C22⋊C4—1582no—
180C2×C42—792no—
181C2×C42—1582no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184#95—794no—
185#95—1582no—
186#95—1582no—
187#95—1582no—
188#95—1582no—
189#95—1582no—
190#95—1582no—
191#95—1582no—
192#95—1581no—
193C24C2×D31611yescharacteristic
194C2×Q8—3162no—
195C2×Q8—1582no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—1582no—
204C22×C4—1581no—
205C22×C4—1581no—
206C22×C4—1584no—
207C22×C4(C2×D79)⋊C412yes—
208C22×C4(C2×D79)⋊C412yes—
209C22×C4C2×C4×D7911yescharacteristic
210C22×C4D4×D7912yes—
211C22×C4C2×C22⋊D7911yescharacteristic
212C22×C4#1192962312yes—
213M16—44no—
214C2×C8—42no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—3162no—
218C4⋊C4—6322no—
219C4⋊C4—1582no—
220C22⋊C4—3162no—
221C22⋊C4—3164no—
222C22⋊C4—3161no—
223C22⋊C4—3161no—
224C42—3162no—
225C42—6322no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23C2×(C2×D79)⋊C411yescharacteristic
230C23#1457913111yescharacteristic
231C23#1457914511yescharacteristic
232Q8—6322no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—3162no—
246C2×C4—3164no—
247C2×C4—3162no—
248C2×C4—3164no—
249C2×C4—3162no—
250C2×C4—3161no—
251C2×C4—3161no—
252C2×C4—3164no—
253C2×C4—3162no—
254C2×C4—3162no—
255C2×C4(C22⋊D79)⋊C412yes—
256C2×C4#1457913312yes—
257C8—82no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22#1909844011yescharacteristic
267C22#1909853612yes—
268C22#1909874312yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—6324no—
274C4—6322no—
275C4—6321no—
276C4—6321no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2#2376378611yescharacteristic
281C2#2376385611yescharacteristic
282C2#2376393511yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1821C2
socleC22×C79#386
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#1821D79
hypocenterC79#9460
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
8. #19098468C4semidirect
10. #19098428C4semidirect
147. C79#9460semidirect

Maximal subgroups

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

subgroupindex
#237678812
#237660032
#237619342
#237612542
C79×#18212
#946079