Finite Group Databasebeta

← #34184032

Subgroups — #34184032

Order 20224.

586 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 127 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#23771739C211yesmaximal, characteristic
2C2×#19098431C211yesmaximal, characteristic
3C2×#19098301C212yesmaximal
4#23761280C212yesmaximal
5C79×#1817C211yesmaximal, characteristic
6C2×#14579140C2211yescharacteristic
7#19098469C412yescomplemented
8#19098461C412yescomplemented
9#19098447—22no—
10#19098436—21no—
11#19098431—24no—
12#19098426—21no—
13#19098423C2211yescharacteristic
14C2×C79⋊#95—21no—
15C2×C79⋊#95—21no—
16C2×C79⋊#95C2212yes—
17#19098301—24no—
18C22×C42×C79C2211yescharacteristic
19C2×C22⋊C8×C79C2212yes—
20#336×C79—22no—
21C22×Dic158C2×C411yescharacteristic
22C23×Dic79—22no—
23#14579140—22no—
24#14579140—22no—
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×C4⋊Dic79C2×C412yes—
27C2×Dic79⋊C4—22no—
28C2×Dic79⋊C4—21no—
29C2×C22⋊Dic79—21no—
30C2×C22⋊Dic79—21no—
31C79⋊#95—24no—
32C79⋊#95—24no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C79⋊#95—21no—
38C79⋊#95D414yes—
39C23×C4×C79D411yescharacteristic
40C23×C4×C79D411yescharacteristic
41C23×C4×C79C2311yescharacteristic
42C22×C8×C79—22no—
43C2×C42×C79—22no—
44C2×C42×C79—22no—
45C2×C42×C79C2×C412yes—
46C2×C42×C79D412yes—
47C22⋊C8×C79—22no—
48C22⋊C8×C79—22no—
49C2×Dic158—22no—
50C2×Dic158—42no—
51C22×Dic79—21no—
52C22×Dic79—22no—
53C22×Dic79—21no—
54C22×Dic79—24no—
55C22×Dic79—44no—
56C22×Dic79—44no—
57C22×Dic79—42no—
58C22×Dic79—42no—
59C4⋊Dic79—22no—
60C4⋊Dic79—42no—
61Dic79⋊C4—44no—
62Dic79⋊C4—42no—
63C22⋊Dic79—41no—
64C22⋊Dic79—41no—
65C22⋊Dic79—42no—
66C24×C79C2×D411yescharacteristic
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—22no—
72C22×C4×C79—22no—
73C22×C4×C79—42no—
74C22×C4×C79—42no—
75C22×C4×C79C22⋊C412yes—
76C22×C4×C79C22⋊C412yes—
77C22×C4×C79C22×C411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79C2×D412yes—
80C22×C4×C79#4112yes—
81C2×C8×C79—44no—
82C2×C8×C79—42no—
83C42×C79—24no—
84C42×C79—24no—
85C23×C79—22no—
86C23×C79—22no—
87C23×C79C2×C22⋊C411yescharacteristic
88C23×C79#12011yescharacteristic
89C23×C79#12311yescharacteristic
90C23×C79#13512yes—
91C23×C79#13712yes—
92C2×C4×C79—22no—
93C2×C4×C79—22no—
94C2×C4×C79—22no—
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79C4×D412yes—
107C2×C4×C79#12112yes—
108C2×Dic79—44no—
109C2×Dic79—44no—
110C2×Dic79—41no—
111C2×Dic79—41no—
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C2×Dic79—44no—
118C2×Dic79—84no—
119C2×Dic79—84no—
120Dic158—82no—
121C8×C79—82no—
122C22×C79—21no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—24no—
127C22×C79—24no—
128C22×C79#38611yescharacteristic
129C22×C79#41012yes—
130C22×C79#41611yescharacteristic
131C22×C79#41911yescharacteristic
132C22×C79#42112yes—
133C22×C79#45212yes—
134C22×C79#48512yes—
135C4×C79—22no—
136C4×C79—22no—
137C4×C79—42no—
138C4×C79—42no—
139Dic79—84no—
140Dic79—81no—
141Dic79—81no—
142Dic79—82no—
143#9153—791nomaximal, Sylow
144C2×C79—22no—
145C2×C79—22no—
146C2×C79#151512yes—
147C2×C79#152112yes—
148C2×C79#187311yescharacteristic
149C2×C79#189711yescharacteristic
150C2×C79#189911yescharacteristic
151#2855—791no—
152C2×#383—791no—
153#1817D7911yescharacteristic
154C2×#328—792no—
155#1294—792no—
156C79#915311yescharacteristic, complemented, Sylow
157C2×C22⋊Q8—791no—
158C22×C42C2×D7911yescharacteristic
159C2×C22⋊C8C2×D7912yes—
160#391—792no—
161#389—792no—
162#387—1582no—
163#385—1581no—
164#383—1584no—
165#381—1581no—
166#380—791no—
167C2×#95—792no—
168C2×#95—1581no—
169C2×#95—1581no—
170#336—22no—
171#328—1584no—
172C22×Q8—791no—
173C23×C4—1582no—
174C23×C4D31611yescharacteristic
175C23×C4D31611yescharacteristic
176C23×C4C22×D7911yescharacteristic
177C22×C8—22no—
178C22⋊Q8—1582no—
179C22⋊Q8—1582no—
180C2×C4⋊C4—792no—
181C2×C4⋊C4—791no—
182C2×C4⋊C4—1582no—
183C2×C4⋊C4—1581no—
184C2×C22⋊C4—1581no—
185C2×C22⋊C4—1581no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C4×D7912yes—
189C2×C42C22⋊D7912yes—
190C22⋊C8—22no—
191C22⋊C8—22no—
192#95—794no—
193#95—1584no—
194#95—1584no—
195#95—1582no—
196#95—1582no—
197#95—1582no—
198#95—1582no—
199#95—1581no—
200C24C2×D31611yescharacteristic
201C2×Q8—3162no—
202C2×Q8—1582no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—3164no—
212C22×C4—3164no—
213C22×C4—3162no—
214C22×C4—3162no—
215C22×C4—1582no—
216C22×C4—1581no—
217C22×C4—1581no—
218C22×C4—1584no—
219C22×C4(C2×D79)⋊C412yes—
220C22×C4(C2×D79)⋊C412yes—
221C22×C4C2×C4×D7911yescharacteristic
222C22×C4D4×D7912yes—
223C22×C4C2×C22⋊D7911yescharacteristic
224C22×C4#1192962312yes—
225C2×C8—44no—
226C2×C8—42no—
227C4⋊C4—3162no—
228C4⋊C4—3164no—
229C4⋊C4—3162no—
230C4⋊C4—1582no—
231C22⋊C4—3161no—
232C22⋊C4—3161no—
233C22⋊C4—3162no—
234C42—24no—
235C42—24no—
236C23—22no—
237C23—22no—
238C23C2×(C2×D79)⋊C411yescharacteristic
239C23#1457913111yescharacteristic
240C23#1457914511yescharacteristic
241C23#1457918312yes—
242C23#1457919412yes—
243Q8—6322no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—3164no—
259C2×C4—3164no—
260C2×C4—3161no—
261C2×C4—3161no—
262C2×C4—3164no—
263C2×C4—3162no—
264C2×C4—3162no—
265C2×C4—3162no—
266C2×C4—3162no—
267C2×C4—3162no—
268C2×C4—6324no—
269C2×C4—6324no—
270C2×C4(C22⋊D79)⋊C412yes—
271C2×C4#1457913312yes—
272C8—82no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22#1909844011yescharacteristic
280C22#1909853612yes—
281C22#1909856011yescharacteristic
282C22#1909857311yescharacteristic
283C22#1909857912yes—
284C22#1909872712yes—
285C22#1909890512yes—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—6324no—
291C4—6322no—
292C4—6321no—
293C4—6321no—
294C2—22no—
295C2—22no—
296C2#2376195212yes—
297C2#2376197812yes—
298C2#2376371011yescharacteristic
299C2#2376384611yescharacteristic
300C2#2376385611yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1817C2
socleC23×C79C2×C22⋊C4
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#1817D79
hypocenterC79#9153
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
7. #19098469C4semidirect
8. #19098461C4semidirect
156. C79#9153semidirect

Maximal subgroups

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

subgroupindex
#237717392
C2×#190984312
C2×#190983012
#237612802
C79×#18172
#915379