Finite Group Databasebeta

← #34184325

Subgroups — #34184325

Order 20224.

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

normal subgroups only

This group has 516 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#23769213C211yesmaximal, characteristic
2#23766003C211yesmaximal, characteristic
3#23761973C212yesmaximal
4#23761268C212yesmaximal
5C2×#324×C79C211yesmaximal, characteristic
6C2×#14579140C2211yescharacteristic
7#19098502C412yescomplemented
8#19098496—22no—
9#19098483C412yescomplemented
10#19098449—24no—
11#19098436—21no—
12#19098426—21no—
13#19098426—21no—
14#19098423C2211yescharacteristic
15#19098423C2212yes—
16C4×C22⋊Dic79—21no—
17#19098301—24no—
18C22×C4⋊C4×C79C2211yescharacteristic
19C2×C22⋊C8×C79C2212yes—
20#324×C79—22no—
21C22×Dic158C2×C411yescharacteristic
22#14579140—22no—
23#14579140—22no—
24C2×C4⋊Dic79—22no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—22no—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—21no—
32C2×C4×Dic79—22no—
33C79⋊#95—22no—
34C79⋊#95—24no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C79⋊#95—22no—
38C79⋊#95C2×C412yes—
39C79⋊#95D414yes—
40C23×C4×C79D411yescharacteristic
41C23×C4×C79D411yescharacteristic
42C23×C4×C79C2311yescharacteristic
43C22×C8×C79—22no—
44C2×C4⋊C4×C79—22no—
45C2×C4⋊C4×C79—22no—
46C2×C4⋊C4×C79C2×C412yes—
47C2×C4⋊C4×C79D412yes—
48C22⋊C8×C79—22no—
49C22⋊C8×C79—22no—
50C2×Dic158—22no—
51C2×Dic158—81no—
52C22×Dic79—21no—
53C22×Dic79—21no—
54C22×Dic79—22no—
55C22×Dic79—24no—
56C4⋊Dic79—22no—
57C4⋊Dic79—82no—
58C4⋊Dic79—81no—
59Dic79⋊C4—42no—
60Dic79⋊C4—82no—
61C22⋊Dic79—41no—
62C22⋊Dic79—41no—
63C22⋊Dic79—42no—
64C22⋊Dic79—42no—
65C22⋊Dic79—42no—
66C4×Dic79—82no—
67C24×C79C2×D411yescharacteristic
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—22no—
72C22×C4×C79—22no—
73C22×C4×C79—21no—
74C22×C4×C79—81no—
75C22×C4×C79—81no—
76C22×C4×C79C22⋊C412yes—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D412yes—
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79#4112yes—
82C2×C8×C79—42no—
83C2×C8×C79—82no—
84C4⋊C4×C79—24no—
85C4⋊C4×C79—24no—
86C23×C79—41no—
87C23×C79—41no—
88C23×C79C2×C22⋊C411yescharacteristic
89C23×C79#12011yescharacteristic
90C23×C79#12311yescharacteristic
91C23×C79#13512yes—
92C23×C79#13712yes—
93C2×C4×C79—22no—
94C2×C4×C79—22no—
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—42no—
98C2×C4×C79—41no—
99C2×C4×C79—41no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—81no—
106C2×C4×C79C4×D412yes—
107C2×C4×C79#12112yes—
108C2×Dic79—44no—
109C2×Dic79—42no—
110C2×Dic79—42no—
111C2×Dic79—42no—
112C2×Dic79—44no—
113C2×Dic79—42no—
114C2×Dic79—41no—
115C2×Dic79—41no—
116C2×Dic79—42no—
117C2×Dic79—44no—
118Dic158—82no—
119C8×C79—82no—
120C22×C79—21no—
121C22×C79—21no—
122C22×C79—21no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—42no—
127C22×C79—42no—
128C22×C79#38611yescharacteristic
129C22×C79#41611yescharacteristic
130C22×C79#41911yescharacteristic
131C22×C79#45212yes—
132C22×C79#48512yes—
133C4×C79—22no—
134C4×C79—22no—
135C4×C79—42no—
136C4×C79—42no—
137Dic79—82no—
138Dic79—84no—
139Dic79—81no—
140Dic79—81no—
141#9202—791nomaximal, Sylow
142C2×C79—21no—
143C2×C79—22no—
144C2×C79—21no—
145C2×C79—22no—
146C2×C79#187311yescharacteristic
147C2×C79#191211yescharacteristic
148C2×C79#191311yescharacteristic
149#2627—791no—
150#2313—791no—
151#1520—792no—
152C2×#324D7911yescharacteristic
153#1289—792no—
154C79#920211yescharacteristic, complemented, Sylow
155C2×C22⋊Q8—791no—
156C22×C4⋊C4C2×D7911yescharacteristic
157C2×C22⋊C8C2×D7912yes—
158#400—792no—
159#398—1582no—
160#395—792no—
161#387—1584no—
162#385—1581no—
163#381—1581no—
164#381—1581no—
165#380—791no—
166#380—792no—
167C4×C22⋊C4—1581no—
168#328—1584no—
169#324—22no—
170C22×Q8—791no—
171C23×C4D31611yescharacteristic
172C23×C4D31611yescharacteristic
173C23×C4C22×D7911yescharacteristic
174C22×C8—22no—
175C22⋊Q8—1582no—
176C22⋊Q8—1582no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—791no—
180C2×C4⋊C4—1581no—
181C2×C4⋊C4—1582no—
182C2×C4⋊C4—1582no—
183C2×C4⋊C4—1581no—
184C2×C4⋊C4C4×D7912yes—
185C2×C4⋊C4C22⋊D7912yes—
186C2×C22⋊C4—1581no—
187C2×C22⋊C4—1582no—
188C2×C22⋊C4—1581no—
189C2×C42—1582no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192#95—792no—
193#95—794no—
194#95—1582no—
195#95—1584no—
196#95—1582no—
197#95—1582no—
198#95—1582no—
199C24C2×D31611yescharacteristic
200C2×Q8—6321no—
201C2×Q8—1582no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—1582no—
211C22×C4—1581no—
212C22×C4—1581no—
213C22×C4—1584no—
214C22×C4(C2×D79)⋊C412yes—
215C22×C4(C2×D79)⋊C412yes—
216C22×C4C2×C4×D7911yescharacteristic
217C22×C4D4×D7912yes—
218C22×C4C2×C22⋊D7911yescharacteristic
219C22×C4#1192962312yes—
220C2×C8—42no—
221C2×C8—82no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—3162no—
225C4⋊C4—6322no—
226C4⋊C4—6322no—
227C4⋊C4—6321no—
228C4⋊C4—1582no—
229C22⋊C4—3161no—
230C22⋊C4—3161no—
231C22⋊C4—3162no—
232C22⋊C4—3162no—
233C22⋊C4—3162no—
234C42—6322no—
235C23—41no—
236C23—41no—
237C23C2×(C2×D79)⋊C411yescharacteristic
238C23#1457913111yescharacteristic
239C23#1457914511yescharacteristic
240C23#1457918312yes—
241C23#1457919412yes—
242Q8—6322no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—3164no—
257C2×C4—3162no—
258C2×C4—3162no—
259C2×C4—3162no—
260C2×C4—3164no—
261C2×C4—3162no—
262C2×C4—3161no—
263C2×C4—3161no—
264C2×C4—3164no—
265C2×C4—3162no—
266C2×C4(C22⋊D79)⋊C412yes—
267C2×C4#1457913312yes—
268C8—82no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—42no—
276C22—42no—
277C22#1909844011yescharacteristic
278C22#1909856011yescharacteristic
279C22#1909857311yescharacteristic
280C22#1909872712yes—
281C22#1909890512yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—6324no—
287C4—6322no—
288C4—6321no—
289C4—6321no—
290C2—22no—
291C2—21no—
292C2—22no—
293C2—21no—
294C2#2376371011yescharacteristic
295C2#2376393511yescharacteristic
296C2#2376394011yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#324×C79C2
socleC22×C79#386
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenterC2×#324D79
hypocenterC79#9202
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
7. #19098502C4semidirect
9. #19098483C4semidirect
154. C79#9202semidirect

Maximal subgroups

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

subgroupindex
#237692132
#237660032
#237619732
#237612682
C2×#324×C792
#920279