Finite Group Databasebeta

← #34185707

Subgroups — #34185707

Order 20224.

618 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 107 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#23768780C211yesmaximal, characteristic
2#23765757C211yesmaximal, characteristic
3#23761921C212yesmaximal
4#23761285C212yesmaximal
5C79×#1463C211yesmaximal, characteristic
6C2×#14579140C2211yescharacteristic
7C2×#14579120—21no—
8#19098493C412yescomplemented
9#19098483C412yescomplemented
10#19098458—22no—
11#19098436—21no—
12#19098436—21no—
13C4⋊C4×Dic79—24no—
14C4×C22⋊Dic79—21no—
15C4×C22⋊Dic79C2211yescharacteristic
16C4×C22⋊Dic79C2212yes—
17#19098301—24no—
18C22×C4⋊C4×C79C2211yescharacteristic
19#407×C79C2212yes—
20#324×C79—22no—
21C22×Dic158C2×C411yescharacteristic
22#14579140—22no—
23#14579140—22no—
24#14579120—28no—
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—22no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—22no—
30C2×C22⋊Dic79—22no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—21no—
33C2×C4×Dic79—22no—
34C2×C4×Dic79—22no—
35C2×C4×Dic79—22no—
36C2×C4×Dic79—21no—
37C79⋊#95—22no—
38C79⋊#95—22no—
39C79⋊#95—22no—
40C79⋊#95C2×C412yes—
41C79⋊#95D414yes—
42C23×C4×C79D411yescharacteristic
43C23×C4×C79D411yescharacteristic
44C23×C4×C79C2311yescharacteristic
45C2×M16×C79—22no—
46C2×C4⋊C4×C79—22no—
47C2×C4⋊C4×C79—22no—
48C2×C4⋊C4×C79C2×C412yes—
49C2×C4⋊C4×C79D412yes—
50C22⋊C8×C79—22no—
51C22⋊C8×C79—22no—
52C2×Dic158—22no—
53C2×Dic158—81no—
54C22×Dic79—22no—
55C22×Dic79—21no—
56C22×Dic79—21no—
57C22×Dic79—24no—
58C4⋊Dic79—28no—
59C4⋊Dic79—22no—
60Dic79⋊C4—44no—
61Dic79⋊C4—42no—
62Dic79⋊C4—82no—
63C22⋊Dic79—24no—
64C22⋊Dic79—28no—
65C22⋊Dic79—41no—
66C22⋊Dic79—41no—
67C4×Dic79—28no—
68C4×Dic79—82no—
69C4×Dic79—81no—
70C4×Dic79—82no—
71C24×C79C2×D411yescharacteristic
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—24no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—81no—
78C22×C4×C79—81no—
79C22×C4×C79C22⋊C412yes—
80C22×C4×C79C22⋊C412yes—
81C22×C4×C79C22×C411yescharacteristic
82C22×C4×C79C2×D412yes—
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79#4112yes—
85M16×C79—82no—
86C2×C8×C79—42no—
87C4⋊C4×C79—24no—
88C4⋊C4×C79—24no—
89C23×C79—22no—
90C23×C79—22no—
91C23×C79—41no—
92C23×C79C2×C22⋊C411yescharacteristic
93C23×C79#12011yescharacteristic
94C23×C79#12311yescharacteristic
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—24no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—22no—
101C2×C4×C79—42no—
102C2×C4×C79—41no—
103C2×C4×C79—41no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—81no—
108C2×C4×C79#10414yes—
109C2×C4×C79C4×D412yes—
110C2×C4×C79#12112yes—
111C2×Dic79—28no—
112C2×Dic79—24no—
113C2×Dic79—44no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—44no—
117C2×Dic79—42no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—42no—
121Dic158—82no—
122C8×C79—82no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—21no—
127C22×C79—21no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—44no—
131C22×C79#38611yescharacteristic
132C22×C79C2×#10412yes—
133C22×C79#45612yes—
134C4×C79—24no—
135C4×C79—22no—
136C4×C79—22no—
137C4×C79—42no—
138Dic79—48no—
139Dic79—44no—
140Dic79—81no—
141Dic79—81no—
142#9408—791nomaximal, Sylow
143C2×C79—22no—
144C2×C79—22no—
145C2×C79—41no—
146C2×C79#188611yescharacteristic
147C2×C79#189611yescharacteristic
148C2×C79#191311yescharacteristic
149#2583—791no—
150#2274—791no—
151#1508—792no—
152#1463D7911yescharacteristic
153#1297—792no—
154C79#940811yescharacteristic, complemented, Sylow
155C2×C22⋊Q8—791no—
156C2×#117—1581no—
157C22×C4⋊C4C2×D7911yescharacteristic
158#407C2×D7912yes—
159#398—792no—
160#395—792no—
161#389—1582no—
162#385—1581no—
163#385—1581no—
164C4×C4⋊C4—1584no—
165C4×C22⋊C4—791no—
166C4×C22⋊C4—792no—
167C4×C22⋊C4—1581no—
168#328—1584no—
169#324—22no—
170C22×Q8—791no—
171C23×C4D31611yescharacteristic
172C23×C4D31611yescharacteristic
173C23×C4C22×D7911yescharacteristic
174C2×M16—22no—
175C22⋊Q8—1582no—
176C22⋊Q8—1582no—
177#117—1588no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—791no—
181C2×C4⋊C4—1582no—
182C2×C4⋊C4—1581no—
183C2×C4⋊C4—1582no—
184C2×C4⋊C4—1582no—
185C2×C4⋊C4C4×D7912yes—
186C2×C4⋊C4C22⋊D7912yes—
187C2×C22⋊C4—1581no—
188C2×C22⋊C4—1582no—
189C2×C22⋊C4—1581no—
190C2×C42—1582no—
191C2×C42—1582no—
192C2×C42—1581no—
193C2×C42—1582no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—792no—
197#95—794no—
198#95—1582no—
199#95—1582no—
200#95—1582no—
201C24C2×D31611yescharacteristic
202C2×Q8—6321no—
203C2×Q8—1582no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—1581no—
212C22×C4—1581no—
213C22×C4—1582no—
214C22×C4—1584no—
215C22×C4(C2×D79)⋊C412yes—
216C22×C4(C2×D79)⋊C412yes—
217C22×C4C2×C4×D7911yescharacteristic
218C22×C4D4×D7912yes—
219C22×C4C2×C22⋊D7911yescharacteristic
220C22×C4#1192962312yes—
221M16—82no—
222C2×C8—42no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—3164no—
226C4⋊C4—3162no—
227C4⋊C4—6322no—
228C4⋊C4—1588no—
229C4⋊C4—1582no—
230C22⋊C4—3161no—
231C22⋊C4—3161no—
232C22⋊C4—1584no—
233C22⋊C4—1588no—
234C42—6322no—
235C42—6321no—
236C42—6322no—
237C42—1588no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23C2×(C2×D79)⋊C411yescharacteristic
242C23#1457913111yescharacteristic
243C23#1457914511yescharacteristic
244Q8—6322no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—81no—
258C2×C4—3164no—
259C2×C4—3162no—
260C2×C4—3164no—
261C2×C4—3162no—
262C2×C4—3162no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4—3162no—
266C2×C4—1588no—
267C2×C4—1584no—
268C2×C4#1457908714yes—
269C2×C4(C22⋊D79)⋊C412yes—
270C2×C4#1457913312yes—
271C8—82no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1909844011yescharacteristic
281C22C2×#1457908712yes—
282C22#1909874812yes—
283C4—24no—
284C4—22no—
285C4—22no—
286C4—42no—
287C4—3168no—
288C4—3164no—
289C4—6321no—
290C4—6321no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#2376379111yescharacteristic
295C2#2376384111yescharacteristic
296C2#2376394011yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1463C2
socleC22×C79#386
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#1463D79
hypocenterC79#9408
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
8. #19098493C4semidirect
9. #19098483C4semidirect
154. C79#9408semidirect

Maximal subgroups

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

subgroupindex
#237687802
#237657572
#237619212
#237612852
C79×#14632
#940879