Finite Group Databasebeta

← #34205068

Subgroups — #34205068

Order 20224.

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

normal subgroups only

This group has 536 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
1C2×#19098358C211yesmaximal, characteristic
2#23763789C211yesmaximal, characteristic
3#23763680C212yesmaximal
4#23761518C212yesmaximal
5C79×#2372C211yesmaximal, characteristic
6#19099038C2211yescharacteristic
7#19098572—21no—
8C2×#14579078C2211yescharacteristic
9C2×#14579076C2212yes—
10#19098436—22no—
11#19098358C414yescomplemented
12#19098329—22no—
13#19098323—21no—
14C2×#124×C79C2211yescharacteristic
15#391×C79—21no—
16#388×C79—22no—
17#386×C79C2212yes—
18C4×C22⋊C4×C79—22no—
19#376×C79—21no—
20C2×#11929608—21no—
21D4×Dic79—42no—
22#14579120—21no—
23C2×Dic79⋊C4—42no—
24C2×C22⋊Dic79—22no—
25C2×C22⋊Dic79—22no—
26#14579085—42no—
27#14579078—22no—
28#14579076—22no—
29#14579076C2×C414yes—
30C79⋊#95—42no—
31C22×Q8×C79D411yescharacteristic
32C22×D4×C79C2311yescharacteristic
33C23×C4×C79D412yes—
34#124×C79—22no—
35#124×C79C2×C412yes—
36#124×C79Q812yes—
37C2×C4⋊C4×C79—21no—
38C2×C22⋊C4×C79—22no—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79D412yes—
41C2×C42×C79—22no—
42C2×C42×C79—21no—
43C2×C42×C79D411yescharacteristic
44#95×C79—21no—
45#95×C79—21no—
46#95×C79—22no—
47#95×C79—22no—
48C22×Dic79—22no—
49C22×Dic79—22no—
50C22×Dic79—42no—
51#11929608—41no—
52#11929608—41no—
53C2×C79⋊C8—41no—
54C4⋊Dic79—22no—
55Dic79⋊C4—84no—
56C22⋊Dic79—24no—
57C22⋊Dic79—42no—
58C22⋊Dic79—42no—
59C22⋊Dic79—41no—
60C4×Dic79—41no—
61C24×C79C2×D412yes—
62C2×Q8×C79—21no—
63C2×Q8×C79—41no—
64C2×Q8×C79—41no—
65C2×Q8×C79C4⋊C412yescomplemented
66C2×D4×C79—41no—
67C2×D4×C79—41no—
68C2×D4×C79C22×C411yescharacteristic
69C2×D4×C79C2×Q811yescharacteristic
70C2×D4×C79#4112yes—
71C22×C4×C79—22no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—22no—
75C22×C4×C79—44no—
76C22×C4×C79—42no—
77C22×C4×C79C2×D411yescharacteristic
78C4⋊C4×C79—81no—
79C22⋊C4×C79—24no—
80C22⋊C4×C79—24no—
81C22⋊C4×C79—44no—
82C22⋊C4×C79—42no—
83C42×C79—21no—
84C42×C79—44no—
85C42×C79—81no—
86C42×C79C4⋊C412yescomplemented
87C23×C79—22no—
88C23×C79—22no—
89C23×C79—42no—
90C23×C79—42no—
91C23×C79C4×D412yes—
92C23×C79#12011yescharacteristic
93C23×C79C22⋊Q812yes—
94Q8×C79—42no—
95Q8×C79—81no—
96D4×C79—41no—
97D4×C79—41no—
98D4×C79—42no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—24no—
102C2×C4×C79—22no—
103C2×C4×C79—22no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—44no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—82no—
114C2×C4×C79C2×C4⋊C411yescharacteristic
115C2×C4×C79#12311yescharacteristic
116C2×Dic79—44no—
117C2×Dic79—42no—
118C2×Dic79—82no—
119C2×Dic79—82no—
120C2×Dic79—82no—
121C2×Dic79—82no—
122C79⋊C8—81no—
123C22×C79—22no—
124C22×C79—22no—
125C22×C79—22no—
126C22×C79—42no—
127C22×C79—42no—
128C22×C79—42no—
129C22×C79—42no—
130C22×C79—82no—
131C22×C79#38511yescharacteristic
132C22×C79B(4,𝔽2)11yescharacteristic
133C22×C79#45811yescharacteristic
134C4×C79—21no—
135C4×C79—21no—
136C4×C79—42no—
137C4×C79—42no—
138C4×C79—44no—
139C4×C79—82no—
140Dic79—84no—
141Dic79—82no—
142#12965—791nomaximal, Sylow
143C2×C79—21no—
144C2×C79—21no—
145C2×C79—42no—
146C2×C79—42no—
147C2×C79#186711yescharacteristic
148C2×C79#219712yes—
149#2372D7911yescharacteristic
150C2×#354—791no—
151#1885—791no—
152#1867—792no—
153#1390—792no—
154C79#1296511yescharacteristic, complemented, Sylow
155C2×#124C2×D7911yescharacteristic
156#518—791no—
157#419—1581no—
158C2×#100—791no—
159C2×#99—792no—
160#391—21no—
161#388—22no—
162#386C2×D7912yes—
163#385—1582no—
164C4×C22⋊C4—22no—
165#376—21no—
166#354—794no—
167#342—1582no—
168#339—1581no—
169C22×Q8C22⋊D7911yescharacteristic
170C22×D4C22×D7911yescharacteristic
171C23×C4C22⋊D7912yes—
172C2×M16—1581no—
173#124—22no—
174#124Dic15812yes—
175#124C4×D7912yes—
176C4×D4—3162no—
177#117—1581no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—3162no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1582no—
183C2×C22⋊C4—1582no—
184C2×C22⋊C4C22⋊D7912yes—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42C22⋊D7911yescharacteristic
188#103—3162no—
189#100—1582no—
190#99—794no—
191#99—1582no—
192#95—21no—
193#95—21no—
194#95—22no—
195#95—22no—
196#95—3162no—
197C24C2×C22⋊D7912yes—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8Dic79⋊C412yescomplemented
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×C4×D7911yescharacteristic
205C2×D4C2×Dic15811yescharacteristic
206C2×D4#1192962412yes—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—3162no—
214C22×C4—1582no—
215C22×C4—1582no—
216C22×C4C2×C22⋊D7911yescharacteristic
217M16—3161no—
218M16—3161no—
219C2×C8—3161no—
220C4⋊C4—81no—
221C4⋊C4—6324no—
222C4⋊C4—1582no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—3162no—
228C22⋊C4—3162no—
229C22⋊C4—3161no—
230C22⋊C4—1584no—
231C42—21no—
232C42—44no—
233C42—81no—
234C42—3161no—
235C42Dic79⋊C412yescomplemented
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23C4×C22⋊D7912yes—
241C23#1457913011yescharacteristic
242C23#1457914012yes—
243Q8—42no—
244Q8—81no—
245D4—41no—
246D4—41no—
247D4—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—3164no—
264C2×C4—3162no—
265C2×C4—6322no—
266C2×C4—6322no—
267C2×C4—6322no—
268C2×C4—6322no—
269C2×C4C2×Dic79⋊C411yescharacteristic
270C2×C4#1457914411yescharacteristic
271C8—6321no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—82no—
280C22#1909843611yescharacteristic
281C22#1909875011yescharacteristic
282C22#1909875311yescharacteristic
283C4—21no—
284C4—21no—
285C4—42no—
286C4—44no—
287C4—42no—
288C4—82no—
289C4—6324no—
290C4—6322no—
291C2—21no—
292C2—21no—
293C2—42no—
294C2—42no—
295C2#2376368111yescharacteristic
296C2#2376539012yes—
subgroupG/H
centerC22#19098750
commutatorC2×D4×C79C22×C4
FrattiniC22×D4C22×D79
FittingC79×#2372C2
socleC22×C79B(4,𝔽2)
prime radical, cosocleC22×D4×C79C23
hypercenter#2372D79
hypocenterC79#12965
LCSG ▷ C2×D4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2372

Complements

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

NH ≅ G/Ntype
11. #19098358C4semidirect
65. C2×Q8×C79C4⋊C4semidirect
86. C42×C79C4⋊C4semidirect
154. C79#12965semidirect
201. C2×Q8Dic79⋊C4semidirect
235. C42Dic79⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#190983582
#237637892
#237636802
#237615182
C79×#23722
#1296579