Finite Group Databasebeta

← #34200480

Subgroups — #34200480

Order 20224.

458 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764531C211yesmaximal, characteristic
2#23762419C212yesmaximal
3#23761424C212yesmaximal, complemented
4#23761330C211yesmaximal, characteristic, complemented
5C79×#2607C211yesmaximal, characteristic, complemented
6#19099125C2211yescharacteristic
7C2×#14579144—21no—
8#19098881—22no—
9C2×#14579075—21no—
10#19098330C2211yescharacteristic
11#19098296—22no—
12#19098287C2212yes—
13#399×C79C2212yes—
14#393×C79—21no—
15#391×C79—22no—
16#387×C79—22no—
17#381×C79C2211yescharacteristic
18C4×C22⋊C4×C79—21no—
19C22×C22⋊D79—21no—
20#14579145—41no—
21#14579144—42no—
22#14579144—42no—
23#14579135—22no—
24#14579131—41no—
25C2×Dic79⋊C4—41no—
26C2×C22⋊Dic79—21no—
27C2×(C2×D79)⋊C4—41no—
28#14579092—42no—
29#14579081—42no—
30#14579075—42no—
31C22⋊(C79⋊C8)—22no—
32C22×Q8×C79—21no—
33C23×C4×C79D411yescharacteristic
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79D412yes—
36C2×C22⋊C4×C79—22no—
37C2×C22⋊C4×C79D411yescharacteristic
38C2×C22⋊C4×C79C2311yescharacteristic
39C2×C42×C79—21no—
40C2×C42×C79—21no—
41#95×C79—22no—
42#95×C79—21no—
43#95×C79—22no—
44#95×C79—22no—
45#95×C79D412yes—
46C23×D79—21no—
47C2×D316—22no—
48C2×C22⋊D79—41no—
49C2×C22⋊D79—41no—
50C2×C22⋊D79—41no—
51C2×C22⋊D79—41no—
52C2×C22⋊D79—41no—
53C2×C22⋊D79—41no—
54C22×Dic79—21no—
55C22×Dic79—41no—
56C2×C79⋊C8—42no—
57C4⋊Dic79—22no—
58Dic79⋊C4—44no—
59C22⋊Dic79—22no—
60C22⋊Dic79—41no—
61(C2×D79)⋊C4—42no—
62(C2×D79)⋊C4—44no—
63C24×C79C2×D411yescharacteristic
64C2×Q8×C79—44no—
65C2×Q8×C79—81no—
66C22×C4×C79—21no—
67C22×C4×C79—22no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—22no—
71C22×C4×C79—41no—
72C22×C4×C79—41no—
73C22×C4×C79SD1612yes—
74C22×C4×C79C2×D412yes—
75C4⋊C4×C79—44no—
76C4⋊C4×C79—42no—
77C22⋊C4×C79—41no—
78C22⋊C4×C79—44no—
79C22⋊C4×C79—41no—
80C42×C79—81no—
81C42×C79—81no—
82C23×C79—21no—
83C23×C79—21no—
84C23×C79—42no—
85C23×C79#12011yescharacteristic
86C23×C79C2×SD1611yescharacteristic
87C23×C79#13711yescharacteristic
88C22×D79—41no—
89C22×D79—41no—
90C22×D79—41no—
91C22×D79—82no—
92Q8×C79—84no—
93C2×C4×C79—24no—
94C2×C4×C79—22no—
95C2×C4×C79—22no—
96C2×C4×C79—42no—
97C2×C4×C79—41no—
98C2×C4×C79—42no—
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—41no—
106C2×C4×C79—42no—
107C2×C4×C79—81no—
108C22⋊D79—81no—
109C22⋊D79—81no—
110C22⋊D79—81no—
111C22⋊D79—81no—
112C2×Dic79—42no—
113C2×Dic79—41no—
114C2×Dic79—41no—
115C2×Dic79—41no—
116C2×Dic79—82no—
117D316—82no—
118C79⋊C8—82no—
119C22×C79—22no—
120C22×C79—21no—
121C22×C79—21no—
122C22×C79—41no—
123C22×C79—41no—
124C22×C79—42no—
125C22×C79—81no—
126C22×C79#44811yescharacteristic
127C22×C79#45311yescharacteristic
128C22×C79B(4,𝔽2)11yescharacteristic
129C2×D79—82no—
130C2×D79—81no—
131C2×D79—82no—
132C2×D79—161no—
133C4×C79—44no—
134C4×C79—42no—
135C4×C79—42no—
136C4×C79—81no—
137C4×C79—82no—
138Dic79—82no—
139Dic79—81no—
140#12122—791nomaximal, Sylow
141C2×C79—21no—
142C2×C79—21no—
143C2×C79—41no—
144C2×C79—41no—
145C2×C79#159611yescharacteristic
146C2×C79#219812yes—
147D79—161no—
148#2607D7911yescharacteristic, complemented
149#2022—791no—
150#1613—792no—
151#1352—792no—
152#1317—791no—
153C79#1212211yescharacteristic, complemented, Sylow
154#534—791no—
155C2×#123—1581no—
156#481—1582no—
157C2×#99—1581no—
158#399C2×D7912yes—
159#393—21no—
160#391—22no—
161#387—22no—
162#381C2×D7911yescharacteristic
163C4×C22⋊C4—21no—
164#342—791no—
165#327—1582no—
166#323—792no—
167C22×Q8—21no—
168C22×D4—1581no—
169C23×C4C22⋊D7911yescharacteristic
170#123—3161no—
171#123—3162no—
172#123—3162no—
173#121—1582no—
174#120—3161no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—3161no—
177C2×C4⋊C4C22⋊D7912yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—3161no—
180C2×C22⋊C4—1581no—
181C2×C22⋊C4C22⋊D7911yescharacteristic
182C2×C22⋊C4C22×D7911yescharacteristic
183C2×C42—21no—
184C2×C42—21no—
185#106—3162no—
186#102—3162no—
187#99—3162no—
188C22⋊C8—1582no—
189#95—22no—
190#95—21no—
191#95—22no—
192#95—22no—
193#95C22⋊D7912yes—
194C24—1581no—
195C24C2×C22⋊D7911yescharacteristic
196C2×Q8—44no—
197C2×Q8—81no—
198C2×D4—3161no—
199C2×D4—3161no—
200C2×D4—3161no—
201C2×D4—3161no—
202C2×D4—3161no—
203C2×D4—3161no—
204C2×D4—1582no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—3161no—
213C22×C4—1581no—
214C22×C4#1192961112yes—
215C22×C4C2×C22⋊D7912yes—
216C2×C8—3162no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—3164no—
220C4⋊C4—1582no—
221C22⋊C4—41no—
222C22⋊C4—44no—
223C22⋊C4—41no—
224C22⋊C4—3162no—
225C22⋊C4—3164no—
226C22⋊C4—3161no—
227C22⋊C4—1582no—
228C42—81no—
229C42—81no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—3161no—
234C23—3161no—
235C23—3161no—
236C23—6322no—
237C23#1457913011yescharacteristic
238C23C2×#1192961111yescharacteristic
239C23#1457919211yescharacteristic
240Q8—84no—
241D4—6322no—
242D4—6321no—
243D4—6321no—
244D4—6321no—
245D4—6321no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—81no—
261C2×C4—3162no—
262C2×C4—3161no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4—6322no—
266C8—6322no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—6322no—
275C22—6321no—
276C22—6322no—
277C22—12641no—
278C22#1909870211yescharacteristic
279C22#1909873111yescharacteristic
280C22#1909875011yescharacteristic
281C4—44no—
282C4—42no—
283C4—42no—
284C4—81no—
285C4—82no—
286C4—6322no—
287C4—6321no—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2—12641no—
293C2#2376232311yescharacteristic
294C2#2376539612yes—
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2607C2
socleC22×C79B(4,𝔽2)
prime radical#19099125C22
hypercenter#2607D79
hypocenterC79#12122
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
3. #23761424C2semidirect
4. #23761330C2semidirect
5. C79×#2607C2semidirect
148. #2607D79semidirect
153. C79#12122semidirect

Maximal subgroups

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

subgroupindex
#237645312
#237624192
#237614242
#237613302
C79×#26072
#1212279