Finite Group Databasebeta

← #34186018

Subgroups — #34186018

Order 20224.

554 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 103 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#23767888C211yesmaximal, characteristic
2#23763480C211yesmaximal, characteristic, complemented
3#23761928C212yesmaximal
4#23761252C212yesmaximal, complemented
5C79×#2313C211yesmaximal, characteristic, complemented
6C2×#14579135C2211yescharacteristic
7#19098527C2212yes—
8#19098489—22no—
9#19098464C412yescomplemented
10#19098442—21no—
11#19098442—21no—
12#19098428C412yescomplemented
13C4×C22⋊Dic79C2211yescharacteristic
14#19098332—22no—
15#19098296—24no—
16C22×C4⋊C4×C79C2211yescharacteristic
17#387×C79—24no—
18#381×C79C2212yes—
19#380×C79—21no—
20C4×C22⋊C4×C79—21no—
21C22×D316C2×C411yescharacteristic
22C22×C22⋊D79—21no—
23C2×#11929608—22no—
24#14579135—22no—
25#14579135—22no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C22⋊Dic79—21no—
28C2×(C2×D79)⋊C4—21no—
29C2×(C2×D79)⋊C4—22no—
30C2×(C2×D79)⋊C4—22no—
31C2×C4×Dic79C2×C412yes—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)—22no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—21no—
37C23×C4×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79C2311yescharacteristic
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79C2×C412yes—
44C2×C4⋊C4×C79D412yes—
45C2×C22⋊C4×C79—22no—
46C2×C42×C79—22no—
47#95×C79—22no—
48#95×C79—22no—
49#95×C79—22no—
50#95×C79D414yes—
51C23×D79—21no—
52C2×D316—22no—
53C2×D316—42no—
54C2×C22⋊D79—44no—
55C2×C22⋊D79—41no—
56C2×C22⋊D79—41no—
57C22×Dic79—21no—
58C22×Dic79—22no—
59#11929608—44no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—22no—
62C22⋊Dic79—42no—
63(C2×D79)⋊C4—42no—
64(C2×D79)⋊C4—82no—
65(C2×D79)⋊C4—82no—
66C4×Dic79—42no—
67C24×C79C2×D411yescharacteristic
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—24no—
73C22×C4×C79—42no—
74C22×C4×C79—81no—
75C22×C4×C79—81no—
76C22×C4×C79C22⋊C412yes—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C22×C4×C79C2×D412yes—
81C22×C4×C79#4112yes—
82C4⋊C4×C79—24no—
83C4⋊C4×C79—24no—
84C4⋊C4×C79—82no—
85C22⋊C4×C79—44no—
86C42×C79—82no—
87C23×C79—22no—
88C23×C79—22no—
89C23×C79—41no—
90C23×C79C2×C22⋊C411yescharacteristic
91C23×C79#12011yescharacteristic
92C23×C79#12311yescharacteristic
93C22×D79—42no—
94C22×D79—42no—
95C22×D79—41no—
96C22×D79—42no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—22no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—44no—
108C2×C4×C79—44no—
109C2×C4×C79—44no—
110C2×C4×C79—81no—
111C2×C4×C79—81no—
112C2×C4×C79C4×D412yes—
113C2×C4×C79#12112yes—
114C22⋊D79—82no—
115C22⋊D79—82no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—41no—
120C2×Dic79—42no—
121D316—82no—
122C79⋊C8—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×C79#41012yes—
133C22×C79#45412yes—
134C2×D79—84no—
135C2×D79—81no—
136C2×D79—81no—
137C2×D79—81no—
138C4×C79—22no—
139C4×C79—22no—
140C4×C79—42no—
141C4×C79—42no—
142C4×C79—84no—
143Dic79—82no—
144Dic79—81no—
145#9465—791nomaximal, Sylow
146C2×C79—22no—
147C2×C79—22no—
148C2×C79—41no—
149C2×C79#188811yescharacteristic
150C2×C79#189911yescharacteristic
151C2×C79#191111yescharacteristic
152D79—161no—
153#2504—791no—
154#2313D7911yescharacteristic, complemented
155#1821—791no—
156#1510—792no—
157#1284—792no—
158C79#946511yescharacteristic, complemented, Sylow
159C2×#121—791no—
160C22×C4⋊C4C2×D7911yescharacteristic
161#407—792no—
162#397—1582no—
163#390—792no—
164#387—24no—
165#386—1581no—
166#386—1581no—
167#382—792no—
168#381C2×D7912yes—
169#380—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4—791no—
172#343—1582no—
173#327—1584no—
174C22×D4—791no—
175C22×D4—1581no—
176C23×C4C22⋊D7911yescharacteristic
177C23×C4C22⋊D7911yescharacteristic
178C23×C4C22×D7911yescharacteristic
179C2×M16—1582no—
180#121—1582no—
181#121—1582no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—791no—
186C2×C4⋊C4C4×D7912yes—
187C2×C4⋊C4D31612yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—1581no—
190C2×C22⋊C4—1581no—
191C2×C22⋊C4—1582no—
192C2×C22⋊C4—1582no—
193C2×C42—22no—
194C2×C42—792no—
195C22⋊C8—1582no—
196C22⋊C8—1582no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—1582no—
201#95—1582no—
202#95—1581no—
203#95C22⋊D7914yes—
204C24—1581no—
205C24C2×C22⋊D7911yescharacteristic
206C2×D4—3162no—
207C2×D4—3164no—
208C2×D4—3161no—
209C2×D4—3161no—
210C2×D4—1582no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—1581no—
220C22×C4—1582no—
221C22×C4(C2×D79)⋊C412yes—
222C22×C4(C2×D79)⋊C412yes—
223C22×C4C2×C4×D7911yescharacteristic
224C22×C4C2×C22⋊D7912yes—
225C22×C4C2×D31611yescharacteristic
226C22×C4#1192962412yes—
227M16—3164no—
228C2×C8—3162no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—82no—
232C4⋊C4—1582no—
233C22⋊C4—44no—
234C22⋊C4—3162no—
235C22⋊C4—3162no—
236C22⋊C4—6322no—
237C22⋊C4—6322no—
238C42—82no—
239C42—3162no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—3162no—
244C23—3162no—
245C23—3161no—
246C23—3162no—
247C23C2×(C2×D79)⋊C411yescharacteristic
248C23#1457913011yescharacteristic
249C23#1457914411yescharacteristic
250D4—6322no—
251D4—6322no—
252D4—6322no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—3162no—
269C2×C4—3162no—
270C2×C4—3162no—
271C2×C4—3161no—
272C2×C4—3162no—
273C2×C4C4×C22⋊D7912yes—
274C2×C4#1457913512yes—
275C8—6322no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—6324no—
285C22—6321no—
286C22—6321no—
287C22—6321no—
288C22#1909844211yescharacteristic
289C22#1909853612yes—
290C22#1909873612yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—84no—
296C4—6322no—
297C4—6321no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—12641no—
302C2#2376380311yescharacteristic
303C2#2376385811yescharacteristic
304C2#2376393211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2313C2
socleC22×C79#386
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenter#2313D79
hypocenterC79#9465
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
2. #23763480C2semidirect
4. #23761252C2semidirect
5. C79×#2313C2semidirect
9. #19098464C4semidirect
12. #19098428C4semidirect
154. #2313D79semidirect
158. C79#9465semidirect

Maximal subgroups

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

subgroupindex
#237678882
#237634802
#237619282
#237612522
C79×#23132
#946579