Finite Group Databasebeta

← #34186013

Subgroups — #34186013

Order 20224.

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

normal subgroups only

This group has 610 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#23767867C211yesmaximal, characteristic
2#23763480C211yesmaximal, characteristic, complemented
3#23761928C212yesmaximal
4#23761252C212yesmaximal, complemented
5C79×#2313C211yesmaximal, characteristic, complemented
6C2×#14579144—21no—
7C2×#14579144—21no—
8C2×#14579135C2211yescharacteristic
9#19098527C2212yes—
10#19098498—22no—
11#19098464C412yescomplemented
12#19098432C412yescomplemented
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#14579144—44no—
25#14579144—44no—
26#14579135—22no—
27#14579135—22no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×Dic79⋊C4—22no—
30C2×Dic79⋊C4—22no—
31C2×C22⋊Dic79—21no—
32C2×(C2×D79)⋊C4—22no—
33C2×(C2×D79)⋊C4—22no—
34C2×(C2×D79)⋊C4—21no—
35C2×C4×Dic79C2×C412yes—
36C22⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—22no—
38C79⋊#95—21no—
39C23×C4×C79D411yescharacteristic
40C23×C4×C79D411yescharacteristic
41C23×C4×C79C2311yescharacteristic
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79—22no—
45C2×C4⋊C4×C79C2×C412yes—
46C2×C4⋊C4×C79D412yes—
47C2×C22⋊C4×C79—22no—
48C2×C42×C79—22no—
49#95×C79—22no—
50#95×C79—22no—
51#95×C79—22no—
52#95×C79D414yes—
53C23×D79—21no—
54C2×D316—22no—
55C2×D316—42no—
56C2×C22⋊D79—42no—
57C2×C22⋊D79—42no—
58C2×C22⋊D79—41no—
59C2×C22⋊D79—41no—
60C22×Dic79—22no—
61C22×Dic79—21no—
62#11929608—44no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—22no—
65Dic79⋊C4—44no—
66Dic79⋊C4—44no—
67C22⋊Dic79—24no—
68(C2×D79)⋊C4—44no—
69(C2×D79)⋊C4—44no—
70(C2×D79)⋊C4—42no—
71C4×Dic79—42no—
72C24×C79C2×D411yescharacteristic
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—24no—
78C22×C4×C79—42no—
79C22×C4×C79—42no—
80C22×C4×C79—42no—
81C22×C4×C79C22⋊C412yes—
82C22×C4×C79C22⋊C412yes—
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D412yes—
86C22×C4×C79#4112yes—
87C4⋊C4×C79—24no—
88C4⋊C4×C79—24no—
89C4⋊C4×C79—82no—
90C22⋊C4×C79—44no—
91C42×C79—82no—
92C23×C79—22no—
93C23×C79—22no—
94C23×C79—41no—
95C23×C79C2×C22⋊C411yescharacteristic
96C23×C79#12011yescharacteristic
97C23×C79#12311yescharacteristic
98C22×D79—41no—
99C22×D79—41no—
100C22×D79—42no—
101C22×D79—41no—
102C22×D79—42no—
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—42no—
108C2×C4×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—44no—
114C2×C4×C79—44no—
115C2×C4×C79—44no—
116C2×C4×C79—81no—
117C2×C4×C79—81no—
118C2×C4×C79C4×D412yes—
119C2×C4×C79#12112yes—
120C22⋊D79—82no—
121C22⋊D79—82no—
122C2×Dic79—24no—
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128D316—82no—
129C79⋊C8—82no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—44no—
138C22×C79#38611yescharacteristic
139C22×C79#41012yes—
140C22×C79#45312yes—
141C2×D79—82no—
142C2×D79—82no—
143C2×D79—81no—
144C2×D79—81no—
145C2×D79—81no—
146C4×C79—22no—
147C4×C79—22no—
148C4×C79—42no—
149C4×C79—42no—
150C4×C79—84no—
151Dic79—44no—
152Dic79—81no—
153#9464—791nomaximal, Sylow
154C2×C79—22no—
155C2×C79—22no—
156C2×C79—41no—
157C2×C79#188711yescharacteristic
158C2×C79#189911yescharacteristic
159C2×C79#191211yescharacteristic
160D79—161no—
161#2501—791no—
162#2313D7911yescharacteristic, complemented
163#1821—791no—
164#1510—792no—
165#1284—792no—
166C79#946411yescharacteristic, complemented, Sylow
167C2×#123—1581no—
168C2×#123—1581no—
169C2×#121—791no—
170C22×C4⋊C4C2×D7911yescharacteristic
171#407—792no—
172#399—1582no—
173#390—792no—
174#387—24no—
175#384—792no—
176#381C2×D7912yes—
177#380—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—791no—
180#343—1582no—
181#327—1584no—
182C22×D4—791no—
183C22×D4—1581no—
184C23×C4C22⋊D7911yescharacteristic
185C23×C4C22⋊D7911yescharacteristic
186C23×C4C22×D7911yescharacteristic
187C2×M16—1582no—
188#123—3164no—
189#123—3164no—
190#121—1582no—
191#121—1582no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—791no—
196C2×C4⋊C4—1582no—
197C2×C4⋊C4—1582no—
198C2×C4⋊C4C4×D7912yes—
199C2×C4⋊C4D31612yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—1582no—
202C2×C22⋊C4—1582no—
203C2×C22⋊C4—1581no—
204C2×C22⋊C4—1581no—
205C2×C42—22no—
206C2×C42—792no—
207C22⋊C8—1582no—
208C22⋊C8—1582no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—1581no—
213#95C22⋊D7914yes—
214C24—1581no—
215C24C2×C22⋊D7911yescharacteristic
216C2×D4—3162no—
217C2×D4—3162no—
218C2×D4—3162no—
219C2×D4—3161no—
220C2×D4—3161no—
221C2×D4—1582no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—1581no—
231C22×C4—1582no—
232C22×C4(C2×D79)⋊C412yes—
233C22×C4(C2×D79)⋊C412yes—
234C22×C4C2×C4×D7911yescharacteristic
235C22×C4C2×C22⋊D7912yes—
236C22×C4C2×D31611yescharacteristic
237C22×C4#1192962412yes—
238M16—3164no—
239C2×C8—3162no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—82no—
243C4⋊C4—3164no—
244C4⋊C4—3164no—
245C4⋊C4—1582no—
246C22⋊C4—44no—
247C22⋊C4—3164no—
248C22⋊C4—3164no—
249C22⋊C4—3162no—
250C22⋊C4—1584no—
251C42—82no—
252C42—3162no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—3161no—
257C23—3161no—
258C23—3162no—
259C23—3161no—
260C23—3162no—
261C23C2×(C2×D79)⋊C411yescharacteristic
262C23#1457913011yescharacteristic
263C23#1457914411yescharacteristic
264D4—6322no—
265D4—6322no—
266D4—6322no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—3162no—
283C2×C4—3162no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—1584no—
288C2×C4C4×C22⋊D7912yes—
289C2×C4#1457913512yes—
290C8—6322no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—6322no—
300C22—6322no—
301C22—6321no—
302C22—6321no—
303C22—6321no—
304C22#1909844211yescharacteristic
305C22#1909853612yes—
306C22#1909873112yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—84no—
312C4—3164no—
313C4—6321no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—12641no—
318C2#2376379811yescharacteristic
319C2#2376385811yescharacteristic
320C2#2376393711yescharacteristic
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#9464
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
11. #19098464C4semidirect
12. #19098432C4semidirect
162. #2313D79semidirect
166. C79#9464semidirect

Maximal subgroups

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

subgroupindex
#237678672
#237634802
#237619282
#237612522
C79×#23132
#946479