Finite Group Databasebeta

← #34180807

Subgroups — #34180807

Order 20224.

784 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 131 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#23762114C212yesmaximal, complemented
2#23762027C212yesmaximal, complemented
3#23761734C212yesmaximal, complemented
4C2×#540×C79C211yesmaximal, characteristic
5#19098620C2214yes—
6#19098307—24no—
7#19098305—24no—
8#19098289—22no—
9#19098287—22no—
10#540×C79—24no—
11#540×C79—24no—
12C2×#127×C79C2211yescharacteristic
13C2×#125×C79C2211yescharacteristic
14C2×#123×C79—22no—
15C2×#121×C79—22no—
16C2×#121×C79C412yes—
17C2×C4×D4×C79C412yes—
18C2×#117×C79C2211yescharacteristic
19C4⋊(C79⋊C8)—24no—
20C22⋊(C79⋊C8)—24no—
21#14579072—24no—
22C22×D4×C79—22no—
23C22×D4×C79—22no—
24C22×D4×C79C2×C412yes—
25C23×C4×C79—22no—
26C23×C4×C79C2×C411yescharacteristic
27#127×C79—22no—
28#127×C79D414yes—
29#125×C79—22no—
30#125×C79D414yes—
31#123×C79—216no—
32#121×C79—216no—
33#121×C79—24no—
34#121×C79—24no—
35C4×D4×C79—24no—
36C4×D4×C79—24no—
37#117×C79—22no—
38#117×C79—22no—
39C2×C4⋊C4×C79—22no—
40C2×C4⋊C4×C79C2×C412yes—
41C2×C4⋊C4×C79D412yes—
42C2×C22⋊C4×C79—22no—
43C2×C22⋊C4×C79—22no—
44C2×C22⋊C4×C79D412yes—
45C2×C42×C79C2×C411yescharacteristic
46C2×C42×C79C2311yescharacteristic
47C2×C79⋊C8—44no—
48C24×C79—22no—
49C24×C79C22⋊C411yescharacteristic
50C2×D4×C79—28no—
51C2×D4×C79—22no—
52C2×D4×C79—28no—
53C2×D4×C79—44no—
54C2×D4×C79—84no—
55C2×D4×C79—84no—
56C2×D4×C79C22⋊C414yes—
57C22×C4×C79—22no—
58C22×C4×C79—28no—
59C22×C4×C79—21no—
60C22×C4×C79—21no—
61C22×C4×C79—42no—
62C22×C4×C79—84no—
63C22×C4×C79C22⋊C411yescharacteristic
64C22×C4×C79C22⋊C412yes—
65C22×C4×C79C22×C411yescharacteristic
66C22×C4×C79C2×D412yes—
67C4⋊C4×C79—22no—
68C4⋊C4×C79—28no—
69C4⋊C4×C79—24no—
70C4⋊C4×C79C22⋊C414yes—
71C22⋊C4×C79—28no—
72C22⋊C4×C79—28no—
73C22⋊C4×C79—24no—
74C42×C79—21no—
75C42×C79—21no—
76C42×C79C2×D412yes—
77C42×C79C2×D412yes—
78C23×C79—21no—
79C23×C79—24no—
80C23×C79—22no—
81C23×C79—44no—
82C23×C79—44no—
83C23×C79—42no—
84C23×C79—44no—
85C23×C79C2×C22⋊C411yescharacteristic
86D4×C79—44no—
87D4×C79—44no—
88D4×C79—88no—
89D4×C79—88no—
90C2×C4×C79—22no—
91C2×C4×C79—22no—
92C2×C4×C79—21no—
93C2×C4×C79—24no—
94C2×C4×C79—22no—
95C2×C4×C79—42no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79—44no—
99C2×C4×C79—42no—
100C2×C4×C79—88no—
101C2×C4×C79C2×C22⋊C411yescharacteristic
102C2×C4×C79C2×C22⋊C411yescharacteristic
103C2×C4×C79#12014yes—
104C79⋊C8—84no—
105C22×C79—22no—
106C22×C79—44no—
107C22×C79—42no—
108C22×C79—44no—
109C22×C79—41no—
110C22×C79—44no—
111C22×C79—82no—
112C22×C79—88no—
113C22×C79#37911yescharacteristic
114C22×C79#41312yes—
115C4×C79—24no—
116C4×C79—21no—
117C4×C79—21no—
118C4×C79—44no—
119C4×C79—42no—
120C4×C79—41no—
121#8526—791nomaximal, Sylow
122C2×C79—22no—
123C2×C79—81no—
124C2×C79—84no—
125C2×C79#178511yescharacteristic
126C2×C79#179512yes—
127C2×#540D7911yescharacteristic
128#1555—792no—
129#1534—792no—
130#1464—792no—
131C79#852611yescharacteristic, complemented, Sylow
132#540—24no—
133#540—24no—
134C2×#127C2×D7911yescharacteristic
135C2×#125C2×D7911yescharacteristic
136C2×#123—22no—
137C2×#121—22no—
138C2×#121Dic7912yes—
139C2×C4×D4Dic7912yes—
140C2×#117C2×D7911yescharacteristic
141#432—794no—
142#331—1584no—
143#330—1584no—
144#324—1582no—
145#323—1582no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×Dic7912yes—
149C23×C4—22no—
150C23×C4C2×Dic7911yescharacteristic
151#127—22no—
152#127C22⋊D7914yes—
153#125—22no—
154#125C22⋊D7914yes—
155#123—216no—
156#121—216no—
157#121—24no—
158#121—24no—
159C4×D4—24no—
160C4×D4—24no—
161#117—22no—
162#117—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C2×Dic7912yes—
165C2×C4⋊C4C22⋊D7912yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4C22⋊D7912yes—
169C2×C42C2×Dic7911yescharacteristic
170C2×C42C22×D7911yescharacteristic
171C4⋊C8—1584no—
172C22⋊C8—1584no—
173#97—1584no—
174C24—22no—
175C24C22⋊Dic7911yescharacteristic
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—44no—
180C2×D4—84no—
181C2×D4—84no—
182C2×D4C22⋊Dic7914yes—
183C22×C4—22no—
184C22×C4—28no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—84no—
189C22×C4C22⋊Dic7911yescharacteristic
190C22×C4C22⋊Dic7912yes—
191C22×C4C22×Dic7911yescharacteristic
192C22×C4C2×C22⋊D7912yes—
193C2×C8—3164no—
194C4⋊C4—22no—
195C4⋊C4—28no—
196C4⋊C4—24no—
197C4⋊C4C22⋊Dic7914yes—
198C22⋊C4—28no—
199C22⋊C4—28no—
200C22⋊C4—24no—
201C42—21no—
202C42—21no—
203C42C2×C22⋊D7912yes—
204C42C2×C22⋊D7912yes—
205C23—21no—
206C23—24no—
207C23—22no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23—44no—
212C23C2×C22⋊Dic7911yescharacteristic
213D4—44no—
214D4—44no—
215D4—88no—
216D4—88no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—88no—
228C2×C4C2×C22⋊Dic7911yescharacteristic
229C2×C4C2×C22⋊Dic7911yescharacteristic
230C2×C4#1457913014yes—
231C8—6324no—
232C22—22no—
233C22—41no—
234C22—44no—
235C22—44no—
236C22—44no—
237C22—42no—
238C22—82no—
239C22—88no—
240C22#1909841911yescharacteristic
241C22#1909854712yes—
242C4—21no—
243C4—21no—
244C4—24no—
245C4—41no—
246C4—44no—
247C4—42no—
248C2—22no—
249C2—84no—
250C2—81no—
251C2#2376332811yescharacteristic
252C2#2376337012yes—
subgroupG/H
centerC22#19098419
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC2×#540×C79C2
socleC22×C79#379
cosocleC2×C42×C79C23
hypercenterC2×#540D79
hypocenterC79#8526
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. #23762114C2semidirect
2. #23762027C2semidirect
3. #23761734C2semidirect
131. C79#8526semidirect

Maximal subgroups

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

subgroupindex
#237621142
#237620272
#237617342
C2×#540×C792
#852679