Finite Group Databasebeta

← #34183304

Subgroups — #34183304

Order 20224.

572 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 135 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#23766003C211yesmaximal, characteristic
2#23763490C212yesmaximal
3C2×#19098289C211yesmaximal, characteristic
4#23761279C212yesmaximal
5C2×#400×C79C211yesmaximal, characteristic
6C2×C22⋊(C79⋊C8)C2212yes—
7#19098449—24no—
8#19098426—21no—
9#19098423C2212yes—
10C4×C22⋊Dic79—21no—
11#19098325—24no—
12#19098289—22no—
13C22×C4⋊C4×C79C2211yescharacteristic
14C22×C4⋊C4×C79C2211yescharacteristic
15#400×C79—22no—
16#400×C79C414yescomplemented
17C2×#95×C79—21no—
18C2×#95×C79—21no—
19C2×#95×C79C2211yescharacteristic
20C22×C79⋊C8—22no—
21C2×C4⋊Dic79D414yes—
22C2×C22⋊Dic79—22no—
23C2×C4×Dic79—22no—
24C22⋊(C79⋊C8)—22no—
25C22⋊(C79⋊C8)—22no—
26C79⋊#95—22no—
27C79⋊#95—24no—
28C79⋊#95—22no—
29C23×C4×C79—21no—
30C23×C4×C79—21no—
31C23×C4×C79D411yescharacteristic
32C23×C4×C79D411yescharacteristic
33C23×C4×C79C2311yescharacteristic
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79—22no—
36C2×C4⋊C4×C79—22no—
37C2×C4⋊C4×C79—22no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79C2×C412yes—
40C2×C4⋊C4×C79C2×C412yes—
41C2×C4⋊C4×C79D412yes—
42#95×C79—24no—
43#95×C79—24no—
44#95×C79—21no—
45#95×C79C2×C412yes—
46C22×Dic79—24no—
47C2×C79⋊C8—44no—
48C2×C79⋊C8—42no—
49C4⋊Dic79—44no—
50C22⋊Dic79—42no—
51C22⋊Dic79—42no—
52C4×Dic79—82no—
53C24×C79C2×D411yescharacteristic
54C22×C4×C79—21no—
55C22×C4×C79—21no—
56C22×C4×C79—21no—
57C22×C4×C79—22no—
58C22×C4×C79—22no—
59C22×C4×C79—21no—
60C22×C4×C79—22no—
61C22×C4×C79—22no—
62C22×C4×C79—42no—
63C22×C4×C79—42no—
64C22×C4×C79—42no—
65C22×C4×C79—42no—
66C22×C4×C79—44no—
67C22×C4×C79—41no—
68C22×C4×C79—41no—
69C22×C4×C79—42no—
70C22×C4×C79C22⋊C412yes—
71C22×C4×C79C22⋊C412yes—
72C22×C4×C79C22×C411yescharacteristic
73C22×C4×C79C2×D412yes—
74C22×C4×C79C2×D411yescharacteristic
75C22×C4×C79#4112yes—
76C4⋊C4×C79—24no—
77C4⋊C4×C79—24no—
78C4⋊C4×C79—24no—
79C4⋊C4×C79—42no—
80C23×C79—22no—
81C23×C79—22no—
82C23×C79C2×C22⋊C411yescharacteristic
83C23×C79#12011yescharacteristic
84C23×C79#12311yescharacteristic
85C23×C79#13512yes—
86C23×C79#13712yes—
87C2×C4×C79—24no—
88C2×C4×C79—22no—
89C2×C4×C79—22no—
90C2×C4×C79—22no—
91C2×C4×C79—22no—
92C2×C4×C79—41no—
93C2×C4×C79—41no—
94C2×C4×C79—42no—
95C2×C4×C79—42no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—42no—
101C2×C4×C79—44no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—81no—
105C2×C4×C79—84no—
106C2×C4×C79—82no—
107C2×C4×C79—82no—
108C2×C4×C79C4×D412yes—
109C2×C4×C79#12112yes—
110C2×Dic79—44no—
111C2×Dic79—44no—
112C2×Dic79—44no—
113C79⋊C8—82no—
114C22×C79—24no—
115C22×C79—24no—
116C22×C79—21no—
117C22×C79—21no—
118C22×C79—21no—
119C22×C79—21no—
120C22×C79#38611yescharacteristic
121C22×C79#41611yescharacteristic
122C22×C79#41911yescharacteristic
123C22×C79#45212yes—
124C22×C79#45412yes—
125C22×C79#45812yes—
126C22×C79#48512yes—
127C4×C79—22no—
128C4×C79—22no—
129C4×C79—42no—
130C4×C79—42no—
131C4×C79—82no—
132C4×C79—82no—
133Dic79—84no—
134#9022—791nomaximal, Sylow
135C2×C79—21no—
136C2×C79—21no—
137C2×C79—22no—
138C2×C79#161514yes—
139C2×C79#187311yescharacteristic
140C2×C79#188811yescharacteristic
141C2×C79#189011yescharacteristic
142C2×#400D7911yescharacteristic
143#2313—791no—
144#1824—792no—
145C2×#324—791no—
146#1294—792no—
147C79#902211yescharacteristic, complemented, Sylow
148C22×C4⋊C4C2×D7911yescharacteristic
149C22×C4⋊C4C2×D7911yescharacteristic
150C2×C22⋊C8—792no—
151#400—22no—
152#400Dic7914yescomplemented
153#387—1584no—
154#381—1581no—
155#380—792no—
156C4×C22⋊C4—1581no—
157C2×#95—21no—
158C2×#95—21no—
159C2×#95C2×D7911yescharacteristic
160#340—1584no—
161#324—1582no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22⋊D7911yescharacteristic
165C23×C4C22⋊D7911yescharacteristic
166C23×C4C22×D7911yescharacteristic
167C22×C8—1582no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—794no—
174C2×C4⋊C4C2×Dic7912yes—
175C2×C4⋊C4C2×Dic7912yes—
176C2×C4⋊C4C22⋊D7912yes—
177C2×C22⋊C4—1582no—
178C2×C42—1582no—
179C22⋊C8—1582no—
180C22⋊C8—1582no—
181#95—24no—
182#95—24no—
183#95—21no—
184#95—1582no—
185#95—1584no—
186#95—1582no—
187#95C2×Dic7912yes—
188C24C2×C22⋊D7911yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—1584no—
206C22×C4C22⋊Dic7912yes—
207C22×C4C22⋊Dic7912yes—
208C22×C4C22×Dic7911yescharacteristic
209C22×C4D4×D7912yes—
210C22×C4C2×C22⋊D7911yescharacteristic
211C22×C4#1192962312yes—
212C2×C8—3164no—
213C2×C8—3162no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—42no—
218C4⋊C4—3164no—
219C22⋊C4—3162no—
220C22⋊C4—3162no—
221C42—6322no—
222C23—22no—
223C23—22no—
224C23C2×C22⋊Dic7911yescharacteristic
225C23#1457912911yescharacteristic
226C23#1457914211yescharacteristic
227C23#1457918112yes—
228C23#1457919212yes—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—3164no—
251C2×C4—3164no—
252C2×C4—3164no—
253C2×C4D4×Dic7912yes—
254C2×C4#1457913312yes—
255C8—6322no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—24no—
261C22—24no—
262C22#1909844311yescharacteristic
263C22#1909855911yescharacteristic
264C22#1909857211yescharacteristic
265C22#1909872312yes—
266C22#1909873512yes—
267C22#1909875212yes—
268C22#1909890112yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—82no—
275C4—6324no—
276C2—21no—
277C2—22no—
278C2—21no—
279C2#2376242914yes—
280C2#2376371311yescharacteristic
281C2#2376380411yescharacteristic
282C2#2376381411yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#400×C79C2
socleC23×C79#120
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenterC2×#400D79
hypocenterC79#9022
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
16. #400×C79C4semidirect
147. C79#9022semidirect
152. #400Dic79semidirect

Maximal subgroups

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

subgroupindex
#237660032
#237634902
C2×#190982892
#237612792
C2×#400×C792
#902279