Finite Group Databasebeta

← #34101220

Subgroups — #34101220

Order 20096.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#14578943C211yesmaximal, characteristic
2C22×#14578941C211yesmaximal, characteristic
3C2×#19097769C211yesmaximal, characteristic, complemented
4C2×#19097694C211yesmaximal, characteristic, complemented
5#23757174C211yesmaximal, characteristic
6#23757027C211yesmaximal, characteristic, complemented
7C2×#103×C157C211yesmaximal, characteristic, complemented
8C22×Dic314C2211yescharacteristic
9C22×D628C2211yescharacteristic
10C2×#14578943—22no—
11C2×#14578943—22no—
12C2×#14578943—22no—
13C2×#14578941—22no—
14C2×#14578941—22no—
15C2×#14578941—22no—
16C22×#11929434C2211yescharacteristic
17#19097769—22no—
18#19097769—22no—
19C2×#14578933—22no—
20C2×#14578931—22no—
21C2×C4×Dic157C2211yescharacteristic
22#19097694—22no—
23C22×Q8×C157C2211yescharacteristic
24C22×C8×C157C2211yescharacteristic
25C2×C4⋊C4×C157C2211yescharacteristic
26#103×C157—22no—
27C23×D157—21no—
28C2×Dic314—21no—
29C2×Dic314—21no—
30C2×Dic314—21no—
31C2×Dic314—21no—
32C2×Dic314—21no—
33C2×Dic314D412yes—
34C2×D628—21no—
35C2×D628—21no—
36C2×D628—21no—
37C2×D628—21no—
38C2×D628—21no—
39C2×D628D412yes—
40C22×Dic157—21no—
41C22×Dic157D412yes—
42#14578943—28no—
43#14578941—28no—
44C2×#11929434—21no—
45C2×#11929434—21no—
46C2×#11929434D412yes—
47#14578933—28no—
48#14578931—44no—
49C4×Dic157—21no—
50C4×Dic157—21no—
51C2×Q8×C157—21no—
52C2×Q8×C157—21no—
53C2×Q8×C157—21no—
54C2×Q8×C157—21no—
55C2×Q8×C157—21no—
56C2×Q8×C157D412yes—
57C22×C4×C157—21no—
58C22×C4×C157—21no—
59C22×C4×C157C2311yescharacteristic
60C2×C8×C157—21no—
61C2×C8×C157—21no—
62C2×C8×C157D412yes—
63C4⋊C4×C157—21no—
64C4⋊C4×C157—21no—
65C22×D157—22no—
66C22×D157—41no—
67C22×D157—41no—
68C22×D157—42no—
69C22×D157—42no—
70Dic314—22no—
71Dic314—22no—
72Dic314—41no—
73Dic314—41no—
74D628—22no—
75D628—22no—
76D628—41no—
77D628—41no—
78C2×Dic157—22no—
79C2×Dic157—22no—
80C2×Dic157—22no—
81C2×Dic157—41no—
82C2×Dic157—41no—
83C2×Dic157SD1614yescomplemented
84#11929434—22no—
85C23×C157C2×D411yescharacteristic
86Q8×C157—22no—
87Q8×C157—22no—
88Q8×C157—41no—
89Q8×C157—41no—
90C2×C4×C157—22no—
91C2×C4×C157—41no—
92C2×C4×C157—41no—
93C2×C4×C157—41no—
94C2×C4×C157—42no—
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C2×C4×C157C2×D411yescharacteristic
100C2×C4×C157#4111yescharacteristic
101C8×C157—22no—
102C22×C157#12011yescharacteristic
103C22×C157#12111yescharacteristic
104C22×C157#12711yescharacteristic
105C22×C157C2×SD1612yes—
106C22×C157#13712yes—
107C2×D157—42no—
108C2×D157—44no—
109C2×D157—82no—
110C2×D157—81no—
111C2×D157—81no—
112C4×C157—42no—
113C4×C157—81no—
114C4×C157#12011yescharacteristic
115C4×C157#12011yescharacteristic
116C4×C157#12111yescharacteristic
117C4×C157#12111yescharacteristic
118Dic157—24no—
119Dic157—42no—
120C2×C157#39211yescharacteristic
121C2×C157#44812yes—
122C2×C157#46112yes—
123C2×C157#49211yescharacteristic
124C2×C157#49711yescharacteristic
125D157—82no—
126C157#201311yescharacteristic, complemented, Sylow
127#2013—1571nomaximal, Sylow
128C22×SD16—1571no—
129C22×SD16—1571no—
130C2×C4⋊Q8—1571no—
131C2×#103—1571no—
132C2×#103D15711yescharacteristic, complemented
133#390—1571no—
134#336—1571no—
135C22×Q8—1571no—
136C22×Q8C2×D15711yescharacteristic
137C22×D4—1571no—
138C2×SD16—3142no—
139C2×SD16—3142no—
140C2×SD16—3142no—
141C2×SD16—3142no—
142C2×SD16—3142no—
143C2×SD16—3142no—
144C22×C8—1571no—
145C22×C8C2×D15711yescharacteristic
146C4⋊Q8—3142no—
147C4⋊Q8—3142no—
148C2×C4⋊C4—3142no—
149C2×C4⋊C4C2×D15711yescharacteristic
150C2×C22⋊C4—3142no—
151C2×C42—1571no—
152#103—22no—
153#103—3142no—
154C24—3141no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—3141no—
161C2×Q8—3141no—
162C2×Q8—3141no—
163C2×Q8—3141no—
164C2×Q8—3141no—
165C2×Q8—1572no—
166C2×Q8D62812yes—
167C2×D4—3141no—
168C2×D4—3141no—
169C2×D4—3141no—
170C2×D4—3141no—
171C2×D4—3141no—
172C2×D4—1572no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—3141no—
176C22×C4—1572no—
177C22×C4C22×D15711yescharacteristic
178SD16—3148no—
179SD16—3148no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—3141no—
183C2×C8—3141no—
184C2×C8—1572no—
185C2×C8C22⋊D15712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—3148no—
189C22⋊C4—6284no—
190C42—3141no—
191C42—3141no—
192C23—3142no—
193C23—6281no—
194C23—6281no—
195C23—6282no—
196C23—6282no—
197C23D4×D15711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—3142no—
203Q8—3142no—
204Q8—6281no—
205Q8—6281no—
206D4—3142no—
207D4—3142no—
208D4—6281no—
209D4—6281no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—3142no—
216C2×C4—3142no—
217C2×C4—3142no—
218C2×C4—6281no—
219C2×C4—6281no—
220C2×C4—1574no—
221C2×C4D4×D15711yescharacteristic
222C2×C4D4×D15711yescharacteristic
223C2×C4D4×D15711yescharacteristic
224C2×C4C2×C22⋊D15711yescharacteristic
225C2×C4C2×D62811yescharacteristic
226C2×C4#1457895411yescharacteristic
227C8—22no—
228C8—3142no—
229C22—6282no—
230C22—6284no—
231C22—12561no—
232C22—12561no—
233C22—12562no—
234C22#1909774111yescharacteristic
235C22#1909774411yescharacteristic
236C22#1909776711yescharacteristic
237C22SD16×D15712yes—
238C22#1909780312yes—
239C4—42no—
240C4—81no—
241C4—3144no—
242C4—6282no—
243C4#1909773911yescharacteristic
244C4#1909774111yescharacteristic
245C4#1909774411yescharacteristic
246C4#1909774511yescharacteristic
247C2—12562no—
248C2#2375718211yescharacteristic
249C2#2375741312yes—
250C2#2375747712yes—
251C2#2375764411yescharacteristic
252C2#2375766611yescharacteristic
subgroupG/H
centerC23D4×D157
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#103×C157C2
socleC23×C157C2×D4
prime radicalC22×D628C22
hypercenterC2×#103D157
hypocenterC157#2013
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#19097769C2semidirect
4. C2×#19097694C2semidirect
6. #23757027C2semidirect
7. C2×#103×C157C2semidirect
83. C2×Dic157SD16semidirect
126. C157#2013semidirect
132. C2×#103D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789432
C22×#145789412
C2×#190977692
C2×#190976942
#237571742
#237570272
C2×#103×C1572
#2013157