Finite Group Databasebeta

← #34101330

Subgroups — #34101330

Order 20096.

316 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#19097747C211yesmaximal, characteristic
2C2×#19097702C211yesmaximal, characteristic, complemented
3C2×#19097695C211yesmaximal, characteristic, complemented
4C2×#19097683C211yesmaximal, characteristic
5#23757130C211yesmaximal, characteristic
6#23757036C211yesmaximal, characteristic, complemented
7C2×#103×C157C211yesmaximal, characteristic, complemented
8C22×C4×D157C2211yescharacteristic
9C22×#11929434C2211yescharacteristic
10#19097747—22no—
11#19097747—22no—
12C2×C4⋊Dic157C2211yescharacteristic
13C2×C4⋊Dic157C2211yescharacteristic
14C2×#14578933—21no—
15C2×#14578931—21no—
16C2×#14578931—21no—
17#19097702—22no—
18#19097695—22no—
19#19097683—22no—
20#19097678—21no—
21C22×Q8×C157C2211yescharacteristic
22C22×C8×C157C2211yescharacteristic
23C2×C4⋊C4×C157C2211yescharacteristic
24#103×C157—22no—
25C23×D157D411yescharacteristic
26C22×Dic157—21no—
27C22×Dic157—21no—
28C22×Dic157D411yescharacteristic
29C2×C4×D157—21no—
30C2×C4×D157—21no—
31C2×C4×D157—21no—
32C2×C4×D157—21no—
33C2×C4×D157—21no—
34C2×C4×D157—21no—
35C2×#11929434—21no—
36C2×#11929434—21no—
37C2×#11929434D412yes—
38C4⋊Dic157—21no—
39C4⋊Dic157—21no—
40C4⋊Dic157—21no—
41C4⋊Dic157D412yes—
42#14578933—24no—
43#14578931—24no—
44#14578931—42no—
45C2×Q8×C157—21no—
46C2×Q8×C157—21no—
47C2×Q8×C157—21no—
48C2×Q8×C157—21no—
49C2×Q8×C157—21no—
50C2×Q8×C157D412yes—
51C22×C4×C157—21no—
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54C2×C8×C157—21no—
55C2×C8×C157—21no—
56C2×C8×C157—21no—
57C4⋊C4×C157—21no—
58C4⋊C4×C157—21no—
59C22×D157—21no—
60C22×D157—21no—
61C22×D157—21no—
62C22×D157—21no—
63C22×D157—21no—
64C22×D157—21no—
65C22×D157SD1612yes—
66C2×Dic157—21no—
67C2×Dic157—22no—
68C2×Dic157—21no—
69C2×Dic157—21no—
70C2×Dic157—41no—
71C2×Dic157—41no—
72C2×Dic157—41no—
73C2×Dic157—41no—
74C2×Dic157—41no—
75C4×D157—41no—
76C4×D157—41no—
77C4×D157—41no—
78C4×D157—41no—
79#11929434—22no—
80C23×C157C2×D411yescharacteristic
81Q8×C157—22no—
82Q8×C157—22no—
83Q8×C157—41no—
84Q8×C157—41no—
85C2×C4×C157—22no—
86C2×C4×C157—41no—
87C2×C4×C157—41no—
88C2×C4×C157—41no—
89C2×C4×C157—41no—
90C2×C4×C157—41no—
91C2×C4×C157C2×D411yescharacteristic
92C2×C4×C157C2×D411yescharacteristic
93C2×C4×C157C2×D411yescharacteristic
94C2×C4×C157#4111yescharacteristic
95C2×C4×C157#4111yescharacteristic
96C2×C4×C157#4111yescharacteristic
97C8×C157—41no—
98C22×C157#12011yescharacteristic
99C22×C157#12111yescharacteristic
100C22×C157#12311yescharacteristic
101C22×C157C2×SD1611yescharacteristic
102C22×C157#13511yescharacteristic
103C22×C157#13711yescharacteristic
104C22×C157#13711yescharacteristic
105C2×D157—22no—
106C2×D157—22no—
107C2×D157—22no—
108C2×D157—41no—
109C2×D157—41no—
110C2×D157—41no—
111C2×D157—41no—
112C4×C157—42no—
113C4×C157—81no—
114C4×C157#12111yescharacteristic
115C4×C157#12111yescharacteristic
116C4×C157#12311yescharacteristic
117C4×C157#12411yescharacteristic
118Dic157—41no—
119Dic157—42no—
120Dic157—81no—
121C2×C157#39411yescharacteristic
122C2×C157#44811yescharacteristic
123C2×C157#45211yescharacteristic
124C2×C157#46511yescharacteristic
125C2×C157#47011yescharacteristic
126C2×C157#48311yescharacteristic
127C2×C157#48511yescharacteristic
128D157—42no—
129C157#203511yescharacteristic, complemented, Sylow
130#2035—1571nomaximal, Sylow
131C2×C22⋊Q8—1571no—
132C2×#106—1571no—
133C2×#103—1571no—
134C2×#103D15711yescharacteristic, complemented
135C2×C22⋊C8—1571no—
136#380—1571no—
137#340—1571no—
138C22×Q8C2×D15711yescharacteristic
139C23×C4—1571no—
140C22×C8—1571no—
141C22×C8C2×D15711yescharacteristic
142C22⋊Q8—3142no—
143C22⋊Q8—3142no—
144C2×C4⋊C4—3141no—
145C2×C4⋊C4—1571no—
146C2×C4⋊C4—1571no—
147C2×C4⋊C4C2×D15711yescharacteristic
148C2×C22⋊C4—3141no—
149C2×C22⋊C4—3141no—
150#106—3142no—
151#103—22no—
152#103—3142no—
153C22⋊C8—3142no—
154#95—3141no—
155C24—1571no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8C22⋊D15712yes—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—3141no—
165C22×C4—3141no—
166C22×C4—3141no—
167C22×C4—3141no—
168C22×C4—3141no—
169C22×C4—3141no—
170C22×C4—3141no—
171C22×C4—3141no—
172C22×C4—1571no—
173C22×C4C22×D15711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—3141no—
178C2×C8—3141no—
179C2×C8—1572no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—3144no—
183C4⋊C4—3141no—
184C4⋊C4—3141no—
185C4⋊C4—3141no—
186C4⋊C4—1572no—
187C22⋊C4—3144no—
188C22⋊C4—6282no—
189C23—3141no—
190C23—3141no—
191C23—3141no—
192C23—3141no—
193C23—3141no—
194C23—3141no—
195C23—1572no—
196C23D4×D15711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—3142no—
208C2×C4—3141no—
209C2×C4—3141no—
210C2×C4—3141no—
211C2×C4—6281no—
212C2×C4—6281no—
213C2×C4—6281no—
214C2×C4—6281no—
215C2×C4—6281no—
216C2×C4—6281no—
217C2×C4—6281no—
218C2×C4—6281no—
219C2×C4—6281no—
220C2×C4D4×D15711yescharacteristic
221C2×C4D4×D15711yescharacteristic
222C2×C4C2×C22⋊D15711yescharacteristic
223C2×C4#1457895311yescharacteristic
224C2×C4#1457895411yescharacteristic
225C2×C4#1457895411yescharacteristic
226C8—41no—
227C8—3142no—
228C22—3142no—
229C22—3142no—
230C22—3142no—
231C22—6281no—
232C22—6281no—
233C22—6281no—
234C22—6281no—
235C22#1909773911yescharacteristic
236C22#1909774411yescharacteristic
237C22#1909775311yescharacteristic
238C22SD16×D15711yescharacteristic
239C22#1909779411yescharacteristic
240C22#1909780011yescharacteristic
241C22#1909780311yescharacteristic
242C4—42no—
243C4—81no—
244C4—6282no—
245C4—6281no—
246C4—12561no—
247C4#1909774311yescharacteristic
248C4#1909774411yescharacteristic
249C4#1909775411yescharacteristic
250C4#1909775711yescharacteristic
251C2—6282no—
252C2#2375718811yescharacteristic
253C2#2375741011yescharacteristic
254C2#2375743411yescharacteristic
255C2#2375750511yescharacteristic
256C2#2375753011yescharacteristic
257C2#2375760211yescharacteristic
258C2#2375761311yescharacteristic
subgroupG/H
centerC23D4×D157
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#103×C157C2
socleC23×C157C2×D4
prime radicalC23×D157D4
hypercenterC2×#103D157
hypocenterC157#2035
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
2. C2×#19097702C2semidirect
3. C2×#19097695C2semidirect
6. #23757036C2semidirect
7. C2×#103×C157C2semidirect
129. C157#2035semidirect
134. C2×#103D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977472
C2×#190977022
C2×#190976952
C2×#190976832
#237571302
#237570362
C2×#103×C1572
#2035157