Finite Group Databasebeta

← #34100332

Subgroups — #34100332

Order 20096.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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×#19097729C211yesmaximal, characteristic
2C2×#19097702C211yesmaximal, characteristic, complemented
3C2×#19097683C211yesmaximal, characteristic
4#23757130C211yesmaximal, characteristic
5#23757036C211yesmaximal, characteristic, complemented
6#23757035C211yesmaximal, characteristic, complemented
7C2×#107×C157C211yesmaximal, characteristic, complemented
8C22×C4×D157C2211yescharacteristic
9C22×#11929434C2211yescharacteristic
10#19097729—22no—
11#19097729—22no—
12C2×C4⋊Dic157C2211yescharacteristic
13C2×C4⋊Dic157C2211yescharacteristic
14C2×#14578931—21no—
15C2×#14578931—21no—
16C2×C4×Dic157—21no—
17#19097702—22no—
18#19097683C414yescomplemented
19#19097678—21no—
20C22×C8×C157C2211yescharacteristic
21C2×C4⋊C4×C157C2211yescharacteristic
22C2×C4⋊C4×C157C2211yescharacteristic
23#107×C157—22no—
24C23×D157D411yescharacteristic
25C22×Dic157—21no—
26C22×Dic157—21no—
27C22×Dic157D411yescharacteristic
28C2×C4×D157—21no—
29C2×C4×D157—21no—
30C2×C4×D157—21no—
31C2×C4×D157—21no—
32C2×C4×D157C2×C412yes—
33C2×C4×D157Q812yes—
34C2×#11929434—21no—
35C2×#11929434—21no—
36C2×#11929434C2×C412yes—
37C4⋊Dic157—21no—
38C4⋊Dic157—21no—
39C4⋊Dic157—21no—
40C4⋊Dic157D412yes—
41#14578931—24no—
42#14578931—42no—
43C4×Dic157—24no—
44C22×C4×C157—21no—
45C22×C4×C157—21no—
46C22×C4×C157C2311yescharacteristic
47C2×C8×C157—21no—
48C2×C8×C157—21no—
49C2×C8×C157C2×C412yes—
50C4⋊C4×C157—21no—
51C4⋊C4×C157—21no—
52C4⋊C4×C157—21no—
53C4⋊C4×C157D412yes—
54C22×D157—21no—
55C22×D157—21no—
56C22×D157—21no—
57C22×D157—21no—
58C22×D157—21no—
59C22×D157—21no—
60C22×D157C4⋊C412yes—
61C2×Dic157—21no—
62C2×Dic157—21no—
63C2×Dic157—22no—
64C2×Dic157—41no—
65C2×Dic157—41no—
66C2×Dic157—41no—
67C2×Dic157—41no—
68C2×Dic157—41no—
69C2×Dic157C4⋊C412yes—
70C4×D157—22no—
71C4×D157—22no—
72C4×D157—22no—
73C4×D157—22no—
74#11929434—22no—
75C23×C157C2×D411yescharacteristic
76C2×C4×C157—22no—
77C2×C4×C157—41no—
78C2×C4×C157—41no—
79C2×C4×C157—41no—
80C2×C4×C157—41no—
81C2×C4×C157—41no—
82C2×C4×C157C22×C411yescharacteristic
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×Q811yescharacteristic
86C2×C4×C157#4111yescharacteristic
87C2×C4×C157#4111yescharacteristic
88C8×C157—22no—
89C22×C157C2×C4⋊C411yescharacteristic
90C22×C157#12011yescharacteristic
91C22×C157#12311yescharacteristic
92C22×C157#13511yescharacteristic
93C22×C157#13511yescharacteristic
94C22×C157AGL(1,ℤ/8)11yescharacteristic
95C22×C157#13711yescharacteristic
96C2×D157—22no—
97C2×D157—22no—
98C2×D157—22no—
99C2×D157—41no—
100C2×D157—41no—
101C2×D157—41no—
102C2×D157—41no—
103C4×C157—42no—
104C4×C157—81no—
105C4×C157C4×D411yescharacteristic
106C4×C157C4×D411yescharacteristic
107C4×C157C22⋊Q811yescharacteristic
108C4×C157C22⋊Q811yescharacteristic
109Dic157—22no—
110Dic157—42no—
111Dic157—81no—
112C2×C157#38511yescharacteristic
113C2×C157#42711yescharacteristic
114C2×C157#42811yescharacteristic
115C2×C157#44911yescharacteristic
116C2×C157#45211yescharacteristic
117C2×C157#48211yescharacteristic
118C2×C157#48511yescharacteristic
119D157—42no—
120C157#185411yescharacteristic, complemented, Sylow
121#1854—1571nomaximal, Sylow
122C2×#117—1571no—
123C2×#107D15711yescharacteristic, complemented
124C2×#106—1571no—
125C2×C22⋊C8—1571no—
126#380—1571no—
127#340—1571no—
128#340—1571no—
129C23×C4—1571no—
130C22×C8—1571no—
131C22×C8C2×D15711yescharacteristic
132#117—3142no—
133#117—3142no—
134C2×C4⋊C4—1571no—
135C2×C4⋊C4—1571no—
136C2×C4⋊C4C2×D15711yescharacteristic
137C2×C4⋊C4C2×D15711yescharacteristic
138C2×C22⋊C4—3141no—
139C2×C22⋊C4—3141no—
140C2×C42—3141no—
141#107—22no—
142#106—3142no—
143C22⋊C8—1574no—
144#95—3141no—
145C24—1571no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3141no—
149C22×C4—3141no—
150C22×C4—3141no—
151C22×C4—3141no—
152C22×C4—3141no—
153C22×C4—3141no—
154C22×C4—1571no—
155C22×C4—1572no—
156C22×C4—1572no—
157C22×C4C22×D15711yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—3141no—
161C2×C8—3141no—
162C2×C8—1572no—
163C2×C8C4×D15712yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—3141no—
168C4⋊C4—3141no—
169C4⋊C4—3141no—
170C4⋊C4—1572no—
171C4⋊C4C22⋊D15712yes—
172C22⋊C4—3144no—
173C22⋊C4—6282no—
174C42—3144no—
175C23—3141no—
176C23—3141no—
177C23—3141no—
178C23—3141no—
179C23—3141no—
180C23—3141no—
181C23—1572no—
182C23D4×D15711yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3141no—
190C2×C4—3141no—
191C2×C4—3142no—
192C2×C4—3142no—
193C2×C4—3142no—
194C2×C4—3142no—
195C2×C4—3142no—
196C2×C4—6281no—
197C2×C4—6281no—
198C2×C4—6281no—
199C2×C4—6281no—
200C2×C4—6281no—
201C2×C4—1572no—
202C2×C4C2×C4×D15711yescharacteristic
203C2×C4D4×D15711yescharacteristic
204C2×C4C2×C22⋊D15711yescharacteristic
205C2×C4Q8×D15711yescharacteristic
206C2×C4#1457895411yescharacteristic
207C2×C4#1457895511yescharacteristic
208C8—22no—
209C8—3142no—
210C22—3142no—
211C22—3142no—
212C22—3142no—
213C22—6281no—
214C22—6281no—
215C22—6281no—
216C22—6281no—
217C22C4⋊C4×D15711yescharacteristic
218C22#1909773911yescharacteristic
219C22#1909775611yescharacteristic
220C22#1909779011yescharacteristic
221C22#1909779411yescharacteristic
222C22#1909779511yescharacteristic
223C22#1909780311yescharacteristic
224C4—42no—
225C4—81no—
226C4—3142no—
227C4—6282no—
228C4—12561no—
229C4C4×C22⋊D15711yescharacteristic
230C4#1909773311yescharacteristic
231C4#1909774711yescharacteristic
232C4#1909774911yescharacteristic
233C2—6282no—
234C2#2375714411yescharacteristic
235C2#2375731011yescharacteristic
236C2#2375731411yescharacteristic
237C2#2375741511yescharacteristic
238C2#2375743411yescharacteristic
239C2#2375760011yescharacteristic
240C2#2375761611yescharacteristic
subgroupG/H
centerC23D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#107×C157C2
socleC23×C157C2×D4
prime radicalC23×D157D4
cosocleC22×C4×C157C23
hypercenterC2×#107D157
hypocenterC157#1854
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#19097702C2semidirect
5. #23757036C2semidirect
6. #23757035C2semidirect
7. C2×#107×C157C2semidirect
18. #19097683C4semidirect
120. C157#1854semidirect
123. C2×#107D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977292
C2×#190977022
C2×#190976832
#237571302
#237570362
#237570352
C2×#107×C1572
#1854157