Finite Group Databasebeta

← #34100667

Subgroups — #34100667

Order 20096.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#23757842C211yesmaximal, characteristic
2#23757747C211yesmaximal, characteristic, complemented
3#23757291C211yesmaximal, characteristic, complemented
4#23757289C211yesmaximal, characteristic, complemented
5C2×#19097687C211yesmaximal, characteristic, complemented
6#23757029C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×D628C2211yescharacteristic
10C2×#14578938C2211yescharacteristic
11#19097768C412yescomplemented
12#19097759C412yescomplemented
13#19097746—22no—
14#19097741—22no—
15C4×D628—22no—
16#19097733—22no—
17#19097730C2211yescharacteristic
18C22⋊C4×D157—22no—
19#19097699—22no—
20#19097697—22no—
21#19097687—22no—
22C2×M16×C157C2211yescharacteristic
23#117×C157C2211yescharacteristic
24#117×C157C2211yescharacteristic
25C23×D157D412yes—
26#14578954—21no—
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30C2×Dic314C2×C411yescharacteristic, complemented
31C2×D628—24no—
32C2×D628—21no—
33C2×D628—21no—
34C2×D628—21no—
35C2×D628—21no—
36C2×D628C2×C412yes—
37C2×D628C2×C411yescharacteristic, complemented
38C2×D628D412yes—
39C2×C22⋊D157—21no—
40C2×C4×D157—24no—
41C2×C4×D157—21no—
42#14578938—21no—
43#14578938—21no—
44C2×#11929434—21no—
45C4⋊Dic157—21no—
46C22⋊Dic157—21no—
47#14578931—24no—
48#14578931—24no—
49C4×Dic157D412yescomplemented
50C22×C4×C157C2311yescharacteristic
51M16×C157—21no—
52M16×C157—21no—
53C2×C8×C157—21no—
54C4⋊C4×C157—21no—
55C4⋊C4×C157—21no—
56C22⋊C4×C157—21no—
57C22⋊C4×C157—21no—
58C42×C157C2×C412yescomplemented
59C42×C157D412yescomplemented
60C22×D157—22no—
61C22×D157—22no—
62C22×D157—21no—
63C22×D157—82no—
64C22×D157C22⋊C414yes—
65Dic314—21no—
66Dic314—21no—
67D628—22no—
68D628—22no—
69D628—21no—
70D628—21no—
71D628—22no—
72D628—22no—
73D628—84no—
74C22⋊D157—41no—
75C22⋊D157—41no—
76C2×Dic157—21no—
77C2×Dic157—22no—
78C4×D157—41no—
79C4×D157—41no—
80C4×D157—84no—
81#11929434—41no—
82C23×C157C2×D411yescharacteristic
83C2×C4×C157—22no—
84C2×C4×C157—22no—
85C2×C4×C157C22×C411yescharacteristic
86C2×C4×C157C2×D411yescharacteristic
87C2×C4×C157C2×D411yescharacteristic
88C2×C4×C157C2×D411yescharacteristic
89C2×C4×C157#4111yescharacteristic
90C2×C4×C157#4111yescharacteristic
91C8×C157—41no—
92C22×C157—41no—
93C22×C157C2×C22⋊C411yescharacteristic
94C22×C157#12011yescharacteristic
95C22×C157#12311yescharacteristic
96C2×D157—24no—
97C2×D157—44no—
98C2×D157—44no—
99C2×D157—42no—
100C2×D157—41no—
101C2×D157—82no—
102C4×C157—42no—
103C4×C157—42no—
104C4×C157C4×D411yescharacteristic
105C4×C157C4×D411yescharacteristic
106C4×C157#12111yescharacteristic
107C4×C157#12111yescharacteristic
108Dic157—42no—
109Dic157—41no—
110C2×C157—21no—
111C2×C157—21no—
112C2×C157—21no—
113C2×C157#38611yescharacteristic
114D157—44no—
115D157—81no—
116C157#191111yescharacteristic, complemented, Sylow
117#1911—1571nomaximal, Sylow
118#535—1571no—
119#518—1571no—
120#421—1571no—
121#421—1571no—
122C2×#100—1571no—
123H(3,ℤ/4)—1571no—
124H(3,ℤ/4)D15711yescharacteristic, complemented
125C2×#41—1571no—
126C22×D4—1571no—
127C2×M16—1571no—
128C2×M16C2×D15711yescharacteristic
129#127—1572no—
130#124—1572no—
131#121—3142no—
132#120—3142no—
133C4×D4—3142no—
134C4×D4—3142no—
135#117—1571no—
136#117C2×D15711yescharacteristic
137#117C2×D15711yescharacteristic
138C2×C22⋊C4—3142no—
139#104—3142no—
140#104—3142no—
141#100—3142no—
142C24—1572no—
143#41—3141no—
144#41—3141no—
145#41—3141no—
146#41—3141no—
147C2×Q8—1571no—
148C2×D4—3144no—
149C2×D4—3141no—
150C2×D4—3141no—
151C2×D4—3141no—
152C2×D4—3141no—
153C2×D4—3141no—
154C2×D4—1571no—
155C2×D4—1572no—
156C2×D4—1572no—
157C22×C4—3144no—
158C22×C4—3141no—
159C22×C4C22×D15711yescharacteristic
160M16—21no—
161M16—21no—
162M16—3141no—
163M16—3141no—
164C2×C8—21no—
165C2×C8—3141no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—3141no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—3144no—
172C22⋊C4—3144no—
173C22⋊C4—3141no—
174C42—1572no—
175C42C4×D15712yescomplemented
176C42D62812yescomplemented
177C23—3142no—
178C23—3142no—
179C23—3141no—
180C23—1574no—
181C23—12562no—
182C23D4×D15711yescharacteristic
183Q8—3141no—
184Q8—3141no—
185D4—3142no—
186D4—3141no—
187D4—3142no—
188D4—3141no—
189D4—3142no—
190D4—3142no—
191D4—6281no—
192D4—6281no—
193D4—12564no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3141no—
197C2×C4—3142no—
198C2×C4—6281no—
199C2×C4—6281no—
200C2×C4—12564no—
201C2×C4C2×C4×D15711yescharacteristic
202C2×C4D4×D15711yescharacteristic
203C2×C4D4×D15711yescharacteristic
204C2×C4C2×D62811yescharacteristic
205C2×C4#1457895411yescharacteristic
206C2×C4#1457895511yescharacteristic
207C8—41no—
208C8—6281no—
209C22—41no—
210C22—3144no—
211C22—6284no—
212C22—6284no—
213C22—6282no—
214C22—6281no—
215C22—12562no—
216C22C22⋊C4×D15711yescharacteristic
217C22#1909774111yescharacteristic
218C22#1909775611yescharacteristic
219C4—42no—
220C4—42no—
221C4—6282no—
222C4—6281no—
223C4#1909773311yescharacteristic
224C4C4×D62811yescharacteristic
225C4#1909774411yescharacteristic
226C4#1909774611yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—6284no—
231C2—12561no—
232C2#2375715111yescharacteristic
subgroupG/H
centerC2#23757151
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingH(3,ℤ/4)×C157C2
socleC2×C157#386
prime radical#23757842C2
cosocleC22×C4×C157C23
hypercenterH(3,ℤ/4)D157
hypocenterC157#1911
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #23757747C2semidirect
3. #23757291C2semidirect
4. #23757289C2semidirect
5. C2×#19097687C2semidirect
6. #23757029C2semidirect
7. H(3,ℤ/4)×C157C2semidirect
11. #19097768C4semidirect
12. #19097759C4semidirect
30. C2×Dic314C2×C4semidirect
37. C2×D628C2×C4semidirect
49. C4×Dic157D4semidirect
58. C42×C157C2×C4semidirect
59. C42×C157D4semidirect
116. C157#1911semidirect
124. H(3,ℤ/4)D157semidirect
175. C42C4×D157semidirect
176. C42D628semidirect

Maximal subgroups

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

subgroupindex
#237578422
#237577472
#237572912
#237572892
C2×#190976872
#237570292
H(3,ℤ/4)×C1572
#1911157