Finite Group Databasebeta

← #34100665

Subgroups — #34100665

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#23757837C211yesmaximal, characteristic
2#23757747C211yesmaximal, characteristic, complemented
3#23757290C211yesmaximal, characteristic, complemented
4C2×#19097687C211yesmaximal, characteristic, complemented
5#23757030C211yesmaximal, characteristic, complemented
6#23757029C211yesmaximal, characteristic, complemented
7#421×C157C211yesmaximal, characteristic, complemented
8C22×D628C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10#19097767C412yescomplemented
11#19097760C412yescomplemented
12#19097745—22no—
13#19097739—22no—
14C4×D628—22no—
15#19097733—22no—
16#19097730C2211yescharacteristic
17#19097730C2211yescharacteristic
18C22⋊C4×D157—22no—
19#19097698—22no—
20#19097687—22no—
21C2×#41×C157C2211yescharacteristic
22C2×M16×C157C2211yescharacteristic
23#117×C157C2211yescharacteristic
24#104×C157—22no—
25C23×D157D412yes—
26C2×D628—21no—
27C2×D628—21no—
28C2×D628—21no—
29C2×D628—21no—
30C2×D628C2×C412yes—
31C2×D628D412yes—
32C2×C22⋊D157—24no—
33C2×C4×D157—24no—
34#14578938—21no—
35#14578938—21no—
36C2×#11929434—21no—
37C4⋊Dic157—21no—
38C4⋊Dic157—21no—
39C22⋊Dic157—21no—
40C22⋊Dic157—21no—
41#14578931—24no—
42#14578931—24no—
43C4×Dic157C2×C412yescomplemented
44C4×Dic157D412yescomplemented
45#41×C157—21no—
46#41×C157—21no—
47#41×C157—21no—
48#41×C157—21no—
49C2×Q8×C157C2×C411yescharacteristic, complemented
50C2×D4×C157—21no—
51C2×D4×C157C2×C411yescharacteristic, complemented
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54M16×C157—21no—
55M16×C157—21no—
56C2×C8×C157—21no—
57C4⋊C4×C157—21no—
58C22⋊C4×C157—21no—
59C42×C157D412yescomplemented
60C22×D157—22no—
61C22×D157—22no—
62C22×D157—82no—
63C22×D157C22⋊C414yes—
64D628—22no—
65D628—22no—
66D628—22no—
67D628—22no—
68C22⋊D157—84no—
69C2×Dic157—22no—
70C2×Dic157—22no—
71C4×D157—84no—
72#11929434—41no—
73C23×C157—21no—
74C23×C157C2×D411yescharacteristic
75Q8×C157—21no—
76Q8×C157—21no—
77D4×C157—21no—
78D4×C157—21no—
79D4×C157—41no—
80D4×C157—41no—
81C2×C4×C157—21no—
82C2×C4×C157—22no—
83C2×C4×C157—41no—
84C2×C4×C157—41no—
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×C157—42no—
94C22×C157—41no—
95C22×C157C2×C22⋊C411yescharacteristic
96C22×C157#12011yescharacteristic
97C22×C157#12311yescharacteristic
98C2×D157—24no—
99C2×D157—44no—
100C2×D157—44no—
101C2×D157—82no—
102C4×C157—42no—
103C4×C157—41no—
104C4×C157C4×D411yescharacteristic
105C4×C157C4×D411yescharacteristic
106C4×C157#12111yescharacteristic
107C4×C157#12111yescharacteristic
108Dic157—42no—
109Dic157—42no—
110C2×C157—21no—
111C2×C157—21no—
112C2×C157—21no—
113C2×C157—81no—
114C2×C157#38611yescharacteristic
115D157—44no—
116C157#191111yescharacteristic, complemented, Sylow
117#1911—1571nomaximal, Sylow
118#535—1571no—
119#518—1571no—
120#421—1571no—
121#421D15711yescharacteristic, complemented
122C2×#100—1571no—
123H(3,ℤ/4)—1571no—
124H(3,ℤ/4)—1571no—
125C2×#41C2×D15711yescharacteristic
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#117—1571no—
137#117C2×D15711yescharacteristic
138C2×C22⋊C4—3142no—
139#104—22no—
140#104—3142no—
141#100—3142no—
142C24—1572no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8C4×D15711yescharacteristic, complemented
148C2×D4—21no—
149C2×D4—3144no—
150C2×D4—3141no—
151C2×D4—3141no—
152C2×D4—3141no—
153C2×D4—3141no—
154C2×D4—1572no—
155C2×D4—1572no—
156C2×D4C4×D15711yescharacteristic, complemented
157C22×C4—21no—
158C22×C4—3144no—
159C22×C4C22×D15711yescharacteristic
160M16—21no—
161M16—21no—
162M16—3141no—
163M16—3141no—
164C2×C8—21no—
165C2×C8—3141no—
166C4⋊C4—21no—
167C4⋊C4—3141no—
168C4⋊C4—3141no—
169C22⋊C4—21no—
170C22⋊C4—3144no—
171C22⋊C4—3144no—
172C22⋊C4—3141no—
173C22⋊C4—3141no—
174C42—1572no—
175C42—1572no—
176C42C22⋊D15712yescomplemented
177C23—21no—
178C23—3142no—
179C23—3142no—
180C23—1574no—
181C23—12562no—
182C23D4×D15711yescharacteristic
183Q8—21no—
184Q8—21no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—3142no—
190D4—3142no—
191D4—3142no—
192D4—3142no—
193D4—12564no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—3142no—
199C2×C4—3142no—
200C2×C4—12564no—
201C2×C4C2×C4×D15711yescharacteristic
202C2×C4D4×D15711yescharacteristic
203C2×C4D4×D15711yescharacteristic
204C2×C4C2×C22⋊D15711yescharacteristic
205C2×C4#1457895311yescharacteristic
206C2×C4#1457895411yescharacteristic
207C8—41no—
208C8—6281no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—3144no—
213C22—6284no—
214C22—6284no—
215C22—12562no—
216C22C22⋊C4×D15711yescharacteristic
217C22#1909773911yescharacteristic
218C22#1909775311yescharacteristic
219C4—42no—
220C4—41no—
221C4—6282no—
222C4—6282no—
223C4#1909773111yescharacteristic
224C4C4×C22⋊D15711yescharacteristic
225C4#1909774211yescharacteristic
226C4#1909774411yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—81no—
231C2—6284no—
232C2#2375714911yescharacteristic
subgroupG/H
centerC2#23757149
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#421×C157C2
socleC2×C157#386
prime radical#23757837C2
cosocleC22×C4×C157C23
hypercenter#421D157
hypocenterC157#1911
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #23757747C2semidirect
3. #23757290C2semidirect
4. C2×#19097687C2semidirect
5. #23757030C2semidirect
6. #23757029C2semidirect
7. #421×C157C2semidirect
10. #19097767C4semidirect
11. #19097760C4semidirect
43. C4×Dic157C2×C4semidirect
44. C4×Dic157D4semidirect
49. C2×Q8×C157C2×C4semidirect
51. C2×D4×C157C2×C4semidirect
59. C42×C157D4semidirect
116. C157#1911semidirect
121. #421D157semidirect
147. C2×Q8C4×D157semidirect
156. C2×D4C4×D157semidirect
176. C42C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578372
#237577472
#237572902
C2×#190976872
#237570302
#237570292
#421×C1572
#1911157