Finite Group Databasebeta

← #34099057

Subgroups — #34099057

Order 20096.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#23757898C211yesmaximal, characteristic, complemented
2#23757424C211yesmaximal, characteristic, complemented
3#23757181C211yesmaximal, characteristic
4#23757011C211yesmaximal, characteristic, complemented
5#23757010C211yesmaximal, characteristic, complemented
6#23756996C211yesmaximal, characteristic, complemented
7#450×C157C211yesmaximal, characteristic, complemented
8C22×C22⋊D157—21no—
9C2×#14578942—21no—
10#19097751—21no—
11#19097750C2211yescharacteristic
12#19097749—21no—
13#19097749—21no—
14#19097748—21no—
15#19097747—21no—
16C2×C22⋊Dic157—21no—
17C2×#14578931C2211yescharacteristic
18C22⋊C4×D157—21no—
19#19097692—21no—
20C22⋊#11929434C2211yescharacteristic
21#19097678C2211yescharacteristic
22C22×D4×C157C2211yescharacteristic
23C2×SD16×C157—21no—
24C22⋊Q8×C157C2211yescharacteristic
25#102×C157—21no—
26C22⋊C8×C157C2211yescharacteristic
27C23×D157D411yescharacteristic
28C2×Dic314—21no—
29C2×Dic314D411yescharacteristic, complemented
30C2×C22⋊D157—42no—
31C2×C22⋊D157—42no—
32C2×C22⋊D157—42no—
33C22×Dic157—21no—
34C22×Dic157D411yescharacteristic
35C2×C4×D157—21no—
36C2×C4×D157—21no—
37#14578942—42no—
38C2×#11929434—21no—
39C4⋊Dic157—21no—
40C4⋊Dic157D411yescharacteristic, complemented
41#14578933—21no—
42#14578933—21no—
43#14578933—21no—
44C22⋊Dic157—21no—
45C22⋊Dic157—42no—
46#14578931—21no—
47#14578931—21no—
48#14578931—21no—
49#14578931—21no—
50C24×C157—21no—
51C2×Q8×C157D411yescharacteristic, complemented
52C2×D4×C157—21no—
53C2×D4×C157—21no—
54C2×D4×C157—42no—
55C22×C4×C157C2311yescharacteristic
56SD16×C157—42no—
57C2×C8×C157—21no—
58C4⋊C4×C157—21no—
59C4⋊C4×C157D411yescharacteristic, complemented
60C22⋊C4×C157—21no—
61C22×D157—21no—
62C22×D157—21no—
63C22×D157—41no—
64C22×D157—41no—
65C22×D157SD1612yescomplemented
66Dic314—41no—
67Dic314—81no—
68C22⋊D157—48no—
69C2×Dic157—22no—
70C2×Dic157—21no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—41no—
74C2×Dic157—41no—
75C2×Dic157—41no—
76C4×D157—81no—
77C4×D157—81no—
78#11929434—41no—
79C23×C157—22no—
80C23×C157—41no—
81C23×C157—41no—
82C23×C157—41no—
83C23×C157—41no—
84C23×C157—41no—
85C23×C157—41no—
86C23×C157C2×D411yescharacteristic
87Q8×C157—41no—
88D4×C157—22no—
89D4×C157—22no—
90D4×C157—41no—
91D4×C157—41no—
92C2×C4×C157—21no—
93C2×C4×C157—21no—
94C2×C4×C157—41no—
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2×D411yescharacteristic
97C8×C157—41no—
98C22×C157—21no—
99C22×C157—21no—
100C22×C157—42no—
101C22×C157—42no—
102C22×C157—42no—
103C22×C157—81no—
104C22×C157—81no—
105C22×C157—81no—
106C22×C157—81no—
107C22×C157#12011yescharacteristic
108C22×C157C2×SD1611yescharacteristic
109C22×C157#13711yescharacteristic
110C2×D157—22no—
111C2×D157—41no—
112C2×D157—41no—
113C2×D157—42no—
114C2×D157—81no—
115C4×C157—21no—
116C4×C157—21no—
117C4×C157—41no—
118C4×C157—41no—
119Dic157—42no—
120Dic157—41no—
121Dic157—41no—
122Dic157—41no—
123C2×C157—21no—
124C2×C157—21no—
125C2×C157—82no—
126C2×C157#44811yescharacteristic
127C2×C157#45311yescharacteristic
128C2×C157B(4,𝔽2)11yescharacteristic
129D157—42no—
130C157#159611yescharacteristic, complemented, Sylow
131#1596—1571nomaximal, Sylow
132#543—1571no—
133#450—1571no—
134#450D15711yescharacteristic, complemented
135#392—1571no—
136#328—1571no—
137#328—1571no—
138#323—1571no—
139C22×D4—3141no—
140C22×D4C2×D15711yescharacteristic
141C2×SD16—21no—
142C2×SD16—3141no—
143C22⋊Q8—3141no—
144C22⋊Q8—3141no—
145C22⋊Q8—3141no—
146C22⋊Q8—3141no—
147C22⋊Q8—3141no—
148C22⋊Q8—1571no—
149C22⋊Q8C2×D15711yescharacteristic
150C2×C22⋊C4—3141no—
151C2×C22⋊C4—3141no—
152C2×C22⋊C4—1571no—
153#102—21no—
154#102—3141no—
155C22⋊C8—1571no—
156C22⋊C8C2×D15711yescharacteristic
157#95—1571no—
158C24—21no—
159C24—1571no—
160C2×Q8—3141no—
161C2×Q8—1571no—
162C2×Q8C22⋊D15711yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—6282no—
167C2×D4—6282no—
168C2×D4—6282no—
169C22×C4—3141no—
170C22×C4—3141no—
171C22×C4—3141no—
172C22×C4—1571no—
173C22×C4C22×D15711yescharacteristic
174SD16—42no—
175SD16—6282no—
176C2×C8—21no—
177C2×C8—3141no—
178C4⋊C4—21no—
179C4⋊C4—3141no—
180C4⋊C4—3141no—
181C4⋊C4—3141no—
182C4⋊C4—3141no—
183C4⋊C4—1571no—
184C4⋊C4C22⋊D15711yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—3141no—
187C22⋊C4—3141no—
188C22⋊C4—3141no—
189C22⋊C4—3141no—
190C22⋊C4—3141no—
191C22⋊C4—6282no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—3141no—
200C23—3141no—
201C23—6281no—
202C23—6281no—
203C23—1572no—
204C23D4×D15711yescharacteristic
205Q8—41no—
206Q8—6281no—
207Q8—12561no—
208D4—22no—
209D4—22no—
210D4—41no—
211D4—41no—
212D4—6288no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—3142no—
217C2×C4—3141no—
218C2×C4—3141no—
219C2×C4—3141no—
220C2×C4—6281no—
221C2×C4—6281no—
222C2×C4—6281no—
223C2×C4—12561no—
224C2×C4—12561no—
225C2×C4D4×D15711yescharacteristic
226C2×C4C2×C22⋊D15711yescharacteristic
227C8—41no—
228C8—6281no—
229C22—21no—
230C22—21no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—3142no—
239C22—6282no—
240C22—6281no—
241C22—6281no—
242C22—12561no—
243C22#1909773911yescharacteristic
244C22SD16×D15711yescharacteristic
245C22#1909780011yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—6282no—
251C4—6281no—
252C4—6281no—
253C4—6281no—
254C2—21no—
255C2—21no—
256C2—82no—
257C2—6282no—
258C2#2375741011yescharacteristic
259C2#2375743911yescharacteristic
260C2#2375745911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#450×C157C2
socleC22×C157#120
prime radical#23757181C2
hypercenter#450D157
hypocenterC157#1596
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23757898C2semidirect
2. #23757424C2semidirect
4. #23757011C2semidirect
5. #23757010C2semidirect
6. #23756996C2semidirect
7. #450×C157C2semidirect
29. C2×Dic314D4semidirect
40. C4⋊Dic157D4semidirect
51. C2×Q8×C157D4semidirect
59. C4⋊C4×C157D4semidirect
65. C22×D157SD16semidirect
130. C157#1596semidirect
134. #450D157semidirect
162. C2×Q8C22⋊D157semidirect
184. C4⋊C4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578982
#237574242
#237571812
#237570112
#237570102
#237569962
#450×C1572
#1596157