Finite Group Databasebeta

← #34099060

Subgroups — #34099060

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#23757426C211yesmaximal, characteristic, complemented
3#23757423C211yesmaximal, characteristic, complemented
4#23757183C211yesmaximal, characteristic
5#23757009C211yesmaximal, characteristic, complemented
6#23756996C211yesmaximal, characteristic, complemented
7#328×C157C211yesmaximal, characteristic, complemented
8C22×D628—21no—
9C22×D628C2211yescharacteristic
10C2×#14578943—21no—
11C2×#14578941—21no—
12#19097751—21no—
13#19097750C2211yescharacteristic
14#19097749—21no—
15#19097749—21no—
16#19097748—21no—
17#19097747—21no—
18C2×#14578931—21no—
19C2×#14578931C2211yescharacteristic
20C22⋊C4×D157—21no—
21#19097693—21no—
22#19097691—21no—
23C22⋊#11929434C2211yescharacteristic
24C22⋊Q8×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26#95×C157C2211yescharacteristic
27C23×D157—21no—
28C23×D157D411yescharacteristic
29C2×Dic314—21no—
30C2×Dic314D411yescharacteristic, complemented
31C2×D628—21no—
32C2×D628—21no—
33C2×D628—42no—
34C2×D628—42no—
35C2×D628—42no—
36C2×D628—42no—
37C22×Dic157D411yescharacteristic
38C2×C4×D157—21no—
39C2×C4×D157—21no—
40#14578943—42no—
41#14578941—42no—
42C2×#11929434—21no—
43C4⋊Dic157—21no—
44C4⋊Dic157D411yescharacteristic, complemented
45#14578933—21no—
46#14578933—21no—
47#14578933—21no—
48C22⋊Dic157—21no—
49#14578931—21no—
50#14578931—21no—
51#14578931—21no—
52#14578931—21no—
53#14578931—42no—
54C2×Q8×C157D411yescharacteristic, complemented
55C22×C4×C157—21no—
56C22×C4×C157C2311yescharacteristic
57C2×C8×C157—21no—
58C4⋊C4×C157—21no—
59C4⋊C4×C157D411yescharacteristic, complemented
60C22⋊C4×C157—21no—
61C22×D157—21no—
62C22×D157—21no—
63C22×D157—22no—
64C22×D157—41no—
65C22×D157—41no—
66C22×D157—41no—
67C22×D157—41no—
68C22×D157—41no—
69C22×D157—41no—
70C22×D157—41no—
71C22×D157—41no—
72C22×D157SD1612yescomplemented
73Dic314—41no—
74Dic314—81no—
75D628—22no—
76D628—22no—
77D628—41no—
78D628—41no—
79D628—48no—
80C2×Dic157—21no—
81C2×Dic157—21no—
82C2×Dic157—21no—
83C2×Dic157—41no—
84C4×D157—81no—
85C4×D157—81no—
86#11929434—41no—
87C23×C157C2×D411yescharacteristic
88Q8×C157—41no—
89C2×C4×C157—22no—
90C2×C4×C157—21no—
91C2×C4×C157—21no—
92C2×C4×C157—41no—
93C2×C4×C157—41no—
94C2×C4×C157—41no—
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2×D411yescharacteristic
97C8×C157—41no—
98C22×C157—21no—
99C22×C157—21no—
100C22×C157#12011yescharacteristic
101C22×C157C2×SD1611yescharacteristic
102C22×C157#13711yescharacteristic
103C2×D157—22no—
104C2×D157—41no—
105C2×D157—41no—
106C2×D157—42no—
107C2×D157—42no—
108C2×D157—42no—
109C2×D157—42no—
110C2×D157—81no—
111C2×D157—81no—
112C2×D157—81no—
113C2×D157—81no—
114C2×D157—81no—
115C4×C157—21no—
116C4×C157—21no—
117C4×C157—42no—
118C4×C157—41no—
119C4×C157—41no—
120Dic157—41no—
121Dic157—41no—
122Dic157—41no—
123C2×C157—21no—
124C2×C157—21no—
125C2×C157#44811yescharacteristic
126C2×C157#45311yescharacteristic
127C2×C157B(4,𝔽2)11yescharacteristic
128D157—42no—
129D157—82no—
130C157#159611yescharacteristic, complemented, Sylow
131#1596—1571nomaximal, Sylow
132#543—1571no—
133#450—1571no—
134#450—1571no—
135#392—1571no—
136#328—1571no—
137#328D15711yescharacteristic, complemented
138#323—1571no—
139C22×D4—3141no—
140C22×D4—1571no—
141C2×SD16—3141no—
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—3141no—
154#102—3141no—
155C22⋊C8—1571no—
156C22⋊C8C2×D15711yescharacteristic
157#95C2×D15711yescharacteristic
158C24—3141no—
159C24—1571no—
160C2×Q8—3141no—
161C2×Q8—1571no—
162C2×Q8D62811yescharacteristic, complemented
163C2×D4—3141no—
164C2×D4—3141no—
165C2×D4—6282no—
166C2×D4—6282no—
167C2×D4—6282no—
168C2×D4—6282no—
169C22×C4—21no—
170C22×C4—3141no—
171C22×C4—3141no—
172C22×C4—1571no—
173C22×C4C22×D15711yescharacteristic
174SD16—6282no—
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⋊C4D62811yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—3141no—
187C22⋊C4—3141no—
188C22⋊C4—3141no—
189C22⋊C4—3141no—
190C22⋊C4—3141no—
191C22⋊C4—6282no—
192C23—3142no—
193C23—3141no—
194C23—3141no—
195C23—6281no—
196C23—6281no—
197C23—6281no—
198C23—6281no—
199C23—6281no—
200C23—6281no—
201C23—6281no—
202C23—6281no—
203C23—1572no—
204C23D4×D15711yescharacteristic
205Q8—41no—
206Q8—6281no—
207Q8—12561no—
208D4—3142no—
209D4—3142no—
210D4—6281no—
211D4—6281no—
212D4—6288no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—3141no—
220C2×C4—3141no—
221C2×C4—3141no—
222C2×C4—6281no—
223C2×C4—12561no—
224C2×C4—12561no—
225C2×C4D4×D15711yescharacteristic
226C2×C4C2×D62811yescharacteristic
227C8—41no—
228C8—6281no—
229C22—21no—
230C22—21no—
231C22—3142no—
232C22—6282no—
233C22—6282no—
234C22—6282no—
235C22—6281no—
236C22—6281no—
237C22—6282no—
238C22—12561no—
239C22—12561no—
240C22—12561no—
241C22—12561no—
242C22—12561no—
243C22#1909774111yescharacteristic
244C22SD16×D15711yescharacteristic
245C22#1909780311yescharacteristic
246C4—21no—
247C4—21no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—6281no—
252C4—6281no—
253C4—6281no—
254C2—21no—
255C2—21no—
256C2—6282no—
257C2—12562no—
258C2#2375741311yescharacteristic
259C2#2375744211yescharacteristic
260C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#328×C157C2
socleC22×C157#120
prime radical#23757183C2
hypercenter#328D157
hypocenterC157#1596
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #23757898C2semidirect
2. #23757426C2semidirect
3. #23757423C2semidirect
5. #23757009C2semidirect
6. #23756996C2semidirect
7. #328×C157C2semidirect
30. C2×Dic314D4semidirect
44. C4⋊Dic157D4semidirect
54. C2×Q8×C157D4semidirect
59. C4⋊C4×C157D4semidirect
72. C22×D157SD16semidirect
130. C157#1596semidirect
137. #328D157semidirect
162. C2×Q8D628semidirect
184. C4⋊C4D628semidirect

Maximal subgroups

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

subgroupindex
#237578982
#237574262
#237574232
#237571832
#237570092
#237569962
#328×C1572
#1596157