Finite Group Databasebeta

← #34099089

Subgroups — #34099089

Order 20096.

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

normal subgroups only

This group has 308 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#23757831C211yesmaximal, characteristic
2#23757414C211yesmaximal, characteristic, complemented
3#23757411C211yesmaximal, characteristic, complemented
4#23757186C211yesmaximal, characteristic, complemented
5#23757006C211yesmaximal, characteristic, complemented
6#23756996C211yesmaximal, characteristic, complemented
7#327×C157C211yesmaximal, characteristic, complemented
8C22×Dic314C2211yescharacteristic, complemented
9C2×D4×D157—21no—
10C2×#14578943—21no—
11C2×#14578942—21no—
12#19097756—21no—
13#19097753—21no—
14#19097745C2211yescharacteristic
15#19097743—21no—
16#19097741—21no—
17#19097739—21no—
18C2×#14578931—21no—
19C2×#14578931C2211yescharacteristic
20#19097696—21no—
21#19097694—21no—
22C22⋊#11929434C2211yescharacteristic, complemented
23#19097678—21no—
24#121×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic, complemented
26#95×C157C2211yescharacteristic, complemented
27C23×D157—21no—
28C23×D157D411yescharacteristic
29C2×Dic314—21no—
30C2×Dic314—21no—
31C2×Dic314—42no—
32C2×D628—21no—
33C2×D628D411yescharacteristic
34C2×C22⋊D157—21no—
35C2×C22⋊D157—21no—
36C2×C22⋊D157—21no—
37D4×D157—42no—
38D4×D157—42no—
39C22×Dic157—21no—
40C22×Dic157D411yescharacteristic
41C2×C4×D157—21no—
42#14578943—42no—
43#14578942—42no—
44C2×#11929434—21no—
45C4⋊Dic157D411yescharacteristic
46#14578933—21no—
47C22⋊Dic157—21no—
48#14578931—21no—
49#14578931—21no—
50#14578931—21no—
51#14578931—21no—
52#14578931—42no—
53C2×D4×C157—21no—
54C2×D4×C157D411yescharacteristic
55C22×C4×C157—21no—
56C22×C4×C157C2311yescharacteristic
57C2×C8×C157—21no—
58C4⋊C4×C157D411yescharacteristic
59C22⋊C4×C157—21no—
60C22×D157—21no—
61C22×D157—21no—
62C22×D157—21no—
63C22×D157—42no—
64C22×D157—41no—
65C22×D157—41no—
66C22×D157—81no—
67C22×D157—81no—
68C22×D157SD1612yescomplemented
69Dic314—22no—
70Dic314—22no—
71Dic314—41no—
72Dic314—41no—
73D628—41no—
74D628—42no—
75C22⋊D157—42no—
76C22⋊D157—41no—
77C22⋊D157—41no—
78C22⋊D157—81no—
79C2×Dic157—22no—
80C2×Dic157—21no—
81C2×Dic157—21no—
82C2×Dic157—41no—
83C2×Dic157—41no—
84C2×Dic157—41no—
85C4×D157—42no—
86#11929434—41no—
87C23×C157—21no—
88C23×C157C2×D411yescharacteristic
89D4×C157—41no—
90D4×C157—41no—
91D4×C157—41no—
92C2×C4×C157—21no—
93C2×C4×C157—41no—
94C2×C4×C157—41no—
95C2×C4×C157—41no—
96C2×C4×C157—41no—
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C8×C157—41no—
100C22×C157—21no—
101C22×C157—21no—
102C22×C157—41no—
103C22×C157—41no—
104C22×C157—41no—
105C22×C157#12011yescharacteristic
106C22×C157C2×SD1611yescharacteristic
107C22×C157AGL(1,ℤ/8)11yescharacteristic
108C2×D157—22no—
109C2×D157—41no—
110C2×D157—41no—
111C2×D157—42no—
112C2×D157—41no—
113C2×D157—41no—
114C2×D157—41no—
115C2×D157—81no—
116C2×D157—84no—
117C4×C157—21no—
118C4×C157—21no—
119C4×C157—41no—
120C4×C157—81no—
121Dic157—41no—
122Dic157—41no—
123Dic157—42no—
124C2×C157—21no—
125C2×C157—21no—
126C2×C157—81no—
127C2×C157#45011yescharacteristic
128C2×C157#45511yescharacteristic
129C2×C157B(4,𝔽2)11yescharacteristic
130D157—42no—
131D157—81no—
132C157#160111yescharacteristic, complemented, Sylow
133#1601—1571nomaximal, Sylow
134#534—1571no—
135#448—1571no—
136#448—1571no—
137#393—1571no—
138#327—1571no—
139#327D15711yescharacteristic, complemented
140#323—1571no—
141C22×Q8—1571no—
142C22×D4—3141no—
143C2×SD16—3141no—
144C2×SD16—3141no—
145#123—3141no—
146#123—3141no—
147#121—3141no—
148#121—1571no—
149#121C2×D15711yescharacteristic
150#120—3141no—
151#120—3141no—
152C2×C22⋊C4—3141no—
153C2×C22⋊C4—1571no—
154#103—3141no—
155#103—3141no—
156C22⋊C8—1571no—
157C22⋊C8C2×D15711yescharacteristic, complemented
158#95—3141no—
159#95C2×D15711yescharacteristic, complemented
160C24—3141no—
161C24—1571no—
162C2×Q8—3141no—
163C2×Q8—3141no—
164C2×Q8—6282no—
165C2×D4—21no—
166C2×D4—3141no—
167C2×D4—3141no—
168C2×D4—3141no—
169C2×D4—3141no—
170C2×D4—6282no—
171C2×D4—6282no—
172C2×D4—1571no—
173C2×D4D62811yescharacteristic
174C22×C4—21no—
175C22×C4—3141no—
176C22×C4—3141no—
177C22×C4—1571no—
178C22×C4C22×D15711yescharacteristic
179SD16—6282no—
180SD16—6282no—
181C2×C8—21no—
182C2×C8—3141no—
183C4⋊C4—3141no—
184C4⋊C4—1571no—
185C4⋊C4D62811yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—3141no—
188C22⋊C4—3141no—
189C22⋊C4—3141no—
190C22⋊C4—3141no—
191C22⋊C4—3141no—
192C22⋊C4—6282no—
193C23—21no—
194C23—3141no—
195C23—3141no—
196C23—3141no—
197C23—6281no—
198C23—6281no—
199C23—6282no—
200C23—1572no—
201C23—12561no—
202C23—12561no—
203C23D4×D15711yescharacteristic
204Q8—3142no—
205Q8—3142no—
206Q8—6281no—
207Q8—6281no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—6281no—
212D4—6282no—
213D4—6282no—
214D4—6281no—
215D4—6281no—
216D4—12561no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—3142no—
223C2×C4—3141no—
224C2×C4—3141no—
225C2×C4—6282no—
226C2×C4—6281no—
227C2×C4—6281no—
228C2×C4—6281no—
229C2×C4D4×D15711yescharacteristic
230C2×C4C2×D62811yescharacteristic
231C8—41no—
232C8—6281no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—3142no—
239C22—6282no—
240C22—6281no—
241C22—6281no—
242C22—6281no—
243C22—6281no—
244C22—6281no—
245C22—12561no—
246C22—12564no—
247C22#1909774111yescharacteristic
248C22SD16×D15711yescharacteristic
249C22#1909779511yescharacteristic
250C4—21no—
251C4—21no—
252C4—41no—
253C4—81no—
254C4—6281no—
255C4—6281no—
256C4—6282no—
257C2—21no—
258C2—21no—
259C2—81no—
260C2—6282no—
261C2—12561no—
262C2#2375742511yescharacteristic
263C2#2375745211yescharacteristic
264C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#327×C157C2
socleC22×C157#120
prime radical#23757831C2
hypercenter#327D157
hypocenterC157#1601
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #23757414C2semidirect
3. #23757411C2semidirect
4. #23757186C2semidirect
5. #23757006C2semidirect
6. #23756996C2semidirect
7. #327×C157C2semidirect
8. C22×Dic314C22semidirect
22. C22⋊#11929434C22semidirect
25. C22⋊C8×C157C22semidirect
26. #95×C157C22semidirect
68. C22×D157SD16semidirect
132. C157#1601semidirect
139. #327D157semidirect
157. C22⋊C8C2×D157semidirect
159. #95C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578312
#237574142
#237574112
#237571862
#237570062
#237569962
#327×C1572
#1601157