Finite Group Databasebeta

← #34099056

Subgroups — #34099056

Order 20096.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#23757835C211yesmaximal, characteristic, complemented
2#23757409C212yesmaximal, complemented
3#23757182C211yesmaximal, characteristic, complemented
4#23757008C212yesmaximal, complemented
5#323×C157C211yesmaximal, characteristic, complemented
6C22×D628C2211yescharacteristic, complemented
7C22×C22⋊D157—21no—
8C22×C22⋊D157—21no—
9C2×D1256—22no—
10#19097755—22no—
11#19097745—21no—
12#19097745C2212yes—
13#19097741—22no—
14C2×#14578931—21no—
15#19097693—22no—
16#19097678C2211yescharacteristic, complemented
17C2×C22⋊C4×C157C2211yescharacteristic
18C22⋊C8×C157C2212yescomplemented
19C23×D157—21no—
20C23×D157—21no—
21C2×D628—21no—
22C2×D628—22no—
23C2×D628—42no—
24C2×D628D412yescomplemented
25C2×C22⋊D157—22no—
26C2×C22⋊D157—22no—
27C2×C22⋊D157—42no—
28C2×C22⋊D157—42no—
29C2×C22⋊D157—42no—
30C2×C22⋊D157—42no—
31C2×C22⋊D157—42no—
32C22×Dic157—21no—
33C22×Dic157—21no—
34D1256—44no—
35C4⋊Dic157—21no—
36C4⋊Dic157D412yescomplemented
37#14578931—22no—
38#14578931—22no—
39#14578931—42no—
40C24×C157D411yescharacteristic
41C22×C4×C157D411yescharacteristic
42C22×C4×C157C2311yescharacteristic
43C2×C8×C157—22no—
44C22⋊C4×C157—22no—
45C22×D157—22no—
46C22×D157—22no—
47C22×D157—41no—
48C22×D157—42no—
49C22×D157—41no—
50C22×D157—41no—
51C22×D157—41no—
52C22×D157—41no—
53C22×D157—41no—
54C22×D157—81no—
55C22×D157—81no—
56D628—24no—
57D628—42no—
58D628—42no—
59D628—81no—
60C22⋊D157—44no—
61C22⋊D157—48no—
62C22⋊D157—42no—
63C22⋊D157—42no—
64C2×Dic157—22no—
65C2×Dic157—22no—
66C2×Dic157—41no—
67C2×Dic157—42no—
68C2×Dic157—41no—
69C23×C157—21no—
70C23×C157—21no—
71C23×C157—41no—
72C23×C157—41no—
73C23×C157D812yescomplemented
74C23×C157C2×D411yescharacteristic
75C2×C4×C157—21no—
76C2×C4×C157—41no—
77C2×C4×C157—41no—
78C2×C4×C157C2×D412yes—
79C8×C157—42no—
80C22×C157—21no—
81C22×C157—21no—
82C22×C157—22no—
83C22×C157—42no—
84C22×C157—41no—
85C22×C157—41no—
86C22×C157—81no—
87C22×C157#12011yescharacteristic
88C22×C157C2×D811yescharacteristic
89C22×C157AGL(1,ℤ/8)11yescharacteristic
90C2×D157—42no—
91C2×D157—42no—
92C2×D157—42no—
93C2×D157—42no—
94C2×D157—42no—
95C2×D157—42no—
96C2×D157—81no—
97C2×D157—81no—
98C2×D157—84no—
99C2×D157—81no—
100C2×D157—81no—
101C4×C157—22no—
102C4×C157—41no—
103Dic157—42no—
104Dic157—42no—
105C2×C157—21no—
106C2×C157—21no—
107C2×C157—42no—
108C2×C157#44711yescharacteristic
109C2×C157#45311yescharacteristic
110C2×C157B(4,𝔽2)11yescharacteristic
111D157—82no—
112D157—82no—
113C157#159511yescharacteristic, complemented, Sylow
114#1595—1571nomaximal, Sylow
115#534—1571no—
116#447—1572no—
117#392—1571no—
118#327—1572no—
119#323D15711yescharacteristic, complemented
120C22×D4—3141no—
121C22×D4—3141no—
122C22×D4—1571no—
123C2×D8—3142no—
124#123—3142no—
125#121—3141no—
126#121—1572no—
127#120—3142no—
128C2×C22⋊C4—3141no—
129C2×C22⋊C4C2×D15711yescharacteristic
130#102—3142no—
131C22⋊C8C2×D15712yescomplemented
132#95—1571no—
133C24—3141no—
134C24—3141no—
135C24D62811yescharacteristic
136C2×D4—3141no—
137C2×D4—3142no—
138C2×D4—3142no—
139C2×D4—3142no—
140C2×D4—6282no—
141C2×D4—6282no—
142C2×D4—6282no—
143C2×D4—6282no—
144C2×D4—6282no—
145C2×D4—6282no—
146C2×D4—1572no—
147C22×C4—3141no—
148C22×C4—3141no—
149C22×C4D62811yescharacteristic
150C22×C4C22×D15711yescharacteristic
151D8—6284no—
152C2×C8—22no—
153C4⋊C4—3141no—
154C4⋊C4—1572no—
155C22⋊C4—22no—
156C22⋊C4—3142no—
157C22⋊C4—3142no—
158C22⋊C4—6282no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—3142no—
164C23—3142no—
165C23—6281no—
166C23—6281no—
167C23—6281no—
168C23—6282no—
169C23—6281no—
170C23—6281no—
171C23—6281no—
172C23—12561no—
173C23—12561no—
174C23D125612yescomplemented
175C23C2×D62811yescharacteristic
176D4—3144no—
177D4—6282no—
178D4—6282no—
179D4—6284no—
180D4—6288no—
181D4—6282no—
182D4—6282no—
183D4—12561no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—3142no—
188C2×C4—3142no—
189C2×C4—6281no—
190C2×C4—6282no—
191C2×C4—6281no—
192C2×C4D4×D15712yes—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—6282no—
202C22—6282no—
203C22—6282no—
204C22—6282no—
205C22—6282no—
206C22—6282no—
207C22—12561no—
208C22—12561no—
209C22—12564no—
210C22—12561no—
211C22—12561no—
212C22#1909774111yescharacteristic
213C22C2×D125611yescharacteristic
214C22#1909779911yescharacteristic
215C4—22no—
216C4—41no—
217C4—6282no—
218C4—6282no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—12562no—
223C2—12562no—
224C2#2375740911yescharacteristic
225C2#2375744311yescharacteristic
226C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#323×C157C2
socleC22×C157#120
hypercenter#323D157
hypocenterC157#1595
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #23757835C2semidirect
2. #23757409C2semidirect
3. #23757182C2semidirect
4. #23757008C2semidirect
5. #323×C157C2semidirect
6. C22×D628C22semidirect
16. #19097678C22semidirect
18. C22⋊C8×C157C22semidirect
24. C2×D628D4semidirect
36. C4⋊Dic157D4semidirect
73. C23×C157D8semidirect
113. C157#1595semidirect
119. #323D157semidirect
131. C22⋊C8C2×D157semidirect
174. C23D1256semidirect

Maximal subgroups

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

subgroupindex
#237578352
#237574092
#237571822
#237570082
#323×C1572
#1595157