Finite Group Databasebeta

← #34100416

Subgroups — #34100416

Order 20096.

374 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 89 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#23757748C211yesmaximal, characteristic
2C2×#19097686C211yesmaximal, characteristic
3#23757146C212yesmaximal, complemented
4#23757039C212yesmaximal, complemented
5#386×C157C211yesmaximal, characteristic
6D4×Dic157—22no—
7D4×Dic157—22no—
8#19097730—21no—
9C2×#14578933—22no—
10C2×C22⋊Dic157—21no—
11C2×C22⋊Dic157C2212yes—
12C2×C22⋊Dic157C2212yes—
13#19097686C414yescomplemented
14#19097678—22no—
15C22×D4×C157C2211yescharacteristic
16C23×C4×C157C2211yescharacteristic
17C2×C22⋊C4×C157—21no—
18C2×C22⋊C4×C157C2211yescharacteristic
19#95×C157—21no—
20C22×Dic157—21no—
21C22×Dic157—21no—
22C22×Dic157—22no—
23C22×Dic157D412yes—
24C22×Dic157D412yes—
25C4⋊Dic157—21no—
26C4⋊Dic157—21no—
27#14578933—44no—
28C22⋊Dic157—22no—
29C22⋊Dic157—22no—
30C22⋊Dic157—22no—
31C22⋊Dic157—22no—
32C22⋊Dic157C2×C414yescomplemented
33C4×Dic157—22no—
34C24×C157D411yescharacteristic
35C24×C157C2311yescharacteristic
36C2×D4×C157—22no—
37C2×D4×C157—22no—
38C2×D4×C157C2×C412yes—
39C2×D4×C157Q812yes—
40C22×C4×C157—21no—
41C22×C4×C157—22no—
42C22×C4×C157—21no—
43C22×C4×C157—42no—
44C22×C4×C157D411yescharacteristic
45C22⋊C4×C157—22no—
46C22⋊C4×C157—42no—
47C2×Dic157—24no—
48C2×Dic157—22no—
49C2×Dic157—22no—
50C2×Dic157—42no—
51C2×Dic157—42no—
52C2×Dic157—42no—
53C2×Dic157—42no—
54C2×Dic157—81no—
55C2×Dic157—81no—
56C23×C157—21no—
57C23×C157—21no—
58C23×C157—41no—
59C23×C157—41no—
60C23×C157—41no—
61C23×C157—41no—
62C23×C157C4⋊C412yescomplemented
63C23×C157C22×C411yescharacteristic
64C23×C157C2×D412yes—
65C23×C157C2×D411yescharacteristic
66C23×C157C2×Q811yescharacteristic
67C23×C157#4112yes—
68D4×C157—24no—
69D4×C157—24no—
70C2×C4×C157—21no—
71C2×C4×C157—21no—
72C2×C4×C157—22no—
73C2×C4×C157—24no—
74C2×C4×C157—41no—
75C2×C4×C157—42no—
76C2×C4×C157—41no—
77C2×C4×C157—41no—
78C2×C4×C157—41no—
79C2×C4×C157—41no—
80C2×C4×C157C4⋊C412yescomplemented
81C22×C157—21no—
82C22×C157—22no—
83C22×C157—21no—
84C22×C157—22no—
85C22×C157—22no—
86C22×C157—41no—
87C22×C157—41no—
88C22×C157—42no—
89C22×C157—42no—
90C22×C157—41no—
91C22×C157—41no—
92C22×C157—81no—
93C22×C157C2×C4⋊C411yescharacteristic
94C22×C157C4×D412yes—
95C22×C157#12011yescharacteristic
96C22×C157C22⋊Q812yes—
97C22×C157#12311yescharacteristic
98C4×C157—22no—
99C4×C157—42no—
100C4×C157—81no—
101Dic157—44no—
102Dic157—42no—
103Dic157—42no—
104C2×C157—22no—
105C2×C157—21no—
106C2×C157—21no—
107C2×C157—22no—
108C2×C157—42no—
109C2×C157#38511yescharacteristic
110C2×C157B(4,𝔽2)11yescharacteristic
111C2×C157#45811yescharacteristic
112C157#186711yescharacteristic, complemented, Sylow
113#1867—1571nomaximal, Sylow
114#518—1571no—
115C2×#99—1571no—
116#386D15711yescharacteristic
117#385—1572no—
118#342—1572no—
119C22×D4C2×D15711yescharacteristic
120C23×C4C2×D15711yescharacteristic
121C4×D4—3142no—
122C4×D4—3142no—
123#117—3141no—
124C2×C4⋊C4—3142no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—3141no—
127C2×C22⋊C4—1572no—
128C2×C22⋊C4—1572no—
129C2×C22⋊C4C2×D15711yescharacteristic
130#99—1574no—
131#95—21no—
132#95—3142no—
133C24C22⋊D15711yescharacteristic
134C24C22×D15711yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4C4×D15712yes—
138C2×D4Dic31412yes—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—42no—
143C22×C4—3142no—
144C22×C4—3141no—
145C22×C4—3141no—
146C22×C4—1572no—
147C22×C4—1572no—
148C22×C4C22⋊D15711yescharacteristic
149C4⋊C4—3141no—
150C4⋊C4—3141no—
151C4⋊C4—6284no—
152C22⋊C4—22no—
153C22⋊C4—42no—
154C22⋊C4—3142no—
155C22⋊C4—3142no—
156C22⋊C4—3142no—
157C22⋊C4—3142no—
158C22⋊C4—1574no—
159C42—3142no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23#1457893312yescomplemented
167C23C2×C4×D15711yescharacteristic
168C23D4×D15712yes—
169C23C2×C22⋊D15711yescharacteristic
170C23C2×Dic31411yescharacteristic
171C23#1457895312yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—3144no—
185C2×C4—3142no—
186C2×C4—3142no—
187C2×C4—6282no—
188C2×C4—6282no—
189C2×C4—6282no—
190C2×C4—6282no—
191C2×C4—12561no—
192C2×C4—12561no—
193C2×C4#1457893312yescomplemented
194C22—21no—
195C22—22no—
196C22—21no—
197C22—22no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×#1457893311yescharacteristic
207C22#1909773112yes—
208C22#1909773911yescharacteristic
209C22#1909774812yes—
210C22#1909775211yescharacteristic
211C4—22no—
212C4—42no—
213C4—81no—
214C4—6284no—
215C4—6282no—
216C4—6282no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#2375714511yescharacteristic
223C2#2375745911yescharacteristic
224C2#2375746211yescharacteristic
subgroupG/H
centerC22#19097739
commutatorC23×C157C22×C4
FrattiniC24C22×D157
Fitting#386×C157C2
socleC22×C157#120
prime radicalC22×D4×C157C22
cosocleC24×C157C23
hypercenter#386D157
hypocenterC157#1867
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #23757146C2semidirect
4. #23757039C2semidirect
13. #19097686C4semidirect
32. C22⋊Dic157C2×C4semidirect
62. C23×C157C4⋊C4semidirect
80. C2×C4×C157C4⋊C4semidirect
112. C157#1867semidirect
166. C23#14578933semidirect
193. C2×C4#14578933semidirect

Maximal subgroups

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

subgroupindex
#237577482
C2×#190976862
#237571462
#237570392
#386×C1572
#1867157