Finite Group Databasebeta

← #34100089

Subgroups — #34100089

Order 20096.

620 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#19097816C211yesmaximal, characteristic
2C2×#19097699C212yesmaximal, complemented
3#23757272C212yesmaximal, complemented
4C2×#19097688C211yesmaximal, characteristic, complemented
5#390×C157C211yesmaximal, characteristic, complemented
6#19097816—22no—
7#19097816—24no—
8#19097816C414yescomplemented
9C2×#14578954C2212yes—
10C2×#14578953—22no—
11C22×C22⋊D157—22no—
12C2×D4×D157—22no—
13C2×#14578938C2212yescomplemented
14#19097699—24no—
15#19097694—22no—
16#19097691—22no—
17#19097688—22no—
18C22×D4×C157C2211yescharacteristic
19C2×C4⋊C4×C157C2211yescharacteristic, complemented
20C2×C22⋊C4×C157—22no—
21C2×C42×C157C2211yescharacteristic, complemented
22C23×D157—22no—
23#14578954—22no—
24#14578954—22no—
25#14578954C2×C414yes—
26#14578954D414yes—
27#14578953—28no—
28#14578953—28no—
29C2×Dic314D412yes—
30C2×D628D412yes—
31C2×C22⋊D157—24no—
32C2×C22⋊D157—28no—
33C2×C22⋊D157—28no—
34C2×C22⋊D157—22no—
35D4×D157—28no—
36D4×D157—28no—
37C22×Dic157—22no—
38C2×C4×D157—22no—
39#14578938—22no—
40#14578938—22no—
41C2×#11929434—22no—
42C24×C157D412yes—
43C2×D4×C157—22no—
44C2×D4×C157—22no—
45C2×D4×C157C2×C412yes—
46C2×D4×C157D412yes—
47C22×C4×C157—21no—
48C22×C4×C157—21no—
49C22×C4×C157C2311yescharacteristic
50C4⋊C4×C157—22no—
51C22⋊C4×C157—44no—
52C42×C157—22no—
53C22×D157—22no—
54C22×D157—28no—
55C22×D157—82no—
56C22×D157—82no—
57Dic314—22no—
58Dic314C22⋊C414yescomplemented
59D628—22no—
60D628C22⋊C414yescomplemented
61C22⋊D157—216no—
62C22⋊D157—24no—
63C22⋊D157—24no—
64C2×Dic157—28no—
65C2×Dic157—22no—
66C4×D157—24no—
67C4×D157—24no—
68#11929434—42no—
69C23×C157—22no—
70C23×C157—22no—
71C23×C157—42no—
72C23×C157—42no—
73C23×C157C22⋊C414yes—
74C23×C157C2×D411yescharacteristic
75D4×C157—24no—
76D4×C157—24no—
77C2×C4×C157—22no—
78C2×C4×C157—41no—
79C2×C4×C157—41no—
80C2×C4×C157—42no—
81C2×C4×C157—41no—
82C2×C4×C157C22×C411yescharacteristic
83C2×C4×C157C2×D412yes—
84C2×C4×C157C2×D412yes—
85C2×C4×C157C2×D411yescharacteristic
86C22×C157—21no—
87C22×C157—21no—
88C22×C157—24no—
89C22×C157—44no—
90C22×C157—42no—
91C22×C157—42no—
92C22×C157—82no—
93C22×C157C2×C22⋊C411yescharacteristic
94C22×C157#12011yescharacteristic
95C22×C157#12011yescharacteristic
96C2×D157—24no—
97C2×D157—42no—
98C2×D157—42no—
99C2×D157—88no—
100C4×C157—42no—
101C4×C157—81no—
102C4×C157C2×C22⋊C412yes—
103C4×C157#12012yes—
104Dic157—24no—
105C2×C157—21no—
106C2×C157—21no—
107C2×C157—44no—
108C2×C157#37911yescharacteristic
109C2×C157#45311yescharacteristic
110C2×C157#45511yescharacteristic
111D157—44no—
112C157#179211yescharacteristic, complemented, Sylow
113#1792—1571nomaximal, Sylow
114C2×H(5,𝔽2)—1571no—
115C2×#104—1572no—
116#417—1572no—
117C2×#100—1571no—
118#390D15711yescharacteristic, complemented
119H(5,𝔽2)—3144no—
120H(5,𝔽2)—3142no—
121H(5,𝔽2)—1574no—
122C2×#41—3142no—
123C2×#41—1572no—
124C22×D4—3142no—
125C22×D4—3142no—
126C22×D4C2×D15711yescharacteristic
127C2×M16—1572no—
128C2×C4⋊C4C2×D15711yescharacteristic, complemented
129C2×C22⋊C4—22no—
130C2×C42C2×D15711yescharacteristic, complemented
131#104—3144no—
132#103—3142no—
133#102—3142no—
134#100—3142no—
135C24—3142no—
136C24C22⋊D15712yes—
137#41—3148no—
138#41—3148no—
139#41—3142no—
140#41—3142no—
141#41—1574no—
142#41—1574no—
143C2×Q8—1572no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—3148no—
147C2×D4—3148no—
148C2×D4—3144no—
149C2×D4—3148no—
150C2×D4—3142no—
151C2×D4—3148no—
152C2×D4—1572no—
153C2×D4C4×D15712yes—
154C2×D4D62812yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—3142no—
158C22×C4—3142no—
159C22×C4C22×D15711yescharacteristic
160M16—3142no—
161M16—3142no—
162C2×C8—3142no—
163C4⋊C4—22no—
164C22⋊C4—44no—
165C42—22no—
166C23—22no—
167C23—22no—
168C23—42no—
169C23—42no—
170C23—3148no—
171C23—3142no—
172C23—12562no—
173C23—12562no—
174C23#1457893114yes—
175C23C2×C22⋊D15711yescharacteristic
176Q8—3142no—
177Q8—1574no—
178D4—24no—
179D4—24no—
180D4—31416no—
181D4—3144no—
182D4—3144no—
183D4—3142no—
184D4—1574no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—3148no—
191C2×C4—3142no—
192C2×C4—3144no—
193C2×C4—3144no—
194C2×C4C2×C4×D15711yescharacteristic
195C2×C4D4×D15712yes—
196C2×C4D4×D15712yes—
197C2×C4C2×D62811yescharacteristic
198C8—6282no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—82no—
206C22—3144no—
207C22—6282no—
208C22—6282no—
209C22—12568no—
210C22C2×#1457893111yescharacteristic
211C22#1909773911yescharacteristic
212C22#1909773911yescharacteristic
213C4—42no—
214C4—81no—
215C4—3144no—
216C4C22⋊C4×D15712yes—
217C4#1909774112yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—6284no—
222C2#2375712811yescharacteristic
223C2#2375744011yescharacteristic
224C2#2375745011yescharacteristic
subgroupG/H
centerC22#19097739
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#390×C157C2
socleC22×C157#120
prime radicalC2×#19097816C2
cosocleC22×C4×C157C23
hypercenter#390D157
hypocenterC157#1792
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#19097699C2semidirect
3. #23757272C2semidirect
4. C2×#19097688C2semidirect
5. #390×C157C2semidirect
8. #19097816C4semidirect
13. C2×#14578938C22semidirect
19. C2×C4⋊C4×C157C22semidirect
21. C2×C42×C157C22semidirect
58. Dic314C22⋊C4semidirect
60. D628C22⋊C4semidirect
112. C157#1792semidirect
118. #390D157semidirect
128. C2×C4⋊C4C2×D157semidirect
130. C2×C42C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978162
C2×#190976992
#237572722
C2×#190976882
#390×C1572
#1792157