Finite Group Databasebeta

← #34109713

Subgroups — #34109713

Order 20096.

520 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C22×C8×D157C211yesmaximal, characteristic, complemented
2#23757780C212yesmaximal, complemented
3#23757511C212yesmaximal, complemented
4#23757506C212yesmaximal, complemented
5#23757499C214yesmaximal, complemented
6#23757311C211yesmaximal, characteristic, complemented
7#23757269C212yesmaximal, complemented
8C2×#135×C157C211yesmaximal, characteristic, complemented
9C22×C4×D157C2211yescharacteristic
10C22×#11929434C2211yescharacteristic
11C2×C8×D157—24no—
12C2×C8×D157C2212yes—
13C2×C8×D157C2212yes—
14#19097752—22no—
15#19097747C2214yes—
16#19097742C2214yes—
17#19097739—22no—
18C4×C22⋊D157—24no—
19#19097730C2212yes—
20#19097705C2212yescomplemented
21#19097703C2212yescomplemented
22#19097695C2214yescomplemented
23#19097692C2214yescomplemented
24C2×#41×C157C2212yes—
25#135×C157—24no—
26C2×Q16×C157C2211yescharacteristic, complemented
27C2×SD16×C157C2212yescomplemented
28C2×D8×C157C2211yescharacteristic, complemented
29C22×C8×C157C2211yescharacteristic
30C23×D157D411yescharacteristic
31C2×C22⋊D157—24no—
32C22×Dic157D411yescharacteristic
33C2×C4×D157—22no—
34C2×C4×D157—22no—
35C2×C4×D157D412yes—
36C2×C4×D157C2312yes—
37C2×#11929434—22no—
38C2×#11929434C2312yes—
39C8×D157—24no—
40C8×D157—24no—
41C4⋊Dic157C2314yes—
42#14578933—24no—
43C22⋊Dic157—24no—
44C22⋊Dic157—22no—
45#14578931—24no—
46C4×Dic157—22no—
47#41×C157—24no—
48#41×C157—24no—
49C2×Q8×C157C2312yes—
50C2×D4×C157—22no—
51C2×D4×C157C2312yes—
52C22×C4×C157—22no—
53C22×C4×C157C2311yescharacteristic
54Q16×C157—22no—
55SD16×C157—24no—
56D8×C157—22no—
57C2×C8×C157D414yescomplemented
58C2×C8×C157C2311yescharacteristic
59C2×C8×C157C2311yescharacteristic
60C22×D157—22no—
61C22×D157—24no—
62C22×D157C2×D412yes—
63C22⋊D157—48no—
64C2×Dic157—22no—
65C2×Dic157—24no—
66C2×Dic157C2×D412yes—
67C4×D157—24no—
68C4×D157—22no—
69C4×D157—22no—
70#11929434—22no—
71C23×C157—22no—
72C23×C157C2×D411yescharacteristic
73Q8×C157—22no—
74Q8×C157—22no—
75D4×C157—22no—
76D4×C157—22no—
77D4×C157—44no—
78C2×C4×C157—22no—
79C2×C4×C157—44no—
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2×D412yes—
82C2×C4×C157#4112yes—
83C2×C4×C157C2411yescharacteristic
84C8×C157C2×D412yes—
85C8×C157C2×D412yes—
86C22×C157—22no—
87C22×C157—42no—
88C22×C157—44no—
89C22×C157#12112yes—
90C22×C157C22×D411yescharacteristic
91C2×D157—28no—
92C2×D157—22no—
93C2×D157—22no—
94C2×D157#13514yescomplemented
95C4×C157—42no—
96C4×C157#12112yes—
97C4×C157C22×D411yescharacteristic
98C4×C157C2×#4111yescharacteristic
99Dic157—22no—
100Dic157—44no—
101C2×C157—22no—
102C2×C157—82no—
103C2×C157C2×#12111yescharacteristic
104C2×C157C2×#13512yes—
105D157—24no—
106C157#305011yescharacteristic, complemented, Sylow
107#3050—1571nomaximal, Sylow
108C2×#135D15711yescharacteristic, complemented
109C23×C8—1571no—
110#525—1572no—
111#467—1572no—
112#466—1572no—
113#465—1574no—
114#427—1571no—
115#416—1572no—
116C2×#41C2×D15712yes—
117C23×C4—1571no—
118#135—24no—
119C2×Q16C2×D15711yescharacteristic, complemented
120C2×SD16C2×D15712yescomplemented
121C2×D8C2×D15711yescharacteristic, complemented
122C22×C8—3144no—
123C22×C8—1571no—
124C22×C8—1572no—
125C22×C8—1572no—
126C22×C8C2×D15711yescharacteristic
127#123—3142no—
128C22⋊Q8—1574no—
129#121—1574no—
130#120—3142no—
131C4×D4—3144no—
132#117—1572no—
133#107—1572no—
134#106—1572no—
135#103—1574no—
136#102—1574no—
137C24—1571no—
138#41—24no—
139#41—24no—
140C2×Q8C22×D15712yes—
141C2×D4—22no—
142C2×D4—3144no—
143C2×D4C22×D15712yes—
144C22×C4—22no—
145C22×C4—3142no—
146C22×C4—3142no—
147C22×C4—1571no—
148C22×C4—1572no—
149C22×C4—1572no—
150C22×C4C22×D15711yescharacteristic
151Q16—22no—
152SD16—24no—
153D8—22no—
154C2×C8—3142no—
155C2×C8—3144no—
156C2×C8—3144no—
157C2×C8—1572no—
158C2×C8C22⋊D15714yescomplemented
159C2×C8C22×D15711yescharacteristic
160C2×C8C22×D15711yescharacteristic
161C4⋊C4—3144no—
162C4⋊C4—1574no—
163C22⋊C4—3144no—
164C22⋊C4—3142no—
165C22⋊C4—3144no—
166C42—3142no—
167C23—22no—
168C23—3144no—
169C23—3142no—
170C23—1572no—
171C23D4×D15711yescharacteristic
172Q8—22no—
173Q8—22no—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—6288no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—3142no—
181C2×C4—3144no—
182C2×C4—3142no—
183C2×C4—3142no—
184C2×C4—3144no—
185C2×C4—1572no—
186C2×C4D4×D15711yescharacteristic
187C2×C4C2×C22⋊D15712yes—
188C2×C4#1457895312yes—
189C2×C4C23×D15711yescharacteristic
190C8—3142no—
191C8C2×C22⋊D15712yes—
192C8C2×C22⋊D15712yes—
193C22—22no—
194C22—42no—
195C22—44no—
196C22—3148no—
197C22—3142no—
198C22—3142no—
199C22—1574no—
200C22#1909774212yes—
201C22C2×D4×D15711yescharacteristic
202C4—42no—
203C4—3142no—
204C4—6284no—
205C4#1909774212yes—
206C4C22×C22⋊D15711yescharacteristic
207C4C2×#1457895311yescharacteristic
208C2—22no—
209C2—82no—
210C2—3144no—
211C2C2×#1909774211yescharacteristic
212C2#135×D15712yes—
subgroupG/H
centerC2×C4D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#135×C157C2
socleC22×C157C22×D4
hypercenterC2×#135D157
hypocenterC157#3050
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×C8×D157C2semidirect
2. #23757780C2semidirect
3. #23757511C2semidirect
4. #23757506C2semidirect
5. #23757499C2semidirect
6. #23757311C2semidirect
7. #23757269C2semidirect
8. C2×#135×C157C2semidirect
20. #19097705C22semidirect
21. #19097703C22semidirect
22. #19097695C22semidirect
23. #19097692C22semidirect
26. C2×Q16×C157C22semidirect
27. C2×SD16×C157C22semidirect
28. C2×D8×C157C22semidirect
57. C2×C8×C157D4semidirect
94. C2×D157#135semidirect
106. C157#3050semidirect
108. C2×#135D157semidirect
119. C2×Q16C2×D157semidirect
120. C2×SD16C2×D157semidirect
121. C2×D8C2×D157semidirect
158. C2×C8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D1572
#237577802
#237575112
#237575062
#237574992
#237573112
#237572692
C2×#135×C1572
#3050157