Finite Group Databasebeta

← #34110383

Subgroups — #34110383

Order 20096.

394 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 121 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#23757747C211yesmaximal, characteristic
2#23757611C212yesmaximal, complemented
3#23757570C212yesmaximal, complemented
4#23757475C212yesmaximal, complemented
5#23757423C212yesmaximal
6#23757371C212yesmaximal, complemented
7C4×#14578941C212yesmaximal, complemented
8#23757334C211yesmaximal, characteristic, complemented
9#541×C157C211yesmaximal, characteristic, complemented
10C22×D628C2211yescharacteristic
11C2×#14578941C2212yes—
12C2×#14578941C2212yes—
13C4×D628C2212yes—
14C4×D628C2212yes—
15#19097733—22no—
16#19097730C2211yescharacteristic
17C22⋊C4×D157—22no—
18#19097704C2212yescomplemented
19#19097702C2212yescomplemented
20C4⋊#11929434C2212yescomplemented
21#19097695C2212yescomplemented
22#19097695C2212yescomplemented
23#19097691C2212yes—
24#19097691C2212yes—
25C22⋊#11929434C2212yes—
26#19097682C2211yescharacteristic, complemented
27C4×#11929434C2211yescharacteristic, complemented
28C4⋊Q8×C157C2211yescharacteristic, complemented
29#126×C157—22no—
30#125×C157—22no—
31#125×C157C2211yescharacteristic, complemented
32C22⋊Q8×C157C2212yes—
33C4×Q8×C157C2212yescomplemented
34#117×C157C2211yescharacteristic
35C23×D157—21no—
36C2×D628—24no—
37C2×D628—21no—
38C2×D628C2312yes—
39C2×C4×D157—22no—
40C2×C4×D157—22no—
41#14578941—24no—
42#14578941—24no—
43C2×#11929434C2312yes—
44C2×#11929434C2312yes—
45C4⋊Dic157C2312yes—
46C22⋊Dic157—21no—
47#14578931—22no—
48#14578931—22no—
49C4×Dic157—21no—
50C2×Q8×C157C2312yes—
51C22×C4×C157C2311yescharacteristic
52C4⋊C4×C157—22no—
53C4⋊C4×C157—22no—
54C4⋊C4×C157—22no—
55C4⋊C4×C157—22no—
56C4⋊C4×C157D412yes—
57C4⋊C4×C157C2312yes—
58C22⋊C4×C157—22no—
59C22⋊C4×C157D412yes—
60C42×C157—22no—
61C42×C157C2311yescharacteristic
62C42×C157C2311yescharacteristic
63C22×D157—22no—
64C22×D157—24no—
65C22×D157—82no—
66D628—24no—
67D628—22no—
68D628#4114yes—
69C2×Dic157—22no—
70C4×D157—82no—
71C4×D157—82no—
72#11929434—22no—
73#11929434—22no—
74C23×C157C2×D411yescharacteristic
75Q8×C157—22no—
76Q8×C157—22no—
77C2×C4×C157—22no—
78C2×C4×C157—22no—
79C2×C4×C157—22no—
80C2×C4×C157C2×D412yes—
81C2×C4×C157C2×D412yes—
82C2×C4×C157C2×D411yescharacteristic
83C2×C4×C157C2411yescharacteristic
84C22×C157—21no—
85C22×C157—22no—
86C22×C157C22×D411yescharacteristic
87C2×D157—24no—
88C2×D157—42no—
89C2×D157—42no—
90C2×D157—84no—
91C4×C157—21no—
92C4×C157—22no—
93C4×C157—22no—
94C4×C157—42no—
95C4×C157—42no—
96C4×C157C2×#4112yes—
97Dic157—42no—
98C2×C157—41no—
99C2×C157#52511yescharacteristic
100C2×C157#57712yes—
101D157—44no—
102C157#312311yescharacteristic, complemented, Sylow
103#3123—1571nomaximal, Sylow
104#541D15711yescharacteristic, complemented
105#518—1571no—
106#485—1572no—
107#478—1572no—
108#461—1572no—
109#450—1572no—
110#440—1572no—
111C4×SD16—1572no—
112#433—1571no—
113C22×D4—1571no—
114C2×SD16—1572no—
115C2×SD16—1572no—
116C4⋊Q8C2×D15711yescharacteristic, complemented
117#126—22no—
118#125—22no—
119#125C2×D15711yescharacteristic, complemented
120C22⋊Q8C2×D15712yes—
121C4×Q8C2×D15712yescomplemented
122C4×D4—3142no—
123C4×D4—1572no—
124C4×D4—1572no—
125#117—1571no—
126#117C2×D15711yescharacteristic
127C2×C22⋊C4—3142no—
128#107—1572no—
129#106—1572no—
130C4⋊C8—1572no—
131#103—1572no—
132#103—1572no—
133#102—1572no—
134#102—1572no—
135C22⋊C8—1572no—
136#97—1571no—
137C4×C8—1571no—
138C24—3141no—
139C2×Q8C22×D15712yes—
140C2×D4—3144no—
141C2×D4—3141no—
142C2×D4—1572no—
143C22×C4—3142no—
144C22×C4—3142no—
145C22×C4C22×D15711yescharacteristic
146SD16—3144no—
147SD16—3144no—
148C2×C8—1572no—
149C2×C8—1572no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—1572no—
155C4⋊C4C22⋊D15712yes—
156C4⋊C4C22×D15712yes—
157C22⋊C4—22no—
158C22⋊C4—3141no—
159C22⋊C4—3142no—
160C22⋊C4—3142no—
161C22⋊C4C22⋊D15712yes—
162C42—22no—
163C42—3141no—
164C42C22×D15711yescharacteristic
165C42C22×D15711yescharacteristic
166C23—3144no—
167C23—3142no—
168C23—12562no—
169C23C2×C22⋊D15711yescharacteristic
170Q8—22no—
171Q8—22no—
172D4—3144no—
173D4—3142no—
174D4—1574no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—3142no—
179C2×C4—12562no—
180C2×C4—12562no—
181C2×C4C2×C22⋊D15712yes—
182C2×C4C2×C22⋊D15711yescharacteristic
183C2×C4C2×C22⋊D15712yes—
184C2×C4C23×D15711yescharacteristic
185C8—3142no—
186C8—3142no—
187C22—22no—
188C22—21no—
189C22—3144no—
190C22—6282no—
191C22—6282no—
192C22—12564no—
193C22C22×C22⋊D15711yescharacteristic
194C4—22no—
195C4—22no—
196C4—21no—
197C4—42no—
198C4—42no—
199C4—6282no—
200C4#41×D15712yes—
201C2—41no—
202C2—6284no—
203C2#2375778011yescharacteristic
204C2#2375808112yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#541×C157C2
socleC22×C157C22×D4
prime radicalC22×D628C22
hypercenter#541D157
hypocenterC157#3123
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #23757611C2semidirect
3. #23757570C2semidirect
4. #23757475C2semidirect
6. #23757371C2semidirect
7. C4×#14578941C2semidirect
8. #23757334C2semidirect
9. #541×C157C2semidirect
18. #19097704C22semidirect
19. #19097702C22semidirect
20. C4⋊#11929434C22semidirect
21. #19097695C22semidirect
22. #19097695C22semidirect
26. #19097682C22semidirect
27. C4×#11929434C22semidirect
28. C4⋊Q8×C157C22semidirect
31. #125×C157C22semidirect
33. C4×Q8×C157C22semidirect
102. C157#3123semidirect
104. #541D157semidirect
116. C4⋊Q8C2×D157semidirect
119. #125C2×D157semidirect
121. C4×Q8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237577472
#237576112
#237575702
#237574752
#237574232
#237573712
C4×#145789412
#237573342
#541×C1572
#3123157