Finite Group Databasebeta

← #34110379

Subgroups — #34110379

Order 20096.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757878C211yesmaximal, characteristic, complemented
2#23757597C212yesmaximal, complemented
3#23757573C212yesmaximal, complemented
4#23757468C212yesmaximal, complemented
5#23757407C212yesmaximal, complemented
6#23757383C212yesmaximal, complemented
7C4×#14578939C212yesmaximal, complemented
8#23757334C211yesmaximal, characteristic, complemented
9#518×C157C211yesmaximal, characteristic, complemented
10C2×#14578939C2212yes—
11C2×#14578939C2212yes—
12#19097768C2211yescharacteristic, complemented
13#19097763C2211yescharacteristic, complemented
14#19097756—22no—
15#19097745C2212yes—
16#19097744—22no—
17C4×D628C2212yescomplemented
18#19097704C2212yescomplemented
19#19097702C2212yescomplemented
20C4⋊#11929434C2212yescomplemented
21#19097692C2212yes—
22#19097692C2212yes—
23#19097691C2212yescomplemented
24#19097691C2212yescomplemented
25C22⋊#11929434C2212yes—
26#19097682C2211yescharacteristic, complemented
27C4×#11929434C2211yescharacteristic, complemented
28C22×D4×C157C2211yescharacteristic
29C4×D4×C157—22no—
30C4×D4×C157C2212yes—
31C4×D4×C157C2212yes—
32#117×C157C2211yescharacteristic
33#117×C157C2211yescharacteristic
34C2×C22⋊C4×C157—22no—
35C2×D628—22no—
36C2×D628C2312yes—
37C2×C22⋊D157—22no—
38C2×C4×D157—22no—
39#14578939—24no—
40#14578939—24no—
41C2×#11929434C2312yes—
42C2×#11929434C2312yes—
43C4⋊Dic157C2312yes—
44#14578933—22no—
45#14578931—22no—
46#14578931—22no—
47C24×C157—21no—
48C2×D4×C157—24no—
49C2×D4×C157—21no—
50C2×D4×C157C2312yes—
51C22×C4×C157—22no—
52C22×C4×C157—22no—
53C22×C4×C157C2311yescharacteristic
54C4⋊C4×C157D412yes—
55C4⋊C4×C157C2312yes—
56C22⋊C4×C157—21no—
57C22⋊C4×C157—22no—
58C22⋊C4×C157—22no—
59C22⋊C4×C157D412yes—
60C42×C157—21no—
61C42×C157C2311yescharacteristic
62C42×C157C2311yescharacteristic
63C22×D157—22no—
64D628—22no—
65D628—22no—
66D628—82no—
67C22⋊D157—82no—
68C2×Dic157—22no—
69C4×D157—82no—
70#11929434—22no—
71#11929434—22no—
72C23×C157—24no—
73C23×C157—22no—
74C23×C157—82no—
75C23×C157C2×D411yescharacteristic
76D4×C157—24no—
77D4×C157—22no—
78D4×C157#4114yes—
79C2×C4×C157—22no—
80C2×C4×C157—22no—
81C2×C4×C157—82no—
82C2×C4×C157—82no—
83C2×C4×C157C2×D412yes—
84C2×C4×C157C2×D412yes—
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157C2411yescharacteristic
87C22×C157—21no—
88C22×C157—24no—
89C22×C157—22no—
90C22×C157—42no—
91C22×C157—42no—
92C22×C157—84no—
93C22×C157C22×D411yescharacteristic
94C2×D157—42no—
95C2×D157—42no—
96C2×D157—42no—
97C4×C157—21no—
98C4×C157—22no—
99C4×C157—22no—
100C4×C157—42no—
101C4×C157C2×#4112yes—
102Dic157—42no—
103C2×C157—41no—
104C2×C157—44no—
105C2×C157#52511yescharacteristic
106C2×C157#57612yes—
107D157—82no—
108C157#312211yescharacteristic, complemented, Sylow
109#3122—1571nomaximal, Sylow
110#540—1571no—
111#518D15711yescharacteristic, complemented
112#482—1572no—
113#478—1572no—
114#459—1572no—
115#447—1572no—
116#442—1572no—
117C4×D8—1572no—
118#433—1571no—
119C22×D4C2×D15711yescharacteristic
120C2×D8—1572no—
121C2×D8—1572no—
122#127—1571no—
123#125—1571no—
124#123—3142no—
125#121—3142no—
126#121—1572no—
127C4×D4—22no—
128C4×D4—1572no—
129C4×D4C2×D15712yes—
130C4×D4C2×D15712yes—
131#117C2×D15711yescharacteristic
132#117C2×D15711yescharacteristic
133C2×C22⋊C4—22no—
134#107—1572no—
135#106—1572no—
136C4⋊C8—1572no—
137#102—1572no—
138#102—1572no—
139#102—1572no—
140#102—1572no—
141C22⋊C8—1572no—
142#97—1571no—
143C4×C8—1571no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—3142no—
148C2×D4—3142no—
149C2×D4—1572no—
150C2×D4C22×D15712yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—3142no—
154C22×C4C22×D15711yescharacteristic
155D8—3144no—
156D8—3144no—
157C2×C8—1572no—
158C2×C8—1572no—
159C4⋊C4—3142no—
160C4⋊C4—1572no—
161C4⋊C4C22⋊D15712yes—
162C4⋊C4C22×D15712yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—3142no—
167C22⋊C4—3142no—
168C22⋊C4C22⋊D15712yes—
169C42—21no—
170C42C22×D15711yescharacteristic
171C42C22×D15711yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—3142no—
176C23C2×C22⋊D15711yescharacteristic
177D4—24no—
178D4—22no—
179D4—3142no—
180D4—3142no—
181D4—12562no—
182D4—12562no—
183D4#1457895414yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—3142no—
189C2×C4—12562no—
190C2×C4C2×C22⋊D15711yescharacteristic
191C2×C4C2×C22⋊D15712yes—
192C2×C4C2×C22⋊D15712yes—
193C2×C4C23×D15711yescharacteristic
194C8—3142no—
195C8—3142no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—6282no—
203C22—6282no—
204C22—6282no—
205C22C22×C22⋊D15711yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—6282no—
211C4C2×#1457895412yes—
212C2—41no—
213C2—44no—
214C2—12562no—
215C2#2375778411yescharacteristic
216C2#2375807612yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#518×C157C2
socleC22×C157C22×D4
hypercenter#518D157
hypocenterC157#3122
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #23757878C2semidirect
2. #23757597C2semidirect
3. #23757573C2semidirect
4. #23757468C2semidirect
5. #23757407C2semidirect
6. #23757383C2semidirect
7. C4×#14578939C2semidirect
8. #23757334C2semidirect
9. #518×C157C2semidirect
12. #19097768C22semidirect
13. #19097763C22semidirect
17. C4×D628C22semidirect
18. #19097704C22semidirect
19. #19097702C22semidirect
20. C4⋊#11929434C22semidirect
23. #19097691C22semidirect
24. #19097691C22semidirect
26. #19097682C22semidirect
27. C4×#11929434C22semidirect
108. C157#3122semidirect
111. #518D157semidirect

Maximal subgroups

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

subgroupindex
#237578782
#237575972
#237575732
#237574682
#237574072
#237573832
C4×#145789392
#237573342
#518×C1572
#3122157