Finite Group Databasebeta

← #34113548

Subgroups — #34113548

Order 20096.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#23757872C216yesmaximal, complemented
2#23757863C216yesmaximal, complemented
3#23757835C213yesmaximal, complemented
4#23757809C216yesmaximal, complemented
5#23757786C216yesmaximal, complemented
6C2×#19097731C213yesmaximal, complemented
7#543×C157C211yesmaximal, characteristic, complemented
8C2×#14578955C2212yescomplemented
9C22×C22⋊D157C2213yes—
10C23×Dic157C2211yescharacteristic
11C22×C4×D157C2213yescomplemented
12#19097766C22112yescomplemented
13#19097762C2216yescomplemented
14#19097760C2216yescomplemented
15#19097756C22112yescomplemented
16#19097755C22112yescomplemented
17#19097747C2216yescomplemented
18#19097746C2216yescomplemented
19#19097745C2216yes—
20#19097745C2216yes—
21#19097741C22112yescomplemented
22Q8×Dic157C2212yescomplemented
23#19097733C2216yescomplemented
24#19097731C22112yescomplemented
25#19097731C22112yescomplemented
26#19097729C2216yescomplemented
27C2×#14578933C2213yescomplemented
28C2×#14578931C2213yescomplemented
29C2×C4×Dic157C2213yescomplemented
30C22⋊Q8×C157C2216yescomplemented
31C22⋊Q8×C157C2216yescomplemented
32C2×C22⋊C4×C157C2213yes—
33C23×D157C2313yes—
34#14578955—44no—
35C2×D628C2316yes—
36C2×D628C2316yes—
37C2×C22⋊D157—212no—
38C2×C22⋊D157C23112yes—
39C22×Dic157—42no—
40C22×Dic157C2313yes—
41C22×Dic157C2313yes—
42C22×Dic157C2313yes—
43C2×C4×D157—46no—
44C2×C4×D157C2316yes—
45C2×C4×D157C2316yes—
46C4⋊Dic157C2316yes—
47C4⋊Dic157C2316yes—
48#14578933C23112yes—
49#14578931C23112yes—
50#14578931C23112yes—
51#14578931C23112yes—
52C4×Dic157C2316yes—
53C4×Dic157C2316yes—
54C24×C157C2311yescharacteristic
55C2×Q8×C157C2312yes—
56C2×Q8×C157C2312yes—
57C22×C4×C157C2313yes—
58C22×C4×C157C2313yes—
59C4⋊C4×C157C2316yes—
60C4⋊C4×C157C2316yes—
61C22⋊C4×C157C23112yes—
62C22×D157—26no—
63C22×D157—43no—
64C22×D157—43no—
65C22×D157C2416yes—
66D628—46no—
67D628—46no—
68C22⋊D157—224no—
69C2×Dic157—26no—
70C2×Dic157—26no—
71C2×Dic157—43no—
72C2×Dic157C2413yes—
73C2×Dic157C2416yes—
74C2×Dic157C2411yescharacteristic
75C4×D157—212no—
76C4×D157—46no—
77C23×C157—41no—
78C23×C157—41no—
79C23×C157#4114yescomplemented
80C23×C157C2413yes—
81Q8×C157—42no—
82Q8×C157—42no—
83C2×C4×C157—43no—
84C2×C4×C157—43no—
85C2×C4×C157C2416yes—
86C2×C4×C157C2416yes—
87C22×C157—23no—
88C22×C157—23no—
89C22×C157—44no—
90C22×C157C2×#4116yes—
91C22×C157C2511yescharacteristic
92C2×D157—26no—
93C2×D157—26no—
94C2×D157—26no—
95C2×D157—412no—
96C4×C157—26no—
97C4×C157—26no—
98Dic157—26no—
99Dic157—23no—
100Dic157H(5,𝔽2)12yes—
101C2×C157—26no—
102C2×C157C22×#4111yescharacteristic
103C2×C157C2×H(5,𝔽2)11yescharacteristic
104C2×C157#58511yescharacteristic
105D157—46no—
106C157#345911yescharacteristic, complemented, Sylow
107#3459—1571nomaximal, Sylow
108#543D15711yescharacteristic, complemented
109#539—1576no—
110#538—1576no—
111#534—1573no—
112#529—1576no—
113#525—1576no—
114C2×C4×D4—1573no—
115C2×#41—1572no—
116C22×D4—1573no—
117C23×C4—1571no—
118C23×C4—1573no—
119#126—15712no—
120#125—1576no—
121#124—1576no—
122#123—15712no—
123#123—15712no—
124C22⋊Q8—1576no—
125C22⋊Q8C2×D15716yescomplemented
126C22⋊Q8C2×D15716yescomplemented
127#121—1576no—
128#121—1576no—
129#121—1576no—
130#120—15712no—
131C4×Q8—1572no—
132C4×D4—1576no—
133C4×D4—15712no—
134C4×D4—15712no—
135#117—1576no—
136C2×C4⋊C4—1573no—
137C2×C22⋊C4—1573no—
138C2×C22⋊C4C2×D15713yes—
139C2×C42—1573no—
140C24—1573no—
141C24C22×D15711yescharacteristic
142#41—6284no—
143C2×Q8C22×D15712yes—
144C2×Q8C22×D15712yes—
145C2×D4—31412no—
146C2×D4—15712no—
147C2×D4—1576no—
148C2×D4—1576no—
149C22×C4—6286no—
150C22×C4—6282no—
151C22×C4—1573no—
152C22×C4—1573no—
153C22×C4—1576no—
154C22×C4—1573no—
155C22×C4—1576no—
156C22×C4C22×D15713yes—
157C22×C4C22×D15713yes—
158C4⋊C4—15712no—
159C4⋊C4—1576no—
160C4⋊C4—1576no—
161C4⋊C4C22×D15716yes—
162C4⋊C4C22×D15716yes—
163C22⋊C4—15712no—
164C22⋊C4—15712no—
165C22⋊C4—15712no—
166C22⋊C4C22×D157112yes—
167C42—1576no—
168C42—1576no—
169C23—41no—
170C23—41no—
171C23—3146no—
172C23—6283no—
173C23—6283no—
174C23—1576no—
175C23#1457895514yescomplemented
176C23C23×D15713yes—
177Q8—42no—
178Q8—42no—
179D4—31424no—
180D4—6286no—
181D4—6286no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—31412no—
185C2×C4—3146no—
186C2×C4—3146no—
187C2×C4—6286no—
188C2×C4—6283no—
189C2×C4—1576no—
190C2×C4—1573no—
191C2×C4—1571no—
192C2×C4C23×D15716yes—
193C2×C4C23×D15716yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—3146no—
198C22—3146no—
199C22—3146no—
200C22—62812no—
201C22C2×#1457895516yes—
202C22C24×D15711yescharacteristic
203C4—26no—
204C4—26no—
205C4—3146no—
206C4—3143no—
207C4—1572no—
208C2—26no—
209C2—6286no—
210C2C22×#1457895511yescharacteristic
211C2H(5,𝔽2)×D15711yescharacteristic
212C2#2375810811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#543×C157C2
hypercenter#543D157
hypocenterC157#3459
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #23757872C2semidirect
2. #23757863C2semidirect
3. #23757835C2semidirect
4. #23757809C2semidirect
5. #23757786C2semidirect
6. C2×#19097731C2semidirect
7. #543×C157C2semidirect
8. C2×#14578955C22semidirect
11. C22×C4×D157C22semidirect
12. #19097766C22semidirect
13. #19097762C22semidirect
14. #19097760C22semidirect
15. #19097756C22semidirect
16. #19097755C22semidirect
17. #19097747C22semidirect
18. #19097746C22semidirect
21. #19097741C22semidirect
22. Q8×Dic157C22semidirect
23. #19097733C22semidirect
24. #19097731C22semidirect
25. #19097731C22semidirect
26. #19097729C22semidirect
27. C2×#14578933C22semidirect
28. C2×#14578931C22semidirect
29. C2×C4×Dic157C22semidirect
30. C22⋊Q8×C157C22semidirect
31. C22⋊Q8×C157C22semidirect
79. C23×C157#41semidirect
106. C157#3459semidirect
108. #543D157semidirect
125. C22⋊Q8C2×D157semidirect
126. C22⋊Q8C2×D157semidirect
175. C23#14578955semidirect

Maximal subgroups

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

subgroupindex
#237578722
#237578632
#237578352
#237578092
#237577862
C2×#190977312
#543×C1572
#3459157