Finite Group Databasebeta

← #30647963

Subgroups — #30647963

Order 16256.

738 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×Dic254C211yesmaximal, characteristic, complemented
2C2×#16861239C212yesmaximal, complemented
3#22550031C214yesmaximal, complemented
4#22550014C214yesmaximal, complemented
5#22549884C212yesmaximal, complemented
6#22549840C211yesmaximal, characteristic, complemented
7#525×C127C211yesmaximal, characteristic
8C22×Dic254—28no—
9C22×Dic254—22no—
10C22×Dic254—21no—
11C22×Dic254C2214yes—
12C22×Dic254C2212yes—
13C23×Dic127—21no—
14#16861239—28no—
15C2×#14523238C2214yescomplemented
16C2×#14523236C2214yescomplemented
17C2×#14523232C2212yes—
18#16861132C2218yescomplemented
19C22⋊(C127⋊C8)C2214yes—
20C2×#41×C127C2211yescharacteristic
21C23×C4×C127C2211yescharacteristic
22#123×C127—21no—
23C22⋊Q8×C127C2212yes—
24#121×C127C2212yes—
25#120×C127—21no—
26C4×D4×C127—22no—
27#117×C127C2211yescharacteristic
28C2×Dic254—28no—
29C2×Dic254—216no—
30C2×Dic254—216no—
31C2×Dic254—24no—
32C2×Dic254—24no—
33C2×Dic254—22no—
34C2×Dic254D418yescomplemented
35C2×Dic254C2314yes—
36C22×Dic127—28no—
37C22×Dic127—22no—
38C22×Dic127—24no—
39#14523238—28no—
40#14523236—28no—
41#14523232—24no—
42C2×C127⋊C8C2314yes—
43C24×C127D411yescharacteristic
44#41×C127—24no—
45C2×Q8×C127C2311yescharacteristic
46C2×D4×C127—22no—
47C2×D4×C127—21no—
48C2×D4×C127C2311yescharacteristic
49C22×C4×C127—24no—
50C22×C4×C127—21no—
51C22×C4×C127D412yes—
52C22×C4×C127D411yescharacteristic
53C22×C4×C127C2312yes—
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—22no—
56C4⋊C4×C127C2312yes—
57C22⋊C4×C127—21no—
58C22⋊C4×C127—22no—
59C22⋊C4×C127—22no—
60C42×C127—21no—
61Dic254—216no—
62Dic254—28no—
63Dic254—24no—
64Dic254C2×D418yes—
65C2×Dic127—216no—
66C2×Dic127—28no—
67C2×Dic127—24no—
68C127⋊C8—24no—
69C23×C127—22no—
70C23×C127—24no—
71C23×C127—21no—
72C23×C127C2×D411yescharacteristic
73C23×C127C2×D412yes—
74Q8×C127—22no—
75D4×C127—22no—
76D4×C127—42no—
77D4×C127—44no—
78C2×C4×C127—22no—
79C2×C4×C127—24no—
80C2×C4×C127—24no—
81C2×C4×C127—22no—
82C2×C4×C127—21no—
83C2×C4×C127—42no—
84C2×C4×C127C2×D414yes—
85C2×C4×C127C2×D412yes—
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2411yescharacteristic
88C22×C127—28no—
89C22×C127—22no—
90C22×C127—22no—
91C22×C127—22no—
92C22×C127—41no—
93C22×C127—42no—
94C22×C127#12012yes—
95C22×C127#13714yescomplemented
96C22×C127C22×D411yescharacteristic
97C4×C127—22no—
98C4×C127—41no—
99C4×C127—42no—
100C4×C127#12012yes—
101C4×C127C22×D412yes—
102Dic127—28no—
103C2×C127—22no—
104C2×C127—24no—
105C2×C127—81no—
106C2×C127C2×#12011yescharacteristic
107C2×C127C2×#13712yes—
108#3004—1271nomaximal, Sylow
109C127#300411yescharacteristic, complemented, Sylow
110C23×Q8—1271no—
111C2×#137—1272no—
112#525D12711yescharacteristic
113#451—1274no—
114#448—1274no—
115#419—1272no—
116#407—1271no—
117C2×#41C2×D12711yescharacteristic
118C22×Q8—1272no—
119C22×Q8—1274no—
120C22×Q8—2548no—
121C22×Q8—2542no—
122C22×Q8—2541no—
123C23×C4—2541no—
124C23×C4C2×D12711yescharacteristic
125#137—2548no—
126C2×Q16—1274no—
127C2×SD16—1274no—
128C2×M16—1272no—
129#123—21no—
130C22⋊Q8C2×D12712yes—
131#121C2×D12712yes—
132#120—21no—
133C4×D4—22no—
134#117C2×D12711yescharacteristic
135#103—1278no—
136C22⋊C8—1274no—
137C24C22⋊D12711yescharacteristic
138#41—24no—
139C2×Q8—1278no—
140C2×Q8—1274no—
141C2×Q8—25416no—
142C2×Q8—2548no—
143C2×Q8—25416no—
144C2×Q8—2544no—
145C2×Q8—2544no—
146C2×Q8—2542no—
147C2×Q8C22×D12711yescharacteristic
148C2×D4—22no—
149C2×D4—21no—
150C2×D4C22×D12711yescharacteristic
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—2548no—
154C22×C4—2544no—
155C22×C4—2542no—
156C22×C4C22⋊D12712yes—
157C22×C4C22⋊D12711yescharacteristic
158C22×C4C22×D12712yes—
159C22×C4C22×D12711yescharacteristic
160Q16—2548no—
161SD16—2548no—
162M16—2544no—
163C2×C8—1274no—
164C4⋊C4—22no—
165C4⋊C4C22×D12712yes—
166C22⋊C4—21no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C42—21no—
170C23—24no—
171C23—22no—
172C23—21no—
173C23C2×C22⋊D12712yes—
174C23C2×C22⋊D12711yescharacteristic
175Q8—22no—
176Q8—1278no—
177Q8—25416no—
178Q8—2544no—
179Q8—2548no—
180D4—22no—
181D4—44no—
182D4—42no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—25416no—
190C2×C4—2544no—
191C2×C4—2548no—
192C2×C4D4×D12714yes—
193C2×C4C2×C22⋊D12712yes—
194C2×C4C2×C22⋊D12711yescharacteristic
195C2×C4C23×D12711yescharacteristic
196C8—2544no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—41no—
202C22—42no—
203C22#1686117712yes—
204C22#1686123914yescomplemented
205C22C22×C22⋊D12711yescharacteristic
206C4—22no—
207C4—41no—
208C4—42no—
209C4—2548no—
210C4#1686117712yes—
211C4C2×D4×D12712yes—
212C2—24no—
213C2—22no—
214C2—81no—
215C2C2×#1686117711yescharacteristic
216C2C2×#1686123912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#525×C127C2
socleC22×C127C22×D4
hypercenter#525D127
hypocenterC127#3004
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C23×Dic254C2semidirect
2. C2×#16861239C2semidirect
3. #22550031C2semidirect
4. #22550014C2semidirect
5. #22549884C2semidirect
6. #22549840C2semidirect
15. C2×#14523238C22semidirect
16. C2×#14523236C22semidirect
18. #16861132C22semidirect
34. C2×Dic254D4semidirect
95. C22×C127#137semidirect
109. C127#3004semidirect
204. C22#16861239semidirect

Maximal subgroups

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

subgroupindex
C23×Dic2542
C2×#168612392
#225500312
#225500142
#225498842
#225498402
#525×C1272
#3004127