Finite Group Databasebeta

← #30648659

Subgroups — #30648659

Order 16256.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16861168C211yesmaximal, characteristic, complemented
2#22550214C214yesmaximal, complemented
3#22550197C214yesmaximal, complemented
4#22549919C212yesmaximal, complemented
5#22549877C212yesmaximal, complemented
6#22549840C211yesmaximal, characteristic, complemented
7#525×C127C211yesmaximal, characteristic
8C2×#14523232C2212yes—
9#16861168—22no—
10#16861168C2214yes—
11C2×C4⋊Dic127C2212yes—
12C2×C22⋊Dic127—21no—
13C2×C4×Dic127—21no—
14#16861142C2214yescomplemented
15#16861140C2214yescomplemented
16#16861133C2214yescomplemented
17#16861130C2214yescomplemented
18C22⋊(C127⋊C8)C2214yes—
19C2×#41×C127C2211yescharacteristic
20C23×C4×C127C2211yescharacteristic
21#123×C127—21no—
22C22⋊Q8×C127C2212yes—
23#121×C127C2212yes—
24#120×C127—21no—
25C4×D4×C127—22no—
26#117×C127C2211yescharacteristic
27C22×Dic127—22no—
28#14523232—24no—
29C2×C127⋊C8C2314yes—
30C4⋊Dic127—22no—
31C4⋊Dic127C2314yes—
32C22⋊Dic127—24no—
33C4×Dic127—24no—
34C24×C127D411yescharacteristic
35#41×C127—24no—
36C2×Q8×C127C2311yescharacteristic
37C2×D4×C127—22no—
38C2×D4×C127—21no—
39C2×D4×C127C2311yescharacteristic
40C22×C4×C127—24no—
41C22×C4×C127—21no—
42C22×C4×C127D412yes—
43C22×C4×C127D411yescharacteristic
44C22×C4×C127C2312yes—
45C22×C4×C127C2311yescharacteristic
46C4⋊C4×C127—22no—
47C4⋊C4×C127C2312yes—
48C22⋊C4×C127—21no—
49C22⋊C4×C127—22no—
50C22⋊C4×C127—22no—
51C42×C127—21no—
52C2×Dic127—24no—
53C2×Dic127—44no—
54C127⋊C8—24no—
55C23×C127—22no—
56C23×C127—21no—
57C23×C127—42no—
58C23×C127C2×D412yes—
59C23×C127C2×D411yescharacteristic
60Q8×C127—22no—
61D4×C127—22no—
62D4×C127—44no—
63D4×C127—42no—
64C2×C4×C127—22no—
65C2×C4×C127—24no—
66C2×C4×C127—24no—
67C2×C4×C127—22no—
68C2×C4×C127—21no—
69C2×C4×C127—42no—
70C2×C4×C127C2×D412yes—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127#4114yes—
73C2×C4×C127C2411yescharacteristic
74C22×C127—22no—
75C22×C127—22no—
76C22×C127—22no—
77C22×C127—22no—
78C22×C127—44no—
79C22×C127—41no—
80C22×C127—42no—
81C22×C127#12312yes—
82C22×C127C22×D411yescharacteristic
83C4×C127—22no—
84C4×C127—41no—
85C4×C127—42no—
86C4×C127#12312yes—
87C4×C127C2×#4112yes—
88Dic127—44no—
89C2×C127—22no—
90C2×C127—42no—
91C2×C127—81no—
92C2×C127C2×#12311yescharacteristic
93C2×C127#57512yes—
94#3093—1271nomaximal, Sylow
95C127#309311yescharacteristic, complemented, Sylow
96#525D12711yescharacteristic
97C2×#117—1271no—
98#485—1274no—
99#482—1274no—
100#428—1272no—
101#417—1272no—
102#407—1271no—
103C2×#41C2×D12711yescharacteristic
104C23×C4C2×D12711yescharacteristic
105C2×M16—1272no—
106#123—21no—
107C22⋊Q8C2×D12712yes—
108#121C2×D12712yes—
109#120—21no—
110C4×D4—22no—
111#117—1274no—
112#117—2542no—
113#117C2×D12711yescharacteristic
114C2×C4⋊C4—1272no—
115C2×C22⋊C4—2541no—
116C2×C42—2541no—
117#107—1274no—
118#106—1274no—
119#103—1274no—
120#102—1274no—
121C22⋊C8—1274no—
122C24C22⋊D12711yescharacteristic
123#41—24no—
124C2×Q8C22×D12711yescharacteristic
125C2×D4—22no—
126C2×D4—21no—
127C2×D4C22×D12711yescharacteristic
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—2542no—
131C22×C4C22⋊D12711yescharacteristic
132C22×C4C22⋊D12712yes—
133C22×C4C22×D12712yes—
134C22×C4C22×D12711yescharacteristic
135M16—2544no—
136C2×C8—1274no—
137C4⋊C4—22no—
138C4⋊C4—1274no—
139C4⋊C4—2542no—
140C4⋊C4C22×D12712yes—
141C22⋊C4—21no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—2544no—
145C42—21no—
146C42—2544no—
147C23—22no—
148C23—21no—
149C23—42no—
150C23C2×C22⋊D12712yes—
151C23C2×C22⋊D12711yescharacteristic
152Q8—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—5084no—
163C2×C4—2544no—
164C2×C4C2×C22⋊D12712yes—
165C2×C4C2×C22⋊D12711yescharacteristic
166C2×C4#1452324714yes—
167C2×C4C23×D12711yescharacteristic
168C8—2544no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—41no—
175C22—42no—
176C22#1686119012yes—
177C22C22×C22⋊D12711yescharacteristic
178C4—22no—
179C4—41no—
180C4—42no—
181C4—5084no—
182C4#1686119012yes—
183C4C2×#1452324712yes—
184C2—22no—
185C2—42no—
186C2—81no—
187C2C2×#1686119011yescharacteristic
188C2#2255067412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#525×C127C2
socleC22×C127C22×D4
hypercenter#525D127
hypocenterC127#3093
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. C2×#16861168C2semidirect
2. #22550214C2semidirect
3. #22550197C2semidirect
4. #22549919C2semidirect
5. #22549877C2semidirect
6. #22549840C2semidirect
14. #16861142C22semidirect
15. #16861140C22semidirect
16. #16861133C22semidirect
17. #16861130C22semidirect
95. C127#3093semidirect

Maximal subgroups

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

subgroupindex
C2×#168611682
#225502142
#225501972
#225499192
#225498772
#225498402
#525×C1272
#3093127