Finite Group Databasebeta

← #30648661

Subgroups — #30648661

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
1#22550387C211yesmaximal, characteristic
2#22550217C214yesmaximal, complemented
3#22550200C214yesmaximal, complemented
4#22549916C212yesmaximal, complemented
5#22549875C212yesmaximal, complemented
6#22549840C211yesmaximal, characteristic, complemented
7C2×#117×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2212yes—
10#16861192—21no—
11#16861188C2212yes—
12#16861183C2212yes—
13#16861178—21no—
14C4×C22⋊D127—22no—
15#16861168C2211yescharacteristic
16#16861142C2214yescomplemented
17#16861140C2214yescomplemented
18#16861132C2214yescomplemented
19#16861129C2214yescomplemented
20C22⋊(C127⋊C8)C2214yes—
21C23×C4×C127C2211yescharacteristic
22#117×C127—22no—
23#117×C127C2214yes—
24C2×C4⋊C4×C127C2212yes—
25C2×C22⋊C4×C127—21no—
26C2×C42×C127—21no—
27#14523248—24no—
28C2×Dic254C2311yescharacteristic
29C2×D508C2311yescharacteristic
30C2×C22⋊D127—21no—
31C2×C22⋊D127—22no—
32C2×C4×D127—21no—
33#14523232—24no—
34C2×C127⋊C8C2314yes—
35C4⋊Dic127C2312yes—
36Dic127⋊C4—22no—
37C22⋊Dic127—22no—
38C22⋊Dic127—21no—
39(C2×D127)⋊C4—22no—
40C4×Dic127—21no—
41C24×C127D411yescharacteristic
42C22×C4×C127—24no—
43C22×C4×C127—22no—
44C22×C4×C127D411yescharacteristic
45C22×C4×C127D412yes—
46C22×C4×C127C2312yes—
47C22×C4×C127C2311yescharacteristic
48C4⋊C4×C127—22no—
49C4⋊C4×C127C2314yes—
50C22⋊C4×C127—24no—
51C42×C127—24no—
52C22×D127—21no—
53Dic254—22no—
54D508—22no—
55C22⋊D127—44no—
56C22⋊D127—42no—
57C2×Dic127—22no—
58C2×Dic127—21no—
59C4×D127—42no—
60C127⋊C8—24no—
61C23×C127—22no—
62C23×C127—42no—
63C23×C127C2×D412yes—
64C23×C127C2×D411yescharacteristic
65C2×C4×C127—22no—
66C2×C4×C127—24no—
67C2×C4×C127—24no—
68C2×C4×C127—24no—
69C2×C4×C127—44no—
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#12312yes—
80C22×C127C22×D411yescharacteristic
81C2×D127—41no—
82C2×D127—42no—
83C4×C127—22no—
84C4×C127—44no—
85C4×C127#12312yes—
86C4×C127C2×#4112yes—
87Dic127—41no—
88Dic127—42no—
89C2×C127—22no—
90C2×C127—42no—
91C2×C127C2×#12311yescharacteristic
92C2×C127#57512yes—
93D127—81no—
94#3093—1271nomaximal, Sylow
95C127#309311yescharacteristic, complemented, Sylow
96#525—1271no—
97C2×#117D12711yescharacteristic, complemented
98#485—1274no—
99#482—1274no—
100#428—1272no—
101#417—1272no—
102#407—1271no—
103C2×#41—1271no—
104C23×C4C2×D12711yescharacteristic
105C2×M16—1272no—
106#123—2541no—
107C22⋊Q8—1272no—
108#121—1272no—
109#120—2541no—
110C4×D4—2542no—
111#117—22no—
112#117—1271no—
113#117C2×D12714yes—
114C2×C4⋊C4C2×D12712yes—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117#107—1274no—
118#106—1274no—
119#103—1274no—
120#102—1274no—
121C22⋊C8—1274no—
122C24C22⋊D12711yescharacteristic
123#41—2544no—
124C2×Q8—1271no—
125C2×D4—1271no—
126C2×D4—2542no—
127C2×D4—2541no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—2541no—
131C22×C4C22⋊D12711yescharacteristic
132C22×C4C22⋊D12712yes—
133C22×C4C22×D12712yes—
134C22×C4C22×D12711yescharacteristic
135M16—2544no—
136C2×C8—1274no—
137C4⋊C4—22no—
138C4⋊C4—1272no—
139C4⋊C4—2542no—
140C4⋊C4C22×D12714yes—
141C22⋊C4—24no—
142C22⋊C4—2541no—
143C22⋊C4—2542no—
144C22⋊C4—2542no—
145C42—24no—
146C42—2541no—
147C23—22no—
148C23—42no—
149C23—2541no—
150C23C2×C22⋊D12712yes—
151C23C2×C22⋊D12711yescharacteristic
152Q8—2542no—
153D4—5084no—
154D4—5082no—
155D4—2542no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—44no—
161C2×C4—5082no—
162C2×C4—2542no—
163C2×C4—2541no—
164C2×C4C2×C22⋊D12712yes—
165C2×C4C2×C22⋊D12711yescharacteristic
166C2×C4#1452324814yes—
167C2×C4C23×D12711yescharacteristic
168C8—2544no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—5081no—
175C22—5082no—
176C22#1686119212yes—
177C22C22×C22⋊D12711yescharacteristic
178C4—22no—
179C4—44no—
180C4—5081no—
181C4—5082no—
182C4#1686119212yes—
183C4C2×#1452324812yes—
184C2—22no—
185C2—42no—
186C2—10161no—
187C2C2×#1686119211yescharacteristic
188C2#2255067412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#117×C127C2
socleC22×C127C22×D4
prime radical#22550387C2
hypercenterC2×#117D127
hypocenterC127#3093
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22550217C2semidirect
3. #22550200C2semidirect
4. #22549916C2semidirect
5. #22549875C2semidirect
6. #22549840C2semidirect
7. C2×#117×C127C2semidirect
16. #16861142C22semidirect
17. #16861140C22semidirect
18. #16861132C22semidirect
19. #16861129C22semidirect
95. C127#3093semidirect
97. C2×#117D127semidirect

Maximal subgroups

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

subgroupindex
#225503872
#225502172
#225502002
#225499162
#225498752
#225498402
C2×#117×C1272
#3093127