Finite Group Databasebeta

← #30637665

Subgroups — #30637665

Order 16256.

310 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550009C212yesmaximal, complemented
2#22549793C211yesmaximal, characteristic, complemented
3#22549610C212yesmaximal, complemented
4#22549602C211yesmaximal, characteristic, complemented
5#537×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C2×#14523233—22no—
8C2×(C2×D127)⋊C4—21no—
9C2×(C2×D127)⋊C4—21no—
10#16861129—22no—
11C22⋊(C127⋊C8)C2212yes—
12C127⋊#95C2211yescharacteristic
13C2×#41×C127—21no—
14C22⋊Q8×C127—22no—
15#121×C127—22no—
16#121×C127—21no—
17#121×C127C2212yes—
18C2×C4⋊C4×C127C2211yescharacteristic
19C23×D127—21no—
20C2×D508—22no—
21C2×D508—42no—
22C22×Dic127—21no—
23#14523233—44no—
24C2×C127⋊C8—22no—
25(C2×D127)⋊C4—42no—
26(C2×D127)⋊C4—42no—
27#41×C127—42no—
28#41×C127—42no—
29C2×Q8×C127—21no—
30C2×D4×C127—21no—
31C2×D4×C127—22no—
32C2×D4×C127—21no—
33C2×D4×C127D412yes—
34C22×C4×C127—22no—
35C22×C4×C127D411yescharacteristic
36C22×C4×C127D411yescharacteristic
37C22×C4×C127C2311yescharacteristic
38C4⋊C4×C127—21no—
39C4⋊C4×C127—22no—
40C4⋊C4×C127D412yes—
41C22⋊C4×C127—22no—
42C22⋊C4×C127—22no—
43C22×D127—22no—
44C22×D127—41no—
45C22×D127—41no—
46C22×D127—41no—
47C22×D127—41no—
48C22×D127—41no—
49C22×D127—41no—
50D508—24no—
51D508—42no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C2×Dic127—41no—
55C127⋊C8—42no—
56C23×C127—22no—
57C23×C127C2×D411yescharacteristic
58Q8×C127—42no—
59D4×C127—42no—
60D4×C127—42no—
61D4×C127—42no—
62D4×C127—42no—
63D4×C127—81no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—22no—
68C2×C4×C127—41no—
69C2×C4×C127—44no—
70C2×C4×C127—41no—
71C2×C4×C127D812yescomplemented
72C2×C4×C127C2×D412yes—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—42no—
76C22×C127—42no—
77C22×C127—42no—
78C22×C127#12011yescharacteristic
79C22×C127C2×D811yescharacteristic
80C22×C127AGL(1,ℤ/8)11yescharacteristic
81C2×D127—42no—
82C2×D127—42no—
83C2×D127—42no—
84C2×D127—81no—
85C2×D127—81no—
86C2×D127—81no—
87C2×D127—81no—
88C4×C127—22no—
89C4×C127—22no—
90C4×C127—42no—
91C4×C127—41no—
92Dic127—81no—
93C2×C127—21no—
94C2×C127—21no—
95C2×C127—82no—
96C2×C127#44711yescharacteristic
97C2×C127#45411yescharacteristic
98C2×C127B(4,𝔽2)11yescharacteristic
99D127—82no—
100#1598—1271nomaximal, Sylow
101C127#159811yescharacteristic, complemented, Sylow
102#537D12711yescharacteristic, complemented
103#447—1272no—
104#394—1271no—
105#327—1272no—
106#324—1271no—
107C2×#41—21no—
108C22×D4—1271no—
109C2×D8—2542no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D12712yes—
114C2×C4⋊C4C2×D12711yescharacteristic
115C2×C22⋊C4—2541no—
116C2×C22⋊C4—2541no—
117#102—2542no—
118C22⋊C8—1272no—
119#95—1271no—
120C24—2541no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—5082no—
128C2×D4—2542no—
129C2×D4C22⋊D12712yes—
130C22×C4—22no—
131C22×C4—2541no—
132C22×C4C22⋊D12711yescharacteristic
133C22×C4C22⋊D12711yescharacteristic
134C22×C4C22×D12711yescharacteristic
135D8—5084no—
136C2×C8—2542no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D12712yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—5082no—
143C22⋊C4—5082no—
144C23—22no—
145C23—5081no—
146C23—5081no—
147C23—5081no—
148C23—5081no—
149C23—5081no—
150C23—5081no—
151C23—2542no—
152C23C2×C22⋊D12711yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—5082no—
160D4—2544no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—5081no—
169C2×C4—5081no—
170C2×C4—5081no—
171C2×C4#1452323312yescomplemented
172C2×C4C2×C22⋊D12712yes—
173C8—5082no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—10161no—
180C22—10161no—
181C22—10161no—
182C22—10161no—
183C22—5082no—
184C22—5082no—
185C22—5082no—
186C22#1686117811yescharacteristic
187C22C2×#1452323311yescharacteristic
188C22#1686123511yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—42no—
193C4—10161no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—10162no—
198C2#2255001011yescharacteristic
199C2#2255004911yescharacteristic
200C2#2255006311yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#537×C127C2
socleC22×C127#120
hypercenter#537D127
hypocenterC127#1598
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #22550009C2semidirect
2. #22549793C2semidirect
3. #22549610C2semidirect
4. #22549602C2semidirect
5. #537×C127C2semidirect
71. C2×C4×C127D8semidirect
101. C127#1598semidirect
102. #537D127semidirect
171. C2×C4#14523233semidirect

Maximal subgroups

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

subgroupindex
#225500092
#225497932
#225496102
#225496022
#537×C1272
#1598127