Finite Group Databasebeta

← #30637670

Subgroups — #30637670

Order 16256.

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

normal subgroups only

This group has 266 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#22550026C212yesmaximal
2#22549793C211yesmaximal, characteristic
3#22549613C212yesmaximal, complemented
4#22549602C211yesmaximal, characteristic, complemented
5#544×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C2×#14523235—22no—
8C2×(C2×D127)⋊C4—21no—
9C2×(C2×D127)⋊C4—21no—
10#16861129—22no—
11C22⋊(C127⋊C8)C2212yes—
12C127⋊#95C2211yescharacteristic
13C4⋊Q8×C127—22no—
14#125×C127—22no—
15C22⋊Q8×C127—21no—
16C22⋊Q8×C127C2212yes—
17#117×C127—21no—
18C2×C4⋊C4×C127C2211yescharacteristic
19C23×D127—21no—
20C2×D508—22no—
21C2×D508—42no—
22C22×Dic127—21no—
23#14523235—44no—
24C2×C127⋊C8—22no—
25(C2×D127)⋊C4—42no—
26(C2×D127)⋊C4—42no—
27C2×Q8×C127—21no—
28C2×Q8×C127D412yes—
29C22×C4×C127D411yescharacteristic
30C22×C4×C127D411yescharacteristic
31C22×C4×C127C2311yescharacteristic
32C4⋊C4×C127—21no—
33C4⋊C4×C127—21no—
34C4⋊C4×C127—21no—
35C4⋊C4×C127—22no—
36C4⋊C4×C127—22no—
37C4⋊C4×C127—22no—
38C4⋊C4×C127D412yes—
39C22⋊C4×C127—22no—
40C42×C127—22no—
41C22×D127—22no—
42C22×D127—41no—
43C22×D127—41no—
44C22×D127—41no—
45C22×D127—41no—
46C22×D127—41no—
47C22×D127—41no—
48D508—24no—
49D508—42no—
50C2×Dic127—41no—
51C2×Dic127—41no—
52C2×Dic127—41no—
53C127⋊C8—42no—
54C23×C127C2×D411yescharacteristic
55Q8×C127—42no—
56Q8×C127—81no—
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—21no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127SD1612yescomplemented
65C2×C4×C127C2×D412yes—
66C22×C127—21no—
67C22×C127—21no—
68C22×C127#12011yescharacteristic
69C22×C127C2×SD1611yescharacteristic
70C22×C127#13711yescharacteristic
71C2×D127—42no—
72C2×D127—42no—
73C2×D127—42no—
74C2×D127—81no—
75C2×D127—81no—
76C2×D127—81no—
77C2×D127—81no—
78C4×C127—22no—
79C4×C127—22no—
80C4×C127—42no—
81C4×C127—41no—
82C4×C127—42no—
83Dic127—81no—
84C2×C127—21no—
85C2×C127—21no—
86C2×C127#44811yescharacteristic
87C2×C127#45411yescharacteristic
88C2×C127B(4,𝔽2)11yescharacteristic
89D127—82no—
90#1599—1271nomaximal, Sylow
91C127#159911yescharacteristic, complemented, Sylow
92#544D12711yescharacteristic, complemented
93#450—1272no—
94#394—1271no—
95#328—1272no—
96#324—1271no—
97C22×D4—1271no—
98C2×SD16—2542no—
99C4⋊Q8—22no—
100#125—22no—
101C22⋊Q8—21no—
102C22⋊Q8C2×D12712yes—
103#117—21no—
104C2×C4⋊C4C2×D12711yescharacteristic
105C2×C22⋊C4—2541no—
106C2×C22⋊C4—2541no—
107#102—2542no—
108C22⋊C8—1272no—
109#95—1271no—
110C24—2541no—
111C2×Q8—21no—
112C2×Q8C22⋊D12712yes—
113C2×D4—5082no—
114C2×D4—2542no—
115C22×C4—2541no—
116C22×C4C22⋊D12711yescharacteristic
117C22×C4C22⋊D12711yescharacteristic
118C22×C4C22×D12711yescharacteristic
119SD16—5084no—
120C2×C8—2542no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4C22⋊D12712yes—
128C22⋊C4—22no—
129C22⋊C4—5082no—
130C22⋊C4—5082no—
131C42—22no—
132C23—5081no—
133C23—5081no—
134C23—5081no—
135C23—5081no—
136C23—5081no—
137C23—5081no—
138C23—2542no—
139C23C2×C22⋊D12711yescharacteristic
140Q8—42no—
141Q8—81no—
142D4—5082no—
143D4—2544no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—5081no—
152C2×C4—5081no—
153C2×C4—5081no—
154C2×C4#1452323512yescomplemented
155C2×C4C2×C22⋊D12712yes—
156C8—5082no—
157C22—21no—
158C22—21no—
159C22—10161no—
160C22—10161no—
161C22—10161no—
162C22—10161no—
163C22—5082no—
164C22—5082no—
165C22—5082no—
166C22#1686117811yescharacteristic
167C22C2×#1452323511yescharacteristic
168C22#1686124011yescharacteristic
169C4—22no—
170C4—22no—
171C4—42no—
172C4—41no—
173C4—42no—
174C4—10161no—
175C2—21no—
176C2—21no—
177C2—10162no—
178C2#2255001511yescharacteristic
179C2#2255004911yescharacteristic
180C2#2255006311yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#544×C127C2
socleC22×C127#120
prime radicalC22×D508C22
hypercenter#544D127
hypocenterC127#1599
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
3. #22549613C2semidirect
4. #22549602C2semidirect
5. #544×C127C2semidirect
64. C2×C4×C127SD16semidirect
91. C127#1599semidirect
92. #544D127semidirect
154. C2×C4#14523235semidirect

Maximal subgroups

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

subgroupindex
#225500262
#225497932
#225496132
#225496022
#544×C1272
#1599127