Finite Group Databasebeta

← #30636726

Subgroups — #30636726

Order 16256.

222 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 35 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#22549785C211yesmaximal, characteristic
2#22549600C211yesmaximal, characteristic, complemented
3#342×C127C211yesmaximal, characteristic, complemented
4C22×D508C411yescharacteristic, complemented
5C22×C22⋊D127—21no—
6C2×(C2×D127)⋊C4—21no—
7C22⋊(C127⋊C8)—21no—
8C127⋊#95C411yescharacteristic, complemented
9C2×C22⋊C4×C127—21no—
10C2×C22⋊C4×C127C2211yescharacteristic
11C23×D127—21no—
12C2×D508—41no—
13C2×D508—42no—
14C2×C22⋊D127—42no—
15C2×C22⋊D127—42no—
16C2×C22⋊D127—41no—
17C2×C22⋊D127—41no—
18C22×Dic127—21no—
19C2×C127⋊C8—41no—
20(C2×D127)⋊C4—42no—
21C24×C127D411yescharacteristic
22C22×C4×C127—41no—
23C22×C4×C127C2×C411yescharacteristic
24C22×C4×C127D411yescharacteristic
25C22⋊C4×C127—22no—
26C22⋊C4×C127—41no—
27C22⋊C4×C127—41no—
28C22×D127—22no—
29C22×D127—41no—
30C22×D127—41no—
31C22×D127—41no—
32C22×D127—41no—
33C22×D127—41no—
34C22×D127—41no—
35D508—42no—
36D508—42no—
37C22⋊D127—44no—
38C22⋊D127—44no—
39C2×Dic127—22no—
40C2×Dic127—41no—
41C2×Dic127—41no—
42C127⋊C8—81no—
43C23×C127—21no—
44C23×C127—21no—
45C23×C127—41no—
46C23×C127—41no—
47C23×C127C22⋊C411yescharacteristic
48C23×C127D811yescharacteristic
49C23×C127SD1611yescharacteristic
50C2×C4×C127—21no—
51C2×C4×C127—21no—
52C2×C4×C127—41no—
53C2×C4×C127—41no—
54C2×C4×C127—42no—
55C2×C4×C127—81no—
56C2×C4×C127—81no—
57C22×C127—21no—
58C22×C127—21no—
59C22×C127—21no—
60C22×C127—21no—
61C22×C127—41no—
62C22×C127—41no—
63C22×C127—41no—
64C22×C127—41no—
65C22×C127—81no—
66C22×C127#9911yescharacteristic
67C22×C127#10211yescharacteristic
68C22×C127#10411yescharacteristic
69C2×D127—42no—
70C2×D127—42no—
71C2×D127—42no—
72C2×D127—81no—
73C2×D127—81no—
74C2×D127—81no—
75C2×D127—81no—
76C4×C127—41no—
77C4×C127—41no—
78C4×C127—82no—
79Dic127—42no—
80C2×C127—21no—
81C2×C127—21no—
82C2×C127—41no—
83C2×C127—41no—
84C2×C127#32711yescharacteristic
85C2×C127AΣL(1,𝔽16)11yescharacteristic
86C2×C127#35311yescharacteristic
87D127—82no—
88#1343—1271nomaximal, Sylow
89C127#134311yescharacteristic, complemented, Sylow
90#392—1271no—
91#342D12711yescharacteristic, complemented
92#323—1271no—
93C22×D4—1271no—
94C22×D4—2541no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—2541no—
97C2×C22⋊C4C2×D12711yescharacteristic
98C22⋊C8—2541no—
99#95—1271no—
100C24—2541no—
101C24C22⋊D12711yescharacteristic
102C2×D4—5081no—
103C2×D4—5082no—
104C2×D4—5082no—
105C2×D4—5081no—
106C2×D4—5081no—
107C2×D4—5082no—
108C22×C4—41no—
109C22×C4—2541no—
110C22×C4C4×D12711yescharacteristic
111C22×C4D50811yescharacteristic
112C2×C8—5081no—
113C22⋊C4—22no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—5082no—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23—5081no—
122C23—5081no—
123C23—5081no—
124C23—5081no—
125C23—5081no—
126C23—5081no—
127C23—2542no—
128C23(C2×D127)⋊C411yescharacteristic
129C23#1452323311yescharacteristic
130C23#1452323511yescharacteristic
131D4—5082no—
132D4—5082no—
133D4—5084no—
134D4—5084no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—5081no—
143C2×C4—5081no—
144C2×C4—2542no—
145C8—10161no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22—81no—
155C22—10161no—
156C22—10161no—
157C22—10161no—
158C22—10161no—
159C22—5082no—
160C22—5082no—
161C22—5082no—
162C22#1686112311yescharacteristic
163C22#1686112911yescharacteristic
164C22#1686113711yescharacteristic
165C4—41no—
166C4—41no—
167C4—82no—
168C4—5082no—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—10162no—
174C2#2254960911yescharacteristic
175C2#2254966211yescharacteristic
176C2#2254966811yescharacteristic
subgroupG/H
centerC22#16861123
commutatorC22×C4×C127C2×C4
FrattiniC2×C22⋊C4C2×D127
Fitting#342×C127C2
socleC22×C127#99
prime radical#22549785C2
cosocleC2×C22⋊C4×C127C22
hypercenter#342D127
hypocenterC127#1343
LCSG ▷ C22×C4×C127 ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #22549600C2semidirect
3. #342×C127C2semidirect
4. C22×D508C4semidirect
8. C127⋊#95C4semidirect
89. C127#1343semidirect
91. #342D127semidirect

Maximal subgroups

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

subgroupindex
#225497852
#225496002
#342×C1272
#1343127