Finite Group Databasebeta

← #30637432

Subgroups — #30637432

Order 16256.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550424C211yesmaximal, characteristic
2#22549932C211yesmaximal, characteristic, complemented
3#22549928C211yesmaximal, characteristic, complemented
4#22549623C211yesmaximal, characteristic, complemented
5#22549621C211yesmaximal, characteristic, complemented
6#22549620C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9#16861209C2211yescharacteristic
10#16861196—21no—
11#16861180—21no—
12#16861180C412yes—
13D4×Dic127—21no—
14D4×Dic127C412yes—
15C42×D127C2211yescharacteristic
16C4⋊(C127⋊C8)C2211yescharacteristic
17C4⋊(C127⋊C8)C2211yescharacteristic
18(C2×D127)⋊C8—21no—
19(C2×D127)⋊C8—21no—
20#16861119—21no—
21C8×Dic127—21no—
22#127×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523247—42no—
26#14523247—42no—
27C2×Dic254—21no—
28C2×C22⋊D127—22no—
29C22×Dic127—22no—
30C2×C4×D127C2×C411yescharacteristic
31C2×C4×D127D411yescharacteristic
32C2×C4×D127D411yescharacteristic
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127C2×C412yes—
37C22⋊Dic127—22no—
38C22⋊Dic127—22no—
39C4×Dic127C2×C411yescharacteristic
40C4×Dic127D411yescharacteristic
41C4×Dic127D411yescharacteristic
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127C2×C412yes—
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C42×C127C2311yescharacteristic
48C22×D127C22⋊C411yescharacteristic
49Dic254—41no—
50Dic254—41no—
51C22⋊D127—44no—
52C2×Dic127—22no—
53C2×Dic127—44no—
54C2×Dic127C22⋊C411yescharacteristic
55C2×Dic127C22⋊C411yescharacteristic
56C2×Dic127C22⋊C411yescharacteristic
57C4×D127—21no—
58C4×D127—21no—
59C4×D127—21no—
60C4×D127—21no—
61C4×D127—41no—
62C127⋊C8—41no—
63C127⋊C8—41no—
64C23×C127—22no—
65D4×C127—42no—
66D4×C127—41no—
67D4×C127—41no—
68D4×C127—41no—
69D4×C127—41no—
70C2×C4×C127C22×C411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C8×C127—41no—
74C8×C127—41no—
75C22×C127—42no—
76C22×C127—42no—
77C22×C127—42no—
78C22×C127C2×C22⋊C411yescharacteristic
79C2×D127—21no—
80C2×D127—21no—
81C2×D127—21no—
82C4×C127—21no—
83C4×C127#13511yescharacteristic
84C4×C127#13511yescharacteristic
85C4×C127AGL(1,ℤ/8)11yescharacteristic
86C4×C127AGL(1,ℤ/8)11yescharacteristic
87Dic127—21no—
88Dic127—21no—
89Dic127—21no—
90Dic127—42no—
91C2×C127—82no—
92C2×C127#41311yescharacteristic
93C2×C127#41611yescharacteristic
94C2×C127#41811yescharacteristic
95D127—41no—
96#1553—1271nomaximal, Sylow
97C127#155311yescharacteristic, complemented, Sylow
98#532—1271no—
99#432—1271no—
100#431—1271no—
101#332—1271no—
102#332—1271no—
103#331—1271no—
104#331D12711yescharacteristic, complemented
105C2×#41—2541no—
106C4⋊Q8—1271no—
107#127C2×D12711yescharacteristic
108#124—2541no—
109#121—1272no—
110#121—2541no—
111C4×D4—1272no—
112C4×D4—2541no—
113C2×C42—1271no—
114C4⋊C8—1271no—
115C4⋊C8—1271no—
116C4⋊C8C2×D12711yescharacteristic
117C4⋊C8C2×D12711yescharacteristic
118C22⋊C8—2541no—
119C22⋊C8—2541no—
120#97—2541no—
121C4×C8—2541no—
122#41—5082no—
123#41—5082no—
124C2×Q8—2541no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—2542no—
128C2×D4C4×D12712yes—
129C22×C4—1271no—
130C22×C4—1271no—
131C22×C4—1271no—
132C22×C4—2542no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2541no—
136C2×C8—2541no—
137C4⋊C4—1272no—
138C4⋊C4—2541no—
139C22⋊C4—2542no—
140C22⋊C4—2542no—
141C42—1271no—
142C42—1271no—
143C42—1271no—
144C42C22×D12711yescharacteristic
145C23—22no—
146C23—1271no—
147Q8—5081no—
148Q8—5081no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—5084no—
155C2×C4—1271no—
156C2×C4—1271no—
157C2×C4—1271no—
158C2×C4—5081no—
159C2×C4—5084no—
160C2×C4—2542no—
161C2×C4—2541no—
162C2×C4—2541no—
163C2×C4—2541no—
164C2×C4—2541no—
165C2×C4C2×C4×D12711yescharacteristic
166C2×C4D4×D12711yescharacteristic
167C2×C4D4×D12711yescharacteristic
168C8—41no—
169C8—41no—
170C8—5081no—
171C8—5081no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—2541no—
176C22—2541no—
177C22—2541no—
178C22C22⋊C4×D12711yescharacteristic
179C4—21no—
180C4—5082no—
181C4—2541no—
182C4—2541no—
183C4—2541no—
184C4#1686122811yescharacteristic
185C4#1686123011yescharacteristic
186C4#1686123311yescharacteristic
187C4#1686123611yescharacteristic
188C2—82no—
189C2—5081no—
190C2#2254985711yescharacteristic
191C2#2254986911yescharacteristic
192C2#2254987911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550424C2
cosocleC42×C127C23
hypercenter#331D127
hypocenterC127#1553
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22549932C2semidirect
3. #22549928C2semidirect
4. #22549623C2semidirect
5. #22549621C2semidirect
6. #22549620C2semidirect
7. #331×C127C2semidirect
97. C127#1553semidirect
104. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225504242
#225499322
#225499282
#225496232
#225496212
#225496202
#331×C1272
#1553127