Finite Group Databasebeta

← #30637434

Subgroups — #30637434

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#22550427C211yesmaximal, characteristic
2#22549932C211yesmaximal, characteristic, complemented
3#22549928C211yesmaximal, characteristic, complemented
4#22549622C211yesmaximal, characteristic, complemented
5#22549619C211yesmaximal, characteristic, complemented
6#22549619C211yesmaximal, characteristic, complemented
7#332×C127C211yesmaximal, characteristic, complemented
8C2×#14523249—21no—
9#16861206C2211yescharacteristic
10#16861198—21no—
11#16861184—21no—
12#16861184C412yes—
13#16861171—21no—
14#16861171C412yes—
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—
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25#14523249—42no—
26#14523249—42no—
27C2×D508—22no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508C2×C412yes—
31C2×C4×D127—22no—
32C2×C4×D127C2×C411yescharacteristic
33C2×C4×D127D411yescharacteristic
34C2×C4×D127D411yescharacteristic
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37(C2×D127)⋊C4—22no—
38(C2×D127)⋊C4—22no—
39C4×Dic127C2×C411yescharacteristic
40C4×Dic127D411yescharacteristic
41C4×Dic127D411yescharacteristic
42C2×Q8×C127—21no—
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127C2×C412yes—
47C42×C127C2311yescharacteristic
48C22×D127—22no—
49C22×D127C22⋊C411yescharacteristic
50D508—42no—
51D508—44no—
52D508—41no—
53D508—41no—
54D508—41no—
55D508—41no—
56C2×Dic127C22⋊C411yescharacteristic
57C2×Dic127C22⋊C411yescharacteristic
58C2×Dic127C22⋊C411yescharacteristic
59C4×D127—21no—
60C4×D127—21no—
61C4×D127—21no—
62C4×D127—21no—
63C4×D127—41no—
64C4×D127—44no—
65C127⋊C8—41no—
66C127⋊C8—41no—
67Q8×C127—41no—
68Q8×C127—41no—
69C2×C4×C127—22no—
70C2×C4×C127C22×C411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C8×C127—41no—
74C8×C127—41no—
75C22×C127C2×C22⋊C411yescharacteristic
76C2×D127—21no—
77C2×D127—21no—
78C2×D127—21no—
79C2×D127—42no—
80C2×D127—42no—
81C2×D127—42no—
82C4×C127—21no—
83C4×C127—42no—
84C4×C127#13511yescharacteristic
85C4×C127#13511yescharacteristic
86C4×C127AGL(1,ℤ/8)11yescharacteristic
87C4×C127AGL(1,ℤ/8)11yescharacteristic
88Dic127—21no—
89Dic127—21no—
90Dic127—21no—
91C2×C127#41311yescharacteristic
92C2×C127#41611yescharacteristic
93C2×C127#41811yescharacteristic
94D127—41no—
95D127—82no—
96#1553—1271nomaximal, Sylow
97C127#155311yescharacteristic, complemented, Sylow
98#532—1271no—
99#432—1271no—
100#431—1271no—
101#332—1271no—
102#332D12711yescharacteristic, complemented
103#331—1271no—
104#331—1271no—
105C2×#41—2541no—
106C4⋊Q8C2×D12711yescharacteristic
107#127—1271no—
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—21no—
125C2×D4—1272no—
126C2×D4—2542no—
127C2×D4—2541no—
128C2×D4—2541no—
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—21no—
138C4⋊C4C4×D12712yes—
139C22⋊C4—2542no—
140C22⋊C4—2542no—
141C42—1271no—
142C42—1271no—
143C42—1271no—
144C42C22×D12711yescharacteristic
145C23—1271no—
146C23—2542no—
147Q8—41no—
148Q8—41no—
149D4—5082no—
150D4—5084no—
151D4—5081no—
152D4—5081no—
153D4—5081no—
154D4—5081no—
155C2×C4—22no—
156C2×C4—1271no—
157C2×C4—1271no—
158C2×C4—1271no—
159C2×C4—5081no—
160C2×C4—5084no—
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—5082no—
173C22—5082no—
174C22—5082no—
175C22—2541no—
176C22—2541no—
177C22—2541no—
178C22C22⋊C4×D12711yescharacteristic
179C4—21no—
180C4—42no—
181C4—2541no—
182C4—2541no—
183C4—2541no—
184C4#1686123011yescharacteristic
185C4#1686123211yescharacteristic
186C4#1686123311yescharacteristic
187C4#1686123611yescharacteristic
188C2—10162no—
189C2—5081no—
190C2#2254985911yescharacteristic
191C2#2254987111yescharacteristic
192C2#2254987911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#332×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550427C2
cosocleC42×C127C23
hypercenter#332D127
hypocenterC127#1553
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22549932C2semidirect
3. #22549928C2semidirect
4. #22549622C2semidirect
5. #22549619C2semidirect
6. #22549619C2semidirect
7. #332×C127C2semidirect
97. C127#1553semidirect
102. #332D127semidirect

Maximal subgroups

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

subgroupindex
#225504272
#225499322
#225499282
#225496222
#225496192
#332×C1272
#1553127