Finite Group Databasebeta

← #30637404

Subgroups — #30637404

Order 16256.

260 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 73 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#22549898C211yesmaximal, characteristic, complemented
3C4⋊C8×D127C211yesmaximal, characteristic, complemented
4#22549623C211yesmaximal, characteristic, complemented
5#22549621C211yesmaximal, characteristic, complemented
6#22549620C211yesmaximal, characteristic, complemented
7#331×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C2×#14523231—21no—
10C2×C8×D127—21no—
11#16861209C2211yescharacteristic
12#16861196—21no—
13#16861180—21no—
14#16861180C412yes—
15D4×Dic127—21no—
16D4×Dic127C412yes—
17C42×D127C2211yescharacteristic
18C4⋊(C127⋊C8)C2211yescharacteristic
19C4⋊(C127⋊C8)C2211yescharacteristic
20Dic127⋊C8—21no—
21Dic127⋊C8—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
33#14523231—42no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C8×D127—42no—
37C4⋊Dic127—21no—
38C4⋊Dic127C2×C412yes—
39C22⋊Dic127—22no—
40C22⋊Dic127—22no—
41C4×Dic127C2×C411yescharacteristic
42C4×Dic127D411yescharacteristic
43C4×Dic127D411yescharacteristic
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127C2×C412yes—
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C42×C127C2311yescharacteristic
50C22×D127C22⋊C411yescharacteristic
51Dic254—41no—
52Dic254—41no—
53C22⋊D127—44no—
54C2×Dic127—22no—
55C2×Dic127—44no—
56C2×Dic127C22⋊C411yescharacteristic
57C2×Dic127C22⋊C411yescharacteristic
58C2×Dic127C22⋊C411yescharacteristic
59C4×D127—21no—
60C4×D127—21no—
61C4×D127—41no—
62C4×D127D812yes—
63C4×D127SD1612yes—
64C127⋊C8—41no—
65C127⋊C8—41no—
66C23×C127—22no—
67D4×C127—42no—
68D4×C127—41no—
69D4×C127—41no—
70D4×C127—41no—
71D4×C127—41no—
72C2×C4×C127C22×C411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C8×C127—41no—
76C8×C127—41no—
77C22×C127—42no—
78C22×C127—42no—
79C22×C127—42no—
80C22×C127C2×C22⋊C411yescharacteristic
81C2×D127—21no—
82C2×D127—21no—
83C2×D127#10212yes—
84C4×C127—21no—
85C4×C127C2×D811yescharacteristic
86C4×C127C2×SD1611yescharacteristic
87C4×C127AGL(1,ℤ/8)11yescharacteristic
88C4×C127#13711yescharacteristic
89Dic127—21no—
90Dic127—21no—
91Dic127—42no—
92Dic127#10212yes—
93C2×C127—82no—
94C2×C127#41311yescharacteristic
95C2×C127C2×#10211yescharacteristic
96C2×C127#41711yescharacteristic
97D127—22no—
98#1547—1271nomaximal, Sylow
99C127#154711yescharacteristic, complemented, Sylow
100#532—1271no—
101#423—1271no—
102C2×C4⋊C8—1271no—
103#332—1271no—
104#332—1271no—
105#331—1271no—
106#331D12711yescharacteristic, complemented
107C2×#41—2541no—
108C2×M16—2541no—
109C22×C8—2541no—
110C4⋊Q8—1271no—
111#127C2×D12711yescharacteristic
112#124—2541no—
113#121—1272no—
114#121—2541no—
115C4×D4—1272no—
116C4×D4—2541no—
117C2×C42—1271no—
118C4⋊C8—1271no—
119C4⋊C8—1271no—
120C4⋊C8—2541no—
121C4⋊C8—2541no—
122C4⋊C8C2×D12711yescharacteristic
123C4⋊C8C2×D12711yescharacteristic
124#41—5082no—
125#41—5082no—
126C2×Q8—2541no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—2542no—
130C2×D4C4×D12712yes—
131C22×C4—1271no—
132C22×C4—1271no—
133C22×C4—1271no—
134C22×C4—2542no—
135M16—5082no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—5082no—
139C2×C8—2541no—
140C2×C8—2541no—
141C4⋊C4—1272no—
142C4⋊C4—2541no—
143C22⋊C4—2542no—
144C22⋊C4—2542no—
145C42—1271no—
146C42—1271no—
147C42—1271no—
148C42C22×D12711yescharacteristic
149C23—22no—
150C23—1271no—
151Q8—5081no—
152Q8—5081no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—5084no—
159C2×C4—1271no—
160C2×C4—1272no—
161C2×C4—1272no—
162C2×C4—1271no—
163C2×C4—1271no—
164C2×C4—5081no—
165C2×C4—5084no—
166C2×C4—2542no—
167C2×C4—2541no—
168C2×C4—2541no—
169C2×C4C2×C4×D12711yescharacteristic
170C2×C4D4×D12711yescharacteristic
171C2×C4D4×D12711yescharacteristic
172C8—41no—
173C8—41no—
174C8—5081no—
175C8—5081no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—1272no—
180C22—2541no—
181C22—2541no—
182C22C22⋊C4×D12711yescharacteristic
183C4—21no—
184C4—1272no—
185C4—5082no—
186C4—2541no—
187C4—2541no—
188C4D8×D12711yescharacteristic
189C4SD16×D12711yescharacteristic
190C4#1686123311yescharacteristic
191C4#1686123811yescharacteristic
192C2—82no—
193C2—2542no—
194C2#2254985711yescharacteristic
195C2#102×D12711yescharacteristic
196C2#2254987411yescharacteristic
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#1547
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. #22549898C2semidirect
3. C4⋊C8×D127C2semidirect
4. #22549623C2semidirect
5. #22549621C2semidirect
6. #22549620C2semidirect
7. #331×C127C2semidirect
99. C127#1547semidirect
106. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225504242
#225498982
C4⋊C8×D1272
#225496232
#225496212
#225496202
#331×C1272
#1547127