Finite Group Databasebeta

← #30637336

Subgroups — #30637336

Order 16256.

216 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 59 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#22550411C211yesmaximal, characteristic
2#22549931C211yesmaximal, characteristic, complemented
3#22549828C211yesmaximal, characteristic, complemented
4#22549618C211yesmaximal, characteristic, complemented
5#22549615C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic, complemented
7#330×C127C211yesmaximal, characteristic, complemented
8C2×#14523231—21no—
9#16861204—21no—
10#16861197C2211yescharacteristic
11#16861194—21no—
12#16861189C411yescharacteristic
13#16861184C411yescharacteristic
14#16861174C411yescharacteristic
15#16861171—21no—
16#16861171C411yescharacteristic
17#16861167—21no—
18C42×D127C2211yescharacteristic
19Dic127⋊C8—21no—
20(C2×D127)⋊C8—21no—
21#16861120C2211yescharacteristic
22#16861120C2211yescharacteristic
23C8×Dic127—21no—
24#125×C127C2211yescharacteristic
25#97×C127C2211yescharacteristic
26#97×C127C2211yescharacteristic
27C2×Dic254C2×C411yescharacteristic
28C2×D508—21no—
29C2×D508C2×C411yescharacteristic
30C2×C4×D127—21no—
31C2×C4×D127C2×C411yescharacteristic
32C2×C4×D127D411yescharacteristic
33C2×C4×D127D411yescharacteristic
34#14523231—24no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127—21no—
38Dic127⋊C4—21no—
39(C2×D127)⋊C4—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44C4×Dic127—21no—
45C4×Dic127C2×C411yescharacteristic
46C4×Dic127D411yescharacteristic
47C4×Dic127D411yescharacteristic
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127C2×C411yescharacteristic
53C4⋊C4×C127C2×C411yescharacteristic
54C42×C127C2311yescharacteristic
55C22×D127—21no—
56C22×D127C22⋊C411yescharacteristic
57Dic254—41no—
58D508—41no—
59D508—42no—
60C2×Dic127—21no—
61C2×Dic127C22⋊C411yescharacteristic
62C2×Dic127C22⋊C411yescharacteristic
63C2×Dic127C22⋊C411yescharacteristic
64C4×D127—22no—
65C4×D127—41no—
66C4×D127—41no—
67C4×D127—42no—
68C127⋊C8—22no—
69C127⋊C8—41no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127C22×C411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C8×C127—22no—
76C8×C127—41no—
77C22×C127C2×C22⋊C411yescharacteristic
78C2×D127—21no—
79C2×D127—21no—
80C2×D127—21no—
81C2×D127—41no—
82C2×D127—41no—
83C2×D127—41no—
84C4×C127—21no—
85C4×C127—21no—
86C4×C127—21no—
87C4×C127—41no—
88C4×C127—41no—
89Dic127—21no—
90Dic127—21no—
91Dic127—41no—
92Dic127#10412yes—
93C2×C127#41311yescharacteristic
94C2×C127C2×#10411yescharacteristic
95C2×C127#42111yescharacteristic
96D127—41no—
97D127—81no—
98#1529—1271nomaximal, Sylow
99C127#152911yescharacteristic, complemented, Sylow
100#529—1271no—
101#432—1271no—
102C4×M16—1271no—
103#330—1271no—
104#330D12711yescharacteristic, complemented
105#329—1271no—
106#329—1271no—
107C2×M16—2541no—
108#126—2541no—
109#125C2×D12711yescharacteristic
110#124—1271no—
111#123—2541no—
112C22⋊Q8—1271no—
113#121—1271no—
114C4×Q8—1271no—
115C4×D4—1271no—
116C4×D4—2541no—
117#117—2541no—
118C2×C42—1271no—
119C4⋊C8—2541no—
120C22⋊C8—2541no—
121#97—1271no—
122#97—1271no—
123#97C2×D12711yescharacteristic
124#97C2×D12711yescharacteristic
125C4×C8—2541no—
126C2×Q8—1271no—
127C2×D4—1271no—
128C2×D4—2541no—
129C22×C4—1271no—
130C22×C4—1271no—
131C22×C4—1271no—
132C22×C4—2541no—
133M16—2544no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—2541no—
137C2×C8—2541no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C4⋊C4C4×D12711yescharacteristic
143C4⋊C4C4×D12711yescharacteristic
144C22⋊C4—2541no—
145C22⋊C4—2541no—
146C22⋊C4—2541no—
147C22⋊C4—2541no—
148C22⋊C4—2541no—
149C42—1271no—
150C42—1271no—
151C42—1271no—
152C42—2541no—
153C42C22×D12711yescharacteristic
154C23—1271no—
155C23—2541no—
156Q8—5081no—
157D4—5081no—
158D4—5082no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—1271no—
162C2×C4—1271no—
163C2×C4—1271no—
164C2×C4—5081no—
165C2×C4—5081no—
166C2×C4—5082no—
167C2×C4—2542no—
168C2×C4—2541no—
169C2×C4C2×C4×D12711yescharacteristic
170C2×C4D4×D12711yescharacteristic
171C2×C4D4×D12711yescharacteristic
172C8—22no—
173C8—41no—
174C8—5081no—
175C8—2542no—
176C22—5081no—
177C22—5081no—
178C22—5081no—
179C22—2541no—
180C22—2541no—
181C22—2541no—
182C22C22⋊C4×D12711yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—41no—
188C4—1272no—
189C4—5081no—
190C4—2541no—
191C4—2541no—
192C2—10161no—
193C2—5081no—
194C2#2254985911yescharacteristic
195C2#104×D12711yescharacteristic
196C2#2254989111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#330×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861184C4
cosocleC42×C127C23
hypercenter#330D127
hypocenterC127#1529
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #22549931C2semidirect
3. #22549828C2semidirect
4. #22549618C2semidirect
5. #22549615C2semidirect
6. #22549615C2semidirect
7. #330×C127C2semidirect
99. C127#1529semidirect
104. #330D127semidirect

Maximal subgroups

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

subgroupindex
#225504112
#225499312
#225498282
#225496182
#225496152
#330×C1272
#1529127