Finite Group Databasebeta

← #30637317

Subgroups — #30637317

Order 16256.

234 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 61 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#22549828C211yesmaximal, characteristic, complemented
3#97×D127C211yesmaximal, characteristic, complemented
4#22549618C211yesmaximal, characteristic, complemented
5#22549615C211yesmaximal, characteristic, complemented
6#22549615C211yesmaximal, characteristic, complemented
7#330×C127C211yesmaximal, characteristic, complemented
8C2×#14523231—21no—
9C2×C8×D127—21no—
10#16861204—21no—
11#16861197C2211yescharacteristic
12#16861194—21no—
13#16861189C411yescharacteristic
14#16861184C411yescharacteristic
15#16861174C411yescharacteristic
16#16861171—21no—
17#16861171C411yescharacteristic
18#16861167—21no—
19C42×D127C2211yescharacteristic
20#16861120C2211yescharacteristic
21#16861120C2211yescharacteristic
22#16861119—21no—
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—
37C8×D127—24no—
38C4⋊Dic127—21no—
39Dic127⋊C4—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45C4×Dic127—21no—
46C4×Dic127C2×C411yescharacteristic
47C4×Dic127D411yescharacteristic
48C4×Dic127D411yescharacteristic
49C2×C8×C127—21no—
50C2×C8×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127C2×C411yescharacteristic
54C4⋊C4×C127C2×C411yescharacteristic
55C42×C127C2311yescharacteristic
56C22×D127—21no—
57C22×D127C22⋊C411yescharacteristic
58Dic254—41no—
59D508—41no—
60D508—42no—
61C2×Dic127—21no—
62C2×Dic127C22⋊C411yescharacteristic
63C2×Dic127C22⋊C411yescharacteristic
64C2×Dic127C22⋊C411yescharacteristic
65C4×D127—22no—
66C4×D127—22no—
67C4×D127—41no—
68C4×D127—42no—
69C127⋊C8—22no—
70C127⋊C8—22no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127C22×C411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C8×C127—22no—
77C8×C127—22no—
78C22×C127C2×C22⋊C411yescharacteristic
79C2×D127—21no—
80C2×D127—21no—
81C2×D127—41no—
82C2×D127—41no—
83C2×D127—41no—
84C2×D127#10412yes—
85C4×C127—21no—
86C4×C127—21no—
87C4×C127—21no—
88C4×C127—41no—
89C4×C127—41no—
90Dic127—21no—
91Dic127—21no—
92Dic127—41no—
93Dic127#10412yes—
94C2×C127#41311yescharacteristic
95C2×C127C2×#10411yescharacteristic
96C2×C127C2×#10411yescharacteristic
97D127—22no—
98D127—81no—
99#1524—1271nomaximal, Sylow
100C127#152411yescharacteristic, complemented, Sylow
101#529—1271no—
102C4×M16—1271no—
103C2×#97—1271no—
104#330—1271no—
105#330D12711yescharacteristic, complemented
106#329—1271no—
107#329—1271no—
108C2×M16—2541no—
109C22×C8—2541no—
110#126—2541no—
111#125C2×D12711yescharacteristic
112#124—1271no—
113#123—2541no—
114C22⋊Q8—1271no—
115#121—1271no—
116C4×Q8—1271no—
117C4×D4—1271no—
118C4×D4—2541no—
119#117—2541no—
120C2×C42—1271no—
121#97—1271no—
122#97—1271no—
123#97—2541no—
124#97C2×D12711yescharacteristic
125#97C2×D12711yescharacteristic
126C4×C8—2541no—
127C2×Q8—1271no—
128C2×D4—1271no—
129C2×D4—2541no—
130C22×C4—1271no—
131C22×C4—1271no—
132C22×C4—1271no—
133C22×C4—2541no—
134M16—2544no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—2544no—
138C2×C8—2541no—
139C2×C8—2541no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—2541no—
143C4⋊C4—2541no—
144C4⋊C4C4×D12711yescharacteristic
145C4⋊C4C4×D12711yescharacteristic
146C22⋊C4—2541no—
147C22⋊C4—2541no—
148C22⋊C4—2541no—
149C22⋊C4—2541no—
150C22⋊C4—2541no—
151C42—1271no—
152C42—1271no—
153C42—1271no—
154C42—2541no—
155C42C22×D12711yescharacteristic
156C23—1271no—
157C23—2541no—
158Q8—5081no—
159D4—5081no—
160D4—5082no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—1271no—
164C2×C4—1271no—
165C2×C4—1271no—
166C2×C4—5081no—
167C2×C4—5082no—
168C2×C4—2542no—
169C2×C4—2542no—
170C2×C4—2541no—
171C2×C4C2×C4×D12711yescharacteristic
172C2×C4D4×D12711yescharacteristic
173C2×C4D4×D12711yescharacteristic
174C8—22no—
175C8—22no—
176C8—2542no—
177C8—2542no—
178C22—1272no—
179C22—5081no—
180C22—5081no—
181C22—5081no—
182C22—2541no—
183C22—2541no—
184C22C22⋊C4×D12711yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—41no—
189C4—41no—
190C4—1272no—
191C4—5081no—
192C4—2541no—
193C4—2541no—
194C2—10161no—
195C2—2542no—
196C2#2254985911yescharacteristic
197C2#104×D12711yescharacteristic
198C2#104×D12711yescharacteristic
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#1524
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. #22549828C2semidirect
3. #97×D127C2semidirect
4. #22549618C2semidirect
5. #22549615C2semidirect
6. #22549615C2semidirect
7. #330×C127C2semidirect
100. C127#1524semidirect
105. #330D127semidirect

Maximal subgroups

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

subgroupindex
#225504112
#225498282
#97×D1272
#225496182
#225496152
#330×C1272
#1524127