Finite Group Databasebeta

← #30637763

Subgroups — #30637763

Order 16256.

208 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550473C211yesmaximal, characteristic
2#22550212C211yesmaximal, characteristic, complemented
3#22550195C211yesmaximal, characteristic
4#22549815C211yesmaximal, characteristic, complemented
5#22549614C211yesmaximal, characteristic, complemented
6#22549611C211yesmaximal, characteristic
7#324×C127C211yesmaximal, characteristic, complemented
8#16861204—21no—
9#16861199—21no—
10#16861194—21no—
11#16861192—21no—
12#16861188C2211yescharacteristic
13#16861187—21no—
14#16861183C2211yescharacteristic
15C4×C22⋊D127—21no—
16C2×C4⋊Dic127C2211yescharacteristic
17#16861143—21no—
18#16861141—21no—
19#16861134—21no—
20#16861131—21no—
21C127⋊#95—21no—
22C127⋊#95—21no—
23C127⋊#95C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26C22⋊C8×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—21no—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C2×C4×D127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D411yescharacteristic
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46C4×Dic127—21no—
47C22×C4×C127D411yescharacteristic
48C22×C4×C127D411yescharacteristic
49C22×C4×C127C2311yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C22×D127—21no—
55Dic254—41no—
56D508—41no—
57C22⋊D127—41no—
58C22⋊D127—41no—
59C2×Dic127—21no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C2×Dic127—42no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C4×D127—81no—
68C23×C127C2×D411yescharacteristic
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C8×C127—41no—
78C8×C127—41no—
79C22×C127—21no—
80C22×C127—21no—
81C22×C127#12011yescharacteristic
82C22×C127#13511yescharacteristic
83C22×C127#13711yescharacteristic
84C2×D127—41no—
85C2×D127—41no—
86C2×D127—41no—
87C4×C127—21no—
88C4×C127—21no—
89C4×C127—41no—
90C4×C127—41no—
91Dic127—41no—
92Dic127—41no—
93Dic127—41no—
94Dic127—81no—
95Dic127—81no—
96C2×C127—21no—
97C2×C127—21no—
98C2×C127#45211yescharacteristic
99C2×C127#45411yescharacteristic
100C2×C127#45811yescharacteristic
101D127—81no—
102#1615—1271nomaximal, Sylow
103C127#161511yescharacteristic, complemented, Sylow
104#539—1271no—
105#484—1271no—
106#481—1271no—
107#400—1271no—
108#328—1271no—
109#327—1271no—
110#324D12711yescharacteristic, complemented
111#126—2541no—
112#125—2541no—
113#123—2541no—
114#123—2541no—
115C22⋊Q8—1271no—
116C22⋊Q8—2541no—
117#121—1271no—
118C4×D4—2541no—
119C2×C4⋊C4—1271no—
120C2×C4⋊C4C2×D12711yescharacteristic
121#107—2541no—
122#106—2541no—
123#103—2541no—
124#102—2541no—
125C22⋊C8C2×D12711yescharacteristic
126C22⋊C8C2×D12711yescharacteristic
127#95—1271no—
128#95—2541no—
129#95—2541no—
130C2×Q8—1271no—
131C2×D4—1271no—
132C2×D4—2541no—
133C22×C4—2541no—
134C22×C4—2541no—
135C22×C4—2541no—
136C22×C4D50811yescharacteristic
137C22×C4D50811yescharacteristic
138C22×C4C22×D12711yescharacteristic
139C2×C8—21no—
140C2×C8—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—1271no—
144C4⋊C4—1271no—
145C4⋊C4—2541no—
146C4⋊C4—2541no—
147C4⋊C4—2541no—
148C4⋊C4—2541no—
149C4⋊C4—2541no—
150C4⋊C4—2541no—
151C22⋊C4—2541no—
152C22⋊C4—2541no—
153C22⋊C4—2541no—
154C22⋊C4—2541no—
155C22⋊C4—2541no—
156C42—2541no—
157C23—2541no—
158C23C2×D50811yescharacteristic
159Q8—5081no—
160D4—5081no—
161D4—5081no—
162D4—5081no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—10161no—
170C2×C4—5081no—
171C2×C4—5081no—
172C2×C4—5082no—
173C2×C4—5081no—
174C2×C4—5081no—
175C2×C4—2541no—
176C2×C4—2541no—
177C2×C4—2541no—
178C2×C4D4×D12711yescharacteristic
179C2×C4D4×D12711yescharacteristic
180C8—41no—
181C8—41no—
182C22—21no—
183C22—21no—
184C22—5081no—
185C22—5081no—
186C22—5081no—
187C22#1686117911yescharacteristic
188C22#1686123111yescharacteristic
189C22#1686124211yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—10161no—
195C4—10161no—
196C4—5081no—
197C4—5081no—
198C4—5081no—
199C2—21no—
200C2—21no—
201C2—10161no—
202C2#2255004011yescharacteristic
203C2#2255005111yescharacteristic
204C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127#120
prime radical#16861183C22
hypercenter#324D127
hypocenterC127#1615
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #22550212C2semidirect
4. #22549815C2semidirect
5. #22549614C2semidirect
7. #324×C127C2semidirect
103. C127#1615semidirect
110. #324D127semidirect

Maximal subgroups

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

subgroupindex
#225504732
#225502122
#225501952
#225498152
#225496142
#225496112
#324×C1272
#1615127