Finite Group Databasebeta

← #30637742

Subgroups — #30637742

Order 16256.

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

normal subgroups only

This group has 232 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#22550464C211yesmaximal, characteristic
2#22550206C211yesmaximal, characteristic, complemented
3#22550189C211yesmaximal, characteristic, complemented
4#22549612C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549600C211yesmaximal, characteristic, complemented
7#397×C127C211yesmaximal, characteristic, complemented
8#16861202—21no—
9#16861195—21no—
10#16861191—21no—
11#16861188C2211yescharacteristic
12#16861183C2211yescharacteristic
13#16861182—21no—
14#16861178—21no—
15C4×C22⋊D127—21no—
16#16861142—21no—
17#16861140—21no—
18#16861132—21no—
19#16861129—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21C22⋊(C127⋊C8)C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C22⋊C4×C127—21no—
24C2×C22⋊C4×C127C2211yescharacteristic
25#95×C127—21no—
26#95×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127D411yescharacteristic
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39C22⋊Dic127—21no—
40C22⋊Dic127—21no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44C4×Dic127—21no—
45C24×C127D411yescharacteristic
46C22×C4×C127—21no—
47C22×C4×C127—21no—
48C22×C4×C127D411yescharacteristic
49C22×C4×C127C2311yescharacteristic
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C22⋊C4×C127—21no—
53C22⋊C4×C127—21no—
54C22⋊C4×C127—42no—
55C22×D127—21no—
56Dic254—41no—
57D508—41no—
58C22⋊D127—41no—
59C22⋊D127—41no—
60C22⋊D127—81no—
61C22⋊D127—81no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C4×D127—81no—
66C127⋊C8—41no—
67C127⋊C8—41no—
68C23×C127—21no—
69C23×C127—21no—
70C23×C127—21no—
71C23×C127—41no—
72C23×C127—41no—
73C23×C127C2×D411yescharacteristic
74C2×C4×C127—21no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—42no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127—41no—
91C22×C127—81no—
92C22×C127#12011yescharacteristic
93C22×C127#13511yescharacteristic
94C22×C127AGL(1,ℤ/8)11yescharacteristic
95C2×D127—41no—
96C2×D127—41no—
97C2×D127—41no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—41no—
101C4×C127—81no—
102C4×C127—81no—
103Dic127—41no—
104Dic127—41no—
105Dic127—41no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—81no—
109C2×C127#44911yescharacteristic
110C2×C127#45511yescharacteristic
111C2×C127#45811yescharacteristic
112D127—81no—
113#1612—1271nomaximal, Sylow
114C127#161211yescharacteristic, complemented, Sylow
115#538—1271no—
116#483—1271no—
117#480—1271no—
118#397D12711yescharacteristic, complemented
119#328—1271no—
120#327—1271no—
121#323—1271no—
122#126—2541no—
123#124—2541no—
124#123—2541no—
125C22⋊Q8—1271no—
126#121—1271no—
127#121—2541no—
128#120—2541no—
129C4×D4—2541no—
130C2×C4⋊C4C2×D12711yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4C2×D12711yescharacteristic
133#107—2541no—
134#106—2541no—
135#103—2541no—
136#102—2541no—
137C22⋊C8—1271no—
138C22⋊C8—1271no—
139#95—21no—
140#95C2×D12711yescharacteristic
141C24C22⋊D12711yescharacteristic
142C2×Q8—1271no—
143C2×D4—1271no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4—2541no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—2541no—
150C22×C4C22⋊D12711yescharacteristic
151C22×C4C22×D12711yescharacteristic
152C2×C8—2541no—
153C2×C8—2541no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—1271no—
157C4⋊C4—1271no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—42no—
163C22⋊C4—2541no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C42—2541no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—2541no—
175C23C2×C22⋊D12711yescharacteristic
176Q8—5081no—
177D4—10161no—
178D4—10161no—
179D4—5081no—
180D4—5081no—
181D4—5081no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—10161no—
192C2×C4—2541no—
193C2×C4—2541no—
194C2×C4—2541no—
195C2×C4D4×D12711yescharacteristic
196C2×C4D4×D12711yescharacteristic
197C8—5081no—
198C8—5081no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22—5081no—
207C22—5081no—
208C22—5081no—
209C22#1686117711yescharacteristic
210C22#1686122911yescharacteristic
211C22#1686123511yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—81no—
217C4—5081no—
218C4—5081no—
219C4—5081no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—10161no—
224C2#2255001911yescharacteristic
225C2#2255005311yescharacteristic
226C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#397×C127C2
socleC22×C127#120
prime radical#22550464C2
hypercenter#397D127
hypocenterC127#1612
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #22550206C2semidirect
3. #22550189C2semidirect
4. #22549612C2semidirect
5. #22549609C2semidirect
6. #22549600C2semidirect
7. #397×C127C2semidirect
114. C127#1612semidirect
118. #397D127semidirect

Maximal subgroups

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

subgroupindex
#225504642
#225502062
#225501892
#225496122
#225496092
#225496002
#397×C1272
#1612127