Finite Group Databasebeta

← #30637299

Subgroups — #30637299

Order 16256.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#16861183C211yesmaximal, characteristic
2#22549736C211yesmaximal, characteristic, complemented
3#22549611C214yesmaximal, complemented
4C2×C22⋊C8×C127C211yesmaximal, characteristic, complemented
5C22×D508C411yescharacteristic, complemented
6C22×C22⋊D127—21no—
7#16861183—21no—
8#16861183—21no—
9#16861183C412yescomplemented
10#16861183C2212yes—
11C2×C4⋊Dic127—21no—
12C2×C4⋊Dic127C411yescharacteristic, complemented
13C2×C22⋊Dic127—21no—
14C2×(C2×D127)⋊C4—21no—
15C127⋊#95C2212yescomplemented
16C23×C4×C127C2211yescharacteristic
17C22×C8×C127—21no—
18C22⋊C8×C127—21no—
19C22⋊C8×C127C2212yescomplemented
20C23×D127—21no—
21C2×D508—21no—
22C2×D508—42no—
23C2×D508C2×C412yes—
24C2×C22⋊D127—22no—
25C2×C22⋊D127—22no—
26C2×C22⋊D127—42no—
27C2×C22⋊D127—42no—
28C22×Dic127—21no—
29C22×Dic127—22no—
30C4⋊Dic127—21no—
31C4⋊Dic127—42no—
32C4⋊Dic127C2×C412yes—
33C22⋊Dic127—42no—
34(C2×D127)⋊C4—22no—
35(C2×D127)⋊C4—22no—
36C24×C127D411yescharacteristic
37C22×C4×C127—42no—
38C22×C4×C127C2×C411yescharacteristic
39C22×C4×C127C2×C411yescharacteristic
40C22×C4×C127D411yescharacteristic
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127C2311yescharacteristic
44C2×C8×C127—22no—
45C2×C8×C127—42no—
46C22×D127—22no—
47C22×D127—42no—
48C22×D127—42no—
49C22×D127—42no—
50D508—42no—
51D508—42no—
52C22⋊D127—44no—
53C22⋊D127—42no—
54C22⋊D127—42no—
55C2×Dic127—22no—
56C2×Dic127—42no—
57C2×Dic127—42no—
58C2×Dic127—42no—
59C2×Dic127—42no—
60C23×C127—41no—
61C23×C127—41no—
62C23×C127C22⋊C411yescharacteristic
63C23×C127C22⋊C411yescharacteristic
64C23×C127D812yes—
65C23×C127SD1612yes—
66C23×C127C2×D411yescharacteristic
67C2×C4×C127—22no—
68C2×C4×C127—22no—
69C2×C4×C127—22no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127C22⋊C411yescharacteristic
74C2×C4×C127C22⋊C411yescharacteristic
75C2×C4×C127C22×C411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C8×C127—42no—
78C22×C127—21no—
79C22×C127—21no—
80C22×C127—21no—
81C22×C127—21no—
82C22×C127—21no—
83C22×C127—21no—
84C22×C127—42no—
85C22×C127—42no—
86C22×C127#10212yes—
87C22×C127#10212yes—
88C22×C127C2×C22⋊C411yescharacteristic
89C22×C127C2×D811yescharacteristic
90C22×C127C2×SD1611yescharacteristic
91C2×D127—42no—
92C2×D127—42no—
93C2×D127—42no—
94C2×D127—84no—
95C4×C127—21no—
96C4×C127—21no—
97C4×C127—21no—
98C4×C127#9912yes—
99Dic127—42no—
100Dic127—82no—
101C2×C127—21no—
102C2×C127—22no—
103C2×C127—22no—
104C2×C127—21no—
105C2×C127C2×#9911yescharacteristic
106C2×C127C2×#10211yescharacteristic
107C2×C127#42111yescharacteristic
108D127—82no—
109#1519—1271nomaximal, Sylow
110C127#151911yescharacteristic, complemented, Sylow
111C2×#121—1271no—
112C2×C22⋊C8D12711yescharacteristic, complemented
113#380—1271no—
114#327—1274no—
115C22×D4—1271no—
116C22×D4—2541no—
117C23×C4C2×D12711yescharacteristic
118C22×C8—21no—
119#121—1272no—
120#121—1272no—
121#121—2541no—
122#121—2541no—
123C2×C4⋊C4—1271no—
124C2×C4⋊C4—2541no—
125C2×C22⋊C4—2541no—
126C2×C22⋊C4—2541no—
127C22⋊C8—21no—
128C22⋊C8C2×D12712yescomplemented
129#95—1272no—
130C24—2541no—
131C24D50811yescharacteristic
132C2×D4—1272no—
133C2×D4—5082no—
134C2×D4—5082no—
135C2×D4—5082no—
136C2×D4—2542no—
137C2×D4—2542no—
138C2×D4—2541no—
139C22×C4—42no—
140C22×C4—2542no—
141C22×C4—2541no—
142C22×C4C4×D12711yescharacteristic
143C22×C4C4×D12711yescharacteristic
144C22×C4C22⋊D12711yescharacteristic
145C22×C4C22⋊D12711yescharacteristic
146C22×C4D50811yescharacteristic
147C22×C4C22×D12711yescharacteristic
148C2×C8—22no—
149C2×C8—42no—
150C4⋊C4—1272no—
151C4⋊C4—5082no—
152C4⋊C4—2541no—
153C22⋊C4—5082no—
154C22⋊C4—2542no—
155C22⋊C4—2542no—
156C23—41no—
157C23—41no—
158C23—5082no—
159C23—5082no—
160C23—5082no—
161C23—2542no—
162C23(C2×D127)⋊C411yescharacteristic
163C23(C2×D127)⋊C411yescharacteristic
164C23D101612yes—
165C23#1452323712yes—
166C23C2×D50811yescharacteristic
167D4—5082no—
168D4—5082no—
169D4—5082no—
170D4—5084no—
171D4—5082no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—5082no—
179C2×C4—5082no—
180C2×C4—5082no—
181C2×C4—5082no—
182C2×C4—2542no—
183C2×C4(C2×D127)⋊C411yescharacteristic
184C2×C4(C2×D127)⋊C411yescharacteristic
185C2×C4C2×C4×D12711yescharacteristic
186C2×C4C2×C22⋊D12711yescharacteristic
187C8—42no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—10164no—
197C22—5082no—
198C22—5082no—
199C22—5082no—
200C22#1686113112yes—
201C22#1686113112yes—
202C22C2×(C2×D127)⋊C411yescharacteristic
203C22C2×D101611yescharacteristic
204C22C2×#1452323711yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—10162no—
209C4—5082no—
210C4#1686112312yes—
211C2—21no—
212C2—22no—
213C2—21no—
214C2—22no—
215C2—10162no—
216C2C2×#1686112311yescharacteristic
217C2C2×#1686113111yescharacteristic
218C2#2254989211yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C22⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#16861183C2
cosocleC22×C4×C127C23
hypercenterC2×C22⋊C8D127
hypocenterC127#1519
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #22549736C2semidirect
3. #22549611C2semidirect
4. C2×C22⋊C8×C127C2semidirect
5. C22×D508C4semidirect
9. #16861183C4semidirect
12. C2×C4⋊Dic127C4semidirect
15. C127⋊#95C22semidirect
19. C22⋊C8×C127C22semidirect
110. C127#1519semidirect
112. C2×C22⋊C8D127semidirect
128. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
#225497362
#225496112
C2×C22⋊C8×C1272
#1519127