Finite Group Databasebeta

← #30637046

Subgroups — #30637046

Order 16256.

332 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 97 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×C4×C22⋊D127C211yesmaximal, characteristic
2C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
3#22549601C212yesmaximal, complemented
4#22549599C212yesmaximal, complemented
5C2×C22⋊C8×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127C411yescharacteristic
7C22×C4×D127C411yescharacteristic
8C22×C127⋊C8—21no—
9C4×C22⋊D127—21no—
10C4×C22⋊D127—21no—
11C4×C22⋊D127—21no—
12C4×C22⋊D127—21no—
13C2×Dic127⋊C4C2211yescharacteristic
14C2×C22⋊Dic127C411yescharacteristic
15C2×(C2×D127)⋊C4C2211yescharacteristic
16C2×C4×Dic127C411yescharacteristic
17C22⋊(C127⋊C8)—21no—
18C22⋊(C127⋊C8)C2212yes—
19C23×C4×C127C2211yescharacteristic
20C22×C8×C127—21no—
21C22⋊C8×C127—21no—
22C22⋊C8×C127C2212yes—
23C23×D127C2×C411yescharacteristic
24C2×C22⋊D127—21no—
25C2×C22⋊D127—21no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127C814yescomplemented
28C22×Dic127C2×C411yescharacteristic
29C22×Dic127C2×C411yescharacteristic
30C22×Dic127C2×C411yescharacteristic
31C2×C4×D127—21no—
32C2×C4×D127—21no—
33C2×C4×D127—42no—
34C2×C127⋊C8—22no—
35C2×C127⋊C8—42no—
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38C22⋊Dic127—21no—
39C22⋊Dic127—21no—
40(C2×D127)⋊C4—21no—
41(C2×D127)⋊C4—21no—
42C4×Dic127—21no—
43C4×Dic127—21no—
44C24×C127C2×C411yescharacteristic
45C22×C4×C127—42no—
46C22×C4×C127C2×C411yescharacteristic
47C22×C4×C127D411yescharacteristic
48C22×C4×C127D411yescharacteristic
49C22×C4×C127D411yescharacteristic
50C22×C4×C127D411yescharacteristic
51C22×C4×C127C2311yescharacteristic
52C2×C8×C127—22no—
53C2×C8×C127—42no—
54C22×D127—21no—
55C22×D127—21no—
56C22×D127—41no—
57C22×D127—41no—
58C22×D127C2×C812yescomplemented
59C22⋊D127—24no—
60C22⋊D127—24no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—41no—
66C2×Dic127—41no—
67C2×Dic127C2×C812yescomplemented
68C4×D127—42no—
69C4×D127—42no—
70C127⋊C8—42no—
71C23×C127—41no—
72C23×C127—41no—
73C23×C127C22⋊C411yescharacteristic
74C23×C127C22⋊C411yescharacteristic
75C23×C127C2×C812yes—
76C23×C127M1612yes—
77C23×C127C22×C411yescharacteristic
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C22⋊C411yescharacteristic
85C2×C4×C127C22⋊C411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C8×C127—42no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—42no—
96C22×C127—42no—
97C22×C127C22⋊C812yes—
98C22×C127C22⋊C812yes—
99C22×C127C2×C22⋊C411yescharacteristic
100C22×C127C22×C811yescharacteristic
101C22×C127C2×M1611yescharacteristic
102C2×D127—22no—
103C2×D127—42no—
104C2×D127—41no—
105C2×D127—41no—
106C2×D127—42no—
107C4×C127—21no—
108C4×C127—21no—
109C4×C127—21no—
110C4×C127#9912yes—
111Dic127—22no—
112Dic127—41no—
113Dic127—41no—
114C2×C127—21no—
115C2×C127—21no—
116C2×C127—22no—
117C2×C127—22no—
118C2×C127C2×C22⋊C811yescharacteristic
119C2×C127C2×#9911yescharacteristic
120C2×C127#41311yescharacteristic
121D127—42no—
122#1459—1271nomaximal, Sylow
123C127#145911yescharacteristic, complemented, Sylow
124C2×C4×D4—1271no—
125C2×C22⋊C8—1271no—
126C2×C22⋊C8D12711yescharacteristic, complemented
127#324—1272no—
128#323—1272no—
129C22×D4—1271no—
130C23×C4—1271no—
131C23×C4C2×D12711yescharacteristic
132C22×C8—21no—
133C22×C8—2541no—
134C4×D4—2541no—
135C4×D4—2541no—
136C4×D4—2541no—
137C4×D4—2541no—
138C2×C4⋊C4—1271no—
139C2×C22⋊C4—1271no—
140C2×C22⋊C4—1271no—
141C2×C42—1271no—
142C22⋊C8—21no—
143C22⋊C8—1272no—
144C22⋊C8—2541no—
145C22⋊C8C2×D12712yes—
146C24—1271no—
147C24C4×D12711yescharacteristic
148C2×D4—1274no—
149C2×D4—2541no—
150C2×D4—2541no—
151C2×D4—2542no—
152C22×C4—42no—
153C22×C4—1271no—
154C22×C4—1271no—
155C22×C4—1271no—
156C22×C4—5082no—
157C22×C4—2541no—
158C22×C4—2541no—
159C22×C4C4×D12711yescharacteristic
160C22×C4C22⋊D12711yescharacteristic
161C22×C4C22⋊D12711yescharacteristic
162C22×C4D50811yescharacteristic
163C22×C4D50811yescharacteristic
164C22×C4C22×D12711yescharacteristic
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—5082no—
168C2×C8—2542no—
169C4⋊C4—2541no—
170C4⋊C4—2541no—
171C22⋊C4—2541no—
172C22⋊C4—2541no—
173C22⋊C4—2541no—
174C22⋊C4—2541no—
175C42—2541no—
176C42—2541no—
177C23—41no—
178C23—41no—
179C23—1272no—
180C23—5081no—
181C23—5081no—
182C23—2541no—
183C23—2541no—
184C23(C2×D127)⋊C411yescharacteristic
185C23(C2×D127)⋊C411yescharacteristic
186C23C8×D12712yes—
187C23#1452323112yes—
188C23C2×C4×D12711yescharacteristic
189D4—2544no—
190D4—2544no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—1272no—
198C2×C4—5082no—
199C2×C4—5082no—
200C2×C4—5081no—
201C2×C4—5081no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4(C2×D127)⋊C411yescharacteristic
207C2×C4(C2×D127)⋊C411yescharacteristic
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4C2×D50811yescharacteristic
210C8—42no—
211C8—5082no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—42no—
219C22—42no—
220C22—5082no—
221C22—5082no—
222C22—5081no—
223C22—5081no—
224C22—2542no—
225C22(C2×D127)⋊C812yes—
226C22(C2×D127)⋊C812yes—
227C22C2×(C2×D127)⋊C411yescharacteristic
228C22C2×C8×D12711yescharacteristic
229C22C2×#1452323111yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—5081no—
234C4—5081no—
235C4—2542no—
236C4#1686112312yes—
237C2—21no—
238C2—21no—
239C2—22no—
240C2—22no—
241C2—5082no—
242C2C2×(C2×D127)⋊C811yescharacteristic
243C2C2×#1686112311yescharacteristic
244C2#2254985811yescharacteristic
subgroupG/H
centerC2×C4(C2×D127)⋊C4
commutatorC22×C127C22×C8
FrattiniC22×C4C22×D127
FittingC2×C22⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×C22⋊D127C4
cosocleC22×C4×C127C23
hypercenterC2×C22⋊C8D127
hypocenterC127#1459
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊(C127⋊C8)C2semidirect
3. #22549601C2semidirect
4. #22549599C2semidirect
5. C2×C22⋊C8×C127C2semidirect
27. C2×C22⋊D127C8semidirect
58. C22×D127C2×C8semidirect
67. C2×Dic127C2×C8semidirect
123. C127#1459semidirect
126. C2×C22⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1272
C2×C22⋊(C127⋊C8)2
#225496012
#225495992
C2×C22⋊C8×C1272
#1459127