Finite Group Databasebeta

← #30640377

Subgroups — #30640377

Order 16256.

280 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550435C211yesmaximal, characteristic
2C2×#16861124C211yesmaximal, characteristic, complemented
3#22549667C212yesmaximal, complemented
4#22549664C212yesmaximal, complemented
5C2×#100×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127C411yescharacteristic, complemented
7#16861192—21no—
8#16861190C411yescharacteristic, complemented
9#16861190C2211yescharacteristic
10#16861181—21no—
11#16861181—21no—
12#16861178—21no—
13#16861177C411yescharacteristic, complemented
14#16861177C2211yescharacteristic
15C2×C22⋊Dic127—21no—
16C2×C22⋊Dic127C411yescharacteristic, complemented
17#16861124—21no—
18#16861124C2212yes—
19C22×D4×C127C2211yescharacteristic
20C2×M16×C127—21no—
21#100×C127—21no—
22#100×C127C2212yes—
23C23×D127C2×C411yescharacteristic, complemented
24C2×C22⋊D127—21no—
25C2×C22⋊D127—21no—
26C2×C22⋊D127—21no—
27C2×C22⋊D127—21no—
28C2×C22⋊D127—41no—
29C2×C22⋊D127—41no—
30C22×Dic127—21no—
31C22×Dic127—41no—
32C22×Dic127C2×C411yescharacteristic, complemented
33Dic127⋊C4—21no—
34Dic127⋊C4—21no—
35C22⋊Dic127—21no—
36C22⋊Dic127—21no—
37C22⋊Dic127—21no—
38C22⋊Dic127—22no—
39C22⋊Dic127—41no—
40C22⋊Dic127C2×C411yescharacteristic, complemented
41C22⋊Dic127C2×C411yescharacteristic, complemented
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44C24×C127C2×C411yescharacteristic
45C24×C127D411yescharacteristic
46C2×D4×C127—41no—
47C2×D4×C127—41no—
48C2×D4×C127C2×C411yescharacteristic
49C2×D4×C127D411yescharacteristic
50C2×D4×C127D411yescharacteristic
51C2×D4×C127C2311yescharacteristic
52C22×C4×C127D411yescharacteristic
53M16×C127—22no—
54M16×C127—41no—
55C2×C8×C127—41no—
56C22×D127—21no—
57C22×D127—41no—
58C22×D127—81no—
59C22⋊D127—42no—
60C22⋊D127—41no—
61C22⋊D127—41no—
62C22⋊D127—81no—
63C22⋊D127—81no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—41no—
68C2×Dic127—42no—
69C2×Dic127—81no—
70C2×Dic127—82no—
71C23×C127—21no—
72C23×C127—21no—
73C23×C127—21no—
74C23×C127—21no—
75C23×C127—81no—
76C23×C127—81no—
77C23×C127C22⋊C411yescharacteristic
78C23×C127C22⋊C411yescharacteristic
79C23×C127C22⋊C411yescharacteristic
80C23×C127C22×C411yescharacteristic
81C23×C127C2×D411yescharacteristic
82D4×C127—41no—
83D4×C127—41no—
84D4×C127—41no—
85D4×C127—41no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127C22⋊C411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C8×C127—42no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—42no—
101C22×C127—81no—
102C22×C127—81no—
103C22×C127—81no—
104C22×C127#9911yescharacteristic
105C22×C127#9911yescharacteristic
106C22×C127C2×C22⋊C411yescharacteristic
107C2×D127—41no—
108C2×D127—41no—
109C2×D127—41no—
110C2×D127—82no—
111C4×C127—21no—
112C4×C127—21no—
113Dic127—41no—
114Dic127—41no—
115Dic127—41no—
116Dic127—82no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127—21no—
120C2×C127—41no—
121C2×C127—41no—
122C2×C127—41no—
123C2×C127—41no—
124C2×C127C2×#9911yescharacteristic
125D127—81no—
126#2121—1271nomaximal, Sylow
127C127#212111yescharacteristic, complemented, Sylow
128#534—1271no—
129C2×#100D12711yescharacteristic, complemented
130C2×#99—1271no—
131#352—1272no—
132AΣL(1,𝔽16)—1272no—
133C22×D4—1271no—
134C22×D4C2×D12711yescharacteristic
135C2×M16—21no—
136#123—1271no—
137#123—1271no—
138#123—2541no—
139#121—2541no—
140#121—2541no—
141#120—1271no—
142#120—1271no—
143#120—2541no—
144C2×C22⋊C4—1271no—
145C2×C22⋊C4—2541no—
146#100—21no—
147#100C2×D12712yes—
148#99—1272no—
149#99—2541no—
150C24—1271no—
151C24C4×D12711yescharacteristic
152C24D50811yescharacteristic
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—5081no—
156C2×D4—5081no—
157C2×D4—2541no—
158C2×D4—2541no—
159C2×D4—2541no—
160C2×D4—2541no—
161C2×D4C4×D12711yescharacteristic
162C2×D4C22⋊D12711yescharacteristic
163C2×D4D50811yescharacteristic
164C2×D4C22×D12711yescharacteristic
165C22×C4—1271no—
166C22×C4—5081no—
167C22×C4—2541no—
168C22×C4C22⋊D12711yescharacteristic
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C4⋊C4—2541no—
173C4⋊C4—2541no—
174C22⋊C4—1271no—
175C22⋊C4—1271no—
176C22⋊C4—5081no—
177C22⋊C4—2542no—
178C22⋊C4—2541no—
179C22⋊C4—2541no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C22⋊C4—2541no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—81no—
188C23—81no—
189C23—10161no—
190C23—5081no—
191C23—2541no—
192C23(C2×D127)⋊C411yescharacteristic
193C23(C2×D127)⋊C411yescharacteristic
194C23(C2×D127)⋊C411yescharacteristic
195C23C2×C4×D12711yescharacteristic
196C23C2×D50811yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—10161no—
202D4—10161no—
203D4—5082no—
204D4—5081no—
205D4—5081no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—10162no—
209C2×C4—10161no—
210C2×C4—5082no—
211C2×C4—5081no—
212C2×C4—2541no—
213C2×C4—2541no—
214C2×C4—2541no—
215C2×C4(C2×D127)⋊C411yescharacteristic
216C2×C4C2×C22⋊D12711yescharacteristic
217C8—42no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—10162no—
232C22—5081no—
233C22—5081no—
234C22—5081no—
235C22#1686112311yescharacteristic
236C22#1686112311yescharacteristic
237C22C2×(C2×D127)⋊C411yescharacteristic
238C4—21no—
239C4—21no—
240C4—10162no—
241C4—5081no—
242C4—5081no—
243C4—5081no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—10161no—
252C2C2×#1686112311yescharacteristic
subgroupG/H
centerC2C2×#16861123
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
FittingC2×#100×C127C2
socleC2×C127C2×#99
prime radical#22550435C2
cosocleC2×D4×C127C23
hypercenterC2×#100D127
hypocenterC127#2121
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#16861124C2semidirect
3. #22549667C2semidirect
4. #22549664C2semidirect
5. C2×#100×C127C2semidirect
6. C22×C22⋊D127C4semidirect
8. #16861190C4semidirect
13. #16861177C4semidirect
16. C2×C22⋊Dic127C4semidirect
23. C23×D127C2×C4semidirect
32. C22×Dic127C2×C4semidirect
40. C22⋊Dic127C2×C4semidirect
41. C22⋊Dic127C2×C4semidirect
127. C127#2121semidirect
129. C2×#100D127semidirect

Maximal subgroups

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

subgroupindex
#225504352
C2×#168611242
#225496672
#225496642
C2×#100×C1272
#2121127