Finite Group Databasebeta

← #30639191

Subgroups — #30639191

Order 16256.

390 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#14523232C211yesmaximal, characteristic, complemented
2C2×#16861183C211yesmaximal, characteristic
3C2×#16861125C212yesmaximal, complemented
4#22549644C212yesmaximal, complemented
5#407×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C22×C22⋊D127C412yes—
8C2×#14523232—24no—
9C2×#14523232—21no—
10C2×#14523232C2212yes—
11C22×C127⋊C8—21no—
12#16861183—21no—
13#16861183—21no—
14#16861183—21no—
15#16861183—21no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×(C2×D127)⋊C4C412yes—
18#16861125—24no—
19C23×C4×C127C2211yescharacteristic
20C2×M16×C127C2212yes—
21C22⋊C8×C127—21no—
22C22⋊C8×C127—21no—
23C23×D127C2×C412yes—
24C2×D508—22no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—22no—
28C2×C22⋊D127—22no—
29C2×C22⋊D127—22no—
30C2×C22⋊D127—44no—
31C22×Dic127C2×C412yes—
32#14523232—24no—
33#14523232—22no—
34#14523232D414yescomplemented
35C2×C127⋊C8—24no—
36C2×C127⋊C8—22no—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39(C2×D127)⋊C4—22no—
40(C2×D127)⋊C4—22no—
41C24×C127C2×C411yescharacteristic
42C22×C4×C127—22no—
43C22×C4×C127—22no—
44C22×C4×C127C2×C411yescharacteristic
45C22×C4×C127D411yescharacteristic
46C22×C4×C127D411yescharacteristic
47C22×C4×C127D411yescharacteristic
48C22×C4×C127D411yescharacteristic
49C22×C4×C127C2311yescharacteristic
50M16×C127—22no—
51M16×C127—22no—
52C2×C8×C127—22no—
53C22×D127—22no—
54C22×D127—24no—
55C22×D127—42no—
56C22×D127—42no—
57D508—41no—
58D508—42no—
59D508—41no—
60C22⋊D127—44no—
61C22⋊D127—44no—
62C2×Dic127—22no—
63C2×Dic127—42no—
64C127⋊C8—24no—
65C23×C127—22no—
66C23×C127—22no—
67C23×C127—41no—
68C23×C127C22⋊C411yescharacteristic
69C23×C127C22⋊C411yescharacteristic
70C23×C127C22×C411yescharacteristic
71C2×C4×C127—22no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127C22⋊C411yescharacteristic
79C2×C4×C127C22⋊C411yescharacteristic
80C2×C4×C127C2×D412yes—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127#4112yes—
84C8×C127—42no—
85C22×C127—21no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—44no—
93C22×C127#10012yescomplemented
94C22×C127C2×C22⋊C411yescharacteristic
95C22×C127C4×D412yes—
96C2×D127—42no—
97C2×D127—42no—
98C2×D127—42no—
99C2×D127—84no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127#12011yescharacteristic
103C4×C127#12112yes—
104C4×C127#12311yescharacteristic
105Dic127—42no—
106C2×C127—22no—
107C2×C127—22no—
108C2×C127—41no—
109C2×C127#38611yescharacteristic
110C2×C127C2×#10011yescharacteristic
111C2×C127#41311yescharacteristic
112D127—82no—
113#1900—1271nomaximal, Sylow
114C127#190011yescharacteristic, complemented, Sylow
115C22×M16—1271no—
116C2×#121—1271no—
117C2×#100—1272no—
118#407D12711yescharacteristic, complemented
119#343—1272no—
120C22×D4—1272no—
121C22×D4—1271no—
122C23×C4C2×D12711yescharacteristic
123C2×M16—1272no—
124C2×M16—2544no—
125C2×M16—2541no—
126C2×M16C2×D12712yes—
127C22×C8—2541no—
128#121—2541no—
129#121—2541no—
130#121—2541no—
131#121—2541no—
132C2×C4⋊C4—1271no—
133C2×C22⋊C4—1272no—
134#100—2544no—
135C22⋊C8—21no—
136C22⋊C8—21no—
137C24—1272no—
138C24C4×D12711yescharacteristic
139C2×D4—5084no—
140C2×D4—2542no—
141C2×D4—2542no—
142C2×D4—2542no—
143C2×D4—2541no—
144C2×D4—2541no—
145C2×D4—2542no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—1272no—
149C22×C4C4×D12711yescharacteristic
150C22×C4C22⋊D12711yescharacteristic
151C22×C4C22⋊D12711yescharacteristic
152C22×C4D50811yescharacteristic
153C22×C4D50811yescharacteristic
154C22×C4C22×D12711yescharacteristic
155M16—22no—
156M16—22no—
157M16—1274no—
158M16—2544no—
159M16—2542no—
160C2×C8—22no—
161C2×C8—2544no—
162C2×C8—2542no—
163C4⋊C4—2541no—
164C4⋊C4—2541no—
165C22⋊C4—2542no—
166C22⋊C4—2542no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—5082no—
171C23—5082no—
172C23—2544no—
173C23—2542no—
174C23(C2×D127)⋊C411yescharacteristic
175C23(C2×D127)⋊C411yescharacteristic
176C23C2×C4×D12711yescharacteristic
177D4—5084no—
178D4—5084no—
179D4—5081no—
180D4—5081no—
181D4—5082no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—5082no—
190C2×C4—2542no—
191C2×C4(C2×D127)⋊C411yescharacteristic
192C2×C4(C2×D127)⋊C411yescharacteristic
193C2×C4D4×D12712yes—
194C2×C4C2×C22⋊D12711yescharacteristic
195C2×C4C2×D50811yescharacteristic
196C2×C4#1452324712yes—
197C8—42no—
198C8—2544no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—44no—
207C22—10164no—
208C22—5082no—
209C22—5082no—
210C22—5082no—
211C22#1686112512yescomplemented
212C22C2×(C2×D127)⋊C411yescharacteristic
213C22(C22⋊D127)⋊C412yes—
214C4—21no—
215C4—21no—
216C4—5082no—
217C4#1686117911yescharacteristic
218C4#1686118112yes—
219C4#1686119311yescharacteristic
220C2—22no—
221C2—22no—
222C2—41no—
223C2—10162no—
224C2#2254975311yescharacteristic
225C2C2×#1686112511yescharacteristic
226C2#2254985811yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#16861183C2
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1900
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#14523232C2semidirect
3. C2×#16861125C2semidirect
4. #22549644C2semidirect
5. #407×C127C2semidirect
34. #14523232D4semidirect
93. C22×C127#100semidirect
114. C127#1900semidirect
118. #407D127semidirect
211. C22#16861125semidirect

Maximal subgroups

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

subgroupindex
C22×#145232322
C2×#168611832
C2×#168611252
#225496442
#407×C1272
#1900127