Finite Group Databasebeta

← #30641326

Subgroups — #30641326

Order 16256.

672 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 189 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×C22⋊Dic127C211yesmaximal, characteristic
2#22549770C214yesmaximal, complemented
3#22549764C214yesmaximal, complemented
4#22549734C212yesmaximal, complemented
5#22549732C211yesmaximal, characteristic
6C22⋊C4×Dic127C212yesmaximal, complemented
7#379×C127C211yesmaximal, characteristic
8C23×Dic127C2211yescharacteristic
9C23×Dic127C2211yescharacteristic
10C2×Dic127⋊C4C2214yescomplemented
11C2×C22⋊Dic127C418yescomplemented
12C2×C22⋊Dic127C2212yes—
13C2×C22⋊Dic127C2212yes—
14C2×C22⋊Dic127C2212yes—
15C2×C22⋊Dic127C2212yes—
16C2×C22⋊Dic127C2212yes—
17C2×C4×Dic127C2214yescomplemented
18C127⋊#95C2214yescomplemented
19C127⋊#95C2214yescomplemented
20C25×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2212yes—
22C2×C22⋊C4×C127C2212yes—
23C2×C22⋊C4×C127C2212yes—
24C22×Dic127—42no—
25C22×Dic127—42no—
26C22×Dic127C2×C414yes—
27C22×Dic127C2×C414yes—
28C22×Dic127C2314yes—
29C22×Dic127C2312yes—
30C22×Dic127C2312yes—
31Dic127⋊C4—44no—
32C22⋊Dic127—24no—
33C22⋊Dic127—42no—
34C22⋊Dic127—42no—
35C22⋊Dic127C2×C418yes—
36C22⋊Dic127C2×C418yes—
37C4×Dic127—44no—
38C24×C127—24no—
39C24×C127—42no—
40C24×C127—42no—
41C24×C127C2×C414yes—
42C24×C127C2311yescharacteristic
43C24×C127C2311yescharacteristic
44C24×C127C2311yescharacteristic
45C22×C4×C127C2314yes—
46C22⋊C4×C127—24no—
47C22⋊C4×C127—42no—
48C22⋊C4×C127—42no—
49C2×Dic127—24no—
50C2×Dic127—24no—
51C2×Dic127—24no—
52C2×Dic127—24no—
53C2×Dic127—24no—
54C2×Dic127—44no—
55C2×Dic127—44no—
56C2×Dic127C22×C414yes—
57C2×Dic127C22×C414yes—
58C23×C127—22no—
59C23×C127—22no—
60C23×C127—28no—
61C23×C127—22no—
62C23×C127—22no—
63C23×C127—22no—
64C23×C127—22no—
65C23×C127—22no—
66C23×C127—22no—
67C23×C127—44no—
68C23×C127—44no—
69C23×C127—416no—
70C23×C127C22×C412yes—
71C23×C127C22×C412yes—
72C23×C127C22×C412yes—
73C23×C127#4114yes—
74C23×C127C2411yescharacteristic
75C2×C4×C127—24no—
76C2×C4×C127—24no—
77C2×C4×C127—24no—
78C22×C127—22no—
79C22×C127—22no—
80C22×C127—28no—
81C22×C127—22no—
82C22×C127—22no—
83C22×C127—22no—
84C22×C127—22no—
85C22×C127—22no—
86C22×C127—22no—
87C22×C127—416no—
88C22×C127—44no—
89C22×C127—44no—
90C22×C127#11714yes—
91C22×C127C23×C411yescharacteristic
92C22×C127C2×#4112yes—
93C22×C127H(5,𝔽2)12yes—
94C22×C127H(5,𝔽2)12yes—
95C4×C127—44no—
96Dic127—24no—
97Dic127—24no—
98Dic127—44no—
99C2×C127—24no—
100C2×C127—42no—
101C2×C127—42no—
102C2×C127C2×#11711yescharacteristic
103C2×C127#51811yescharacteristic
104C2×C127#51811yescharacteristic
105C2×C127#53814yes—
106#2312—1271nomaximal, Sylow
107C127#231211yescharacteristic, complemented, Sylow
108C22×C22⋊C4—1271no—
109#388—1274no—
110#388—1274no—
111#380—1272no—
112#379—1271no—
113#379D12711yescharacteristic
114C4×C22⋊C4—1272no—
115C25C2×D12711yescharacteristic
116C23×C4—1271no—
117C23×C4—1271no—
118C2×C4⋊C4—1274no—
119C2×C22⋊C4—1278no—
120C2×C22⋊C4—1272no—
121C2×C22⋊C4—1272no—
122C2×C22⋊C4—1272no—
123C2×C22⋊C4—1272no—
124C2×C22⋊C4—1272no—
125C2×C22⋊C4C2×D12712yes—
126C2×C22⋊C4C2×D12712yes—
127C2×C22⋊C4C2×D12712yes—
128C2×C42—1274no—
129#95—1274no—
130#95—1274no—
131C24—24no—
132C24—42no—
133C24—42no—
134C24C4×D12714yes—
135C24C22×D12711yescharacteristic
136C24C22×D12711yescharacteristic
137C24C22×D12711yescharacteristic
138C22×C4—1274no—
139C22×C4—1274no—
140C22×C4—1272no—
141C22×C4—1272no—
142C22×C4—1274no—
143C22×C4—5082no—
144C22×C4—5082no—
145C22×C4C22×D12714yes—
146C4⋊C4—5084no—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—1278no—
151C22⋊C4—1278no—
152C22⋊C4—5082no—
153C22⋊C4—5082no—
154C22⋊C4—2544no—
155C42—5084no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—28no—
165C23—44no—
166C23—44no—
167C23—416no—
168C23C2×C4×D12712yes—
169C23C2×C4×D12712yes—
170C23C2×C4×D12712yes—
171C23#1452324714yes—
172C23C23×D12711yescharacteristic
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—1274no—
177C2×C4—1274no—
178C2×C4—5084no—
179C2×C4—5084no—
180C2×C4—2544no—
181C2×C4—2544no—
182C2×C4—2544no—
183C2×C4—2544no—
184C2×C4—2544no—
185C22—22no—
186C22—22no—
187C22—28no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—416no—
195C22—44no—
196C22—44no—
197C22#1686116514yes—
198C22C22×C4×D12711yescharacteristic
199C22C2×#1452324712yes—
200C22#1686125412yes—
201C22#1686125412yes—
202C4—44no—
203C4—5084no—
204C4—2544no—
205C4—2544no—
206C2—24no—
207C2—42no—
208C2—42no—
209C2C2×#1686116511yescharacteristic
210C2#2255034911yescharacteristic
211C2#2255034911yescharacteristic
212C2#2255046114yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#379×C127C2
socle, cosocleC23×C127C24
prime radicalC25×C127C22
hypercenter#379D127
hypocenterC127#2312
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #22549770C2semidirect
3. #22549764C2semidirect
4. #22549734C2semidirect
6. C22⋊C4×Dic127C2semidirect
10. C2×Dic127⋊C4C22semidirect
11. C2×C22⋊Dic127C4semidirect
17. C2×C4×Dic127C22semidirect
18. C127⋊#95C22semidirect
19. C127⋊#95C22semidirect
107. C127#2312semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Dic1272
#225497702
#225497642
#225497342
#225497322
C22⋊C4×Dic1272
#379×C1272
#2312127