Finite Group Databasebeta

← #30641597

Subgroups — #30641597

Order 16256.

604 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 221 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×#16861165C212yesmaximal
2C22×C4⋊Dic127C211yesmaximal, characteristic
3#22549774C212yesmaximal, complemented
4#22549761C214yesmaximal, complemented
5#22549737C212yesmaximal
6#22549736C211yesmaximal, characteristic
7C4×C4⋊Dic127C212yesmaximal, complemented
8C4×C22⋊C4×C127C211yesmaximal, characteristic
9C23×Dic127C2212yes—
10#16861165—28no—
11C2×C4⋊Dic127C418yescomplemented
12C2×C4⋊Dic127C2212yes—
13C2×C4⋊Dic127C2212yes—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4C2214yes—
17C2×C22⋊Dic127C2212yes—
18C2×C4×Dic127C2214yes—
19C127⋊#95C2212yes—
20C127⋊#95C2214yes—
21C127⋊#95C2214yes—
22C23×C4×C127C2211yescharacteristic
23C2×C22⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C42×C127C2212yes—
26#95×C127C2212yes—
27C22×Dic127—24no—
28C22×Dic127—24no—
29C22×Dic127C2×C418yes—
30C22×Dic127C2314yes—
31C22×Dic127C2312yes—
32C22×Dic127C2312yes—
33C4⋊Dic127—24no—
34C4⋊Dic127C2×C418yes—
35C4⋊Dic127C2×C414yes—
36C4⋊Dic127C2×C414yes—
37Dic127⋊C4—28no—
38C22⋊Dic127—24no—
39C4×Dic127—28no—
40C24×C127C2311yescharacteristic
41C22×C4×C127—24no—
42C22×C4×C127C2×C414yes—
43C22×C4×C127C2312yes—
44C22×C4×C127C2311yescharacteristic
45C22×C4×C127C2312yes—
46C22×C4×C127C2311yescharacteristic
47C22⋊C4×C127—22no—
48C22⋊C4×C127D414yes—
49C42×C127—24no—
50C2×Dic127—216no—
51C2×Dic127—22no—
52C2×Dic127—22no—
53C2×Dic127—22no—
54C2×Dic127—22no—
55C2×Dic127—24no—
56C2×Dic127—24no—
57C2×Dic127—24no—
58C2×Dic127C22×C414yes—
59C2×Dic127C22×C414yes—
60C23×C127—22no—
61C23×C127—22no—
62C23×C127C22×C412yes—
63C23×C127C2×D412yes—
64C23×C127#4112yes—
65C23×C127C2411yescharacteristic
66C2×C4×C127—22no—
67C2×C4×C127—22no—
68C2×C4×C127—22no—
69C2×C4×C127—28no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127C22×C412yes—
74C2×C4×C127C22×C412yes—
75C2×C4×C127C2×D414yes—
76C2×C4×C127#4114yes—
77C22×C127—22no—
78C22×C127—22no—
79C22×C127—24no—
80C22×C127—24no—
81C22×C127C4×D414yes—
82C22×C127C23×C411yescharacteristic
83C22×C127C22×D411yescharacteristic
84C22×C127C2×#4112yes—
85C22×C127C2×#4111yescharacteristic
86C22×C1272−1+412yes—
87C4×C127—22no—
88C4×C127—24no—
89C4×C127—42no—
90C4×C127#11714yescomplemented
91Dic127—24no—
92Dic127—24no—
93Dic127—44no—
94C2×C127—24no—
95C2×C127C2×#11711yescharacteristic
96C2×C127C2×C4×D411yescharacteristic
97C2×C127#51911yescharacteristic
98C2×C127#54712yes—
99C2×C127#55212yes—
100#2344—1271nomaximal, Sylow
101C127#234411yescharacteristic, complemented, Sylow
102C2×#117—1272no—
103C22×C4⋊C4—1271no—
104#389—1272no—
105#387—1274no—
106#381—1272no—
107#380—1271no—
108C4×C4⋊C4—1272no—
109C4×C22⋊C4D12711yescharacteristic
110C23×C4—1272no—
111C23×C4C2×D12711yescharacteristic
112#117—2548no—
113C2×C4⋊C4—1278no—
114C2×C4⋊C4—1271no—
115C2×C4⋊C4—1272no—
116C2×C4⋊C4—1271no—
117C2×C4⋊C4—1274no—
118C2×C4⋊C4—1272no—
119C2×C22⋊C4—1272no—
120C2×C22⋊C4C2×D12711yescharacteristic
121C2×C22⋊C4C2×D12711yescharacteristic
122C2×C42—1274no—
123C2×C42C2×D12712yes—
124#95—1274no—
125#95—1272no—
126#95—1274no—
127#95C2×D12712yes—
128C24C22×D12711yescharacteristic
129C22×C4—24no—
130C22×C4—1278no—
131C22×C4—1272no—
132C22×C4—1272no—
133C22×C4—1274no—
134C22×C4—2544no—
135C22×C4—2544no—
136C22×C4C4×D12714yes—
137C22×C4C22×D12712yes—
138C22×C4C22×D12711yescharacteristic
139C22×C4C22×D12712yes—
140C22×C4C22×D12711yescharacteristic
141C4⋊C4—1278no—
142C4⋊C4—1274no—
143C4⋊C4—1274no—
144C4⋊C4—2548no—
145C4⋊C4—2544no—
146C22⋊C4—22no—
147C22⋊C4—2544no—
148C22⋊C4D50814yes—
149C42—24no—
150C42—2548no—
151C23—22no—
152C23—22no—
153C23C2×C4×D12712yes—
154C23C2×D50812yes—
155C23#1452324812yes—
156C23C23×D12711yescharacteristic
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—28no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—1274no—
165C2×C4—1274no—
166C2×C4—2544no—
167C2×C4—2544no—
168C2×C4—2544no—
169C2×C4—2542no—
170C2×C4—2542no—
171C2×C4—2542no—
172C2×C4—2542no—
173C2×C4—25416no—
174C2×C4C2×C4×D12712yes—
175C2×C4C2×C4×D12712yes—
176C2×C4C2×D50814yes—
177C2×C4#1452324714yes—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22C4×D50814yes—
183C22C22×C4×D12711yescharacteristic
184C22C22×D50811yescharacteristic
185C22C2×#1452324712yes—
186C22C2×#1452324811yescharacteristic
187C22#1686125612yes—
188C4—24no—
189C4—22no—
190C4—42no—
191C4—5084no—
192C4—2544no—
193C4—2544no—
194C4#1686116514yescomplemented
195C2—24no—
196C2C2×#1686116511yescharacteristic
197C2C2×C4×D50811yescharacteristic
198C2#2255035311yescharacteristic
199C2#2255053012yes—
200C2#2255056612yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC4×C22⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenterC4×C22⋊C4D127
hypocenterC127#2344
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #22549774C2semidirect
4. #22549761C2semidirect
7. C4×C4⋊Dic127C2semidirect
11. C2×C4⋊Dic127C4semidirect
90. C4×C127#117semidirect
101. C127#2344semidirect
194. C4#16861165semidirect

Maximal subgroups

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

subgroupindex
C2×#168611652
C22×C4⋊Dic1272
#225497742
#225497612
#225497372
#225497362
C4×C4⋊Dic1272
C4×C22⋊C4×C1272
#2344127