Finite Group Databasebeta

← #30641347

Subgroups — #30641347

Order 16256.

600 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 229 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×Dic127⋊C4C211yesmaximal, characteristic
3#22549773C214yesmaximal, complemented
4#22549771C214yesmaximal, complemented
5#22549736C211yesmaximal, characteristic
6#22549734C212yesmaximal
7#380×C127C211yesmaximal, characteristic
8C23×Dic127C2211yescharacteristic
9C23×Dic127C2211yescharacteristic
10#16861165—28no—
11C2×C4⋊Dic127C2212yes—
12C2×Dic127⋊C4C418yescomplemented
13C2×Dic127⋊C4C2214yes—
14C2×Dic127⋊C4C2214yes—
15C2×Dic127⋊C4C2212yes—
16C2×Dic127⋊C4C2212yes—
17C2×C22⋊Dic127C2212yes—
18C2×C4×Dic127C2214yes—
19C127⋊#95C2212yes—
20C127⋊#95C2214yes—
21C23×C4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2212yes—
23C2×C22⋊C4×C127C2212yes—
24#95×C127C2212yes—
25C22×Dic127—42no—
26C22×Dic127C2×C414yes—
27C22×Dic127C2×C414yes—
28C22×Dic127D414yes—
29C22×Dic127Q814yes—
30C22×Dic127C2314yes—
31C22×Dic127C2312yes—
32C22×Dic127C2312yes—
33C4⋊Dic127—42no—
34Dic127⋊C4—28no—
35Dic127⋊C4—44no—
36Dic127⋊C4C2×C418yes—
37Dic127⋊C4C2×C418yes—
38C22⋊Dic127—24no—
39C4×Dic127—28no—
40C24×C127C2311yescharacteristic
41C22×C4×C127—42no—
42C22×C4×C127C2×C414yes—
43C22×C4×C127C2314yes—
44C22×C4×C127C2311yescharacteristic
45C22×C4×C127C2311yescharacteristic
46C4⋊C4×C127—42no—
47C22⋊C4×C127—24no—
48C2×Dic127—24no—
49C2×Dic127—24no—
50C2×Dic127—24no—
51C2×Dic127—24no—
52C2×Dic127—44no—
53C2×Dic127C4⋊C4116yescomplemented
54C2×Dic127C22×C414yes—
55C2×Dic127C22×C414yes—
56C2×Dic127C2×D414yes—
57C2×Dic127C2×Q814yes—
58C23×C127—22no—
59C23×C127—22no—
60C23×C127C22×C412yes—
61C23×C127C2×D412yes—
62C23×C127C2×Q812yes—
63C23×C127C2411yescharacteristic
64C2×C4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127—24no—
67C2×C4×C127—24no—
68C2×C4×C127—24no—
69C2×C4×C127—44no—
70C2×C4×C127C22×C412yes—
71C2×C4×C127C22×C412yes—
72C22×C127—22no—
73C22×C127—22no—
74C22×C127—24no—
75C22×C127—24no—
76C22×C127C2×C4⋊C414yes—
77C22×C127C23×C411yescharacteristic
78C22×C127C22×D411yescharacteristic
79C22×C127C22×Q811yescharacteristic
80C22×C127H(5,𝔽2)12yes—
81C22×C1272−1+412yes—
82C4×C127—22no—
83C4×C127—22no—
84C4×C127—44no—
85Dic127—24no—
86Dic127—44no—
87Dic127C2×C4⋊C418yes—
88C2×C127—24no—
89C2×C127C22×C4⋊C411yescharacteristic
90C2×C127#51811yescharacteristic
91C2×C127#51911yescharacteristic
92C2×C127#53712yes—
93C2×C127#54412yes—
94#2316—1271nomaximal, Sylow
95C127#231611yescharacteristic, complemented, Sylow
96C2×#117—1272no—
97C22×C4⋊C4—1271no—
98#389—1274no—
99#389—1274no—
100#380—1271no—
101#380—1272no—
102#380D12711yescharacteristic
103C23×C4—1271no—
104C23×C4—1271no—
105C23×C4C2×D12711yescharacteristic
106#117—2548no—
107C2×C4⋊C4—1278no—
108C2×C4⋊C4—1272no—
109C2×C4⋊C4—1272no—
110C2×C4⋊C4—1274no—
111C2×C4⋊C4—1272no—
112C2×C4⋊C4—1274no—
113C2×C4⋊C4C2×D12712yes—
114C2×C22⋊C4—1272no—
115C2×C22⋊C4C2×D12712yes—
116C2×C42—1274no—
117#95—1274no—
118#95—1272no—
119#95C2×D12712yes—
120C24C22×D12711yescharacteristic
121C22×C4—42no—
122C22×C4—1274no—
123C22×C4—1274no—
124C22×C4—1274no—
125C22×C4—1272no—
126C22×C4—1274no—
127C22×C4—1272no—
128C22×C4—1274no—
129C22×C4—5082no—
130C22×C4C4×D12714yes—
131C22×C4C22×D12714yes—
132C22×C4C22×D12711yescharacteristic
133C22×C4C22×D12711yescharacteristic
134C4⋊C4—42no—
135C4⋊C4—1278no—
136C4⋊C4—1278no—
137C4⋊C4—5084no—
138C4⋊C4—5082no—
139C4⋊C4—2548no—
140C22⋊C4—24no—
141C22⋊C4—2544no—
142C42—2548no—
143C23—22no—
144C23—22no—
145C23C2×C4×D12712yes—
146C23D4×D12712yes—
147C23Q8×D12712yes—
148C23C23×D12711yescharacteristic
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—44no—
155C2×C4—1274no—
156C2×C4—1274no—
157C2×C4—1274no—
158C2×C4—12716no—
159C2×C4—1274no—
160C2×C4—5084no—
161C2×C4—2544no—
162C2×C4—2544no—
163C2×C4—2544no—
164C2×C4—2544no—
165C2×C4C2×C4×D12712yes—
166C2×C4C2×C4×D12712yes—
167C22—22no—
168C22—22no—
169C22—24no—
170C22—24no—
171C22C4⋊C4×D12714yes—
172C22C22×C4×D12711yescharacteristic
173C22C2×D4×D12711yescharacteristic
174C22C2×Q8×D12711yescharacteristic
175C22#1686125412yes—
176C22#1686125612yes—
177C4—22no—
178C4—22no—
179C4—44no—
180C4—1278no—
181C4—5084no—
182C4—2544no—
183C2—24no—
184C2C2×C4⋊C4×D12711yescharacteristic
185C2#2255034911yescharacteristic
186C2#2255035311yescharacteristic
187C2#2255045312yes—
188C2#2255050512yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#380×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenter#380D127
hypocenterC127#2316
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #22549773C2semidirect
4. #22549771C2semidirect
12. C2×Dic127⋊C4C4semidirect
53. C2×Dic127C4⋊C4semidirect
95. C127#2316semidirect

Maximal subgroups

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

subgroupindex
C2×#168611652
C22×Dic127⋊C42
#225497732
#225497712
#225497362
#225497342
#380×C1272
#2316127