Finite Group Databasebeta

← #30641613

Subgroups — #30641613

Order 16256.

504 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 197 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×#16861168C211yesmaximal, characteristic
2#22549774C212yesmaximal, complemented
3#22549764C214yesmaximal
4#22549740C212yesmaximal, complemented
5#22549736C211yesmaximal, characteristic
6C4×C4⋊Dic127C212yesmaximal, complemented
7#22549721C212yesmaximal, complemented
8C4×C22⋊C4×C127C211yesmaximal, characteristic
9#16861168C418yescomplemented
10C2×C4⋊Dic127C2212yes—
11C2×C4⋊Dic127C2211yescharacteristic
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4C2214yes—
14C2×C22⋊Dic127C2212yes—
15C2×C22⋊Dic127C2212yes—
16C2×C4×Dic127C2214yes—
17C2×C4×Dic127C2212yes—
18C127⋊#95C2214yes—
19C127⋊#95C2214yes—
20C127⋊#95C2212yes—
21C23×C4×C127C2211yescharacteristic
22C2×C22⋊C4×C127C2212yes—
23C2×C42×C127C2212yes—
24#95×C127C2211yescharacteristic
25#95×C127C2211yescharacteristic
26C22×Dic127C2314yes—
27C22×Dic127C2312yes—
28C22×Dic127C2312yes—
29C4⋊Dic127—24no—
30C4⋊Dic127C2×C414yes—
31C4⋊Dic127C2×C414yes—
32Dic127⋊C4—44no—
33C22⋊Dic127—42no—
34C22⋊Dic127C2×C418yes—
35C4×Dic127—28no—
36C4×Dic127C2×C418yes—
37C24×C127C2311yescharacteristic
38C22×C4×C127—24no—
39C22×C4×C127C2×C414yes—
40C22×C4×C127C2312yes—
41C22×C4×C127C2311yescharacteristic
42C22×C4×C127C2312yes—
43C22×C4×C127C2311yescharacteristic
44C22⋊C4×C127—24no—
45C42×C127—24no—
46C2×Dic127—24no—
47C2×Dic127—22no—
48C2×Dic127—22no—
49C2×Dic127—24no—
50C2×Dic127—24no—
51C2×Dic127—24no—
52C2×Dic127C22×C414yes—
53C2×Dic127C22×C414yes—
54C23×C127—22no—
55C23×C127—22no—
56C23×C127—21no—
57C23×C127—21no—
58C23×C127C22×C412yes—
59C23×C127C2411yescharacteristic
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127—22no—
63C2×C4×C127—28no—
64C2×C4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127—22no—
67C2×C4×C127C22×C412yes—
68C2×C4×C127C22×C412yes—
69C2×C4×C127#4114yes—
70C2×C4×C127#4114yes—
71C22×C127—22no—
72C22×C127—22no—
73C22×C127—22no—
74C22×C127—42no—
75C22×C127—42no—
76C22×C127C23×C411yescharacteristic
77C22×C127C2×#4112yes—
78C22×C127C2×#4112yes—
79C22×C127H(5,𝔽2)11yescharacteristic
80C22×C1272−1+411yescharacteristic
81C4×C127—22no—
82C4×C127—24no—
83C4×C127—42no—
84C4×C127#11714yes—
85Dic127—24no—
86Dic127—24no—
87Dic127—44no—
88C2×C127—42no—
89C2×C127C2×#11711yescharacteristic
90C2×C127C4×#4111yescharacteristic
91C2×C127#52011yescharacteristic
92C2×C127#55212yes—
93C2×C127#55512yes—
94#2346—1271nomaximal, Sylow
95C127#234611yescharacteristic, complemented, Sylow
96C2×#117—1271no—
97#389—1272no—
98#388—1274no—
99#382—1272no—
100#380—1271no—
101C4×C4⋊C4—1272no—
102C4×C22⋊C4D12711yescharacteristic
103#376—1272no—
104C23×C4C2×D12711yescharacteristic
105#117—1278no—
106C2×C4⋊C4—1271no—
107C2×C4⋊C4—1271no—
108C2×C4⋊C4—1274no—
109C2×C4⋊C4—1272no—
110C2×C22⋊C4—1272no—
111C2×C22⋊C4—1272no—
112C2×C22⋊C4C2×D12712yes—
113C2×C42—1272no—
114C2×C42—1274no—
115C2×C42C2×D12712yes—
116#95—1272no—
117#95—1274no—
118#95—1274no—
119#95C2×D12711yescharacteristic
120#95C2×D12711yescharacteristic
121C24C22×D12711yescharacteristic
122C22×C4—24no—
123C22×C4—1272no—
124C22×C4—1272no—
125C22×C4—1274no—
126C22×C4C4×D12714yes—
127C22×C4C22×D12712yes—
128C22×C4C22×D12711yescharacteristic
129C22×C4C22×D12712yes—
130C22×C4C22×D12711yescharacteristic
131C4⋊C4—1274no—
132C4⋊C4—1274no—
133C4⋊C4—5084no—
134C4⋊C4—2544no—
135C22⋊C4—24no—
136C22⋊C4—1278no—
137C22⋊C4—5082no—
138C42—24no—
139C42—1278no—
140C42—2548no—
141C23—22no—
142C23—22no—
143C23—21no—
144C23—21no—
145C23C2×C4×D12712yes—
146C23C23×D12711yescharacteristic
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—28no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—1274no—
155C2×C4—1274no—
156C2×C4—2544no—
157C2×C4—2544no—
158C2×C4—2544no—
159C2×C4—2544no—
160C2×C4—2542no—
161C2×C4—2542no—
162C2×C4C2×C4×D12712yes—
163C2×C4C2×C4×D12712yes—
164C2×C4#1452324714yes—
165C2×C4#1452324814yes—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—42no—
170C22—42no—
171C22C22×C4×D12711yescharacteristic
172C22C2×#1452324712yes—
173C22C2×#1452324812yes—
174C22#1686125511yescharacteristic
175C22#1686125611yescharacteristic
176C4—24no—
177C4—22no—
178C4—42no—
179C4—5084no—
180C4—2544no—
181C4—2544no—
182C4#1686116514yes—
183C2—42no—
184C2C2×#1686116511yescharacteristic
185C2C4×#1452324811yescharacteristic
186C2#2255035611yescharacteristic
187C2#2255056612yes—
188C2#2255059112yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC4×C22⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC24×C127C23
hypercenterC4×C22⋊C4D127
hypocenterC127#2346
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22549774C2semidirect
4. #22549740C2semidirect
6. C4×C4⋊Dic127C2semidirect
7. #22549721C2semidirect
9. #16861168C4semidirect
95. C127#2346semidirect

Maximal subgroups

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

subgroupindex
C2×#168611682
#225497742
#225497642
#225497402
#225497362
C4×C4⋊Dic1272
#225497212
C4×C22⋊C4×C1272
#2346127