Finite Group Databasebeta

← #30639176

Subgroups — #30639176

Order 16256.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#22550387C211yesmaximal, characteristic
2C2×#16861168C211yesmaximal, characteristic, complemented
3C2×#16861135C212yesmaximal, complemented
4#22549637C212yesmaximal, complemented
5#407×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2211yescharacteristic
7#16861192—21no—
8#16861188C412yescomplemented
9#16861183C412yescomplemented
10#16861178—21no—
11C4×C22⋊D127—22no—
12#16861168—24no—
13#16861168C2211yescharacteristic
14C2×C4⋊Dic127—21no—
15C2×C22⋊Dic127—21no—
16C2×C4×Dic127C2212yes—
17#16861135—24no—
18C23×C4×C127C2211yescharacteristic
19C2×M16×C127C2212yes—
20C22⋊C8×C127—22no—
21#14523248—22no—
22#14523248—22no—
23C2×Dic254C2×C411yescharacteristic
24C2×D508C2×C411yescharacteristic
25C2×C22⋊D127—22no—
26C2×C22⋊D127—21no—
27C22×Dic127—22no—
28C2×C4×D127—21no—
29C4⋊Dic127—24no—
30C4⋊Dic127C2×C412yes—
31Dic127⋊C4—22no—
32C22⋊Dic127—24no—
33C22⋊Dic127—22no—
34C22⋊Dic127—21no—
35(C2×D127)⋊C4—22no—
36C4×Dic127—22no—
37C4×Dic127—21no—
38C4×Dic127D414yescomplemented
39C24×C127D411yescharacteristic
40C22×C4×C127—22no—
41C22×C4×C127—22no—
42C22×C4×C127C2×C412yes—
43C22×C4×C127D412yes—
44C22×C4×C127D411yescharacteristic
45C22×C4×C127C2311yescharacteristic
46M16×C127—22no—
47M16×C127—22no—
48C2×C8×C127—22no—
49C22×D127—21no—
50Dic254—22no—
51D508—22no—
52C22⋊D127—44no—
53C22⋊D127—41no—
54C22⋊D127—41no—
55C2×Dic127—22no—
56C2×Dic127—21no—
57C2×Dic127—24no—
58C2×Dic127—44no—
59C4×D127—42no—
60C23×C127—21no—
61C23×C127—21no—
62C23×C127—42no—
63C23×C127C22⋊C412yes—
64C23×C127C2×D411yescharacteristic
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—24no—
68C2×C4×C127—24no—
69C2×C4×C127C22⋊C412yes—
70C2×C4×C127C22×C411yescharacteristic
71C2×C4×C127C2×D412yes—
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127#4112yes—
74C8×C127—42no—
75C22×C127—22no—
76C22×C127—22no—
77C22×C127—22no—
78C22×C127—22no—
79C22×C127—42no—
80C22×C127—42no—
81C22×C127C2×C22⋊C411yescharacteristic
82C22×C127#12011yescharacteristic
83C22×C127#12311yescharacteristic
84C2×D127—41no—
85C2×D127—42no—
86C4×C127—22no—
87C4×C127C4×D412yes—
88C4×C127#12112yes—
89Dic127—44no—
90Dic127—41no—
91Dic127—42no—
92C2×C127—21no—
93C2×C127—21no—
94C2×C127—42no—
95C2×C127#38611yescharacteristic
96C2×C127#42112yes—
97D127—81no—
98#1897—1271nomaximal, Sylow
99C127#189711yescharacteristic, complemented, Sylow
100#525—1271no—
101C2×#117—1271no—
102C2×#104—1272no—
103#407D12711yescharacteristic, complemented
104#339—1272no—
105C2×#41—1271no—
106C23×C4C2×D12711yescharacteristic
107C2×M16C2×D12712yes—
108#123—2541no—
109C22⋊Q8—1272no—
110#121—1272no—
111#120—2541no—
112C4×D4—2542no—
113#117—1271no—
114#117—2544no—
115C2×C4⋊C4—2541no—
116C2×C22⋊C4—2541no—
117C2×C42—1272no—
118#104—2544no—
119C22⋊C8—22no—
120C24D50811yescharacteristic
121#41—2542no—
122#41—2542no—
123C2×Q8—1271no—
124C2×D4—1271no—
125C2×D4—2542no—
126C2×D4—2541no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—2542no—
130C22×C4—2541no—
131C22×C4C4×D12712yes—
132C22×C4C22⋊D12712yes—
133C22×C4D50811yescharacteristic
134C22×C4C22×D12711yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—1272no—
139C4⋊C4—2544no—
140C4⋊C4—2542no—
141C22⋊C4—2544no—
142C22⋊C4—2542no—
143C22⋊C4—2541no—
144C22⋊C4—2542no—
145C42—1274no—
146C42—2542no—
147C42—2541no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—2541no—
152C23(C2×D127)⋊C412yes—
153C23C2×D50811yescharacteristic
154Q8—2542no—
155D4—5084no—
156D4—5081no—
157D4—5081no—
158D4—2542no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—5084no—
164C2×C4—5082no—
165C2×C4—2542no—
166C2×C4—2541no—
167C2×C4—2544no—
168C2×C4(C2×D127)⋊C412yes—
169C2×C4C2×C4×D12711yescharacteristic
170C2×C4D4×D12712yes—
171C2×C4C2×C22⋊D12711yescharacteristic
172C2×C4#1452324712yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—5082no—
181C22—5081no—
182C22C2×(C2×D127)⋊C411yescharacteristic
183C22#1686117911yescharacteristic
184C22#1686119311yescharacteristic
185C4—22no—
186C4—5084no—
187C4—5082no—
188C4—5081no—
189C4(C22⋊D127)⋊C412yes—
190C4#1686118112yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—10161no—
195C2#2254975311yescharacteristic
196C2#2254989212yes—
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550387C2
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1897
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16861168C2semidirect
3. C2×#16861135C2semidirect
4. #22549637C2semidirect
5. #407×C127C2semidirect
8. #16861188C4semidirect
9. #16861183C4semidirect
38. C4×Dic127D4semidirect
99. C127#1897semidirect
103. #407D127semidirect

Maximal subgroups

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

subgroupindex
#225503872
C2×#168611682
C2×#168611352
#225496372
#407×C1272
#1897127