Finite Group Databasebeta

← #30639173

Subgroups — #30639173

Order 16256.

442 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16861137C212yesmaximal, complemented
3#22549840C211yesmaximal, characteristic, complemented
4#22549636C212yesmaximal, complemented
5C22×C42×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2211yescharacteristic
7C2×#14523232C2212yes—
8#16861192—21no—
9#16861188C412yescomplemented
10#16861183C412yescomplemented
11#16861178—21no—
12C4×C22⋊D127—22no—
13#16861168C2211yescharacteristic
14#16861137—24no—
15C22⋊(C127⋊C8)—22no—
16C23×C4×C127—21no—
17C23×C4×C127C2211yescharacteristic
18C2×C42×C127—21no—
19C2×C42×C127—24no—
20C2×C42×C127C2212yes—
21#14523248—22no—
22#14523248—22no—
23C2×Dic254C2×C411yescharacteristic, complemented
24C2×D508C2×C411yescharacteristic, complemented
25C2×C22⋊D127—22no—
26C2×C22⋊D127—21no—
27C2×C4×D127—21no—
28#14523232—22no—
29#14523232—22no—
30C2×C127⋊C8—22no—
31C4⋊Dic127C2×C412yescomplemented
32Dic127⋊C4—22no—
33C22⋊Dic127—22no—
34C22⋊Dic127—21no—
35(C2×D127)⋊C4—22no—
36C4×Dic127—21no—
37C24×C127D411yescharacteristic
38C22×C4×C127—24no—
39C22×C4×C127—28no—
40C22×C4×C127—22no—
41C22×C4×C127—22no—
42C22×C4×C127—22no—
43C22×C4×C127C2×C412yes—
44C22×C4×C127D411yescharacteristic
45C22×C4×C127D412yes—
46C22×C4×C127C2311yescharacteristic
47C42×C127—24no—
48C42×C127—22no—
49C42×C127D414yescomplemented
50C22×D127—21no—
51Dic254—22no—
52D508—22no—
53C22⋊D127—44no—
54C22⋊D127—41no—
55C22⋊D127—41no—
56C2×Dic127—22no—
57C2×Dic127—21no—
58C4×D127—42no—
59C127⋊C8—42no—
60C23×C127—24no—
61C23×C127—21no—
62C23×C127—21no—
63C23×C127C22⋊C412yes—
64C23×C127C2×D411yescharacteristic
65C2×C4×C127—216no—
66C2×C4×C127—28no—
67C2×C4×C127—21no—
68C2×C4×C127—24no—
69C2×C4×C127—21no—
70C2×C4×C127—24no—
71C2×C4×C127—24no—
72C2×C4×C127C22⋊C412yes—
73C2×C4×C127C22×C411yescharacteristic
74C2×C4×C127C2×D412yes—
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127#4112yes—
77C22×C127—24no—
78C22×C127—24no—
79C22×C127—22no—
80C22×C127—22no—
81C22×C127—22no—
82C22×C127#10414yescomplemented
83C22×C127C2×C22⋊C411yescharacteristic
84C22×C127#12011yescharacteristic
85C22×C127#12311yescharacteristic
86C2×D127—41no—
87C2×D127—42no—
88C4×C127—28no—
89C4×C127—22no—
90C4×C127C4×D412yes—
91C4×C127#12112yes—
92Dic127—41no—
93Dic127—42no—
94C2×C127—21no—
95C2×C127—24no—
96C2×C127—21no—
97C2×C127#38611yescharacteristic
98C2×C127C2×#10412yes—
99D127—81no—
100#1896—1271nomaximal, Sylow
101C127#189611yescharacteristic, complemented, Sylow
102#525—1271no—
103C22×C42D12711yescharacteristic, complemented
104C2×#104—1272no—
105#407—1271no—
106#339—1272no—
107C2×#41—1271no—
108C23×C4—21no—
109C23×C4C2×D12711yescharacteristic
110C2×M16—1272no—
111#123—2541no—
112C22⋊Q8—1272no—
113#121—1272no—
114#120—2541no—
115C4×D4—2542no—
116#117—1271no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42C2×D12712yes—
120#104—2544no—
121C22⋊C8—2542no—
122C24C22⋊D12711yescharacteristic
123#41—2542no—
124#41—2542no—
125C2×Q8—1271no—
126C2×D4—1271no—
127C2×D4—2542no—
128C2×D4—2541no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—2541no—
135C22×C4C4×D12712yes—
136C22×C4C22⋊D12711yescharacteristic
137C22×C4D50812yes—
138C22×C4C22×D12711yescharacteristic
139M16—2542no—
140M16—2542no—
141C2×C8—2542no—
142C4⋊C4—1272no—
143C4⋊C4—2542no—
144C22⋊C4—2542no—
145C22⋊C4—2541no—
146C22⋊C4—2542no—
147C42—24no—
148C42—22no—
149C42—2541no—
150C42C22⋊D12714yescomplemented
151C23—21no—
152C23—24no—
153C23—21no—
154C23—2541no—
155C23(C2×D127)⋊C412yes—
156C23C2×C22⋊D12711yescharacteristic
157Q8—2542no—
158D4—5081no—
159D4—5084no—
160D4—5081no—
161D4—2542no—
162C2×C4—216no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—5082no—
170C2×C4—2542no—
171C2×C4—2541no—
172C2×C4(C2×D127)⋊C412yes—
173C2×C4C2×C4×D12711yescharacteristic
174C2×C4C2×C22⋊D12712yes—
175C2×C4C2×D50811yescharacteristic
176C2×C4#1452324812yes—
177C8—5082no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—5082no—
184C22—5081no—
185C22#1686113714yescomplemented
186C22C2×(C2×D127)⋊C411yescharacteristic
187C22#1686117811yescharacteristic
188C22#1686119211yescharacteristic
189C4—28no—
190C4—22no—
191C4—5082no—
192C4—5081no—
193C4C4×C22⋊D12712yes—
194C4#1686118312yes—
195C2—21no—
196C2—21no—
197C2—24no—
198C2—10161no—
199C2#2254975511yescharacteristic
200C2C2×#1686113712yes—
subgroupG/H
centerC2×C4C2×D508
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC22×C42×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550387C2
cosocleC22×C4×C127C23
hypercenterC22×C42D127
hypocenterC127#1896
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#16861137C2semidirect
3. #22549840C2semidirect
4. #22549636C2semidirect
5. C22×C42×C127C2semidirect
9. #16861188C4semidirect
10. #16861183C4semidirect
23. C2×Dic254C2×C4semidirect
24. C2×D508C2×C4semidirect
31. C4⋊Dic127C2×C4semidirect
49. C42×C127D4semidirect
82. C22×C127#104semidirect
101. C127#1896semidirect
103. C22×C42D127semidirect
150. C42C22⋊D127semidirect
185. C22#16861137semidirect

Maximal subgroups

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

subgroupindex
#225503872
C2×#168611372
#225498402
#225496362
C22×C42×C1272
#1896127