Finite Group Databasebeta

← #30639217

Subgroups — #30639217

Order 16256.

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

normal subgroups only

This group has 346 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
1C2×#16861235C211yesmaximal, characteristic
2#22549894C211yesmaximal, characteristic, complemented
3#22549881C211yesmaximal, characteristic, complemented
4#22549849C211yesmaximal, characteristic, complemented
5#22549646C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#518×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235—22no—
11C2×#14523236C412yescomplemented
12C2×#14523233C412yescomplemented
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861137—22no—
17#16861130—22no—
18#16861123—22no—
19C22×D4×C127C2211yescharacteristic
20C4×D4×C127—22no—
21C4×D4×C127—22no—
22#117×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24C2×C22⋊C4×C127—22no—
25#14523248—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29C2×Dic254C2×C411yescharacteristic, complemented
30C2×D508C2×C411yescharacteristic, complemented
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33#14523236—24no—
34#14523233—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8C2×C412yescomplemented
41C4⋊Dic127—21no—
42C22⋊Dic127—21no—
43C4×Dic127—21no—
44C24×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127C2×C412yes—
49C2×D4×C127D412yes—
50C2×D4×C127D412yes—
51C22×C4×C127—24no—
52C22×C4×C127C2311yescharacteristic
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C22⋊C4×C127—24no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127D412yes—
58C42×C127—21no—
59C42×C127D412yes—
60C22×D127—21no—
61Dic254—21no—
62Dic254—21no—
63D508—21no—
64D508—21no—
65C22⋊D127—41no—
66C22⋊D127—41no—
67C2×Dic127—21no—
68C2×Dic127—41no—
69C4×D127—41no—
70C4×D127—41no—
71C127⋊C8—22no—
72C127⋊C8—41no—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127—22no—
76C23×C127—82no—
77C23×C127C2×D411yescharacteristic
78D4×C127—22no—
79D4×C127—22no—
80D4×C127—22no—
81D4×C127C22⋊C414yes—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—84no—
85C2×C4×C127C22×C411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C2×C4×C127#4111yescharacteristic
91C22×C127—24no—
92C22×C127—41no—
93C22×C127—44no—
94C22×C127—82no—
95C22×C127—82no—
96C22×C127C4×D411yescharacteristic
97C22×C127#12011yescharacteristic
98C22×C127#12111yescharacteristic
99C2×D127—41no—
100C2×D127—81no—
101C4×C127—42no—
102C4×C127—42no—
103C4×C127C2×C22⋊C411yescharacteristic
104C4×C127C4×D411yescharacteristic
105C4×C127#12111yescharacteristic
106C4×C127#12311yescharacteristic
107Dic127—41no—
108Dic127—81no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—44no—
113C2×C127#38611yescharacteristic
114D127—81no—
115#1904—1271nomaximal, Sylow
116C127#190411yescharacteristic, complemented, Sylow
117C2×AGL(1,ℤ/8)—1271no—
118#518D12711yescharacteristic, complemented
119#421—1271no—
120#418—1271no—
121#410—1271no—
122#344—1271no—
123H(3,ℤ/4)—1271no—
124C2×#41—1271no—
125C22×D4C2×D12711yescharacteristic
126AGL(1,ℤ/8)—2542no—
127AGL(1,ℤ/8)—2542no—
128C2×SD16—1272no—
129C2×D8—1272no—
130C2×M16—1271no—
131C2×M16—1271no—
132C4×D4—22no—
133C4×D4—22no—
134#117—1271no—
135#117C2×D12711yescharacteristic
136#117C2×D12711yescharacteristic
137C2×C22⋊C4—22no—
138#104—2542no—
139#102—2542no—
140#99—2542no—
141C24—21no—
142#41—2541no—
143#41—2541no—
144#41—2541no—
145#41—2541no—
146C2×Q8—1271no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—1271no—
151C2×D4—2541no—
152C2×D4C4×D12712yes—
153C2×D4C22⋊D12712yes—
154C2×D4D50812yes—
155C22×C4—24no—
156C22×C4—2541no—
157C22×C4C22×D12711yescharacteristic
158SD16—2544no—
159D8—2544no—
160M16—2541no—
161M16—2541no—
162M16—2541no—
163M16—2541no—
164C2×C8—1272no—
165C2×C8—2541no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—2541no—
169C22⋊C4—24no—
170C22⋊C4—21no—
171C22⋊C4—2541no—
172C22⋊C4C22⋊D12712yes—
173C42—21no—
174C42—2541no—
175C42C22⋊D12712yes—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—82no—
180C23—2541no—
181C23C2×C22⋊D12711yescharacteristic
182Q8—2541no—
183Q8—2541no—
184D4—22no—
185D4—22no—
186D4—22no—
187D4—5081no—
188D4—5081no—
189D4—2541no—
190D4—2541no—
191D4(C2×D127)⋊C414yes—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—84no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4—2541no—
199C2×C4C2×C4×D12711yescharacteristic
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4C2×C22⋊D12711yescharacteristic
202C2×C4C2×D50811yescharacteristic
203C2×C4#1452324811yescharacteristic
204C2×C4#1452324811yescharacteristic
205C8—5081no—
206C8—2542no—
207C22—24no—
208C22—41no—
209C22—44no—
210C22—82no—
211C22—82no—
212C22—10161no—
213C22—5081no—
214C22C4×C22⋊D12711yescharacteristic
215C22#1686117811yescharacteristic
216C22#1686118311yescharacteristic
217C4—42no—
218C4—42no—
219C4—10161no—
220C4—5081no—
221C4C2×(C2×D127)⋊C411yescharacteristic
222C4C4×C22⋊D12711yescharacteristic
223C4#1686118311yescharacteristic
224C4#1686119211yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—44no—
229C2—10161no—
230C2#2254975511yescharacteristic
subgroupG/H
centerC2#22549755
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#518×C127C2
socleC2×C127#386
prime radicalC2×#16861235C2
cosocleC22×C4×C127C23
hypercenter#518D127
hypocenterC127#1904
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #22549894C2semidirect
3. #22549881C2semidirect
4. #22549849C2semidirect
5. #22549646C2semidirect
6. #22549632C2semidirect
7. #518×C127C2semidirect
11. C2×#14523236C4semidirect
12. C2×#14523233C4semidirect
29. C2×Dic254C2×C4semidirect
30. C2×D508C2×C4semidirect
40. C2×C127⋊C8C2×C4semidirect
116. C127#1904semidirect
118. #518D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225498942
#225498812
#225498492
#225496462
#225496322
#518×C1272
#1904127