Finite Group Databasebeta

← #30640001

Subgroups — #30640001

Order 16256.

274 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 63 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#22549924C211yesmaximal, characteristic, complemented
3#22549894C211yesmaximal, characteristic, complemented
4#22549858C211yesmaximal, characteristic, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861235—22no—
10#16861235—22no—
11C2×#14523236—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C2×#14523232C2211yescharacteristic
16#16861144—22no—
17#16861137—22no—
18#16861127—21no—
19#16861126—22no—
20#16861125—21no—
21C22×D4×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#102×C127—22no—
25#14523248—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29C2×Dic254D411yescharacteristic
30C2×D508D411yescharacteristic
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—
39#14523232D412yes—
40#14523232D412yes—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C24×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127D412yes—
51C22×C4×C127C2311yescharacteristic
52M16×C127—21no—
53M16×C127—21no—
54C2×C8×C127—21no—
55C4⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C42×C127—21no—
58C22×D127—21no—
59Dic254—21no—
60Dic254—21no—
61D508—21no—
62D508—21no—
63C22⋊D127—41no—
64C22⋊D127—41no—
65C2×Dic127—21no—
66C4×D127—41no—
67C4×D127—41no—
68C127⋊C8—22no—
69C127⋊C8—22no—
70C127⋊C8—41no—
71C23×C127—22no—
72C23×C127—41no—
73C23×C127—41no—
74C23×C127—82no—
75C23×C127#4111yescharacteristic
76D4×C127—22no—
77D4×C127—22no—
78D4×C127—41no—
79D4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C8×C127—41no—
88C22×C127—41no—
89C22×C127—42no—
90C22×C127—82no—
91C22×C127—82no—
92C22×C127—82no—
93C22×C127#12111yescharacteristic
94C22×C127#12311yescharacteristic
95C22×C127#12411yescharacteristic
96C2×D127—41no—
97C2×D127—81no—
98C4×C127—81no—
99C4×C127#12011yescharacteristic
100C4×C127#12111yescharacteristic
101C4×C127#12111yescharacteristic
102C4×C127#12311yescharacteristic
103Dic127—41no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—82no—
108C2×C127#39411yescharacteristic
109D127—81no—
110#2048—1271nomaximal, Sylow
111C127#204811yescharacteristic, complemented, Sylow
112C2×AGL(1,ℤ/8)—1271no—
113#430—1271no—
114#421—1271no—
115#418D12711yescharacteristic, complemented
116#413—1271no—
117C2×#100—1271no—
118#344—1271no—
119C2×#41—1271no—
120C22×D4C2×D12711yescharacteristic
121AGL(1,ℤ/8)—2542no—
122AGL(1,ℤ/8)—2542no—
123C2×SD16—2541no—
124C2×D8—2541no—
125C2×M16—1271no—
126C2×M16—1271no—
127C2×M16—1271no—
128C2×M16C2×D12711yescharacteristic
129#117C2×D12711yescharacteristic
130#108—2542no—
131#104—2542no—
132#102—22no—
133#101—2541no—
134#100—2541no—
135#100—2542no—
136C24—21no—
137#41—2541no—
138#41—2541no—
139#41—2541no—
140#41—2541no—
141C2×Q8—1271no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—1271no—
148C2×D4—2541no—
149C2×D4C22⋊D12712yes—
150C22×C4—2541no—
151C22×C4C22×D12711yescharacteristic
152SD16—2544no—
153D8—2544no—
154M16—21no—
155M16—21no—
156M16—1272no—
157M16—1272no—
158M16—2541no—
159M16—2541no—
160M16—2541no—
161M16—2541no—
162C2×C8—21no—
163C2×C8—2541no—
164C2×C8—2541no—
165C2×C8—2541no—
166C4⋊C4—21no—
167C22⋊C4—21no—
168C42—21no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—82no—
173C23—2541no—
174C23#1452324711yescharacteristic
175Q8—2541no—
176Q8—2541no—
177D4—22no—
178D4—22no—
179D4—41no—
180D4—41no—
181D4—5081no—
182D4—5081no—
183D4—2541no—
184D4—2541no—
185C2×C4—41no—
186C2×C4—5081no—
187C2×C4—5081no—
188C2×C4—2541no—
189C2×C4D4×D12711yescharacteristic
190C2×C4D4×D12711yescharacteristic
191C2×C4D4×D12711yescharacteristic
192C2×C4C2×C22⋊D12711yescharacteristic
193C2×C4#1452324811yescharacteristic
194C2×C4#1452324811yescharacteristic
195C8—41no—
196C8—5081no—
197C8—2542no—
198C8—2542no—
199C22—41no—
200C22—42no—
201C22—82no—
202C22—82no—
203C22—82no—
204C22—10161no—
205C22—5081no—
206C22#1686118111yescharacteristic
207C22#1686119111yescharacteristic
208C22#1686119511yescharacteristic
209C4—81no—
210C4—5081no—
211C4#1686117711yescharacteristic
212C4#1686118211yescharacteristic
213C4#1686118211yescharacteristic
214C4#1686119211yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—82no—
219C2—10161no—
220C2#2254979111yescharacteristic
subgroupG/H
centerC2#22549791
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC2×C127#394
prime radicalC2×#16861235C2
hypercenter#418D127
hypocenterC127#2048
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #22549924C2semidirect
3. #22549894C2semidirect
4. #22549858C2semidirect
5. C2×#16861126C2semidirect
6. #22549646C2semidirect
7. #418×C127C2semidirect
111. C127#2048semidirect
115. #418D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225499242
#225498942
#225498582
C2×#168611262
#225496462
#418×C1272
#2048127