Finite Group Databasebeta

← #30639224

Subgroups — #30639224

Order 16256.

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

normal subgroups only

This group has 310 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×#16861240C211yesmaximal, characteristic
2#22549894C211yesmaximal, characteristic
3#22549885C211yesmaximal, characteristic, complemented
4#22549849C211yesmaximal, characteristic
5#22549646C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#519×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240—22no—
11C2×#14523238C412yescomplemented
12C2×#14523235C412yescomplemented
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861137—22no—
17#16861133—22no—
18#16861123—22no—
19C22×Q8×C127C2211yescharacteristic
20C4×Q8×C127—22no—
21C4×Q8×C127—22no—
22#117×C127—22no—
23#117×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25#14523248—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29C2×Dic254C2×C411yescharacteristic
30C2×D508C2×C411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33#14523238—24no—
34#14523235—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39C2×C127⋊C8—21no—
40C2×C127⋊C8C2×C412yes—
41C4⋊Dic127—21no—
42C22⋊Dic127—21no—
43C4×Dic127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127C2×C412yes—
48C2×Q8×C127D412yes—
49C2×Q8×C127D412yes—
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—24no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C22⋊C4×C127—21no—
56C22⋊C4×C127D412yes—
57C42×C127—24no—
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×C127C2×D411yescharacteristic
74Q8×C127—22no—
75Q8×C127—22no—
76Q8×C127—22no—
77Q8×C127C22⋊C414yes—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127C22×C411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C22×C127—41no—
90C22×C127C4×D411yescharacteristic
91C22×C127#12011yescharacteristic
92C22×C127#12111yescharacteristic
93C2×D127—41no—
94C2×D127—81no—
95C4×C127—24no—
96C4×C127—42no—
97C4×C127—42no—
98C4×C127C2×C22⋊C411yescharacteristic
99C4×C127C4×D411yescharacteristic
100C4×C127#12111yescharacteristic
101C4×C127#12311yescharacteristic
102Dic127—41no—
103Dic127—81no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127#38611yescharacteristic
108D127—81no—
109#1905—1271nomaximal, Sylow
110C127#190511yescharacteristic, complemented, Sylow
111C2×#137—1271no—
112#519D12711yescharacteristic, complemented
113#421—1271no—
114#419—1271no—
115#410—1271no—
116#344—1271no—
117H(3,ℤ/4)—1271no—
118C2×#41—1271no—
119C22×Q8C2×D12711yescharacteristic
120#137—2542no—
121#137—2542no—
122C2×Q16—1272no—
123C2×SD16—1272no—
124C2×M16—1271no—
125C2×M16—1271no—
126C4×Q8—22no—
127C4×Q8—22no—
128#117—22no—
129#117—1271no—
130#117C2×D12711yescharacteristic
131#117C2×D12711yescharacteristic
132#104—2542no—
133#103—2542no—
134#99—2542no—
135#41—2541no—
136#41—2541no—
137#41—2541no—
138#41—2541no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—1271no—
143C2×Q8C4×D12712yes—
144C2×Q8C22⋊D12712yes—
145C2×Q8D50812yes—
146C2×D4—1271no—
147C2×D4—2541no—
148C22×C4—21no—
149C22×C4—2541no—
150C22×C4C22×D12711yescharacteristic
151Q16—2544no—
152SD16—2544no—
153M16—2541no—
154M16—2541no—
155M16—2541no—
156M16—2541no—
157C2×C8—1272no—
158C2×C8—2541no—
159C4⋊C4—24no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—2541no—
163C22⋊C4—21no—
164C22⋊C4—2541no—
165C22⋊C4C22⋊D12712yes—
166C42—24no—
167C42—21no—
168C42—2541no—
169C42C22⋊D12712yes—
170C23—2541no—
171C23C2×C22⋊D12711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—22no—
175Q8—2541no—
176Q8—2541no—
177Q8(C2×D127)⋊C414yes—
178D4—5081no—
179D4—5081no—
180D4—2541no—
181D4—2541no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—5081no—
188C2×C4—5081no—
189C2×C4—5081no—
190C2×C4—2541no—
191C2×C4C2×C4×D12711yescharacteristic
192C2×C4C2×C22⋊D12711yescharacteristic
193C2×C4C2×C22⋊D12711yescharacteristic
194C2×C4C2×D50811yescharacteristic
195C2×C4#1452324811yescharacteristic
196C2×C4#1452324811yescharacteristic
197C8—5081no—
198C8—2542no—
199C22—41no—
200C22—10161no—
201C22—5081no—
202C22C4×C22⋊D12711yescharacteristic
203C22#1686117811yescharacteristic
204C22#1686118311yescharacteristic
205C4—24no—
206C4—42no—
207C4—42no—
208C4—10161no—
209C4—5081no—
210C4C2×(C2×D127)⋊C411yescharacteristic
211C4C4×C22⋊D12711yescharacteristic
212C4#1686118311yescharacteristic
213C4#1686119211yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—10161no—
218C2#2254975511yescharacteristic
subgroupG/H
centerC2#22549755
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#519×C127C2
socleC2×C127#386
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#519D127
hypocenterC127#1905
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #519

Complements

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

NH ≅ G/Ntype
3. #22549885C2semidirect
5. #22549646C2semidirect
6. #22549632C2semidirect
7. #519×C127C2semidirect
11. C2×#14523238C4semidirect
12. C2×#14523235C4semidirect
110. C127#1905semidirect
112. #519D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225498942
#225498852
#225498492
#225496462
#225496322
#519×C1272
#1905127