Finite Group Databasebeta

← #30638696

Subgroups — #30638696

Order 16256.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#22550712C211yesmaximal, characteristic
2#22549894C212yesmaximal, complemented
3#22549870C212yesmaximal, complemented
4#22549843C211yesmaximal, characteristic, complemented
5#410×C127C211yesmaximal, characteristic, complemented
6#16861257C412yescomplemented
7#16861256—22no—
8#16861255—22no—
9#16861254—22no—
10#16861254C412yescomplemented
11C2×#14523248—22no—
12C2×#14523248C2212yes—
13#41×D127—24no—
14C2×#14523232C2211yescharacteristic, complemented
15C22×C127⋊C8C2211yescharacteristic, complemented
16#16861137—24no—
17#16861132—22no—
18#16861129—22no—
19C22⋊(C127⋊C8)—22no—
20C2×#41×C127C2211yescharacteristic
21#117×C127C2212yescomplemented
22#99×C127—22no—
23#14523249—24no—
24#14523248—24no—
25#14523248—24no—
26#14523248—24no—
27#14523248—22no—
28#14523248—22no—
29#14523248C2×C414yes—
30#14523248D414yes—
31#14523247—28no—
32#14523247—24no—
33C2×Dic254—22no—
34C2×Dic254D412yes—
35Q8×D127—24no—
36C2×D508—22no—
37C2×D508D412yes—
38C2×C22⋊D127—24no—
39C2×C22⋊D127—22no—
40D4×D127—28no—
41D4×D127—24no—
42C2×C4×D127—24no—
43C2×C4×D127—22no—
44#14523232—22no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8—22no—
47C2×C127⋊C8—21no—
48#41×C127—22no—
49#41×C127—22no—
50C2×Q8×C127C2×C411yescharacteristic
51C2×D4×C127C2×C411yescharacteristic
52C2×D4×C127D412yes—
53C22×C4×C127D412yes—
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—22no—
56C22⋊C4×C127—22no—
57C42×C127—22no—
58C22×D127—24no—
59C22×D127—22no—
60Dic254—24no—
61Dic254—22no—
62Dic254C22⋊C414yes—
63D508—24no—
64D508—22no—
65D508C22⋊C414yes—
66C22⋊D127—28no—
67C22⋊D127—24no—
68C22⋊D127—24no—
69C2×Dic127—24no—
70C2×Dic127—22no—
71C4×D127—28no—
72C4×D127—24no—
73C4×D127—24no—
74C127⋊C8—41no—
75C127⋊C8—41no—
76C23×C127C22⋊C412yes—
77C23×C127C2×D411yescharacteristic
78Q8×C127—22no—
79D4×C127—24no—
80D4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—42no—
84C2×C4×C127C22⋊C412yes—
85C2×C4×C127C22×C411yescharacteristic
86C2×C4×C127C2×D412yes—
87C2×C4×C127C2×D412yes—
88C2×C4×C127C2×D411yescharacteristic
89C22×C127—22no—
90C22×C127—41no—
91C22×C127—42no—
92C22×C127C2×C22⋊C411yescharacteristic
93C22×C127#12012yes—
94C2×D127—24no—
95C2×D127—82no—
96C2×D127—84no—
97C4×C127—22no—
98C4×C127—82no—
99C4×C127C2×C22⋊C412yes—
100C4×C127#12011yescharacteristic
101C4×C127#12011yescharacteristic
102Dic127—24no—
103C2×C127—22no—
104C2×C127—21no—
105C2×C127—42no—
106C2×C127#37911yescharacteristic
107D127—44no—
108#1794—1271nomaximal, Sylow
109C127#179411yescharacteristic, complemented, Sylow
110#585—1271no—
111#421—1272no—
112#416—1272no—
113#410D12711yescharacteristic, complemented
114#408—1271no—
1152−1+4—1272no—
1162−1+4—2542no—
117H(5,𝔽2)—1272no—
118H(5,𝔽2)—2542no—
119H(5,𝔽2)—2542no—
120C2×#41—1272no—
121C2×#41—2544no—
122C2×#41—2542no—
123C2×#41C2×D12711yescharacteristic
124C2×M16—1271no—
125C22×C8—1271no—
126#117C2×D12712yescomplemented
127#104—2544no—
128#103—2542no—
129#102—2542no—
130#99—22no—
131C22⋊C8—2542no—
132#41—22no—
133#41—22no—
134#41—1274no—
135#41—1274no—
136#41—2544no—
137#41—2544no—
138#41—2544no—
139#41—2548no—
140#41—2544no—
141#41—2544no—
142#41—2542no—
143#41—2542no—
144C2×Q8—1272no—
145C2×Q8—2542no—
146C2×Q8—2544no—
147C2×Q8C4×D12711yescharacteristic
148C2×D4—1272no—
149C2×D4—2542no—
150C2×D4—2544no—
151C2×D4—2548no—
152C2×D4—2544no—
153C2×D4—2542no—
154C2×D4C4×D12711yescharacteristic
155C2×D4C22⋊D12712yes—
156C22×C4—2544no—
157C22×C4—2542no—
158C22×C4D50812yes—
159C22×C4C22×D12711yescharacteristic
160M16—2542no—
161C2×C8—2542no—
162C2×C8—2541no—
163C2×C8—2541no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—2544no—
168C23—2542no—
169C23(C2×D127)⋊C412yes—
170C23C2×C22⋊D12711yescharacteristic
171Q8—22no—
172Q8—1274no—
173Q8—2544no—
174Q8—2542no—
175D4—24no—
176D4—22no—
177D4—1274no—
178D4—2544no—
179D4—2548no—
180D4—2544no—
181D4—2544no—
182D4—2542no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—2548no—
187C2×C4—2544no—
188C2×C4—2542no—
189C2×C4—2544no—
190C2×C4—2544no—
191C2×C4(C2×D127)⋊C412yes—
192C2×C4C2×C4×D12711yescharacteristic
193C2×C4D4×D12712yes—
194C2×C4D4×D12712yes—
195C2×C4C2×D50811yescharacteristic
196C8—5081no—
197C8—5081no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—10162no—
202C22—10164no—
203C22—2544no—
204C22C2×(C2×D127)⋊C411yescharacteristic
205C22#1686117712yes—
206C4—22no—
207C4—82no—
208C4—2544no—
209C4C22⋊C4×D12712yes—
210C4#1686117911yescharacteristic
211C4#1686117911yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—5084no—
216C2#2254973111yescharacteristic
subgroupG/H
centerC4#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#379
prime radical#22550712C2
cosocleC22×C4×C127C23
hypercenter#410D127
hypocenterC127#1794
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22549894C2semidirect
3. #22549870C2semidirect
4. #22549843C2semidirect
5. #410×C127C2semidirect
6. #16861257C4semidirect
10. #16861254C4semidirect
14. C2×#14523232C22semidirect
15. C22×C127⋊C8C22semidirect
21. #117×C127C22semidirect
109. C127#1794semidirect
113. #410D127semidirect
126. #117C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225498942
#225498702
#225498432
#410×C1272
#1794127