Finite Group Databasebeta

← #30638679

Subgroups — #30638679

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#22549858C212yesmaximal, complemented
3#22549844C212yesmaximal, complemented
4#22549843C211yesmaximal, characteristic, complemented
5#413×C127C211yesmaximal, characteristic, complemented
6#16861257—21no—
7#16861256—22no—
8#16861255—22no—
9#16861254—21no—
10#16861254C414yes—
11C2×#14523248—22no—
12C2×#14523248C2212yes—
13#41×D127—24no—
14C2×#14523232C2211yescharacteristic, complemented
15C22×C127⋊C8C2211yescharacteristic, complemented
16#16861127—22no—
17#16861125—22no—
18C22⋊(C127⋊C8)—22no—
19(C2×D127)⋊C8—24no—
20C2×#41×C127C2211yescharacteristic
21C2×M16×C127C2212yescomplemented
22#101×C127—21no—
23#100×C127—21no—
24#14523249—24no—
25#14523248—24no—
26#14523248—24no—
27#14523248—24no—
28#14523248—22no—
29#14523248—22no—
30#14523248—22no—
31#14523248—22no—
32#14523247—24no—
33#14523247—28no—
34C2×Dic254—22no—
35C2×Dic254D412yes—
36Q8×D127—24no—
37C2×D508—22no—
38C2×D508D412yes—
39C2×C22⋊D127—24no—
40C2×C22⋊D127C2×C414yes—
41D4×D127—24no—
42D4×D127—28no—
43C2×C4×D127—24no—
44C2×C4×D127D414yes—
45#14523232—22no—
46C2×C127⋊C8—22no—
47C2×C127⋊C8—21no—
48C2×C127⋊C8—21no—
49#41×C127—22no—
50#41×C127—22no—
51C2×Q8×C127D411yescharacteristic
52C2×D4×C127C2×C412yes—
53C2×D4×C127D411yescharacteristic
54C22×C4×C127D412yes—
55C22×C4×C127C2311yescharacteristic
56M16×C127—22no—
57M16×C127—22no—
58C2×C8×C127—22no—
59C22×D127—24no—
60C22×D127C22⋊C414yes—
61Dic254—24no—
62Dic254—22no—
63Dic254—22no—
64D508—24no—
65D508—22no—
66D508—22no—
67C22⋊D127—28no—
68C22⋊D127—24no—
69C22⋊D127—24no—
70C2×Dic127—24no—
71C2×Dic127C22⋊C414yes—
72C4×D127—28no—
73C4×D127—24no—
74C4×D127—24no—
75C127⋊C8—41no—
76C127⋊C8—41no—
77C23×C127C22⋊C412yes—
78C23×C127C22×C411yescharacteristic
79Q8×C127—22no—
80D4×C127—24no—
81D4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127C22⋊C412yes—
85C2×C4×C127C2×D412yes—
86C2×C4×C127C2×D412yes—
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C8×C127—42no—
90C22×C127—22no—
91C22×C127—41no—
92C22×C127—42no—
93C22×C127C2×C22⋊C412yes—
94C22×C127C2×C22⋊C411yescharacteristic
95C2×D127—24no—
96C2×D127—44no—
97C2×D127—84no—
98C4×C127—22no—
99C4×C127#12012yes—
100C4×C127#12011yescharacteristic
101C4×C127#12011yescharacteristic
102Dic127—24no—
103C2×C127—22no—
104C2×C127—21no—
105C2×C127—42no—
106C2×C127#37911yescharacteristic
107D127—44no—
108#1791—1271nomaximal, Sylow
109C127#179111yescharacteristic, complemented, Sylow
110#585—1271no—
111#413—1272no—
112#413D12711yescharacteristic, complemented
113#408—1271no—
114#408—1272no—
1152−1+4—2542no—
1162−1+4—2541no—
117H(5,𝔽2)—1274no—
118H(5,𝔽2)—2542no—
119H(5,𝔽2)—2541no—
120C2×#41—1272no—
121C2×#41—2544no—
122C2×#41—2542no—
123C2×#41C2×D12711yescharacteristic
124C2×M16—1271no—
125C2×M16C2×D12712yescomplemented
126C22×C8—1271no—
127#101—21no—
128#101—2542no—
129#100—21no—
130#100—2542no—
131C22⋊C8—2544no—
132C22⋊C8—2542no—
133#41—22no—
134#41—22no—
135#41—2544no—
136#41—2544no—
137#41—2548no—
138#41—2544no—
139#41—2544no—
140#41—2544no—
141#41—2542no—
142#41—2542no—
143#41—2542no—
144#41—2542no—
145C2×Q8—1272no—
146C2×Q8—2542no—
147C2×Q8—2544no—
148C2×Q8C22⋊D12711yescharacteristic
149C2×D4—1274no—
150C2×D4—1272no—
151C2×D4—2542no—
152C2×D4—2548no—
153C2×D4—2544no—
154C2×D4—2544no—
155C2×D4C4×D12712yes—
156C2×D4C22⋊D12711yescharacteristic
157C22×C4—1274no—
158C22×C4—2544no—
159C22×C4D50812yes—
160C22×C4C22×D12711yescharacteristic
161M16—22no—
162M16—22no—
163M16—2542no—
164C2×C8—22no—
165C2×C8—2542no—
166C2×C8—2541no—
167C2×C8—2541no—
168C23—1274no—
169C23—2544no—
170C23(C2×D127)⋊C412yes—
171C23C2×C4×D12711yescharacteristic
172Q8—22no—
173Q8—2544no—
174Q8—2542no—
175Q8—2542no—
176D4—24no—
177D4—22no—
178D4—2544no—
179D4—2548no—
180D4—2544no—
181D4—2544no—
182D4—2542no—
183D4—2542no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—1274no—
187C2×C4—2548no—
188C2×C4—2544no—
189C2×C4—2544no—
190C2×C4—2544no—
191C2×C4(C2×D127)⋊C412yes—
192C2×C4D4×D12712yes—
193C2×C4D4×D12712yes—
194C2×C4C2×C22⋊D12711yescharacteristic
195C2×C4C2×D50811yescharacteristic
196C8—42no—
197C8—5081no—
198C8—5081no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—10164no—
203C22—5084no—
204C22—2544no—
205C22C22⋊C4×D12712yes—
206C22C2×(C2×D127)⋊C411yescharacteristic
207C4—22no—
208C4—2544no—
209C4#1686117712yes—
210C4#1686117911yescharacteristic
211C4#1686117911yescharacteristic
212C2—21no—
213C2—22no—
214C2—42no—
215C2—5084no—
216C2#2254973111yescharacteristic
subgroupG/H
centerC4#16861179
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#413×C127C2
socleC2×C127#379
prime radical#22550712C2
cosocleC22×C4×C127C23
hypercenter#413D127
hypocenterC127#1791
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #22549858C2semidirect
3. #22549844C2semidirect
4. #22549843C2semidirect
5. #413×C127C2semidirect
14. C2×#14523232C22semidirect
15. C22×C127⋊C8C22semidirect
21. C2×M16×C127C22semidirect
109. C127#1791semidirect
112. #413D127semidirect
125. C2×M16C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225498582
#225498442
#225498432
#413×C1272
#1791127