Finite Group Databasebeta

← #30638694

Subgroups — #30638694

Order 16256.

548 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 121 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#22549893C212yesmaximal, complemented
2#22549872C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic, complemented
4#22549843C211yesmaximal, characteristic, complemented
5#585×C127C211yesmaximal, characteristic
6C2×#14523232C2211yescharacteristic, complemented
7C22×C127⋊C8C2211yescharacteristic, complemented
8#16861168C2212yescomplemented
9#16861136—24no—
10#16861133—22no—
11#16861130—22no—
12#16861124—22no—
13C22⋊(C127⋊C8)—22no—
142−1+4×C127—22no—
152−1+4×C127C412yescomplemented
16H(5,𝔽2)×C127—22no—
17H(5,𝔽2)×C127—22no—
18H(5,𝔽2)×C127C412yescomplemented
19C2×#41×C127—24no—
20C2×#41×C127—22no—
21C2×#41×C127C2212yes—
22C2×#41×C127C2211yescharacteristic
23#14523232—22no—
24C2×C127⋊C8—21no—
25C2×C127⋊C8—22no—
26C2×C127⋊C8—21no—
27C4⋊Dic127—22no—
28C22⋊Dic127—22no—
29C4×Dic127—22no—
30#41×C127—24no—
31#41×C127—28no—
32#41×C127—24no—
33#41×C127—24no—
34#41×C127—24no—
35#41×C127—24no—
36#41×C127—22no—
37#41×C127—22no—
38#41×C127—22no—
39#41×C127—22no—
40#41×C127C2×C414yes—
41#41×C127D414yes—
42C2×Q8×C127—22no—
43C2×Q8×C127—24no—
44C2×Q8×C127C2×C411yescharacteristic
45C2×Q8×C127D412yes—
46C2×D4×C127—28no—
47C2×D4×C127—22no—
48C2×D4×C127—24no—
49C2×D4×C127—22no—
50C2×D4×C127—24no—
51C2×D4×C127C2×C411yescharacteristic
52C2×D4×C127D412yes—
53C2×D4×C127D412yes—
54C22×C4×C127—24no—
55C22×C4×C127—22no—
56C22×C4×C127D412yes—
57C22×C4×C127C2311yescharacteristic
58C2×Dic127—42no—
59C127⋊C8—41no—
60C127⋊C8—41no—
61C23×C127—24no—
62C23×C127—22no—
63C23×C127C22⋊C412yes—
64C23×C127C2×D411yescharacteristic
65Q8×C127—24no—
66Q8×C127—22no—
67Q8×C127—22no—
68Q8×C127C22⋊C414yes—
69D4×C127—28no—
70D4×C127—24no—
71D4×C127—24no—
72D4×C127—24no—
73D4×C127—24no—
74D4×C127—22no—
75D4×C127—22no—
76D4×C127C22⋊C414yes—
77C2×C4×C127—24no—
78C2×C4×C127—28no—
79C2×C4×C127—22no—
80C2×C4×C127—24no—
81C2×C4×C127—24no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
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—24no—
91C22×C127—41no—
92C22×C127—42no—
93C22×C127—82no—
94C22×C127—84no—
95C22×C127C2×C22⋊C411yescharacteristic
96C22×C127#12012yes—
97C4×C127—24no—
98C4×C127—22no—
99C4×C127C2×C22⋊C412yes—
100C4×C127#12011yescharacteristic
101C4×C127#12011yescharacteristic
102Dic127—82no—
103C2×C127—22no—
104C2×C127—21no—
105C2×C127—42no—
106C2×C127—44no—
107C2×C127#37911yescharacteristic
108#1794—1271nomaximal, Sylow
109C127#179411yescharacteristic, complemented, Sylow
110#585D12711yescharacteristic
111#421—1272no—
112#416—1272no—
113#410—1271no—
114#408—1271no—
1152−1+4—22no—
1162−1+4Dic12712yescomplemented
117H(5,𝔽2)—22no—
118H(5,𝔽2)—22no—
119H(5,𝔽2)Dic12712yescomplemented
120C2×#41—24no—
121C2×#41—22no—
122C2×#41C2×D12712yes—
123C2×#41C2×D12711yescharacteristic
124C2×M16—1271no—
125C22×C8—1271no—
126#117—1272no—
127#104—2544no—
128#103—2542no—
129#102—2542no—
130#99—2542no—
131C22⋊C8—2542no—
132#41—24no—
133#41—28no—
134#41—24no—
135#41—24no—
136#41—24no—
137#41—24no—
138#41—22no—
139#41—22no—
140#41—22no—
141#41—22no—
142#41C2×Dic12714yes—
143#41C22⋊D12714yes—
144C2×Q8—22no—
145C2×Q8—24no—
146C2×Q8C2×Dic12711yescharacteristic
147C2×Q8C22⋊D12712yes—
148C2×D4—28no—
149C2×D4—22no—
150C2×D4—24no—
151C2×D4—22no—
152C2×D4—24no—
153C2×D4C2×Dic12711yescharacteristic
154C2×D4C22⋊D12712yes—
155C2×D4C22⋊D12712yes—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4C22⋊D12712yes—
159C22×C4C22×D12711yescharacteristic
160M16—2542no—
161C2×C8—2541no—
162C2×C8—2541no—
163C2×C8—2542no—
164C4⋊C4—2542no—
165C22⋊C4—2542no—
166C42—2542no—
167C23—24no—
168C23—22no—
169C23C22⋊Dic12712yes—
170C23C2×C22⋊D12711yescharacteristic
171Q8—24no—
172Q8—22no—
173Q8—22no—
174Q8C22⋊Dic12714yes—
175D4—28no—
176D4—24no—
177D4—24no—
178D4—24no—
179D4—24no—
180D4—22no—
181D4—22no—
182D4C22⋊Dic12714yes—
183C2×C4—24no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—5082no—
191C2×C4C22⋊Dic12712yes—
192C2×C4C22×Dic12711yescharacteristic
193C2×C4C2×C22⋊D12712yes—
194C2×C4C2×C22⋊D12712yes—
195C2×C4C2×C22⋊D12711yescharacteristic
196C8—5081no—
197C8—5081no—
198C22—22no—
199C22—24no—
200C22—41no—
201C22—42no—
202C22—82no—
203C22—84no—
204C22C2×C22⋊Dic12711yescharacteristic
205C22#1686117812yes—
206C4—24no—
207C4—22no—
208C4—10162no—
209C4C2×C22⋊Dic12712yes—
210C4#1686117811yescharacteristic
211C4#1686117811yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—44no—
216C2#2254973211yescharacteristic
subgroupG/H
centerC4#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#585×C127C2
socleC2×C127#379
cosocleC22×C4×C127C23
hypercenter#585D127
hypocenterC127#1794
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #22549893C2semidirect
2. #22549872C2semidirect
3. #22549850C2semidirect
4. #22549843C2semidirect
6. C2×#14523232C22semidirect
7. C22×C127⋊C8C22semidirect
8. #16861168C22semidirect
15. 2−1+4×C127C4semidirect
18. H(5,𝔽2)×C127C4semidirect
109. C127#1794semidirect
116. 2−1+4Dic127semidirect
119. H(5,𝔽2)Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498932
#225498722
#225498502
#225498432
#585×C1272
#1794127