Finite Group Databasebeta

← #30640380

Subgroups — #30640380

Order 16256.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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#22549860C211yesmaximal, characteristic, complemented
2#22549850C211yesmaximal, characteristic, complemented
3#22549663C214yesmaximal, complemented
4#535×C127C211yesmaximal, characteristic
5C2×#14523232—21no—
6#16861168—21no—
7#16861128—21no—
8#16861126C2212yes—
9#16861124—21no—
10#16861124C2212yes—
11C2×#41×C127C2211yescharacteristic
12C22×D4×C127C411yescharacteristic, complemented
13#127×C127—21no—
14#124×C127—21no—
15#121×C127—21no—
16#121×C127—21no—
17#120×C127C412yescomplemented
18#120×C127C2212yes—
19#117×C127C411yescharacteristic, complemented
20#14523232—22no—
21#14523232—41no—
22C2×C127⋊C8—41no—
23C4⋊Dic127—41no—
24C22⋊Dic127—22no—
25C4×Dic127—41no—
26C24×C127C2×C412yes—
27#41×C127—41no—
28#41×C127—41no—
29C2×Q8×C127D411yescharacteristic
30C2×D4×C127—22no—
31C2×D4×C127—22no—
32C2×D4×C127—21no—
33C2×D4×C127—21no—
34C2×D4×C127—41no—
35C2×D4×C127—41no—
36C2×D4×C127C2×C411yescharacteristic
37C2×D4×C127D411yescharacteristic
38C2×D4×C127C2311yescharacteristic
39C22×C4×C127C2×C411yescharacteristic
40C22×C4×C127D411yescharacteristic
41C22×C4×C127D411yescharacteristic
42C4⋊C4×C127—21no—
43C22⋊C4×C127—22no—
44C22⋊C4×C127—22no—
45C22⋊C4×C127C2×C412yes—
46C42×C127—21no—
47C2×Dic127—42no—
48C127⋊C8—42no—
49C23×C127—22no—
50C23×C127—42no—
51C23×C127—82no—
52C23×C127C22⋊C411yescharacteristic
53C23×C127C22×C411yescharacteristic
54C23×C127C2×D411yescharacteristic
55Q8×C127—41no—
56D4×C127—41no—
57D4×C127—41no—
58D4×C127—41no—
59D4×C127—42no—
60D4×C127—41no—
61D4×C127—41no—
62D4×C127—82no—
63D4×C127—82no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—22no—
71C2×C4×C127C22⋊C411yescharacteristic
72C2×C4×C127C22⋊C411yescharacteristic
73C2×C4×C127C22⋊C411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127—21no—
77C22×C127—21no—
78C22×C127—41no—
79C22×C127—41no—
80C22×C127—41no—
81C22×C127—42no—
82C22×C127—42no—
83C22×C127—42no—
84C22×C127—84no—
85C22×C127C2×C22⋊C411yescharacteristic
86C4×C127—21no—
87C4×C127—21no—
88C4×C127—21no—
89C4×C127—42no—
90C4×C127#9911yescharacteristic
91C4×C127#9911yescharacteristic
92Dic127—82no—
93C2×C127—21no—
94C2×C127—41no—
95C2×C127—41no—
96C2×C127—41no—
97C2×C127—82no—
98C2×C127C2×#9911yescharacteristic
99#2122—1271nomaximal, Sylow
100C127#212211yescharacteristic, complemented, Sylow
101#535D12711yescharacteristic
102#413—1271no—
103#410—1271no—
104AΣL(1,𝔽16)—1274no—
105C2×#41C2×D12711yescharacteristic
106C22×D4Dic12711yescharacteristic, complemented
107C2×M16—2541no—
108#127—21no—
109#124—21no—
110#121—21no—
111#121—21no—
112#120Dic12712yescomplemented
113#120C2×D12712yes—
114#117—2541no—
115#117Dic12711yescharacteristic, complemented
116#101—2541no—
117#100—1272no—
118#99—1272no—
119#99—2541no—
120C24C2×Dic12712yes—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D12711yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4C2×Dic12711yescharacteristic
131C2×D4C22⋊D12711yescharacteristic
132C2×D4C22×D12711yescharacteristic
133C22×C4C2×Dic12711yescharacteristic
134C22×C4C22⋊D12711yescharacteristic
135C22×C4C22⋊D12711yescharacteristic
136M16—5081no—
137M16—2542no—
138C2×C8—5081no—
139C4⋊C4—21no—
140C4⋊C4—5081no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—2542no—
144C22⋊C4C2×Dic12712yes—
145C42—21no—
146C42—5081no—
147C23—22no—
148C23—42no—
149C23—82no—
150C23C22⋊Dic12711yescharacteristic
151C23C22×Dic12711yescharacteristic
152C23C2×C22⋊D12711yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—42no—
158D4—41no—
159D4—41no—
160D4—82no—
161D4—82no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—5082no—
170C2×C4C22⋊Dic12711yescharacteristic
171C2×C4C22⋊Dic12711yescharacteristic
172C2×C4C22⋊Dic12711yescharacteristic
173C2×C4C2×C22⋊D12711yescharacteristic
174C8—5082no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22C2×C22⋊Dic12711yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—42no—
190C4—10162no—
191C4#1686112411yescharacteristic
192C4#1686112411yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—82no—
198C2C2×#1686112411yescharacteristic
subgroupG/H
centerC2C2×#16861124
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
Fitting, prime radical#535×C127C2
socleC2×C127C2×#99
cosocleC2×D4×C127C23
hypercenter#535D127
hypocenterC127#2122
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22549860C2semidirect
2. #22549850C2semidirect
3. #22549663C2semidirect
12. C22×D4×C127C4semidirect
17. #120×C127C4semidirect
19. #117×C127C4semidirect
100. C127#2122semidirect
106. C22×D4Dic127semidirect
112. #120Dic127semidirect
115. #117Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498602
#225498502
#225496632
#535×C1272
#2122127