Finite Group Databasebeta

← #30649030

Subgroups — #30649030

Order 16256.

358 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550488C211yesmaximal, characteristic
2#22550217C212yesmaximal
3#22550183C212yesmaximal, complemented
4#22550086C212yesmaximal, complemented
5#22550032C212yesmaximal
6#22549981C212yesmaximal, complemented
7C4×#14523238C212yesmaximal, complemented
8#22549937C211yesmaximal, characteristic
9#519×C127C211yesmaximal, characteristic
10C2×#14523238C2212yes—
11C2×#14523238C2212yes—
12#16861209C2211yescharacteristic
13#16861202—22no—
14#16861201C2211yescharacteristic
15#16861199—22no—
16#16861188C2212yes—
17C4×Dic254C2212yes—
18#16861142C2212yes—
19#16861140C2212yes—
20C4⋊(C127⋊C8)C2212yes—
21#16861133C2212yes—
22#16861133C2212yes—
23#16861132C2212yes—
24#16861132C2212yes—
25C22⋊(C127⋊C8)C2212yes—
26#16861120C2211yescharacteristic
27C4×C127⋊C8C2211yescharacteristic
28C22×Q8×C127C2211yescharacteristic
29C4×Q8×C127—22no—
30C4×Q8×C127C2212yes—
31C4×Q8×C127C2212yes—
32#117×C127—22no—
33#117×C127C2211yescharacteristic
34#117×C127C2211yescharacteristic
35C2×Dic254C2312yes—
36#14523238—24no—
37#14523238—24no—
38C2×C127⋊C8C2312yes—
39C2×C127⋊C8C2312yes—
40C4⋊Dic127—22no—
41C4⋊Dic127C2312yes—
42Dic127⋊C4—22no—
43Dic127⋊C4—22no—
44Dic127⋊C4—22no—
45C22⋊Dic127—22no—
46C4×Dic127—22no—
47C2×Q8×C127—24no—
48C2×Q8×C127—21no—
49C2×Q8×C127C2312yes—
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—22no—
53C4⋊C4×C127—22no—
54C4⋊C4×C127D412yes—
55C4⋊C4×C127C2312yes—
56C22⋊C4×C127—21no—
57C22⋊C4×C127D412yes—
58C42×C127—22no—
59C42×C127—22no—
60C42×C127—21no—
61C42×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63Dic254—22no—
64Dic254—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C127⋊C8—22no—
68C127⋊C8—22no—
69C23×C127C2×D411yescharacteristic
70Q8×C127—24no—
71Q8×C127—22no—
72Q8×C127#4114yes—
73C2×C4×C127—24no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127C2×D412yes—
78C2×C4×C127C2×D412yes—
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2411yescharacteristic
81C22×C127—21no—
82C22×C127—22no—
83C22×C127C22×D411yescharacteristic
84C4×C127—24no—
85C4×C127—21no—
86C4×C127—22no—
87C4×C127—22no—
88C4×C127—42no—
89C4×C127C2×#4112yes—
90Dic127—42no—
91Dic127—42no—
92C2×C127—41no—
93C2×C127#52511yescharacteristic
94C2×C127#57812yes—
95#3127—1271nomaximal, Sylow
96C127#312711yescharacteristic, complemented, Sylow
97#541—1271no—
98#519D12711yescharacteristic
99#485—1272no—
100#479—1272no—
101#462—1272no—
102#451—1272no—
103#441—1272no—
104C4×Q16—1272no—
105#433—1271no—
106C22×Q8C2×D12711yescharacteristic
107C2×Q16—1272no—
108C2×Q16—1272no—
109C4⋊Q8—1271no—
110#126—2542no—
111#125—1271no—
112#125—2542no—
113C22⋊Q8—1272no—
114C4×Q8—22no—
115C4×Q8—1272no—
116C4×Q8C2×D12712yes—
117C4×Q8C2×D12712yes—
118#117—22no—
119#117C2×D12711yescharacteristic
120#117C2×D12711yescharacteristic
121#107—1272no—
122#106—1272no—
123C4⋊C8—1272no—
124#103—1272no—
125#103—1272no—
126#103—1272no—
127#103—1272no—
128C22⋊C8—1272no—
129#97—1271no—
130C4×C8—1271no—
131C2×Q8—24no—
132C2×Q8—21no—
133C2×Q8—1272no—
134C2×Q8C22×D12712yes—
135C22×C4—21no—
136C22×C4C22×D12711yescharacteristic
137Q16—2544no—
138Q16—2544no—
139C2×C8—1272no—
140C2×C8—1272no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—1272no—
144C4⋊C4—2542no—
145C4⋊C4—2542no—
146C4⋊C4—2542no—
147C4⋊C4—2542no—
148C4⋊C4C22⋊D12712yes—
149C4⋊C4C22×D12712yes—
150C22⋊C4—21no—
151C22⋊C4—2542no—
152C22⋊C4C22⋊D12712yes—
153C42—22no—
154C42—22no—
155C42—21no—
156C42—2542no—
157C42C22×D12711yescharacteristic
158C42C22×D12711yescharacteristic
159C23C2×C22⋊D12711yescharacteristic
160Q8—24no—
161Q8—22no—
162Q8—2542no—
163Q8—2542no—
164Q8#1452324814yes—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—2542no—
170C2×C4—2542no—
171C2×C4C2×C22⋊D12712yes—
172C2×C4C2×C22⋊D12712yes—
173C2×C4C2×C22⋊D12711yescharacteristic
174C2×C4C23×D12711yescharacteristic
175C8—2542no—
176C8—2542no—
177C22—21no—
178C22—22no—
179C22C22×C22⋊D12711yescharacteristic
180C4—24no—
181C4—21no—
182C4—22no—
183C4—22no—
184C4—42no—
185C4—5082no—
186C4—5082no—
187C4C2×#1452324812yes—
188C2—41no—
189C2#2255038711yescharacteristic
190C2#2255069112yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#519×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#519D127
hypocenterC127#3127
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
3. #22550183C2semidirect
4. #22550086C2semidirect
6. #22549981C2semidirect
7. C4×#14523238C2semidirect
96. C127#3127semidirect

Maximal subgroups

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

subgroupindex
#225504882
#225502172
#225501832
#225500862
#225500322
#225499812
C4×#145232382
#225499372
#519×C1272
#3127127