Finite Group Databasebeta

← #30640670

Subgroups — #30640670

Order 16256.

220 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×#16861155C211yesmaximal, characteristic, complemented
2C2×#16861153C211yesmaximal, characteristic, complemented
3#22550117C211yesmaximal, characteristic, complemented
4#22549680C211yesmaximal, characteristic, complemented
5#22549677C211yesmaximal, characteristic, complemented
6C22⋊(C127⋊C16)C211yesmaximal, characteristic, complemented
7C2×#135×C127C211yesmaximal, characteristic
8C2×Dic508C2211yescharacteristic
9#16861188—21no—
10#16861155—22no—
11#16861153—22no—
12C2×C127⋊C16C2211yescharacteristic
13C2×C127⋊C16C2211yescharacteristic
14#16861143C2211yescharacteristic
15#16861134—21no—
16C2×#41×C127—21no—
17#135×C127—22no—
18#135×C127—22no—
19C2×Q16×C127C2211yescharacteristic
20C2×SD16×C127—21no—
21C2×D8×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23C2×Dic254—21no—
24Dic508—21no—
25Dic508—21no—
26C4⋊Dic127—21no—
27Dic127⋊C4—41no—
28C22⋊Dic127—41no—
29C127⋊C16—21no—
30C127⋊C16—21no—
31#41×C127—24no—
32#41×C127—42no—
33C2×Q8×C127—21no—
34C2×D4×C127—21no—
35C2×D4×C127—41no—
36C22×C4×C127—41no—
37C22×C4×C127D411yescharacteristic
38Q16×C127—21no—
39Q16×C127D412yes—
40SD16×C127—24no—
41D8×C127—21no—
42D8×C127D412yes—
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127D411yescharacteristic
46C2×C8×C127C2311yescharacteristic
47Dic254—41no—
48Dic254—41no—
49C2×Dic127—41no—
50C2×Dic127—41no—
51C23×C127—41no—
52C23×C127D811yescharacteristic
53Q8×C127—22no—
54Q8×C127—41no—
55D4×C127—22no—
56D4×C127—44no—
57D4×C127—41no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—44no—
61C2×C4×C127—41no—
62C2×C4×C127D811yescharacteristic
63C2×C4×C127C2×D411yescharacteristic
64C8×C127—21no—
65C8×C127C2×D411yescharacteristic
66C8×C127C2×D411yescharacteristic
67C22×C127—21no—
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—42no—
71C22×C127—81no—
72C22×C127—81no—
73C22×C127C2×D811yescharacteristic
74C4×C127—21no—
75C4×C127—42no—
76C4×C127#12011yescharacteristic
77C4×C127AGL(1,ℤ/8)11yescharacteristic
78Dic127—81no—
79Dic127—81no—
80C2×C127—41no—
81C2×C127—82no—
82C2×C127#44711yescharacteristic
83C2×C127#50811yescharacteristic
84C2×C127#51011yescharacteristic
85#2189—1271nomaximal, Sylow
86C127#218911yescharacteristic, complemented, Sylow
87C2×#135D12711yescharacteristic
88C2×Q32—1271no—
89C2×SD32—1271no—
90#467—1271no—
91#358—1271no—
92#357—1271no—
93C22⋊C16—1271no—
94C2×#41—21no—
95#135—22no—
96#135—22no—
97C2×Q16—1271no—
98C2×Q16C2×D12711yescharacteristic
99C2×SD16—21no—
100C2×D8C2×D12711yescharacteristic
101C22×C8C2×D12711yescharacteristic
102C22⋊Q8—2541no—
103Q32—2542no—
104SD32—2542no—
105C2×C16—1271no—
106C2×C16—1271no—
107#107—1271no—
108#103—2541no—
109#41—24no—
110#41—42no—
111C2×Q8—21no—
112C2×Q8—2541no—
113C2×D4—21no—
114C2×D4—41no—
115C22×C4—41no—
116C22×C4C22⋊D12711yescharacteristic
117Q16—21no—
118Q16—2541no—
119Q16—2541no—
120Q16C22⋊D12712yes—
121SD16—24no—
122D8—21no—
123D8C22⋊D12712yes—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8C22⋊D12711yescharacteristic
127C2×C8C22×D12711yescharacteristic
128C4⋊C4—5081no—
129C4⋊C4—2541no—
130C22⋊C4—5081no—
131C16—2541no—
132C16—2541no—
133C23—41no—
134C23#1452323311yescharacteristic
135Q8—22no—
136Q8—41no—
137Q8—5081no—
138Q8—5081no—
139D4—22no—
140D4—44no—
141D4—41no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—44no—
145C2×C4—41no—
146C2×C4—5081no—
147C2×C4—5081no—
148C2×C4#1452323311yescharacteristic
149C2×C4C2×C22⋊D12711yescharacteristic
150C8—21no—
151C8C2×C22⋊D12711yescharacteristic
152C8C2×C22⋊D12711yescharacteristic
153C22—21no—
154C22—21no—
155C22—21no—
156C22—42no—
157C22—81no—
158C22—81no—
159C22C2×#1452323311yescharacteristic
160C4—21no—
161C4—42no—
162C4—10161no—
163C4—10161no—
164C4#1686117811yescharacteristic
165C4#1686123511yescharacteristic
166C2—41no—
167C2—82no—
168C2#2255001011yescharacteristic
169C2#2255030611yescharacteristic
170C2#2255031611yescharacteristic
subgroupG/H
centerC22C2×#14523233
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting, prime radicalC2×#135×C127C2
socleC22×C127C2×D8
hypercenterC2×#135D127
hypocenterC127#2189
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16861155C2semidirect
2. C2×#16861153C2semidirect
3. #22550117C2semidirect
4. #22549680C2semidirect
5. #22549677C2semidirect
6. C22⋊(C127⋊C16)C2semidirect
86. C127#2189semidirect

Maximal subgroups

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

subgroupindex
C2×#168611552
C2×#168611532
#225501172
#225496802
#225496772
C22⋊(C127⋊C16)2
C2×#135×C1272
#2189127