Finite Group Databasebeta

← #30640706

Subgroups — #30640706

Order 16256.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550693C211yesmaximal, characteristic
2#22550317C211yesmaximal, characteristic, complemented
3C2×#16861155C211yesmaximal, characteristic, complemented
4#22549691C211yesmaximal, characteristic, complemented
5#22549682C211yesmaximal, characteristic, complemented
6#22549660C211yesmaximal, characteristic, complemented
7#473×C127C211yesmaximal, characteristic, complemented
8#16861256—21no—
9#16861242—22no—
10#16861242—21no—
11#16861231—21no—
12#16861231C2211yescharacteristic
13C2×Dic508—21no—
14C2×Dic508C2211yescharacteristic
15#16861155—22no—
16#16861155—21no—
17#16861152—21no—
18#16861149C2211yescharacteristic
19C2×C127⋊C16C2211yescharacteristic
20#137×C127—21no—
21C2×Q16×C127C2211yescharacteristic
22#131×C127C2211yescharacteristic
23#108×C127C2211yescharacteristic
24#101×C127—21no—
25#14523248—22no—
26#14523248—21no—
27#14523247—42no—
28#14523247—41no—
29C2×Dic254—22no—
30C2×Dic254—21no—
31Q8×D127—41no—
32Dic508—22no—
33Dic508—21no—
34Dic508D411yescharacteristic
35Dic508D412yes—
36#14523237—22no—
37#14523237—21no—
38D1016D411yescharacteristic
39C127⋊C16—21no—
40C127⋊C16—21no—
41#41×C127D411yescharacteristic
42C2×Q8×C127—21no—
43Q16×C127—21no—
44Q16×C127—21no—
45Q16×C127—41no—
46SD16×C127—41no—
47M16×C127—21no—
48M16×C127—21no—
49M16×C127D411yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127C2311yescharacteristic
52Dic254—22no—
53Dic254—21no—
54Dic254—42no—
55Dic254—41no—
56D508—21no—
57C22⋊D127—42no—
58C22⋊D127—41no—
59C2×Dic127—42no—
60C2×Dic127—41no—
61C4×D127—42no—
62C4×D127—41no—
63Q8×C127—41no—
64Q8×C127—41no—
65Q8×C127D811yescharacteristic
66D4×C127—21no—
67D4×C127D811yescharacteristic
68C2×C4×C127—21no—
69C2×C4×C127—41no—
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—21no—
72C8×C127—41no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127AGL(1,ℤ/8)11yescharacteristic
77C2×D127—41no—
78C4×C127—21no—
79C4×C127—81no—
80C4×C127#12011yescharacteristic
81C4×C127C2×D811yescharacteristic
82Dic127—42no—
83Dic127—41no—
84C2×C127—21no—
85C2×C127—41no—
86C2×C127#44711yescharacteristic
87D127—81no—
88#2195—1271nomaximal, Sylow
89C127#219511yescharacteristic, complemented, Sylow
90#578—1271no—
91#510—1271no—
92C2×Q32—1271no—
93#473D12711yescharacteristic, complemented
94#362—1271no—
95#359—1271no—
96#350—1271no—
972−1+4—2541no—
98#137—21no—
99#137—2542no—
100#137—2541no—
101#135—1271no—
102#135—2541no—
103C2×Q16—1271no—
104C2×Q16—2541no—
105C2×Q16C2×D12711yescharacteristic
106#131C2×D12711yescharacteristic
107Q32—2542no—
108Q32—2541no—
109SD32—2541no—
110M32—1271no—
111C2×C16—1271no—
112#108C2×D12711yescharacteristic
113#101—21no—
114#41—5082no—
115#41—5081no—
116#41—2542no—
117#41—2541no—
118#41C22⋊D12711yescharacteristic
119C2×Q8—21no—
120C2×Q8—5081no—
121C2×Q8—2542no—
122C2×Q8—2541no—
123Q16—21no—
124Q16—21no—
125Q16—41no—
126Q16—1272no—
127Q16—1271no—
128Q16—2542no—
129Q16—2541no—
130SD16—41no—
131SD16—2542no—
132SD16—2541no—
133D8—1271no—
134M16—21no—
135M16—21no—
136M16C22⋊D12711yescharacteristic
137C2×C8—21no—
138C2×C8C22×D12711yescharacteristic
139C16—2541no—
140C16—2541no—
141Q8—41no—
142Q8—41no—
143Q8—5082no—
144Q8—5081no—
145Q8—2542no—
146Q8—2541no—
147Q8#1452323311yescharacteristic
148D4—21no—
149D4—5082no—
150D4—5081no—
151D4—2541no—
152D4#1452323311yescharacteristic
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—5082no—
156C2×C4—5082no—
157C2×C4—5081no—
158C2×C4—5081no—
159C2×C4C2×C22⋊D12711yescharacteristic
160C8—21no—
161C8—41no—
162C8D4×D12711yescharacteristic
163C8D4×D12711yescharacteristic
164C22—21no—
165C22—5081no—
166C22#1686123411yescharacteristic
167C4—21no—
168C4—81no—
169C4—5082no—
170C4—5081no—
171C4#1686117711yescharacteristic
172C4C2×#1452323311yescharacteristic
173C2—21no—
174C2—41no—
175C2—10161no—
176C2#2255000911yescharacteristic
subgroupG/H
centerC2#22550009
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#473×C127C2
socleC2×C127#447
prime radical#22550693C2
hypercenter#473D127
hypocenterC127#2195
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #473

Complements

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

NH ≅ G/Ntype
2. #22550317C2semidirect
3. C2×#16861155C2semidirect
4. #22549691C2semidirect
5. #22549682C2semidirect
6. #22549660C2semidirect
7. #473×C127C2semidirect
89. C127#2195semidirect
93. #473D127semidirect

Maximal subgroups

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

subgroupindex
#225506932
#225503172
C2×#168611552
#225496912
#225496822
#225496602
#473×C1272
#2195127