Finite Group Databasebeta

← #30637733

Subgroups — #30637733

Order 16256.

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

normal subgroups only

This group has 254 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
1#22550187C212yesmaximal, complemented
2#22549806C211yesmaximal, characteristic, complemented
3#22549610C212yesmaximal, complemented
4#22549602C211yesmaximal, characteristic, complemented
5#537×C127C211yesmaximal, characteristic
6C2×C4⋊Dic127C2211yescharacteristic
7C2×Dic127⋊C4—21no—
8#16861142—22no—
9#16861130—22no—
10C22⋊(C127⋊C8)C2212yes—
11C127⋊#95—21no—
12C127⋊#95C2211yescharacteristic
13C2×#41×C127—21no—
14C22⋊Q8×C127—22no—
15#121×C127—22no—
16#121×C127—21no—
17#121×C127C2212yes—
18C2×C4⋊C4×C127C2211yescharacteristic
19C22×Dic127—21no—
20C22×Dic127—21no—
21C2×C127⋊C8—22no—
22C4⋊Dic127—22no—
23Dic127⋊C4—42no—
24#41×C127—42no—
25#41×C127—42no—
26C2×Q8×C127—21no—
27C2×D4×C127—21no—
28C2×D4×C127—22no—
29C2×D4×C127—21no—
30C2×D4×C127D412yes—
31C22×C4×C127—22no—
32C22×C4×C127D411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127C2311yescharacteristic
35C4⋊C4×C127—21no—
36C4⋊C4×C127—22no—
37C4⋊C4×C127D412yes—
38C22⋊C4×C127—22no—
39C22⋊C4×C127—22no—
40C2×Dic127—22no—
41C2×Dic127—41no—
42C2×Dic127—41no—
43C2×Dic127—41no—
44C2×Dic127—41no—
45C2×Dic127—41no—
46C127⋊C8—42no—
47C23×C127—22no—
48C23×C127C2×D411yescharacteristic
49Q8×C127—42no—
50D4×C127—42no—
51D4×C127—42no—
52D4×C127—42no—
53D4×C127—42no—
54D4×C127—81no—
55C2×C4×C127—21no—
56C2×C4×C127—21no—
57C2×C4×C127—21no—
58C2×C4×C127—22no—
59C2×C4×C127—41no—
60C2×C4×C127—44no—
61C2×C4×C127—41no—
62C2×C4×C127D812yes—
63C2×C4×C127C2×D412yes—
64C22×C127—21no—
65C22×C127—21no—
66C22×C127—42no—
67C22×C127—42no—
68C22×C127—42no—
69C22×C127#12011yescharacteristic
70C22×C127C2×D811yescharacteristic
71C22×C127AGL(1,ℤ/8)11yescharacteristic
72C4×C127—22no—
73C4×C127—22no—
74C4×C127—42no—
75C4×C127—41no—
76Dic127—42no—
77Dic127—81no—
78C2×C127—21no—
79C2×C127—21no—
80C2×C127—82no—
81C2×C127#44711yescharacteristic
82C2×C127#45611yescharacteristic
83C2×C127#45811yescharacteristic
84#1610—1271nomaximal, Sylow
85C127#161011yescharacteristic, complemented, Sylow
86#537D12711yescharacteristic
87#480—1272no—
88#398—1271no—
89#327—1272no—
90#324—1271no—
91C2×#41—21no—
92C22⋊Q8—22no—
93#121—22no—
94#121—21no—
95#121C2×D12712yes—
96C2×C4⋊C4—1271no—
97C2×C4⋊C4—2541no—
98C2×C4⋊C4C2×D12711yescharacteristic
99#107—2542no—
100#102—2542no—
101C22⋊C8—1272no—
102#95—1271no—
103#95—2541no—
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C22⋊D12712yes—
111C22×C4—22no—
112C22×C4—2541no—
113C22×C4—2541no—
114C22×C4C22⋊D12711yescharacteristic
115C22×C4C22⋊D12711yescharacteristic
116C22×C4C22×D12711yescharacteristic
117C2×C8—2542no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—5082no—
121C4⋊C4—2542no—
122C4⋊C4C22⋊D12712yes—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C23—22no—
126C23C2×C22⋊D12711yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—44no—
139C2×C4—41no—
140C2×C4—5081no—
141C2×C4—5081no—
142C2×C4—5081no—
143C2×C4—5081no—
144C2×C4—5081no—
145C2×C4—2542no—
146C2×C4#1452323312yes—
147C2×C4C2×C22⋊D12712yes—
148C8—5082no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#1686117811yescharacteristic
155C22C2×#1452323311yescharacteristic
156C22#1686123511yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—42no—
161C4—10161no—
162C4—5082no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2255001011yescharacteristic
167C2#2255005911yescharacteristic
168C2#2255006611yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#537×C127C2
socleC22×C127#120
hypercenter#537D127
hypocenterC127#1610
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #22550187C2semidirect
2. #22549806C2semidirect
3. #22549610C2semidirect
4. #22549602C2semidirect
85. C127#1610semidirect

Maximal subgroups

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

subgroupindex
#225501872
#225498062
#225496102
#225496022
#537×C1272
#1610127