Finite Group Databasebeta

← #30639956

Subgroups — #30639956

Order 16256.

290 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 71 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×#16861136C212yesmaximal, complemented
2#22549877C212yesmaximal, complemented
3#22549741C211yesmaximal, characteristic, complemented
4#22549645C211yesmaximal, characteristic, complemented
5#537×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7C2×C4⋊Dic127C2211yescharacteristic
8C2×C4×Dic127C2211yescharacteristic
9#16861136—24no—
10#16861133—22no—
11#16861130—22no—
12C127⋊#95—22no—
13C2×#41×C127C2212yes—
14C22⋊Q8×C127—22no—
15#121×C127—22no—
16#121×C127—22no—
17C2×C4⋊C4×C127C2211yescharacteristic
18C22×Dic127—21no—
19C22×Dic127—21no—
20#14523232—22no—
21#14523232—22no—
22C2×C127⋊C8—22no—
23C4⋊Dic127—22no—
24C4×Dic127—21no—
25C4×Dic127D412yescomplemented
26#41×C127—22no—
27#41×C127—22no—
28#41×C127—22no—
29#41×C127—22no—
30C2×Q8×C127D412yes—
31C2×D4×C127—22no—
32C2×D4×C127—22no—
33C2×D4×C127D412yes—
34C22×C4×C127—22no—
35C22×C4×C127D412yes—
36C22×C4×C127C2311yescharacteristic
37C4⋊C4×C127—22no—
38C4⋊C4×C127—22no—
39C22⋊C4×C127—24no—
40C2×Dic127—22no—
41C2×Dic127—41no—
42C2×Dic127—41no—
43C2×Dic127—42no—
44C2×Dic127—41no—
45C127⋊C8—42no—
46C23×C127—22no—
47C23×C127C2×D411yescharacteristic
48Q8×C127—22no—
49Q8×C127—22no—
50D4×C127—22no—
51D4×C127—22no—
52D4×C127—42no—
53D4×C127—42no—
54D4×C127—82no—
55C2×C4×C127—22no—
56C2×C4×C127—22no—
57C2×C4×C127—42no—
58C2×C4×C127—42no—
59C2×C4×C127—42no—
60C2×C4×C127C2×D411yescharacteristic
61C2×C4×C127C2×D412yes—
62C2×C4×C127#4112yes—
63C2×C4×C127#4111yescharacteristic
64C22×C127—21no—
65C22×C127—21no—
66C22×C127—42no—
67C22×C127—42no—
68C22×C127—42no—
69C22×C127#12011yescharacteristic
70C22×C127#12111yescharacteristic
71C22×C127#12411yescharacteristic
72C4×C127—42no—
73C4×C127—42no—
74C4×C127#12112yes—
75C4×C127#12312yes—
76Dic127—42no—
77Dic127—81no—
78C2×C127—21no—
79C2×C127—21no—
80C2×C127—82no—
81C2×C127#39411yescharacteristic
82C2×C127#45411yescharacteristic
83C2×C127#45511yescharacteristic
84#2040—1271nomaximal, Sylow
85C127#204011yescharacteristic, complemented, Sylow
86#537D12711yescharacteristic
87C2×#104—1272no—
88#417—1272no—
89#382—1271no—
90#343—1271no—
91C2×#41C2×D12712yes—
92C2×M16—1272no—
93C22⋊Q8—22no—
94#121—22no—
95#121—22no—
96C2×C4⋊C4—1271no—
97C2×C4⋊C4C2×D12711yescharacteristic
98C2×C42—1271no—
99#104—2544no—
100#103—2542no—
101#102—2542no—
102#95—2542no—
103#41—22no—
104#41—22no—
105#41—22no—
106#41—22no—
107C2×Q8C22⋊D12712yes—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4C22⋊D12712yes—
111C22×C4—22no—
112C22×C4—2541no—
113C22×C4—2541no—
114C22×C4C22⋊D12712yes—
115C22×C4C22×D12711yescharacteristic
116M16—2542no—
117M16—2542no—
118C2×C8—2542no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—2542no—
122C22⋊C4—24no—
123C42—1272no—
124C42—2541no—
125C23—22no—
126C23C2×C22⋊D12711yescharacteristic
127Q8—22no—
128Q8—22no—
129D4—22no—
130D4—22no—
131D4—42no—
132D4—42no—
133D4—82no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—5082no—
140C2×C4—5081no—
141C2×C4—5081no—
142C2×C4—5081no—
143C2×C4—2542no—
144C2×C4D4×D12711yescharacteristic
145C2×C4C2×C22⋊D12712yes—
146C2×C4#1452324711yescharacteristic
147C2×C4#1452324712yes—
148C8—5082no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#1686117811yescharacteristic
155C22#1686118011yescharacteristic
156C22#1686119611yescharacteristic
157C4—42no—
158C4—42no—
159C4—10161no—
160C4—5082no—
161C4#1686118112yes—
162C4#1686119012yes—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2254979011yescharacteristic
167C2#2255004911yescharacteristic
168C2#2255005411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#537×C127C2
socleC22×C127#120
hypercenter#537D127
hypocenterC127#2040
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16861136C2semidirect
2. #22549877C2semidirect
3. #22549741C2semidirect
4. #22549645C2semidirect
25. C4×Dic127D4semidirect
85. C127#2040semidirect

Maximal subgroups

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

subgroupindex
C2×#168611362
#225498772
#225497412
#225496452
#537×C1272
#2040127