Finite Group Databasebeta

← #30640014

Subgroups — #30640014

Order 16256.

266 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 67 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#22549899C211yesmaximal, characteristic, complemented
2C2×#16861136C212yesmaximal, complemented
3#22549860C212yesmaximal, complemented
4C127⋊#338C211yesmaximal, characteristic, complemented
5#575×C127C211yesmaximal, characteristic
6C2×#14523232C2211yescharacteristic
7C2×#14523232C2212yes—
8C2×C4×Dic127C2211yescharacteristic
9Dic127⋊C8—22no—
10#16861136—22no—
11#16861136—22no—
12#16861128—22no—
13#16861126—22no—
14C2×#41×C127C2212yes—
15#137×C127—22no—
16AGL(1,ℤ/8)×C127—22no—
17#135×C127—22no—
18C2×M16×C127C2211yescharacteristic
19C22×Dic127—21no—
20#14523232—22no—
21#14523232—22no—
22#14523232—22no—
23C2×C127⋊C8—21no—
24C2×C127⋊C8—22no—
25C4×Dic127—21no—
26C4×Dic127D411yescharacteristic
27C4×Dic127D411yescharacteristic
28#41×C127—22no—
29#41×C127—22no—
30#41×C127—22no—
31#41×C127—22no—
32C2×Q8×C127D412yes—
33C2×D4×C127—22no—
34C2×D4×C127D412yes—
35C22×C4×C127—22no—
36C22×C4×C127C2311yescharacteristic
37Q16×C127—22no—
38SD16×C127—24no—
39D8×C127—22no—
40M16×C127—22no—
41C2×C8×C127D412yes—
42C2×Dic127—21no—
43C2×Dic127—21no—
44C2×Dic127—42no—
45C127⋊C8—41no—
46C127⋊C8—42no—
47C23×C127—22no—
48C23×C127#4111yescharacteristic
49Q8×C127—22no—
50Q8×C127—22no—
51D4×C127—22no—
52D4×C127—22no—
53D4×C127—42no—
54D4×C127—42no—
55C2×C4×C127—22no—
56C2×C4×C127—42no—
57C2×C4×C127—42no—
58C2×C4×C127C2×D411yescharacteristic
59C2×C4×C127C2×D412yes—
60C2×C4×C127C2×D411yescharacteristic
61C2×C4×C127#4112yes—
62C8×C127—22no—
63C22×C127—41no—
64C22×C127—42no—
65C22×C127—82no—
66C22×C127#12111yescharacteristic
67C22×C127#12312yes—
68C4×C127—42no—
69C4×C127#12011yescharacteristic
70C4×C127#12112yes—
71C4×C127#12411yescharacteristic
72Dic127—22no—
73Dic127—41no—
74C2×C127—21no—
75C2×C127—22no—
76C2×C127—82no—
77C2×C127#39411yescharacteristic
78#2050—1271nomaximal, Sylow
79C127#205011yescharacteristic, complemented, Sylow
80#575D12711yescharacteristic
81#423—1271no—
82C2×#104—1272no—
83#413—1272no—
84#338—1271no—
85C2×#41C2×D12712yes—
86#137—22no—
87AGL(1,ℤ/8)—22no—
88#135—22no—
89C2×M16—1272no—
90C2×M16—1271no—
91C2×M16C2×D12711yescharacteristic
92C2×C42—1271no—
93C4⋊C8—2542no—
94#104—2542no—
95#104—2542no—
96#101—2542no—
97#100—2542no—
98#41—22no—
99#41—22no—
100#41—22no—
101#41—22no—
102C2×Q8C22⋊D12712yes—
103C2×D4—22no—
104C2×D4C22⋊D12712yes—
105C22×C4—22no—
106C22×C4—2541no—
107C22×C4C22×D12711yescharacteristic
108Q16—22no—
109SD16—24no—
110D8—22no—
111M16—22no—
112M16—2542no—
113M16—2542no—
114M16—2542no—
115C2×C8—2541no—
116C2×C8—2542no—
117C2×C8C22⋊D12712yes—
118C42—1271no—
119C42—1271no—
120C42—2541no—
121C23—22no—
122C23#1452324711yescharacteristic
123Q8—22no—
124Q8—22no—
125D4—22no—
126D4—22no—
127D4—42no—
128D4—42no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—5082no—
133C2×C4—2541no—
134C2×C4—2541no—
135C2×C4D4×D12711yescharacteristic
136C2×C4C2×C22⋊D12711yescharacteristic
137C2×C4C2×C22⋊D12712yes—
138C2×C4#1452324712yes—
139C8—22no—
140C8—5081no—
141C8—5082no—
142C22—41no—
143C22—42no—
144C22—82no—
145C22#1686118011yescharacteristic
146C22#1686119012yes—
147C4—42no—
148C4—5081no—
149C4—2542no—
150C4#1686117811yescharacteristic
151C4#1686118112yes—
152C4#1686119611yescharacteristic
153C2—21no—
154C2—22no—
155C2—82no—
156C2#2254979011yescharacteristic
subgroupG/H
centerC4#16861196
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#575×C127C2
socleC2×C127#394
hypercenter#575D127
hypocenterC127#2050
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22549899C2semidirect
2. C2×#16861136C2semidirect
3. #22549860C2semidirect
4. C127⋊#338C2semidirect
79. C127#2050semidirect

Maximal subgroups

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

subgroupindex
#225498992
C2×#168611362
#225498602
C127⋊#3382
#575×C1272
#2050127