Finite Group Databasebeta

← #30639277

Subgroups — #30639277

Order 16256.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550440C211yesmaximal, characteristic
2#22549860C211yesmaximal, characteristic, complemented
3C2×#16861125C212yesmaximal, complemented
4#22549648C212yesmaximal, complemented
5#568×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C2×#14523232C2212yes—
8#16861205—21no—
9#16861198—21no—
10#16861183—22no—
11#16861177C414yes—
12#16861168C2211yescharacteristic
13#16861128—21no—
14#16861126—21no—
15#16861125—24no—
16C2×#41×C127C2211yescharacteristic
17#131×C127—24no—
18C2×M16×C127—22no—
19C2×M16×C127C2212yes—
20C23×D127C2×C412yes—
21C2×D508—21no—
22C2×D508—22no—
23C2×D508—21no—
24C2×D508—22no—
25C2×C22⋊D127—24no—
26#14523232—22no—
27#14523232—22no—
28C2×C127⋊C8—22no—
29C4⋊Dic127—21no—
30C22⋊Dic127C2×C412yes—
31(C2×D127)⋊C4—24no—
32C4×Dic127—21no—
33#41×C127—22no—
34#41×C127—22no—
35C2×Q8×C127D411yescharacteristic
36C2×D4×C127C2×C412yes—
37C2×D4×C127D411yescharacteristic
38C22×C4×C127D412yes—
39C22×C4×C127C2311yescharacteristic
40M16×C127—24no—
41M16×C127—22no—
42M16×C127D414yescomplemented
43C2×C8×C127—24no—
44C2×C8×C127—22no—
45C22×D127—22no—
46C22×D127—24no—
47C22×D127—82no—
48D508—41no—
49D508—42no—
50D508—41no—
51C22⋊D127—84no—
52C2×Dic127—22no—
53C127⋊C8—42no—
54C23×C127C22⋊C412yes—
55C23×C127C22×C411yescharacteristic
56Q8×C127—22no—
57D4×C127—24no—
58D4×C127—22no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127C22⋊C412yes—
62C2×C4×C127C2×D412yes—
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127#4112yes—
66C8×C127—24no—
67C22×C127—22no—
68C22×C127—41no—
69C22×C127—42no—
70C22×C127C2×C22⋊C411yescharacteristic
71C22×C127C4×D412yes—
72C2×D127—44no—
73C2×D127—42no—
74C2×D127—84no—
75C4×C127—22no—
76C4×C127#12011yescharacteristic
77C4×C127#12112yes—
78C4×C127#12311yescharacteristic
79Dic127—42no—
80C2×C127—21no—
81C2×C127—22no—
82C2×C127—42no—
83C2×C127#38611yescharacteristic
84D127—82no—
85#1914—1271nomaximal, Sylow
86C127#191411yescharacteristic, complemented, Sylow
87#568D12711yescharacteristic, complemented
88#535—1271no—
89#413—1271no—
90C2×#100—1272no—
91#344—1272no—
92C2×#41C2×D12711yescharacteristic
93C22×D4—1271no—
94#131—24no—
95C2×M16—22no—
96C2×M16—1272no—
97C2×M16C2×D12712yes—
98#127—2541no—
99#124—2541no—
100#121—2542no—
101#120—1274no—
102#117—1271no—
103#101—2541no—
104#100—2541no—
105#100—2544no—
106C24—1272no—
107#41—22no—
108#41—22no—
109C2×Q8D50811yescharacteristic
110C2×D4—2544no—
111C2×D4—2542no—
112C2×D4—2541no—
113C2×D4—2541no—
114C2×D4—2542no—
115C2×D4C4×D12712yes—
116C2×D4D50811yescharacteristic
117C22×C4C22⋊D12712yes—
118C22×C4C22×D12711yescharacteristic
119M16—24no—
120M16—22no—
121M16—2542no—
122M16—2542no—
123M16C22⋊D12714yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—2542no—
127C4⋊C4—2541no—
128C22⋊C4—1272no—
129C22⋊C4—2544no—
130C42—2541no—
131C23—10162no—
132C23—2544no—
133C23—2542no—
134C23(C2×D127)⋊C412yes—
135C23C2×C4×D12711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—10164no—
140D4—5081no—
141D4—5081no—
142D4—5082no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—2542no—
146C2×C4(C2×D127)⋊C412yes—
147C2×C4C2×C22⋊D12711yescharacteristic
148C2×C4C2×C22⋊D12712yes—
149C2×C4C2×D50811yescharacteristic
150C2×C4#1452324812yes—
151C8—24no—
152C8—5082no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—10164no—
157C22—5084no—
158C22—5082no—
159C22C2×(C2×D127)⋊C411yescharacteristic
160C22C4×C22⋊D12712yes—
161C4—22no—
162C4—5082no—
163C4#1686117811yescharacteristic
164C4#1686118312yes—
165C4#1686119211yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—10162no—
170C2#2254975511yescharacteristic
subgroupG/H
centerC2#22549755
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#386
prime radical#22550440C2
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1914
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #22549860C2semidirect
3. C2×#16861125C2semidirect
4. #22549648C2semidirect
5. #568×C127C2semidirect
42. M16×C127D4semidirect
86. C127#1914semidirect
87. #568D127semidirect
123. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504402
#225498602
C2×#168611252
#225496482
#568×C1272
#1914127