Finite Group Databasebeta

← #30638745

Subgroups — #30638745

Order 16256.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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×C4⋊(C127⋊C8)C211yesmaximal, characteristic
2C2×#16861133C212yesmaximal
3#22549742C211yesmaximal, characteristic
4#22549638C212yesmaximal
5C2×C4×Q8×C127C211yesmaximal, characteristic
6C22×C127⋊C8C2212yes—
7C2×C4⋊Dic127C2212yes—
8C4⋊(C127⋊C8)—22no—
9#16861133—24no—
10C127⋊#95—21no—
11C127⋊#95—21no—
12C22×Q8×C127C2211yescharacteristic
13C4×Q8×C127—22no—
14C4×Q8×C127C414yescomplemented
15C2×C4⋊C4×C127—21no—
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C42×C127—21no—
18C2×C42×C127C2211yescharacteristic
19C22×Dic127—22no—
20C2×C127⋊C8—22no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8—22no—
23C4⋊Dic127—22no—
24C4⋊Dic127—22no—
25C2×Q8×C127—21no—
26C2×Q8×C127—21no—
27C2×Q8×C127—21no—
28C2×Q8×C127C2×C412yes—
29C2×Q8×C127D412yes—
30C2×Q8×C127Q812yes—
31C22×C4×C127—21no—
32C22×C4×C127—21no—
33C22×C4×C127D411yescharacteristic
34C22×C4×C127D411yescharacteristic
35C22×C4×C127C2311yescharacteristic
36C4⋊C4×C127—24no—
37C4⋊C4×C127—21no—
38C4⋊C4×C127C2×C412yes—
39C42×C127—24no—
40C42×C127C2×C412yes—
41C42×C127D412yes—
42C2×Dic127—42no—
43C2×Dic127—42no—
44C2×Dic127—42no—
45C127⋊C8—42no—
46C23×C127C2×D411yescharacteristic
47Q8×C127—22no—
48Q8×C127—22no—
49Q8×C127—22no—
50Q8×C127C4⋊C414yes—
51C2×C4×C127—22no—
52C2×C4×C127—22no—
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—22no—
56C2×C4×C127—22no—
57C2×C4×C127—42no—
58C2×C4×C127C22⋊C412yes—
59C2×C4×C127C22⋊C412yes—
60C2×C4×C127C22×C411yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C2×C4×C127C2×Q811yescharacteristic
64C2×C4×C127#4111yescharacteristic
65C2×C4×C127#4111yescharacteristic
66C22×C127C2×C22⋊C411yescharacteristic
67C22×C127#12111yescharacteristic
68C22×C127C22⋊Q811yescharacteristic
69C22×C127#13512yes—
70C22×C127#13712yes—
71C4×C127—24no—
72C4×C127—22no—
73C4×C127—22no—
74C4×C127—42no—
75C4×C127C2×C4⋊C411yescharacteristic
76C4×C127#11711yescharacteristic
77C4×C127#12111yescharacteristic
78C4×C127C22⋊Q811yescharacteristic
79Dic127—82no—
80C2×C127#38011yescharacteristic
81C2×C127#41611yescharacteristic
82C2×C127#41911yescharacteristic
83C2×C127#46412yes—
84C2×C127#47912yes—
85#1805—1271nomaximal, Sylow
86C127#180511yescharacteristic, complemented, Sylow
87C2×C4×Q8D12711yescharacteristic
88C2×C4⋊C8—1271no—
89C2×#103—1272no—
90#382—1271no—
91#340—1272no—
92C22×Q8C2×D12711yescharacteristic
93C22×C8—1272no—
94C4×Q8—22no—
95C4×Q8Dic12714yescomplemented
96C2×C4⋊C4—21no—
97C2×C4⋊C4—1272no—
98C2×C4⋊C4C2×D12711yescharacteristic
99C2×C42—21no—
100C2×C42C2×D12711yescharacteristic
101C4⋊C8—2542no—
102#103—2544no—
103#95—2541no—
104#95—2541no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8C2×Dic12712yes—
109C2×Q8D50812yes—
110C2×Q8Dic25412yes—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—2542no—
114C22×C4C22⋊D12711yescharacteristic
115C22×C4C22⋊D12711yescharacteristic
116C22×C4C22×D12711yescharacteristic
117C2×C8—2542no—
118C2×C8—2542no—
119C2×C8—2542no—
120C4⋊C4—24no—
121C4⋊C4—21no—
122C4⋊C4—2542no—
123C4⋊C4—2542no—
124C4⋊C4C2×Dic12712yes—
125C42—24no—
126C42C2×Dic12712yes—
127C42C22⋊D12712yes—
128C23C2×C22⋊D12711yescharacteristic
129Q8—22no—
130Q8—22no—
131Q8—22no—
132Q8C4⋊Dic12714yes—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—42no—
140C2×C4—5082no—
141C2×C4—5082no—
142C2×C4—5082no—
143C2×C4C22⋊Dic12712yes—
144C2×C4C22⋊Dic12712yes—
145C2×C4C22×Dic12711yescharacteristic
146C2×C4C2×C22⋊D12711yescharacteristic
147C2×C4C2×D50811yescharacteristic
148C2×C4C2×Dic25411yescharacteristic
149C2×C4#1452324811yescharacteristic
150C2×C4#1452324811yescharacteristic
151C8—5082no—
152C22C2×C22⋊Dic12711yescharacteristic
153C22#1686118311yescharacteristic
154C22#1686118811yescharacteristic
155C22#1686122912yes—
156C22#1686124012yes—
157C4—22no—
158C4—22no—
159C4—24no—
160C4—42no—
161C4—10162no—
162C4C2×C4⋊Dic12711yescharacteristic
163C4#1686116811yescharacteristic
164C4#1686118311yescharacteristic
165C4#1686118811yescharacteristic
166C2#2254973611yescharacteristic
167C2#2254987211yescharacteristic
168C2#2254988511yescharacteristic
169C2#2255009812yes—
170C2#2255018312yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C4×Q8×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenterC2×C4×Q8D127
hypocenterC127#1805
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
14. C4×Q8×C127C4semidirect
86. C127#1805semidirect
95. C4×Q8Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C127⋊C8)2
C2×#168611332
#225497422
#225496382
C2×C4×Q8×C1272
#1805127