Finite Group Databasebeta

← #30638720

Subgroups — #30638720

Order 16256.

336 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 95 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×D4×Dic127C211yesmaximal, characteristic
2#22549899C212yesmaximal, complemented
3C2×#16861126C211yesmaximal, characteristic
4#22549644C212yesmaximal, complemented
5C2×#100×C127C211yesmaximal, characteristic
6C23×Dic127C412yes—
7C2×#14523232C2212yes—
8D4×Dic127—22no—
9D4×Dic127—22no—
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C22⋊Dic127C412yes—
12C2×C4×Dic127C2211yescharacteristic
13Dic127⋊C8—22no—
14Dic127⋊C8—22no—
15#16861126—22no—
16C22×D4×C127C2211yescharacteristic
17C2×M16×C127C2212yes—
18#100×C127—22no—
19C22×Dic127—22no—
20C22×Dic127—22no—
21C22×Dic127—44no—
22C22×Dic127C2×C411yescharacteristic
23C22×Dic127C2×C412yes—
24C22×Dic127C2×C411yescharacteristic
25#14523232—22no—
26#14523232—22no—
27C2×C127⋊C8—22no—
28C4⋊Dic127—22no—
29C22⋊Dic127—24no—
30C4×Dic127D412yes—
31C4×Dic127D412yes—
32C24×C127C2×C412yes—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C2×D4×C127D412yes—
36C2×D4×C127Q812yes—
37C22×C4×C127C2311yescharacteristic
38M16×C127—22no—
39M16×C127—22no—
40C2×C8×C127—22no—
41C2×Dic127—22no—
42C2×Dic127—22no—
43C2×Dic127—41no—
44C2×Dic127—44no—
45C2×Dic127—44no—
46C2×Dic127—42no—
47C2×Dic127C22⋊C412yes—
48C2×Dic127C22⋊C411yescharacteristic
49C2×Dic127C22⋊C411yescharacteristic
50C127⋊C8—42no—
51C23×C127—22no—
52C23×C127—22no—
53C23×C127—42no—
54C23×C127—42no—
55C23×C127C4⋊C414yes—
56C23×C127C22×C411yescharacteristic
57D4×C127—24no—
58D4×C127—24no—
59C2×C4×C127C2×D412yes—
60C2×C4×C127C2×D411yescharacteristic
61C2×C4×C127C2×Q811yescharacteristic
62C2×C4×C127#4112yes—
63C8×C127—42no—
64C22×C127—21no—
65C22×C127—21no—
66C22×C127—24no—
67C22×C127—44no—
68C22×C127—44no—
69C22×C127—42no—
70C22×C127—42no—
71C22×C127C2×C22⋊C411yescharacteristic
72C22×C127C2×C4⋊C411yescharacteristic
73C22×C127#11711yescharacteristic
74C4×C127#12112yes—
75C4×C127C22⋊Q812yes—
76Dic127—21no—
77Dic127—22no—
78Dic127—42no—
79Dic127#10012yescomplemented
80C2×C127—21no—
81C2×C127—21no—
82C2×C127—44no—
83C2×C127#38011yescharacteristic
84C2×C127C2×#10011yescharacteristic
85C2×C127#41311yescharacteristic
86#1800—1271nomaximal, Sylow
87C127#180011yescharacteristic, complemented, Sylow
88C2×C4×D4—1271no—
89#423—1272no—
90C2×#100—1271no—
91C2×#100D12711yescharacteristic
92#343—1272no—
93C22×D4C2×D12711yescharacteristic
94C23×C4—1272no—
95C2×M16—1272no—
96C2×M16C2×D12712yes—
97C4×D4—2542no—
98C4×D4—2542no—
99C2×C4⋊C4—1271no—
100C2×C22⋊C4—1272no—
101C2×C42—1271no—
102C4⋊C8—2542no—
103C4⋊C8—2542no—
104#100—22no—
105#100—2542no—
106C24C4×D12712yes—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4C22⋊D12712yes—
110C2×D4Dic25412yes—
111C22×C4—1271no—
112C22×C4—1271no—
113C22×C4—1272no—
114C22×C4—5084no—
115C22×C4—2542no—
116C22×C4—2542no—
117C22×C4C22×D12711yescharacteristic
118M16—22no—
119M16—22no—
120M16—2542no—
121M16—2542no—
122C2×C8—22no—
123C2×C8—2542no—
124C4⋊C4—2542no—
125C22⋊C4—2544no—
126C42—1272no—
127C42—1272no—
128C23—22no—
129C23—22no—
130C23—42no—
131C23—42no—
132C23Dic127⋊C414yes—
133C23C2×C4×D12711yescharacteristic
134D4—24no—
135D4—24no—
136C2×C4—1271no—
137C2×C4—1272no—
138C2×C4—1271no—
139C2×C4—5084no—
140C2×C4—5084no—
141C2×C4—5082no—
142C2×C4—5081no—
143C2×C4—2542no—
144C2×C4—2542no—
145C2×C4D4×D12712yes—
146C2×C4C2×C22⋊D12711yescharacteristic
147C2×C4C2×Dic25411yescharacteristic
148C2×C4#1452324712yes—
149C8—42no—
150C8—5082no—
151C22—21no—
152C22—21no—
153C22—24no—
154C22—44no—
155C22—44no—
156C22—42no—
157C22—42no—
158C22C22⋊C4×D12711yescharacteristic
159C22C2×Dic127⋊C411yescharacteristic
160C22#1686116511yescharacteristic
161C4—1272no—
162C4—5082no—
163C4—2541no—
164C4—2542no—
165C4#1686118112yes—
166C4#1686118612yes—
167C2—21no—
168C2—21no—
169C2—44no—
170C2#2254973411yescharacteristic
171C2#100×D12711yescharacteristic
172C2#2254985711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1800
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
4. #22549644C2semidirect
79. Dic127#100semidirect
87. C127#1800semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1272
#225498992
C2×#168611262
#225496442
C2×#100×C1272
#1800127