Finite Group Databasebeta

← #30637686

Subgroups — #30637686

Order 16256.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550502C211yesmaximal, characteristic
2#22550032C212yesmaximal, complemented
3#22549612C212yesmaximal, complemented
4#22549600C211yesmaximal, characteristic
5#393×C127C211yesmaximal, characteristic
6C2×#14523238—22no—
7#16861188—21no—
8#16861188C2212yes—
9#16861186—22no—
10#16861186—22no—
11C2×C22⋊Dic127—21no—
12#16861133—22no—
13C22⋊(C127⋊C8)C2212yes—
14C22×Q8×C127C2211yescharacteristic
15C2×C22⋊C4×C127—21no—
16C2×C22⋊C4×C127C2211yescharacteristic
17#95×C127—21no—
18#95×C127C2211yescharacteristic
19C2×Dic254—21no—
20C2×Dic254D412yes—
21C22×Dic127—21no—
22C22×Dic127—21no—
23#14523238—44no—
24C2×C127⋊C8—22no—
25C4⋊Dic127—21no—
26C4⋊Dic127D412yes—
27Dic127⋊C4—22no—
28Dic127⋊C4—22no—
29C22⋊Dic127—22no—
30C22⋊Dic127—22no—
31C24×C127D411yescharacteristic
32C2×Q8×C127—22no—
33C2×Q8×C127—42no—
34C22×C4×C127—21no—
35C22×C4×C127—21no—
36C22×C4×C127D411yescharacteristic
37C22×C4×C127C2311yescharacteristic
38C22⋊C4×C127—22no—
39C22⋊C4×C127—42no—
40Dic254—42no—
41Dic254—81no—
42C2×Dic127—22no—
43C2×Dic127—22no—
44C2×Dic127—81no—
45C2×Dic127—81no—
46C127⋊C8—42no—
47C23×C127—21no—
48C23×C127—21no—
49C23×C127—41no—
50C23×C127—41no—
51C23×C127Q1612yescomplemented
52C23×C127C2×D411yescharacteristic
53Q8×C127—24no—
54Q8×C127—42no—
55C2×C4×C127—22no—
56C2×C4×C127—21no—
57C2×C4×C127—41no—
58C2×C4×C127—41no—
59C2×C4×C127—41no—
60C2×C4×C127—41no—
61C2×C4×C127—41no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127C2×D412yes—
65C22×C127—21no—
66C22×C127—21no—
67C22×C127—22no—
68C22×C127—42no—
69C22×C127—41no—
70C22×C127—41no—
71C22×C127—81no—
72C22×C127#12011yescharacteristic
73C22×C127C2×Q1611yescharacteristic
74C22×C127#13711yescharacteristic
75C4×C127—22no—
76C4×C127—41no—
77C4×C127—42no—
78C4×C127—81no—
79Dic127—42no—
80Dic127—42no—
81C2×C127—21no—
82C2×C127—21no—
83C2×C127—42no—
84C2×C127#45111yescharacteristic
85C2×C127#45511yescharacteristic
86C2×C127B(4,𝔽2)11yescharacteristic
87#1602—1271nomaximal, Sylow
88C127#160211yescharacteristic, complemented, Sylow
89#543—1271no—
90#451—1272no—
91#393D12711yescharacteristic
92#328—1272no—
93#323—1271no—
94C22×Q8C2×D12711yescharacteristic
95C2×Q16—2542no—
96C22⋊Q8—1272no—
97C22⋊Q8—2542no—
98C22⋊Q8—2542no—
99C22⋊Q8—2541no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—2541no—
102C2×C22⋊C4C2×D12711yescharacteristic
103#103—2542no—
104C22⋊C8—1272no—
105#95—21no—
106#95C2×D12711yescharacteristic
107C24C22⋊D12711yescharacteristic
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—1272no—
111C2×Q8—2541no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—2541no—
115C22×C4—2541no—
116C22×C4C22⋊D12711yescharacteristic
117C22×C4C22×D12711yescharacteristic
118Q16—5084no—
119C2×C8—2542no—
120C4⋊C4—1272no—
121C4⋊C4—2542no—
122C4⋊C4—2541no—
123C4⋊C4—2542no—
124C22⋊C4—22no—
125C22⋊C4—42no—
126C22⋊C4—2542no—
127C22⋊C4—2542no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1452323812yescomplemented
133C23C2×C22⋊D12711yescharacteristic
134Q8—24no—
135Q8—42no—
136Q8—10161no—
137Q8—5082no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—10161no—
148C2×C4—10161no—
149C2×C4—2542no—
150C2×C4—2542no—
151C2×C4D4×D12712yes—
152C8—5082no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1686117711yescharacteristic
161C22C2×#1452323811yescharacteristic
162C22#1686123911yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—81no—
167C4—5082no—
168C4—5082no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2255003111yescharacteristic
173C2#2255005311yescharacteristic
174C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#393×C127C2
socleC22×C127#120
prime radicalC24×C127D4
hypercenter#393D127
hypocenterC127#1602
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #22550032C2semidirect
3. #22549612C2semidirect
51. C23×C127Q16semidirect
88. C127#1602semidirect
132. C23#14523238semidirect

Maximal subgroups

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

subgroupindex
#225505022
#225500322
#225496122
#225496002
#393×C1272
#1602127