Finite Group Databasebeta

← #30637059

Subgroups — #30637059

Order 16256.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C22×Dic127⋊C4C211yesmaximal, characteristic
2#22549840C211yesmaximal, characteristic
3#22549601C214yesmaximal, complemented
4#407×C127C211yesmaximal, characteristic
5C23×Dic127C411yescharacteristic
6C23×Dic127C411yescharacteristic
7C2×#14523232—21no—
8C2×Dic127⋊C4—21no—
9C2×Dic127⋊C4—21no—
10C2×Dic127⋊C4—21no—
11C2×Dic127⋊C4—21no—
12C2×Dic127⋊C4C412yes—
13C2×Dic127⋊C4C2212yes—
14C22⋊(C127⋊C8)—21no—
15C22⋊(C127⋊C8)C2212yes—
16C23×C4×C127C2211yescharacteristic
17C2×M16×C127—21no—
18C22⋊C8×C127—21no—
19C22⋊C8×C127C2212yes—
20C22×Dic127—24no—
21C22×Dic127—21no—
22C22×Dic127—21no—
23C22×Dic127—21no—
24C22×Dic127—21no—
25C22×Dic127—42no—
26C22×Dic127C2×C412yes—
27C22×Dic127C2×C412yes—
28#14523232—42no—
29C2×C127⋊C8—22no—
30Dic127⋊C4—22no—
31Dic127⋊C4—22no—
32Dic127⋊C4—22no—
33Dic127⋊C4—22no—
34C24×C127C2×C411yescharacteristic
35C22×C4×C127—42no—
36C22×C4×C127C2×C411yescharacteristic
37C22×C4×C127D411yescharacteristic
38C22×C4×C127D411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C22×C4×C127D411yescharacteristic
41C22×C4×C127C2311yescharacteristic
42M16×C127—42no—
43C2×C8×C127—22no—
44C2×Dic127—24no—
45C2×Dic127—24no—
46C2×Dic127—22no—
47C2×Dic127—22no—
48C2×Dic127—22no—
49C2×Dic127—42no—
50C2×Dic127—42no—
51C2×Dic127—42no—
52C2×Dic127M1614yescomplemented
53C127⋊C8—42no—
54C23×C127—22no—
55C23×C127—22no—
56C23×C127—22no—
57C23×C127C22⋊C411yescharacteristic
58C23×C127C22⋊C411yescharacteristic
59C23×C127C22×C411yescharacteristic
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C22⋊C411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C8×C127—42no—
71C22×C127—28no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127—21no—
78C22×C127#9912yes—
79C22×C127#10112yes—
80C22×C127C2×C22⋊C411yescharacteristic
81C22×C127C2×M1612yes—
82C4×C127—21no—
83C4×C127—21no—
84C4×C127—21no—
85C4×C127—21no—
86Dic127—24no—
87Dic127—42no—
88C2×C127—22no—
89C2×C127—22no—
90C2×C127—22no—
91C2×C127#40711yescharacteristic
92C2×C127C2×#9911yescharacteristic
93C2×C127C2×#10111yescharacteristic
94#1463—1271nomaximal, Sylow
95C127#146311yescharacteristic, complemented, Sylow
96C22×C4⋊C4—1271no—
97#407—1271no—
98#407D12711yescharacteristic
99#324—1274no—
100C23×C4—1271no—
101C23×C4—1271no—
102C23×C4C2×D12711yescharacteristic
103C2×M16—21no—
104C2×M16—2541no—
105C2×C4⋊C4—1272no—
106C2×C4⋊C4—1272no—
107C2×C4⋊C4—2541no—
108C2×C4⋊C4—2541no—
109C2×C4⋊C4—2541no—
110C2×C4⋊C4—2541no—
111C22⋊C8—21no—
112C22⋊C8—1272no—
113C22⋊C8—2541no—
114C22⋊C8C2×D12712yes—
115C24C4×D12711yescharacteristic
116C22×C4—42no—
117C22×C4—1272no—
118C22×C4—1272no—
119C22×C4—5082no—
120C22×C4—2544no—
121C22×C4—2541no—
122C22×C4—2541no—
123C22×C4—2541no—
124C22×C4—2541no—
125C22×C4C4×D12711yescharacteristic
126C22×C4C22⋊D12711yescharacteristic
127C22×C4C22⋊D12711yescharacteristic
128C22×C4D50811yescharacteristic
129C22×C4D50811yescharacteristic
130C22×C4C22×D12711yescharacteristic
131M16—42no—
132M16—5082no—
133C2×C8—22no—
134C2×C8—2542no—
135C4⋊C4—2542no—
136C4⋊C4—2542no—
137C4⋊C4—2542no—
138C4⋊C4—2542no—
139C23—22no—
140C23—22no—
141C23—22no—
142C23(C2×D127)⋊C411yescharacteristic
143C23(C2×D127)⋊C411yescharacteristic
144C23C2×C4×D12711yescharacteristic
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—1274no—
152C2×C4—5082no—
153C2×C4—5082no—
154C2×C4—5082no—
155C2×C4—2542no—
156C2×C4—2544no—
157C2×C4—2544no—
158C2×C4—2542no—
159C2×C4—2542no—
160C2×C4(C2×D127)⋊C411yescharacteristic
161C2×C4(C2×D127)⋊C411yescharacteristic
162C2×C4C2×C22⋊D12711yescharacteristic
163C2×C4C2×D50811yescharacteristic
164C8—42no—
165C8—5082no—
166C22—28no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22#1686112312yes—
174C22#1686112712yes—
175C22C2×(C2×D127)⋊C411yescharacteristic
176C22M16×D12712yes—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—5082no—
182C4—2544no—
183C2—22no—
184C2—22no—
185C2—22no—
186C2#2254983911yescharacteristic
187C2C2×#1686112311yescharacteristic
188C2C2×#1686112711yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC24×C127C2×C4
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1463
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #22549601C2semidirect
52. C2×Dic127M16semidirect
95. C127#1463semidirect

Maximal subgroups

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

subgroupindex
C22×Dic127⋊C42
#225498402
#225496012
#407×C1272
#1463127