Finite Group Databasebeta

← #30637069

Subgroups — #30637069

Order 16256.

300 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 69 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×(C22⋊D127)⋊C4C211yesmaximal, characteristic
2#22549930C212yesmaximal, complemented
3#22549600C211yesmaximal, characteristic, complemented
4#22549599C212yesmaximal, complemented
5#323×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127C411yescharacteristic
7C23×Dic127C411yescharacteristic
8C22×C4×D127C411yescharacteristic
9(C22⋊D127)⋊C4—22no—
10(C22⋊D127)⋊C4—22no—
11C2×Dic127⋊C4C411yescharacteristic
12C2×(C2×D127)⋊C4C2211yescharacteristic
13C2×C4×Dic127C2211yescharacteristic
14Dic127⋊C8—22no—
15C22⋊(C127⋊C8)C2212yes—
16#16861119—22no—
17C2×C22⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C23×D127C2×C411yescharacteristic
20C2×C22⋊D127—22no—
21C2×C22⋊D127—22no—
22C2×C22⋊D127—22no—
23C22×Dic127—21no—
24C22×Dic127—21no—
25C22×Dic127—42no—
26C22×Dic127C2×C411yescharacteristic
27C22×Dic127C2×C411yescharacteristic
28C22×Dic127C2×C411yescharacteristic
29C2×C4×D127—21no—
30C2×C4×D127—21no—
31C2×C4×D127—42no—
32C2×C127⋊C8—22no—
33Dic127⋊C4—22no—
34(C2×D127)⋊C4—22no—
35C4×Dic127D412yes—
36C4×Dic127D412yes—
37C24×C127C2×C411yescharacteristic
38C22×C4×C127C2×C411yescharacteristic
39C22×C4×C127C2311yescharacteristic
40C2×C8×C127—22no—
41C22⋊C4×C127—22no—
42C22×D127—21no—
43C22×D127—21no—
44C22×D127—41no—
45C22×D127—41no—
46C22×D127M1612yes—
47C22⋊D127—28no—
48C2×Dic127—21no—
49C2×Dic127—21no—
50C2×Dic127—22no—
51C2×Dic127—21no—
52C2×Dic127—42no—
53C2×Dic127—42no—
54C2×Dic127—41no—
55C2×Dic127C22⋊C412yes—
56C2×Dic127C22⋊C411yescharacteristic
57C2×Dic127C22⋊C411yescharacteristic
58C4×D127—42no—
59C4×D127—42no—
60C127⋊C8—42no—
61C23×C127—21no—
62C23×C127—21no—
63C23×C127—41no—
64C23×C127—41no—
65C23×C127M1612yes—
66C23×C127C22×C411yescharacteristic
67C2×C4×C127—21no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127C2×D412yes—
71C8×C127—42no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127—22no—
75C22×C127—42no—
76C22×C127—42no—
77C22×C127—41no—
78C22×C127—41no—
79C22×C127C2×C22⋊C411yescharacteristic
80C22×C127C2×M1611yescharacteristic
81C22×C127C2×M1611yescharacteristic
82C2×D127—22no—
83C2×D127—42no—
84C2×D127—41no—
85C2×D127—41no—
86C2×D127—42no—
87C4×C127—22no—
88C4×C127—41no—
89Dic127—22no—
90Dic127—21no—
91Dic127—41no—
92Dic127#10012yes—
93C2×C127—21no—
94C2×C127—21no—
95C2×C127—42no—
96C2×C127#40711yescharacteristic
97C2×C127#41011yescharacteristic
98C2×C127C2×#10011yescharacteristic
99D127—42no—
100#1466—1271nomaximal, Sylow
101C127#146611yescharacteristic, complemented, Sylow
102C2×C4×D4—1271no—
103#432—1272no—
104#323—1271no—
105#323—1272no—
106#323D12711yescharacteristic, complemented
107C22×D4—1271no—
108C23×C4—1271no—
109C23×C4—1271no—
110C4×D4—2542no—
111C4×D4—2542no—
112C2×C4⋊C4—1271no—
113C2×C22⋊C4—1271no—
114C2×C22⋊C4C2×D12711yescharacteristic
115C2×C42—1271no—
116C4⋊C8—2542no—
117C22⋊C8—1272no—
118C22⋊C8C2×D12712yes—
119#97—2542no—
120C24—1271no—
121C24C4×D12711yescharacteristic
122C2×D4—2542no—
123C2×D4—2542no—
124C2×D4—2542no—
125C22×C4—1271no—
126C22×C4—1271no—
127C22×C4—1271no—
128C22×C4—5082no—
129C22×C4—5082no—
130C22×C4—2541no—
131C22×C4—2541no—
132C22×C4—2541no—
133C22×C4—2541no—
134C22×C4C4×D12711yescharacteristic
135C22×C4C22×D12711yescharacteristic
136C2×C8—22no—
137C2×C8—2542no—
138C4⋊C4—2542no—
139C22⋊C4—22no—
140C22⋊C4—2542no—
141C42—1272no—
142C42—1272no—
143C23—21no—
144C23—21no—
145C23—41no—
146C23—41no—
147C23—1272no—
148C23—5081no—
149C23—5081no—
150C23—2541no—
151C23—2541no—
152C23#1452323112yes—
153C23C2×C4×D12711yescharacteristic
154D4—2548no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—1271no—
159C2×C4—1272no—
160C2×C4—1271no—
161C2×C4—5082no—
162C2×C4—5082no—
163C2×C4—5082no—
164C2×C4—5082no—
165C2×C4—5081no—
166C2×C4—2542no—
167C2×C4—2541no—
168C2×C4—2541no—
169C2×C4—2541no—
170C2×C4D4×D12712yes—
171C8—42no—
172C8—5082no—
173C22—21no—
174C22—21no—
175C22—22no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—5082no—
181C22—5082no—
182C22—5081no—
183C22—5081no—
184C22—2542no—
185C22C22⋊C4×D12711yescharacteristic
186C22M16×D12711yescharacteristic
187C22C2×#1452323111yescharacteristic
188C4—22no—
189C4—41no—
190C4—1272no—
191C4—5081no—
192C4—2542no—
193C4—2541no—
194C2—21no—
195C2—21no—
196C2—42no—
197C2—5082no—
198C2#2254983811yescharacteristic
199C2#2254984811yescharacteristic
200C2#100×D12711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×C22⋊D127C4
cosocleC22×C4×C127C23
hypercenter#323D127
hypocenterC127#1466
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22549930C2semidirect
3. #22549600C2semidirect
4. #22549599C2semidirect
5. #323×C127C2semidirect
101. C127#1466semidirect
106. #323D127semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D127)⋊C42
#225499302
#225496002
#225495992
#323×C1272
#1466127