Finite Group Databasebeta

← #30637089

Subgroups — #30637089

Order 16256.

322 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 77 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×#16861183C211yesmaximal, characteristic
2#22549840C211yesmaximal, characteristic, complemented
3#22549599C214yesmaximal, complemented
4C2×C22⋊C8×C127C211yesmaximal, characteristic, complemented
5C22×D508C411yescharacteristic
6C22×C22⋊D127C412yes—
7C2×#14523232—21no—
8#16861183—21no—
9#16861183—21no—
10#16861183—21no—
11#16861183—21no—
12C2×C4⋊Dic127C411yescharacteristic
13C2×(C2×D127)⋊C4C2212yes—
14C22⋊(C127⋊C8)—21no—
15C22⋊(C127⋊C8)C2212yes—
16C23×C4×C127C2211yescharacteristic
17C22×C8×C127—21no—
18C22⋊C8×C127—21no—
19C22⋊C8×C127C2212yes—
20C23×D127C2×C412yes—
21C2×D508—21no—
22C2×D508—21no—
23C2×D508—42no—
24C2×C22⋊D127—22no—
25C2×C22⋊D127—22no—
26C2×C22⋊D127—44no—
27C22×Dic127C2×C412yes—
28#14523232—42no—
29C2×C127⋊C8—22no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32(C2×D127)⋊C4—22no—
33(C2×D127)⋊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
42C2×C8×C127—22no—
43C2×C8×C127—42no—
44C22×D127—22no—
45C22×D127—24no—
46C22×D127—42no—
47C22×D127—42no—
48D508—42no—
49D508—42no—
50C22⋊D127—44no—
51C22⋊D127—44no—
52C2×Dic127—22no—
53C2×Dic127—42no—
54C127⋊C8—42no—
55C23×C127—22no—
56C23×C127—22no—
57C23×C127—41no—
58C23×C127C22⋊C411yescharacteristic
59C23×C127C22⋊C411yescharacteristic
60C23×C127C22×C411yescharacteristic
61C2×C4×C127—22no—
62C2×C4×C127—22no—
63C2×C4×C127—22no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127C22⋊C411yescharacteristic
68C2×C4×C127C22⋊C411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—42no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127—21no—
78C22×C127—21no—
79C22×C127—44no—
80C22×C127#10012yes—
81C22×C127C2×C22⋊C411yescharacteristic
82C22×C127#13112yes—
83C2×D127—42no—
84C2×D127—42no—
85C2×D127—42no—
86C2×D127—84no—
87C4×C127—21no—
88C4×C127—21no—
89C4×C127—21no—
90C4×C127#9912yes—
91Dic127—42no—
92C2×C127—22no—
93C2×C127—22no—
94C2×C127—41no—
95C2×C127#40811yescharacteristic
96C2×C127C2×#9911yescharacteristic
97C2×C127C2×#10011yescharacteristic
98D127—82no—
99#1471—1271nomaximal, Sylow
100C127#147111yescharacteristic, complemented, Sylow
101C2×#121—1271no—
102#407—1271no—
103C2×C22⋊C8D12711yescharacteristic, complemented
104#323—1274no—
105C22×D4—1272no—
106C22×D4—1271no—
107C23×C4C2×D12711yescharacteristic
108C2×M16—2541no—
109C22×C8—21no—
110#121—2541no—
111#121—2541no—
112#121—2541no—
113#121—2541no—
114C2×C4⋊C4—1271no—
115C2×C22⋊C4—1272no—
116C22⋊C8—21no—
117C22⋊C8—1272no—
118C22⋊C8—2541no—
119C22⋊C8C2×D12712yes—
120C24—1272no—
121C24C4×D12711yescharacteristic
122C2×D4—5082no—
123C2×D4—5084no—
124C2×D4—2542no—
125C2×D4—2542no—
126C2×D4—2541no—
127C2×D4—2541no—
128C22×C4—42no—
129C22×C4—1272no—
130C22×C4C4×D12711yescharacteristic
131C22×C4C22⋊D12711yescharacteristic
132C22×C4C22⋊D12711yescharacteristic
133C22×C4D50811yescharacteristic
134C22×C4D50811yescharacteristic
135C22×C4C22×D12711yescharacteristic
136M16—5082no—
137C2×C8—22no—
138C2×C8—42no—
139C2×C8—2542no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C22⋊C4—2542no—
143C22⋊C4—2542no—
144C23—22no—
145C23—22no—
146C23—41no—
147C23—5082no—
148C23—5082no—
149C23—2544no—
150C23—2542no—
151C23(C2×D127)⋊C411yescharacteristic
152C23(C2×D127)⋊C411yescharacteristic
153C23C2×C4×D12711yescharacteristic
154D4—5082no—
155D4—5084no—
156D4—5084no—
157D4—5082no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—5082no—
165C2×C4—2542no—
166C2×C4(C2×D127)⋊C411yescharacteristic
167C2×C4(C2×D127)⋊C411yescharacteristic
168C2×C4C2×C22⋊D12711yescharacteristic
169C2×C4C2×D50811yescharacteristic
170C8—42no—
171C8—5082no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—44no—
180C22—10164no—
181C22—5082no—
182C22—5082no—
183C22—5082no—
184C22#1686112512yes—
185C22C2×(C2×D127)⋊C411yescharacteristic
186C22#1686121512yes—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—5082no—
191C4#1686112312yes—
192C2—22no—
193C2—22no—
194C2—41no—
195C2—10162no—
196C2#2254984211yescharacteristic
197C2C2×#1686112311yescharacteristic
198C2C2×#1686112511yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C22⋊C8×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#16861183C2
cosocleC22×C4×C127C23
hypercenterC2×C22⋊C8D127
hypocenterC127#1471
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #22549840C2semidirect
3. #22549599C2semidirect
4. C2×C22⋊C8×C127C2semidirect
100. C127#1471semidirect
103. C2×C22⋊C8D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
#225498402
#225495992
C2×C22⋊C8×C1272
#1471127