Finite Group Databasebeta

← #30637097

Subgroups — #30637097

Order 16256.

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

normal subgroups only

This group has 214 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×#16861203C211yesmaximal, characteristic
2#22549933C211yesmaximal, characteristic, complemented
3#22549601C212yesmaximal, complemented
4#22549599C212yesmaximal
5#431×C127C211yesmaximal, characteristic, complemented
6#16861203—21no—
7#16861203—21no—
8#16861203—21no—
9#16861203—21no—
10C2×Dic127⋊C4C411yescharacteristic
11C2×Dic127⋊C4C411yescharacteristic
12C2×Dic127⋊C4C2211yescharacteristic
13C2×(C2×D127)⋊C4C411yescharacteristic
14C2×(C2×D127)⋊C4C411yescharacteristic
15C2×(C2×D127)⋊C4C2211yescharacteristic
16C4⋊(C127⋊C8)—21no—
17C22⋊(C127⋊C8)C2212yes—
18#16861120—21no—
19C2×C42×C127C2211yescharacteristic
20C4⋊C8×C127—21no—
21C22⋊C8×C127C2212yes—
22C4×C8×C127—21no—
23C23×D127C2×C411yescharacteristic
24C22×Dic127C2×C411yescharacteristic
25C22×Dic127C2×C411yescharacteristic
26C22×Dic127C2×C411yescharacteristic
27C2×C127⋊C8—22no—
28Dic127⋊C4—21no—
29Dic127⋊C4—21no—
30Dic127⋊C4—21no—
31Dic127⋊C4—21no—
32Dic127⋊C4—21no—
33Dic127⋊C4—21no—
34(C2×D127)⋊C4—21no—
35(C2×D127)⋊C4—21no—
36(C2×D127)⋊C4—21no—
37(C2×D127)⋊C4—21no—
38(C2×D127)⋊C4—21no—
39(C2×D127)⋊C4—21no—
40C22×C4×C127C2×C411yescharacteristic
41C22×C4×C127C2×C411yescharacteristic
42C22×C4×C127C2311yescharacteristic
43C2×C8×C127—22no—
44C42×C127D411yescharacteristic
45C42×C127D411yescharacteristic
46C42×C127D411yescharacteristic
47C42×C127D411yescharacteristic
48C22×D127—21no—
49C22×D127—22no—
50C22×D127—41no—
51C22×D127—41no—
52C2×Dic127—21no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—22no—
56C2×Dic127—41no—
57C2×Dic127—41no—
58C127⋊C8—42no—
59C23×C127C22×C411yescharacteristic
60C2×C4×C127—22no—
61C2×C4×C127—41no—
62C2×C4×C127—41no—
63C2×C4×C127C22⋊C411yescharacteristic
64C2×C4×C127C22⋊C411yescharacteristic
65C2×C4×C127C22⋊C411yescharacteristic
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C8×C127—42no—
70C22×C127—21no—
71C22×C127—21no—
72C22×C127C2×C22⋊C411yescharacteristic
73C22×C127#13112yes—
74C2×D127—41no—
75C2×D127—41no—
76C2×D127—41no—
77C2×D127—82no—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127—21no—
83C4×C127—21no—
84Dic127—41no—
85Dic127—41no—
86Dic127—41no—
87C2×C127—22no—
88C2×C127#40811yescharacteristic
89C2×C127#41011yescharacteristic
90C2×C127#41311yescharacteristic
91D127—81no—
92#1473—1271nomaximal, Sylow
93C127#147311yescharacteristic, complemented, Sylow
94C2×#126—1271no—
95#432—1271no—
96#431D12711yescharacteristic, complemented
97#324—1272no—
98#323—1272no—
99#126—2541no—
100#126—2541no—
101#126—2541no—
102#126—2541no—
103C2×C4⋊C4—1271no—
104C2×C4⋊C4—1271no—
105C2×C4⋊C4—1271no—
106C2×C22⋊C4—1271no—
107C2×C22⋊C4—1271no—
108C2×C22⋊C4—1271no—
109C2×C42C2×D12711yescharacteristic
110C4⋊C8—21no—
111C4⋊C8—2541no—
112C22⋊C8—1272no—
113C22⋊C8C2×D12712yes—
114#97—2541no—
115C4×C8—21no—
116C24—1271no—
117C22×C4—1271no—
118C22×C4—1271no—
119C22×C4—1271no—
120C22×C4C4×D12711yescharacteristic
121C22×C4C4×D12711yescharacteristic
122C22×C4C22×D12711yescharacteristic
123C2×C8—22no—
124C2×C8—2542no—
125C4⋊C4—2541no—
126C4⋊C4—2541no—
127C4⋊C4—2541no—
128C4⋊C4—2541no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C22⋊C4—2541no—
132C22⋊C4—2541no—
133C22⋊C4—2541no—
134C22⋊C4—2541no—
135C22⋊C4—2541no—
136C22⋊C4—2541no—
137C42C22⋊D12711yescharacteristic
138C42C22⋊D12711yescharacteristic
139C42D50811yescharacteristic
140C42D50811yescharacteristic
141C23—5081no—
142C23—5081no—
143C23—2542no—
144C23—2541no—
145C23C2×C4×D12711yescharacteristic
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—5081no—
150C2×C4—5081no—
151C2×C4—2542no—
152C2×C4—2541no—
153C2×C4—2541no—
154C2×C4—2541no—
155C2×C4(C2×D127)⋊C411yescharacteristic
156C2×C4(C2×D127)⋊C411yescharacteristic
157C2×C4(C2×D127)⋊C411yescharacteristic
158C2×C4(C2×D127)⋊C411yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C2×C4C2×D50811yescharacteristic
161C8—42no—
162C8—5082no—
163C22—21no—
164C22—21no—
165C22—10162no—
166C22—5081no—
167C22—5081no—
168C22—5081no—
169C22C2×(C2×D127)⋊C411yescharacteristic
170C22#1686121512yes—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—5081no—
178C4—5081no—
179C4—5081no—
180C2—22no—
181C2—10161no—
182C2#2254984211yescharacteristic
183C2#2254984911yescharacteristic
184C2#2254985811yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#431×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127C2×C4
cosocleC22×C4×C127C23
hypercenter#431D127
hypocenterC127#1473
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #22549933C2semidirect
3. #22549601C2semidirect
5. #431×C127C2semidirect
93. C127#1473semidirect
96. #431D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612032
#225499332
#225496012
#225495992
#431×C1272
#1473127