Finite Group Databasebeta

← #30637217

Subgroups — #30637217

Order 16256.

238 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 73 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#22550410C211yesmaximal, characteristic
2#22549722C211yesmaximal, characteristic, complemented
3#22549614C212yesmaximal, complemented
4#22549611C212yesmaximal
5#431×C127C211yesmaximal, characteristic, complemented
6#16861203—21no—
7#16861201C411yescharacteristic, complemented
8#16861197C411yescharacteristic, complemented
9#16861192—21no—
10#16861188C2211yescharacteristic
11#16861183C2211yescharacteristic
12C4×Dic254C411yescharacteristic, complemented
13C4×D508C411yescharacteristic, complemented
14C4×C22⋊D127—21no—
15#16861166—21no—
16C2×C4×Dic127—21no—
17C127⋊#95—21no—
18C127⋊#95C2212yes—
19C2×C42×C127C2211yescharacteristic
20C4⋊C8×C127—21no—
21C22⋊C8×C127C2212yes—
22C4×C8×C127—21no—
23C2×Dic254C2×C411yescharacteristic
24C2×D508C2×C411yescharacteristic
25C2×C22⋊D127—21no—
26C22×Dic127—22no—
27C2×C4×D127—21no—
28C4⋊Dic127C2×C411yescharacteristic
29C4⋊Dic127C2×C411yescharacteristic
30Dic127⋊C4—21no—
31Dic127⋊C4—21no—
32Dic127⋊C4—21no—
33Dic127⋊C4—21no—
34C22⋊Dic127—21no—
35(C2×D127)⋊C4—21no—
36(C2×D127)⋊C4—21no—
37(C2×D127)⋊C4—21no—
38(C2×D127)⋊C4—21no—
39C4×Dic127—24no—
40C4×Dic127—21no—
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127C2311yescharacteristic
44C2×C8×C127—22no—
45C42×C127C2×C411yescharacteristic
46C42×C127C2×C411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127D411yescharacteristic
49C22×D127—21no—
50Dic254—41no—
51D508—41no—
52C22⋊D127—42no—
53C2×Dic127—24no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—42no—
58C2×Dic127—42no—
59C4×D127—42no—
60C23×C127C2×D411yescharacteristic
61C2×C4×C127—22no—
62C2×C4×C127—22no—
63C2×C4×C127—41no—
64C2×C4×C127C22⋊C411yescharacteristic
65C2×C4×C127C22⋊C411yescharacteristic
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C22⋊C411yescharacteristic
68C2×C4×C127C22×C411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C8×C127—42no—
71C22×C127—21no—
72C22×C127—21no—
73C22×C127C2×C22⋊C411yescharacteristic
74C22×C127#13512yes—
75C2×D127—41no—
76C2×D127—41no—
77C2×D127—41no—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127—21no—
83C4×C127#10412yes—
84Dic127—44no—
85Dic127—41no—
86Dic127—41no—
87Dic127—41no—
88C2×C127—22no—
89C2×C127#41011yescharacteristic
90C2×C127#41611yescharacteristic
91C2×C127C2×#10411yescharacteristic
92D127—81no—
93#1500—1271nomaximal, Sylow
94C127#150011yescharacteristic, complemented, Sylow
95#529—1271no—
96#431D12711yescharacteristic, complemented
97#376—1271no—
98#328—1272no—
99#327—1272no—
100#126—2541no—
101#125—1271no—
102#124—1271no—
103#123—2541no—
104C22⋊Q8—1271no—
105#121—1271no—
106C4×Q8—1271no—
107C4×D4—1271no—
108C4×D4—2541no—
109#117—2541no—
110C2×C42—2541no—
111C2×C42C2×D12711yescharacteristic
112C4⋊C8—21no—
113C22⋊C8C2×D12712yes—
114C4×C8—21no—
115#95—1272no—
116#95—2541no—
117C2×Q8—1271no—
118C2×D4—1271no—
119C2×D4—2541no—
120C22×C4—2542no—
121C22×C4—2541no—
122C22×C4D50811yescharacteristic
123C22×C4D50811yescharacteristic
124C22×C4C22×D12711yescharacteristic
125C2×C8—22no—
126C4⋊C4—1271no—
127C4⋊C4—1271no—
128C4⋊C4—2541no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4—2541no—
132C22⋊C4—2541no—
133C22⋊C4—2541no—
134C22⋊C4—2541no—
135C22⋊C4—2541no—
136C22⋊C4—2541no—
137C42—2544no—
138C42—2541no—
139C42C4×D12711yescharacteristic
140C42C4×D12711yescharacteristic
141C42C22⋊D12711yescharacteristic
142C42C22⋊D12711yescharacteristic
143C23—2541no—
144C23C2×D50811yescharacteristic
145Q8—5081no—
146D4—5081no—
147D4—5082no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—5082no—
152C2×C4—5082no—
153C2×C4—5082no—
154C2×C4—2544no—
155C2×C4—2541no—
156C2×C4—2541no—
157C2×C4—2541no—
158C2×C4(C2×D127)⋊C411yescharacteristic
159C2×C4(C2×D127)⋊C411yescharacteristic
160C2×C4(C2×D127)⋊C411yescharacteristic
161C2×C4(C2×D127)⋊C411yescharacteristic
162C2×C4C2×C4×D12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—5081no—
168C22—5081no—
169C22—5081no—
170C22C2×(C2×D127)⋊C411yescharacteristic
171C22#1686123112yes—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—5084no—
178C4—5081no—
179C4—5081no—
180C4—5081no—
181C4#1686113512yes—
182C2—22no—
183C2—10161no—
184C2#2254984911yescharacteristic
185C2#2254987311yescharacteristic
186C2C2×#1686113511yescharacteristic
subgroupG/H
centerC2×C4(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#431×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861183C22
cosocleC22×C4×C127C23
hypercenter#431D127
hypocenterC127#1500
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #22549722C2semidirect
3. #22549614C2semidirect
5. #431×C127C2semidirect
7. #16861201C4semidirect
8. #16861197C4semidirect
12. C4×Dic254C4semidirect
13. C4×D508C4semidirect
94. C127#1500semidirect
96. #431D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
#225497222
#225496142
#225496112
#431×C1272
#1500127