Finite Group Databasebeta

← #30649021

Subgroups — #30649021

Order 16256.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550481C211yesmaximal, characteristic
2#22550200C212yesmaximal, complemented
3#22550183C212yesmaximal, complemented
4#22550076C212yesmaximal, complemented
5#22550015C212yesmaximal, complemented
6#22549974C212yesmaximal, complemented
7C4×#14523235C212yesmaximal, complemented
8#22549937C211yesmaximal, characteristic, complemented
9#519×C127C211yesmaximal, characteristic, complemented
10C2×#14523235C2212yes—
11C2×#14523235C2212yes—
12#16861206C2211yescharacteristic
13#16861201C2211yescharacteristic
14#16861194—22no—
15#16861183C2212yes—
16#16861182—22no—
17C4×D508C2212yes—
18#16861142C2212yes—
19#16861140C2212yes—
20C4⋊(C127⋊C8)C2212yes—
21#16861133C2212yes—
22#16861133C2212yes—
23#16861129C2212yes—
24#16861129C2212yes—
25C22⋊(C127⋊C8)C2212yes—
26#16861120C2211yescharacteristic
27C4×C127⋊C8C2211yescharacteristic
28C22×Q8×C127C2211yescharacteristic
29C4×Q8×C127—22no—
30C4×Q8×C127C2212yes—
31C4×Q8×C127C2212yes—
32#117×C127—22no—
33#117×C127C2211yescharacteristic
34#117×C127C2211yescharacteristic
35C2×D508—22no—
36C2×D508C2312yes—
37C2×C22⋊D127—22no—
38C2×C4×D127—22no—
39#14523235—24no—
40#14523235—24no—
41C2×C127⋊C8C2312yes—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127C2312yes—
44Dic127⋊C4—22no—
45(C2×D127)⋊C4—22no—
46(C2×D127)⋊C4—22no—
47C2×Q8×C127—24no—
48C2×Q8×C127—21no—
49C2×Q8×C127C2312yes—
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—22no—
53C4⋊C4×C127—22no—
54C4⋊C4×C127D412yes—
55C4⋊C4×C127C2312yes—
56C22⋊C4×C127—21no—
57C22⋊C4×C127D412yes—
58C42×C127—22no—
59C42×C127—22no—
60C42×C127—21no—
61C42×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63C22×D127—22no—
64D508—22no—
65D508—22no—
66D508—82no—
67C22⋊D127—82no—
68C2×Dic127—22no—
69C4×D127—82no—
70C127⋊C8—22no—
71C127⋊C8—22no—
72C23×C127C2×D411yescharacteristic
73Q8×C127—24no—
74Q8×C127—22no—
75Q8×C127#4114yes—
76C2×C4×C127—24no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127C2×D412yes—
81C2×C4×C127C2×D412yes—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2411yescharacteristic
84C22×C127—21no—
85C22×C127—22no—
86C22×C127C22×D411yescharacteristic
87C2×D127—42no—
88C2×D127—42no—
89C2×D127—42no—
90C4×C127—24no—
91C4×C127—21no—
92C4×C127—22no—
93C4×C127—22no—
94C4×C127—42no—
95C4×C127C2×#4112yes—
96Dic127—42no—
97C2×C127—41no—
98C2×C127#52511yescharacteristic
99C2×C127#57712yes—
100D127—82no—
101#3126—1271nomaximal, Sylow
102C127#312611yescharacteristic, complemented, Sylow
103#540—1271no—
104#519D12711yescharacteristic, complemented
105#482—1272no—
106#479—1272no—
107#460—1272no—
108#448—1272no—
109#440—1272no—
110C4×SD16—1272no—
111#433—1271no—
112C22×Q8C2×D12711yescharacteristic
113C2×SD16—1272no—
114C2×SD16—1272no—
115#127—1271no—
116#125—1271no—
117#123—2542no—
118#121—1272no—
119#121—2542no—
120C4×Q8—22no—
121C4×Q8C2×D12712yes—
122C4×Q8C2×D12712yes—
123C4×D4—1272no—
124#117—22no—
125#117C2×D12711yescharacteristic
126#117C2×D12711yescharacteristic
127#107—1272no—
128#106—1272no—
129C4⋊C8—1272no—
130#103—1272no—
131#103—1272no—
132#102—1272no—
133#102—1272no—
134C22⋊C8—1272no—
135#97—1271no—
136C4×C8—1271no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D12712yes—
140C2×D4—1272no—
141C2×D4—2542no—
142C2×D4—2542no—
143C22×C4—21no—
144C22×C4—2542no—
145C22×C4C22×D12711yescharacteristic
146SD16—2544no—
147SD16—2544no—
148C2×C8—1272no—
149C2×C8—1272no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—1272no—
153C4⋊C4—2542no—
154C4⋊C4C22⋊D12712yes—
155C4⋊C4C22×D12712yes—
156C22⋊C4—21no—
157C22⋊C4—2542no—
158C22⋊C4—2542no—
159C22⋊C4C22⋊D12712yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D12711yescharacteristic
164C42C22×D12711yescharacteristic
165C23—2542no—
166C23C2×C22⋊D12711yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1452324814yes—
170D4—10162no—
171D4—10162no—
172D4—2542no—
173D4—2542no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—10162no—
179C2×C4—2542no—
180C2×C4C2×C22⋊D12712yes—
181C2×C4C2×C22⋊D12712yes—
182C2×C4C2×C22⋊D12711yescharacteristic
183C2×C4C23×D12711yescharacteristic
184C8—2542no—
185C8—2542no—
186C22—21no—
187C22—22no—
188C22—5082no—
189C22—5082no—
190C22—5082no—
191C22C22×C22⋊D12711yescharacteristic
192C4—24no—
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—5082no—
198C4C2×#1452324812yes—
199C2—41no—
200C2—10162no—
201C2#2255038711yescharacteristic
202C2#2255068412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#519×C127C2
socleC22×C127C22×D4
prime radical#22550481C2
hypercenter#519D127
hypocenterC127#3126
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #22550200C2semidirect
3. #22550183C2semidirect
4. #22550076C2semidirect
5. #22550015C2semidirect
6. #22549974C2semidirect
7. C4×#14523235C2semidirect
8. #22549937C2semidirect
9. #519×C127C2semidirect
102. C127#3126semidirect
104. #519D127semidirect

Maximal subgroups

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

subgroupindex
#225504812
#225502002
#225501832
#225500762
#225500152
#225499742
C4×#145232352
#225499372
#519×C1272
#3126127