Finite Group Databasebeta

← #30650196

Subgroups — #30650196

Order 16256.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#22550499C212yesmaximal, complemented
2#22550165C212yesmaximal, complemented
3#22550159C212yesmaximal, complemented
4#22550129C211yesmaximal, characteristic
5#22550107C212yesmaximal, complemented
6#22550100C212yesmaximal, complemented
7#22550094C212yesmaximal, complemented
8#22549904C211yesmaximal, characteristic, complemented
9#470×C127C211yesmaximal, characteristic, complemented
10C2×#14523235C2212yes—
11C2×D1016C2211yescharacteristic
12C22×C127⋊C8C2211yescharacteristic
13#16861208C2212yes—
14#16861204—22no—
15#16861198C2212yes—
16#16861195—22no—
17#16861191—22no—
18#16861189—22no—
19#16861183C2212yes—
20Q8×Dic127C2212yes—
21#16861171C2212yes—
22#16861168C2211yescharacteristic
23#16861141C2211yescharacteristic
24#16861140C2212yes—
25Dic127⋊C8C2212yes—
26Dic127⋊C8C2212yes—
27#16861133C2212yes—
28#16861131C2212yes—
29#16861129C2212yes—
30C2×Q16×C127C2211yescharacteristic
31C2×M16×C127C2211yescharacteristic
32C22⋊Q8×C127C2212yes—
33#106×C127C2211yescharacteristic
34#103×C127C2212yes—
35C2×Dic254—22no—
36C2×D508C2312yes—
37C2×C22⋊D127—22no—
38C2×C4×D127—22no—
39#14523235—42no—
40D1016—41no—
41C2×C127⋊C8—41no—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127—22no—
44C4⋊Dic127D412yes—
45C4⋊Dic127C2312yes—
46Dic127⋊C4—22no—
47C22⋊Dic127D411yescharacteristic
48C22⋊Dic127D411yescharacteristic
49(C2×D127)⋊C4—22no—
50(C2×D127)⋊C4—22no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53C4×Dic127—22no—
54C4×Dic127C2312yes—
55C2×Q8×C127C2312yes—
56C22×C4×C127C2311yescharacteristic
57Q16×C127—41no—
58M16×C127—41no—
59C2×C8×C127C2311yescharacteristic
60C2×C8×C127C2311yescharacteristic
61C4⋊C4×C127—22no—
62C4⋊C4×C127C2312yes—
63C22⋊C4×C127—22no—
64C22×D127—22no—
65Dic254—82no—
66D508—22no—
67D508—22no—
68C22⋊D127—82no—
69C2×Dic127—22no—
70C2×Dic127C2×D412yes—
71C2×Dic127C2×D412yes—
72C4×D127—82no—
73C127⋊C8—22no—
74C23×C127C2×D411yescharacteristic
75Q8×C127—22no—
76Q8×C127—22no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2411yescharacteristic
83C8×C127—21no—
84C8×C127—21no—
85C22×C127—21no—
86C22×C127—22no—
87C22×C127C22×D411yescharacteristic
88C2×D127—42no—
89C2×D127—42no—
90C2×D127—42no—
91C4×C127—21no—
92C4×C127—42no—
93C4×C127—42no—
94C4×C127H(5,𝔽2)11yescharacteristic
95C4×C1272−1+411yescharacteristic
96Dic127—22no—
97Dic127—22no—
98Dic127—42no—
99C2×C127—41no—
100C2×C127#53711yescharacteristic
101C2×C127#57712yes—
102D127—82no—
103#3263—1271nomaximal, Sylow
104C127#326311yescharacteristic, complemented, Sylow
105#542—1272no—
106#476—1272no—
107#475—1272no—
108#470D12711yescharacteristic, complemented
109#469—1271no—
110#465—1272no—
111#464—1272no—
112#463—1272no—
113#424—1271no—
114C2×Q16C2×D12711yescharacteristic
115C2×SD16—1272no—
116C2×D8—1271no—
117C2×M16C2×D12711yescharacteristic
118C22×C8—1271no—
119C4⋊Q8—1272no—
120#126—2542no—
121#124—1272no—
122#124—2542no—
123#123—2542no—
124C22⋊Q8—2542no—
125C22⋊Q8C2×D12712yes—
126#121—1272no—
127C4×Q8—1272no—
128C4×D4—1272no—
129#117—1271no—
130#106—1271no—
131#106—1272no—
132#106C2×D12711yescharacteristic
133C4⋊C8—1272no—
134C4⋊C8—1272no—
135#103—1272no—
136#103C2×D12712yes—
137#102—1272no—
138#102—1272no—
139C2×Q8—2542no—
140C2×Q8C22×D12712yes—
141C2×D4—1272no—
142C2×D4—2542no—
143C22×C4—2542no—
144C22×C4C22×D12711yescharacteristic
145Q16—41no—
146SD16—5082no—
147D8—5081no—
148M16—41no—
149C2×C8—1272no—
150C2×C8—5081no—
151C2×C8C22×D12711yescharacteristic
152C2×C8C22×D12711yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—1272no—
155C4⋊C4—1272no—
156C4⋊C4—2542no—
157C4⋊C4—2542no—
158C4⋊C4C22×D12712yes—
159C22⋊C4—22no—
160C22⋊C4—1271no—
161C22⋊C4—1271no—
162C22⋊C4—2542no—
163C22⋊C4—2542no—
164C22⋊C4—2542no—
165C22⋊C4—2542no—
166C42—1272no—
167C42—2542no—
168C23—2542no—
169C23D4×D12711yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—10162no—
173D4—10162no—
174D4—2542no—
175D4—2542no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—1272no—
181C2×C4—1272no—
182C2×C4—10162no—
183C2×C4—2542no—
184C2×C4D4×D12711yescharacteristic
185C2×C4C23×D12711yescharacteristic
186C8—21no—
187C8—21no—
188C8—2542no—
189C22—22no—
190C22—21no—
191C22—5082no—
192C22—5082no—
193C22—5082no—
194C22C2×D4×D12711yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—5082no—
199C4—2542no—
200C4—2542no—
201C4#1686125411yescharacteristic
202C4#1686125611yescharacteristic
203C2—41no—
204C2—10162no—
205C2#2255045311yescharacteristic
206C2#2255068512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#470×C127C2
socleC22×C127C22×D4
prime radical#22550129C2
hypercenter#470D127
hypocenterC127#3263
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #22550499C2semidirect
2. #22550165C2semidirect
3. #22550159C2semidirect
5. #22550107C2semidirect
6. #22550100C2semidirect
7. #22550094C2semidirect
8. #22549904C2semidirect
9. #470×C127C2semidirect
104. C127#3263semidirect
108. #470D127semidirect

Maximal subgroups

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

subgroupindex
#225504992
#225501652
#225501592
#225501292
#225501072
#225501002
#225500942
#225499042
#470×C1272
#3263127