Finite Group Databasebeta

← #30650165

Subgroups — #30650165

Order 16256.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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, complemented
2#22550176C212yesmaximal, complemented
3#22550173C212yesmaximal, complemented
4#22550127C212yesmaximal, complemented
5#22550110C212yesmaximal, complemented
6#22550071C212yesmaximal, complemented
7#22550068C212yesmaximal, complemented
8#22549903C211yesmaximal, characteristic, complemented
9#540×C127C211yesmaximal, characteristic, complemented
10C2×#14523233C2212yes—
11C2×#14523233C2212yes—
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861206C2211yescharacteristic
15#16861201C2211yescharacteristic
16#16861194—22no—
17#16861183C2212yes—
18#16861182—22no—
19C4×D508C2212yes—
20#16861142C2212yes—
21#16861140C2212yes—
22C4⋊(C127⋊C8)C2212yes—
23C4⋊(C127⋊C8)C2212yes—
24#16861130C2212yes—
25#16861130C2212yes—
26#16861129C2212yes—
27#16861129C2212yes—
28#127×C127C2211yescharacteristic
29#125×C127C2211yescharacteristic
30#123×C127—22no—
31#121×C127—22no—
32#121×C127C2212yes—
33C4×D4×C127C2212yes—
34#117×C127C2211yescharacteristic
35C2×D508—22no—
36C2×D508C2312yes—
37C2×C22⋊D127—22no—
38C2×C4×D127—22no—
39#14523233—42no—
40#14523233—42no—
41#14523232—41no—
42C2×C127⋊C8—41no—
43C2×C127⋊C8C2312yes—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127C2312yes—
46Dic127⋊C4—22no—
47(C2×D127)⋊C4—22no—
48(C2×D127)⋊C4—22no—
49C2×D4×C127—22no—
50C2×D4×C127—22no—
51C2×D4×C127C2312yes—
52C22×C4×C127—22no—
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—22no—
55C4⋊C4×C127D412yes—
56C4⋊C4×C127C2312yes—
57C22⋊C4×C127—22no—
58C22⋊C4×C127—22no—
59C22⋊C4×C127D412yes—
60C42×C127C2311yescharacteristic
61C42×C127C2311yescharacteristic
62C22×D127—22no—
63D508—22no—
64D508—22no—
65D508—82no—
66C22⋊D127—82no—
67C2×Dic127—22no—
68C4×D127—82no—
69C127⋊C8—22no—
70C127⋊C8—22no—
71C23×C127—22no—
72C23×C127C2×D411yescharacteristic
73D4×C127—22no—
74D4×C127—22no—
75D4×C127—82no—
76D4×C127—82no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—22no—
80C2×C4×C127—82no—
81C2×C4×C127C2×D412yes—
82C2×C4×C127C2×D412yes—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2411yescharacteristic
85C22×C127—21no—
86C22×C127—22no—
87C22×C127—42no—
88C22×C127—42no—
89C22×C127—42no—
90C22×C127C22×D411yescharacteristic
91C2×D127—42no—
92C2×D127—42no—
93C2×D127—42no—
94C4×C127—21no—
95C4×C127—22no—
96C4×C127—22no—
97C4×C127—42no—
98C4×C127H(5,𝔽2)11yescharacteristic
99C4×C1272−1+411yescharacteristic
100Dic127—42no—
101C2×C127—41no—
102C2×C127—82no—
103C2×C127#53711yescharacteristic
104C2×C127#57612yes—
105D127—82no—
106#3259—1271nomaximal, Sylow
107C127#325911yescharacteristic, complemented, Sylow
108#540—1271no—
109#540D12711yescharacteristic, complemented
110#478—1272no—
111#478—1272no—
112#469—1272no—
113#466—1272no—
114#459—1272no—
115#459—1272no—
116#424—1271no—
117C2×D8—1272no—
118C2×D8—1272no—
119C2×M16—1271no—
120C22×C8—1271no—
121#127—1271no—
122#127C2×D12711yescharacteristic
123#125—1271no—
124#125C2×D12711yescharacteristic
125#123—22no—
126#123—2542no—
127#121—22no—
128#121—1272no—
129#121—2542no—
130#121C2×D12712yes—
131C4×D4—1272no—
132C4×D4C2×D12712yes—
133#117C2×D12711yescharacteristic
134#107—1272no—
135#106—1272no—
136C4⋊C8—1272no—
137C4⋊C8—1272no—
138#102—1272no—
139#102—1272no—
140#102—1272no—
141#102—1272no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—1272no—
145C2×D4—2542no—
146C2×D4—2542no—
147C2×D4C22×D12712yes—
148C22×C4—22no—
149C22×C4—2542no—
150C22×C4C22×D12711yescharacteristic
151D8—5082no—
152D8—5082no—
153M16—5081no—
154C2×C8—1272no—
155C2×C8—1272no—
156C2×C8—5081no—
157C4⋊C4—22no—
158C4⋊C4—1272no—
159C4⋊C4—2542no—
160C4⋊C4C22⋊D12712yes—
161C4⋊C4C22×D12712yes—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—2542no—
165C22⋊C4—2542no—
166C22⋊C4C22⋊D12712yes—
167C42C22×D12711yescharacteristic
168C42C22×D12711yescharacteristic
169C23—22no—
170C23—2542no—
171C23C2×C22⋊D12711yescharacteristic
172D4—22no—
173D4—22no—
174D4—82no—
175D4—82no—
176D4—10162no—
177D4—10162no—
178D4—2542no—
179D4—2542no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—82no—
184C2×C4—10162no—
185C2×C4—2542no—
186C2×C4C2×C22⋊D12712yes—
187C2×C4C2×C22⋊D12712yes—
188C2×C4C2×C22⋊D12711yescharacteristic
189C2×C4C23×D12711yescharacteristic
190C8—2542no—
191C8—2542no—
192C22—22no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—5082no—
198C22—5082no—
199C22—5082no—
200C22C22×C22⋊D12711yescharacteristic
201C4—22no—
202C4—22no—
203C4—21no—
204C4—42no—
205C4—5082no—
206C4#1686125511yescharacteristic
207C4#1686125611yescharacteristic
208C2—41no—
209C2—82no—
210C2—10162no—
211C2#2255045411yescharacteristic
212C2#2255067912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#540×C127C2
socleC22×C127C22×D4
hypercenter#540D127
hypocenterC127#3259
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #22550481C2semidirect
2. #22550176C2semidirect
3. #22550173C2semidirect
4. #22550127C2semidirect
5. #22550110C2semidirect
6. #22550071C2semidirect
7. #22550068C2semidirect
8. #22549903C2semidirect
9. #540×C127C2semidirect
107. C127#3259semidirect
109. #540D127semidirect

Maximal subgroups

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

subgroupindex
#225504812
#225501762
#225501732
#225501272
#225501102
#225500712
#225500682
#225499032
#540×C1272
#3259127