Finite Group Databasebeta

← #30650166

Subgroups — #30650166

Order 16256.

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

normal subgroups only

This group has 370 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#22550479C212yesmaximal, complemented
2#22550178C212yesmaximal, complemented
3#22550175C212yesmaximal, complemented
4#22550129C211yesmaximal, characteristic, complemented
5#22550110C212yesmaximal, complemented
6#22550070C212yesmaximal, complemented
7#22550069C212yesmaximal, complemented
8#22549904C211yesmaximal, characteristic, complemented
9#469×C127C211yesmaximal, characteristic, complemented
10C2×D1016C2211yescharacteristic
11C2×#14523233C2212yes—
12C22×C127⋊C8C2211yescharacteristic
13#16861205C2212yes—
14#16861200C2212yes—
15#16861193—22no—
16#16861191—22no—
17#16861183C2212yes—
18#16861181—22no—
19#16861180—22no—
20D4×Dic127C2212yes—
21#16861171C2212yes—
22#16861168C2211yescharacteristic
23#16861142C2212yes—
24#16861141C2211yescharacteristic
25Dic127⋊C8C2212yes—
26Dic127⋊C8C2212yes—
27#16861131C2212yes—
28#16861130C2212yes—
29#16861129C2212yes—
30C2×D8×C127C2211yescharacteristic
31C2×M16×C127C2211yescharacteristic
32#121×C127C2212yes—
33#106×C127C2211yescharacteristic
34#102×C127C2212yes—
35C2×D508C2312yes—
36C2×C22⋊D127—22no—
37C2×C22⋊D127—22no—
38C2×C22⋊D127—22no—
39C22×Dic127—22no—
40C2×C4×D127—22no—
41D1016—41no—
42#14523233—42no—
43C2×C127⋊C8—41no—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127—22no—
46C4⋊Dic127D412yes—
47C4⋊Dic127C2312yes—
48Dic127⋊C4—22no—
49C22⋊Dic127—22no—
50C22⋊Dic127D411yescharacteristic
51C22⋊Dic127D411yescharacteristic
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—22no—
54C4×Dic127C2312yes—
55C2×D4×C127—22no—
56C2×D4×C127C2312yes—
57C22×C4×C127C2311yescharacteristic
58D8×C127—41no—
59M16×C127—41no—
60C2×C8×C127C2311yescharacteristic
61C2×C8×C127C2311yescharacteristic
62C4⋊C4×C127C2312yes—
63C22⋊C4×C127—22no—
64C22×D127—22no—
65D508—22no—
66D508—22no—
67C22⋊D127—82no—
68C22⋊D127—82no—
69C22⋊D127—82no—
70C2×Dic127—22no—
71C2×Dic127—82no—
72C2×Dic127C2×D412yes—
73C2×Dic127C2×D412yes—
74C4×D127—82no—
75C127⋊C8—22no—
76C23×C127—22no—
77C23×C127C2×D411yescharacteristic
78D4×C127—22no—
79D4×C127—22no—
80D4×C127—82no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—22no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2411yescharacteristic
86C8×C127—21no—
87C8×C127—21no—
88C22×C127—21no—
89C22×C127—22no—
90C22×C127—42no—
91C22×C127—42no—
92C22×C127—42no—
93C22×C127C22×D411yescharacteristic
94C2×D127—42no—
95C2×D127—42no—
96C2×D127—42no—
97C4×C127—21no—
98C4×C127—42no—
99C4×C127H(5,𝔽2)11yescharacteristic
100C4×C1272−1+411yescharacteristic
101Dic127—22no—
102Dic127—22no—
103Dic127—42no—
104C2×C127—41no—
105C2×C127—82no—
106C2×C127#53711yescharacteristic
107C2×C127#57612yes—
108D127—82no—
109#3259—1271nomaximal, Sylow
110C127#325911yescharacteristic, complemented, Sylow
111#540—1272no—
112#478—1272no—
113#478—1272no—
114#469—1271no—
115#469D12711yescharacteristic, complemented
116#466—1272no—
117#459—1272no—
118#459—1272no—
119#424—1271no—
120C2×D8—1272no—
121C2×D8—1271no—
122C2×D8C2×D12711yescharacteristic
123C2×M16C2×D12711yescharacteristic
124C22×C8—1271no—
125#127—1272no—
126#125—1272no—
127#123—2542no—
128#123—2542no—
129#121—1272no—
130#121—2542no—
131#121—2542no—
132#121C2×D12712yes—
133C4×D4—1272no—
134C4×D4—1272no—
135#117—1271no—
136#107—1272no—
137#106—1271no—
138#106C2×D12711yescharacteristic
139C4⋊C8—1272no—
140C4⋊C8—1272no—
141#102—1272no—
142#102—1272no—
143#102—1272no—
144#102C2×D12712yes—
145C2×D4—22no—
146C2×D4—1272no—
147C2×D4—2542no—
148C2×D4—2542no—
149C2×D4—2542no—
150C2×D4C22×D12712yes—
151C22×C4—2542no—
152C22×C4—2542no—
153C22×C4C22×D12711yescharacteristic
154D8—41no—
155D8—5081no—
156D8—5082no—
157M16—41no—
158C2×C8—1272no—
159C2×C8—5081no—
160C2×C8C22×D12711yescharacteristic
161C2×C8C22×D12711yescharacteristic
162C4⋊C4—1272no—
163C4⋊C4—1272no—
164C4⋊C4—2542no—
165C4⋊C4—2542no—
166C4⋊C4C22×D12712yes—
167C22⋊C4—22no—
168C22⋊C4—1271no—
169C22⋊C4—1271no—
170C22⋊C4—2542no—
171C22⋊C4—2542no—
172C22⋊C4—2542no—
173C42—1272no—
174C23—22no—
175C23—2542no—
176C23D4×D12711yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—10162no—
181D4—10162no—
182D4—10162no—
183D4—2542no—
184D4—2542no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—1272no—
189C2×C4—1272no—
190C2×C4—10162no—
191C2×C4—10162no—
192C2×C4—2542no—
193C2×C4D4×D12711yescharacteristic
194C2×C4C23×D12711yescharacteristic
195C8—21no—
196C8—21no—
197C8—2542no—
198C22—22no—
199C22—21no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—5082no—
204C22—5082no—
205C22—5082no—
206C22C2×D4×D12711yescharacteristic
207C4—21no—
208C4—42no—
209C4—5082no—
210C4—2542no—
211C4—2542no—
212C4#1686125411yescharacteristic
213C4#1686125611yescharacteristic
214C2—41no—
215C2—82no—
216C2—10162no—
217C2#2255045311yescharacteristic
218C2#2255068012yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#469×C127C2
socleC22×C127C22×D4
hypercenter#469D127
hypocenterC127#3259
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #22550479C2semidirect
2. #22550178C2semidirect
3. #22550175C2semidirect
4. #22550129C2semidirect
5. #22550110C2semidirect
6. #22550070C2semidirect
7. #22550069C2semidirect
8. #22549904C2semidirect
9. #469×C127C2semidirect
110. C127#3259semidirect
115. #469D127semidirect

Maximal subgroups

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

subgroupindex
#225504792
#225501782
#225501752
#225501292
#225501102
#225500702
#225500692
#225499042
#469×C1272
#3259127