Finite Group Databasebeta

← #30650164

Subgroups — #30650164

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#22550179C212yesmaximal, complemented
3#22550174C212yesmaximal, complemented
4#22550127C212yesmaximal, complemented
5#22550112C211yesmaximal, characteristic, complemented
6#22550070C212yesmaximal, complemented
7#22550069C212yesmaximal, complemented
8#22549902C211yesmaximal, characteristic, complemented
9#466×C127C211yesmaximal, characteristic, complemented
10C2×D1016C2211yescharacteristic
11C2×#14523233C2212yes—
12C2×#14523232C2211yescharacteristic
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#16861143C2211yescharacteristic
24#16861140C2212yes—
25Dic127⋊C8C2212yes—
26Dic127⋊C8C2212yes—
27#16861131C2212yes—
28#16861130C2212yes—
29#16861129C2212yes—
30C2×D8×C127C2211yescharacteristic
31C22×C8×C127C2211yescharacteristic
32#121×C127C2212yes—
33#107×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—
43#14523232—41no—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127—22no—
46C4⋊Dic127D411yescharacteristic
47C4⋊Dic127D411yescharacteristic
48C4⋊Dic127C2312yes—
49Dic127⋊C4—22no—
50C22⋊Dic127—22no—
51C22⋊Dic127D412yes—
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—
59C2×C8×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—21no—
90C22×C127—21no—
91C22×C127—42no—
92C22×C127—42no—
93C22×C127—42no—
94C22×C127C22×D411yescharacteristic
95C2×D127—42no—
96C2×D127—42no—
97C2×D127—42no—
98C4×C127—21no—
99C4×C127—42no—
100C4×C127H(5,𝔽2)11yescharacteristic
101C4×C1272−1+411yescharacteristic
102Dic127—22no—
103Dic127—22no—
104Dic127—42no—
105C2×C127—41no—
106C2×C127—82no—
107C2×C127#53711yescharacteristic
108C2×C127#57611yescharacteristic
109C2×C127#57611yescharacteristic
110D127—82no—
111#3259—1271nomaximal, Sylow
112C127#325911yescharacteristic, complemented, Sylow
113#540—1272no—
114#478—1272no—
115#478—1272no—
116#469—1272no—
117#466—1271no—
118#466D12711yescharacteristic, complemented
119#459—1272no—
120#459—1272no—
121#424—1271no—
122C2×D8—1272no—
123C2×D8—1271no—
124C2×D8C2×D12711yescharacteristic
125C2×M16—1271no—
126C22×C8C2×D12711yescharacteristic
127#127—1272no—
128#125—1272no—
129#123—2542no—
130#123—2542no—
131#121—1272no—
132#121—2542no—
133#121—2542no—
134#121C2×D12712yes—
135C4×D4—1272no—
136C4×D4—1272no—
137#117—1271no—
138#107—1271no—
139#107C2×D12711yescharacteristic
140#106—1272no—
141C4⋊C8—1272no—
142C4⋊C8—1272no—
143#102—1272no—
144#102—1272no—
145#102—1272no—
146#102C2×D12712yes—
147C2×D4—22no—
148C2×D4—1272no—
149C2×D4—2542no—
150C2×D4—2542no—
151C2×D4—2542no—
152C2×D4C22×D12712yes—
153C22×C4—2542no—
154C22×C4—2542no—
155C22×C4C22×D12711yescharacteristic
156D8—41no—
157D8—5081no—
158D8—5082no—
159M16—5081no—
160C2×C8—41no—
161C2×C8—1272no—
162C2×C8C22×D12711yescharacteristic
163C2×C8C22×D12711yescharacteristic
164C4⋊C4—1271no—
165C4⋊C4—1271no—
166C4⋊C4—1272no—
167C4⋊C4—2542no—
168C4⋊C4—2542no—
169C4⋊C4C22×D12712yes—
170C22⋊C4—22no—
171C22⋊C4—1272no—
172C22⋊C4—2542no—
173C22⋊C4—2542no—
174C22⋊C4—2542no—
175C42—1272no—
176C23—22no—
177C23—2542no—
178C23D4×D12711yescharacteristic
179D4—22no—
180D4—22no—
181D4—82no—
182D4—10162no—
183D4—10162no—
184D4—10162no—
185D4—2542no—
186D4—2542no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—1272no—
191C2×C4—1272no—
192C2×C4—10162no—
193C2×C4—10162no—
194C2×C4—2542no—
195C2×C4D4×D12711yescharacteristic
196C2×C4C23×D12711yescharacteristic
197C8—21no—
198C8—21no—
199C8—2542no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—5082no—
207C22—5082no—
208C22—5082no—
209C22C2×D4×D12711yescharacteristic
210C4—21no—
211C4—42no—
212C4—5082no—
213C4—2542no—
214C4—2542no—
215C4#1686125411yescharacteristic
216C4#1686125611yescharacteristic
217C2—41no—
218C2—82no—
219C2—10162no—
220C2#2255045311yescharacteristic
221C2#2255067711yescharacteristic
222C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#466×C127C2
socleC22×C127C22×D4
hypercenter#466D127
hypocenterC127#3259
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #22550479C2semidirect
2. #22550179C2semidirect
3. #22550174C2semidirect
4. #22550127C2semidirect
5. #22550112C2semidirect
6. #22550070C2semidirect
7. #22550069C2semidirect
8. #22549902C2semidirect
9. #466×C127C2semidirect
112. C127#3259semidirect
118. #466D127semidirect

Maximal subgroups

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

subgroupindex
#225504792
#225501792
#225501742
#225501272
#225501122
#225500702
#225500692
#225499022
#466×C1272
#3259127