Finite Group Databasebeta

← #30650107

Subgroups — #30650107

Order 16256.

430 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 113 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#22550421C212yesmaximal, complemented
2#22550165C212yesmaximal, complemented
3#22550164C212yesmaximal, complemented
4#22550129C211yesmaximal, characteristic, complemented
5#22550127C212yesmaximal, complemented
6#22550070C212yesmaximal, complemented
7#22550069C212yesmaximal, complemented
8#22549899C211yesmaximal, characteristic, complemented
9#469×C127C211yesmaximal, characteristic, complemented
10C2×#14523247—22no—
11C2×D1016C2211yescharacteristic
12C2×#14523233C2212yes—
13C2×#14523232C2211yescharacteristic
14#16861208C2212yes—
15#16861205C2212yes—
16#16861195—22no—
17#16861183C2212yes—
18#16861181—22no—
19D4×Dic127C2212yes—
20#16861171C2212yes—
21(C22⋊D127)⋊C4—22no—
22C2×C4×Dic127C2211yescharacteristic
23#16861141C2211yescharacteristic
24#16861140C2212yes—
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—
35#14523247—48no—
36C2×Dic254—22no—
37C2×D508C2312yes—
38C2×C22⋊D127—22no—
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C22×Dic127—22no—
42C22×Dic127D412yes—
43C2×C4×D127—22no—
44D1016—41no—
45#14523233—42no—
46#14523232—41no—
47C2×C127⋊C8C2312yes—
48C4⋊Dic127C2312yes—
49Dic127⋊C4—22no—
50C22⋊Dic127—22no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53C4×Dic127D411yescharacteristic
54C4×Dic127D411yescharacteristic
55C4×Dic127C2312yes—
56C2×D4×C127—22no—
57C2×D4×C127C2312yes—
58C22×C4×C127C2311yescharacteristic
59D8×C127—41no—
60M16×C127—41no—
61C2×C8×C127C2311yescharacteristic
62C2×C8×C127C2311yescharacteristic
63C4⋊C4×C127C2312yes—
64C22⋊C4×C127—22no—
65C22×D127—22no—
66Dic254—44no—
67D508—22no—
68D508—22no—
69C22⋊D127—44no—
70C22⋊D127—44no—
71C22⋊D127—44no—
72C2×Dic127—24no—
73C2×Dic127—22no—
74C2×Dic127—44no—
75C2×Dic127C2×D412yes—
76C2×Dic127C2×D412yes—
77C4×D127—44no—
78C127⋊C8—22no—
79C23×C127—22no—
80C23×C127C2×D411yescharacteristic
81D4×C127—22no—
82D4×C127—22no—
83D4×C127—44no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—22no—
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2411yescharacteristic
89C8×C127—21no—
90C8×C127—21no—
91C22×C127—21no—
92C22×C127—22no—
93C22×C127—42no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127C22×D411yescharacteristic
97C2×D127—42no—
98C2×D127—42no—
99C2×D127—42no—
100C4×C127—21no—
101C4×C127—42no—
102C4×C127H(5,𝔽2)11yescharacteristic
103C4×C1272−1+411yescharacteristic
104Dic127—22no—
105Dic127—42no—
106Dic127AGL(1,ℤ/8)14yes—
107C2×C127—41no—
108C2×C127—82no—
109C2×C127#53711yescharacteristic
110C2×C127C2×AGL(1,ℤ/8)12yes—
111D127—82no—
112#3253—1271nomaximal, Sylow
113C127#325311yescharacteristic, complemented, Sylow
114#532—1272no—
115#476—1272no—
116#476—1272no—
117#469—1271no—
118#469—1272no—
119#469D12711yescharacteristic, complemented
120#459—1272no—
121#459—1272no—
122#423—1271no—
123C2×#41—2542no—
124C2×D8—1272no—
125C2×D8—1271no—
126C2×D8C2×D12711yescharacteristic
127C2×M16—1271no—
128C2×M16C2×D12711yescharacteristic
129C4⋊Q8—1272no—
130#127—1272no—
131#124—2542no—
132#121—1272no—
133#121—2542no—
134#121C2×D12712yes—
135C4×D4—1272no—
136C4×D4—1272no—
137C4×D4—2542no—
138C2×C42—1271no—
139#106—1271no—
140#106—1272no—
141#106C2×D12711yescharacteristic
142C4⋊C8—1272no—
143C4⋊C8—1272no—
144#102—1272no—
145#102—1272no—
146#102—1272no—
147#102C2×D12712yes—
148#41—5088no—
149C2×Q8—2542no—
150C2×D4—22no—
151C2×D4—1272no—
152C2×D4—2542no—
153C2×D4—2542no—
154C2×D4—2542no—
155C2×D4C22×D12712yes—
156C22×C4—1272no—
157C22×C4—2542no—
158C22×C4—2542no—
159C22×C4C22×D12711yescharacteristic
160D8—41no—
161D8—5081no—
162D8—5082no—
163M16—41no—
164M16—5081no—
165C2×C8—1272no—
166C2×C8C22×D12711yescharacteristic
167C2×C8C22×D12711yescharacteristic
168C4⋊C4—1272no—
169C4⋊C4—2542no—
170C4⋊C4C22×D12712yes—
171C22⋊C4—22no—
172C22⋊C4—2542no—
173C22⋊C4—2542no—
174C22⋊C4—2542no—
175C42—1271no—
176C42—1271no—
177C42—1272no—
178C23—22no—
179C23—2542no—
180C23D4×D12711yescharacteristic
181Q8—5084no—
182D4—22no—
183D4—22no—
184D4—44no—
185D4—5084no—
186D4—5084no—
187D4—5084no—
188D4—2542no—
189D4—2542no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—1272no—
194C2×C4—1272no—
195C2×C4—5084no—
196C2×C4—5084no—
197C2×C4—2544no—
198C2×C4—2542no—
199C2×C4D4×D12711yescharacteristic
200C2×C4C23×D12711yescharacteristic
201C8—21no—
202C8—21no—
203C8—2542no—
204C22—22no—
205C22—21no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—5082no—
210C22—5082no—
211C22—5082no—
212C22C2×D4×D12711yescharacteristic
213C4—21no—
214C4—42no—
215C4—1274no—
216C4—5082no—
217C4—2542no—
218C4#1686125411yescharacteristic
219C4#1686125611yescharacteristic
220C2—41no—
221C2—82no—
222C2—10162no—
223C2#2255045311yescharacteristic
224C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#469×C127C2
socleC22×C127C22×D4
hypercenter#469D127
hypocenterC127#3253
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. #22550421C2semidirect
2. #22550165C2semidirect
3. #22550164C2semidirect
4. #22550129C2semidirect
5. #22550127C2semidirect
6. #22550070C2semidirect
7. #22550069C2semidirect
8. #22549899C2semidirect
9. #469×C127C2semidirect
113. C127#3253semidirect
119. #469D127semidirect

Maximal subgroups

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

subgroupindex
#225504212
#225501652
#225501642
#225501292
#225501272
#225500702
#225500692
#225498992
#469×C1272
#3253127