Finite Group Databasebeta

← #30640744

Subgroups — #30640744

Order 16256.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550629C211yesmaximal, characteristic, complemented
2#22550067C211yesmaximal, characteristic
3#22550065C211yesmaximal, characteristic
4#22550056C211yesmaximal, characteristic
5#22549670C211yesmaximal, characteristic, complemented
6#22549667C211yesmaximal, characteristic, complemented
7#352×C127C211yesmaximal, characteristic, complemented
8#16861254C2211yescharacteristic
9#16861242—21no—
10#16861208—21no—
11#16861202—21no—
12#16861202—21no—
13#16861199—21no—
14#16861196C2211yescharacteristic
15#16861193—21no—
16#16861192—21no—
17#16861190C2211yescharacteristic
18#16861188—21no—
19#16861186—21no—
20#16861135—21no—
21#16861124C2211yescharacteristic
22#16861123—21no—
23#16861123—21no—
24#123×C127C2211yescharacteristic
25#100×C127C2211yescharacteristic
26#99×C127C2211yescharacteristic
27#14523248—21no—
28#14523247—41no—
29C2×Dic254—21no—
30C2×Dic254D411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C22⋊D127—21no—
33D4×D127—41no—
34C22×Dic127D411yescharacteristic
35Dic508—41no—
36#14523237—41no—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127—41no—
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45Dic127⋊C4—41no—
46C22⋊Dic127—21no—
47C22⋊Dic127—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C22⋊Dic127D411yescharacteristic
51(C2×D127)⋊C4—41no—
52(C2×D127)⋊C4—41no—
53C4×Dic127—21no—
54C4×Dic127D411yescharacteristic
55C2×D4×C127C2311yescharacteristic
56C22×C4×C127D411yescharacteristic
57M16×C127—21no—
58C4⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—21no—
61C22⋊C4×C127D411yescharacteristic
62C22×D127—21no—
63C22×D127—21no—
64Dic254—21no—
65Dic254—41no—
66Dic254—81no—
67D508—21no—
68C22⋊D127—41no—
69C22⋊D127—41no—
70C22⋊D127—41no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C4×D127—41no—
80C23×C127C2×D411yescharacteristic
81C23×C127C2×D411yescharacteristic
82D4×C127—41no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127C2×D411yescharacteristic
88C8×C127—41no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127#12011yescharacteristic
94C2×D127—41no—
95C2×D127—81no—
96C2×D127—81no—
97C4×C127—21no—
98C4×C127—41no—
99C4×C127—41no—
100C4×C127—81no—
101Dic127—41no—
102Dic127—41no—
103Dic127—41no—
104Dic127—41no—
105Dic127—41no—
106Dic127—81no—
107C2×C127—21no—
108C2×C127—41no—
109C2×C127—41no—
110C2×C127B(4,𝔽2)11yescharacteristic
111D127—81no—
112#2201—1271nomaximal, Sylow
113C127#220111yescharacteristic, complemented, Sylow
114#563—1271no—
115#458—1271no—
116#458—1271no—
117#455—1271no—
118#354—1271no—
119#352—1271no—
120#352D12711yescharacteristic, complemented
121H(5,𝔽2)—1271no—
122#137—2541no—
123C4⋊Q8—2541no—
124#126—2541no—
125#126—2541no—
126#125—2541no—
127#124—1271no—
128#123—1271no—
129#123—2541no—
130#123—2541no—
131#123C2×D12711yescharacteristic
132C22⋊Q8—2541no—
133C22⋊Q8—2541no—
134#104—2541no—
135#100C2×D12711yescharacteristic
136#99—1271no—
137#99—2541no—
138#99—2541no—
139#99C2×D12711yescharacteristic
140#41—5081no—
141#41—2541no—
142C2×Q8—1271no—
143C2×Q8—2541no—
144C2×D4—5081no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4C22×D12711yescharacteristic
148C22×C4—1271no—
149C22×C4D50811yescharacteristic
150Q16—5081no—
151SD16—5081no—
152M16—21no—
153C4⋊C4—21no—
154C4⋊C4—5081no—
155C4⋊C4—5081no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—1271no—
166C22⋊C4—5081no—
167C22⋊C4—5081no—
168C22⋊C4—2541no—
169C22⋊C4—2541no—
170C22⋊C4—2541no—
171C22⋊C4—2541no—
172C22⋊C4D50811yescharacteristic
173C42—1271no—
174C42—2541no—
175C23—2541no—
176C23—2541no—
177C23D4×D12711yescharacteristic
178C23C2×D50811yescharacteristic
179Q8—10161no—
180Q8—5081no—
181Q8—2541no—
182D4—41no—
183D4—5081no—
184D4—5081no—
185D4—5081no—
186D4—2541no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4—2541no—
198C2×C4—2541no—
199C2×C4—2541no—
200C2×C4D4×D12711yescharacteristic
201C8—41no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—10161no—
207C22—10161no—
208C22—5081no—
209C22#1686117911yescharacteristic
210C4—21no—
211C4—41no—
212C4—41no—
213C4—81no—
214C4—10161no—
215C4—5081no—
216C4—5081no—
217C4—5081no—
218C4—5081no—
219C4—5081no—
220C2—21no—
221C2—41no—
222C2—41no—
223C2—10161no—
224C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#352×C127C2
socleC2×C127B(4,𝔽2)
prime radical#16861254C22
hypercenter#352D127
hypocenterC127#2201
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
1. #22550629C2semidirect
5. #22549670C2semidirect
6. #22549667C2semidirect
7. #352×C127C2semidirect
113. C127#2201semidirect
120. #352D127semidirect

Maximal subgroups

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

subgroupindex
#225506292
#225500672
#225500652
#225500562
#225496702
#225496672
#352×C1272
#2201127