Finite Group Databasebeta

← #30640735

Subgroups — #30640735

Order 16256.

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

normal subgroups only

This group has 260 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#22550615C211yesmaximal, characteristic, complemented
2#22550066C211yesmaximal, characteristic, complemented
3#22550044C211yesmaximal, characteristic, complemented
4#22549668C211yesmaximal, characteristic, complemented
5#22549665C211yesmaximal, characteristic, complemented
6#22549663C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C127C211yesmaximal, characteristic, complemented
8#16861234—21no—
9#16861205C2211yescharacteristic
10#16861202—21no—
11#16861191—21no—
12#16861190—21no—
13#16861190C2211yescharacteristic
14#16861182—21no—
15#16861181—21no—
16#16861180—21no—
17#16861178—21no—
18#16861136—21no—
19#16861126C2211yescharacteristic
20#16861124—21no—
21#16861124C2211yescharacteristic
22H(5,𝔽2)×C127C2211yescharacteristic
23#120×C127—21no—
24#120×C127C2211yescharacteristic
25#99×C127—21no—
26#99×C127C2211yescharacteristic
27C2×D508D411yescharacteristic, complemented
28C2×C22⋊D127—21no—
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C22×Dic127D411yescharacteristic, complemented
33C2×C4×D127—21no—
34#14523235—41no—
35#14523233—41no—
36#14523232—21no—
37C4⋊Dic127—21no—
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—41no—
41C22⋊Dic127—21no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45C22⋊Dic127—41no—
46C22⋊Dic127D411yescharacteristic, complemented
47(C2×D127)⋊C4—21no—
48C4×Dic127D411yescharacteristic, complemented
49C24×C127D411yescharacteristic
50#41×C127—21no—
51#41×C127—41no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127C2311yescharacteristic
58C22⋊C4×C127—21no—
59C22⋊C4×C127—21no—
60C22⋊C4×C127—41no—
61C22⋊C4×C127D411yescharacteristic
62C22×D127—21no—
63D508—41no—
64C22⋊D127—81no—
65C22⋊D127—81no—
66C22⋊D127—81no—
67C22⋊D127—81no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—21no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C4×D127—81no—
74C127⋊C8—41no—
75C23×C127—21no—
76C23×C127—21no—
77C23×C127—21no—
78C23×C127—41no—
79C23×C127—81no—
80C23×C127C2×D411yescharacteristic
81C23×C127C2×D411yescharacteristic
82Q8×C127—21no—
83D4×C127—21no—
84D4×C127—41no—
85D4×C127—41no—
86D4×C127—41no—
87D4×C127—41no—
88D4×C127—81no—
89D4×C127—82no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127C2×D411yescharacteristic
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—42no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—81no—
106C22×C127—81no—
107C22×C127—81no—
108C22×C127#12011yescharacteristic
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—41no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127—81no—
116Dic127—41no—
117Dic127—41no—
118Dic127—41no—
119Dic127—81no—
120C2×C127—21no—
121C2×C127—41no—
122C2×C127—41no—
123C2×C127—81no—
124C2×C127—81no—
125C2×C127B(4,𝔽2)11yescharacteristic
126D127—81no—
127#2200—1271nomaximal, Sylow
128C127#220011yescharacteristic, complemented, Sylow
129#560—1271no—
130#458—1271no—
131B(4,𝔽2)D12711yescharacteristic, complemented
132#453—1271no—
133#353—1271no—
134#352—1271no—
135AΣL(1,𝔽16)—1271no—
136H(5,𝔽2)C2×D12711yescharacteristic
137AGL(1,ℤ/8)—2541no—
138#127—1271no—
139#126—2541no—
140#123—1271no—
141#123—2541no—
142#123—2541no—
143#121—2541no—
144#121—2541no—
145#121—2541no—
146#120—21no—
147#120—2541no—
148#120C2×D12711yescharacteristic
149#104—2541no—
150#100—1271no—
151#99—21no—
152#99—1271no—
153#99—2541no—
154#99C2×D12711yescharacteristic
155C24C22⋊D12711yescharacteristic
156#41—21no—
157#41—41no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—1271no—
164C2×D4—2541no—
165C2×D4—2541no—
166C2×D4—2541no—
167C2×D4—2541no—
168C2×D4C22×D12711yescharacteristic
169C22×C4—1271no—
170C22×C4—2541no—
171SD16—5081no—
172D8—5081no—
173M16—2541no—
174C4⋊C4—5081no—
175C4⋊C4—2541no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—41no—
181C22⋊C4—1271no—
182C22⋊C4—5081no—
183C22⋊C4—2541no—
184C22⋊C4—2541no—
185C22⋊C4—2541no—
186C22⋊C4—2541no—
187C22⋊C4—2541no—
188C22⋊C4C22⋊D12711yescharacteristic
189C42—1271no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—2541no—
196C23D4×D12711yescharacteristic
197C23C2×C22⋊D12711yescharacteristic
198Q8—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—81no—
205D4—82no—
206D4—10161no—
207D4—10161no—
208D4—10161no—
209D4—10161no—
210D4—5081no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—10161no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4D4×D12711yescharacteristic
223C8—5081no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—5081no—
237C22—5081no—
238C22—5081no—
239C22#1686117711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—81no—
244C4—10161no—
245C4—5081no—
246C4—5081no—
247C4—5081no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—10161no—
254C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
FittingB(4,𝔽2)×C127C2
socleC2×C127B(4,𝔽2)
hypercenterB(4,𝔽2)D127
hypocenterC127#2200
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22550615C2semidirect
2. #22550066C2semidirect
3. #22550044C2semidirect
4. #22549668C2semidirect
5. #22549665C2semidirect
6. #22549663C2semidirect
7. B(4,𝔽2)×C127C2semidirect
27. C2×D508D4semidirect
32. C22×Dic127D4semidirect
46. C22⋊Dic127D4semidirect
48. C4×Dic127D4semidirect
128. C127#2200semidirect
131. B(4,𝔽2)D127semidirect

Maximal subgroups

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

subgroupindex
#225506152
#225500662
#225500442
#225496682
#225496652
#225496632
B(4,𝔽2)×C1272
#2200127