Finite Group Databasebeta

← #30640718

Subgroups — #30640718

Order 16256.

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

normal subgroups only

This group has 258 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#22550557C211yesmaximal, characteristic, complemented
2#22550066C211yesmaximal, characteristic, complemented
3#22550054C211yesmaximal, characteristic, complemented
4#22549670C211yesmaximal, characteristic, complemented
5#22549665C211yesmaximal, characteristic, complemented
6#22549663C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C127C211yesmaximal, characteristic
8#16861239—21no—
9#16861202—21no—
10#16861196C2211yescharacteristic
11#16861190—21no—
12#16861190—21no—
13#16861190C2211yescharacteristic
14#16861186—21no—
15D4×Dic127—21no—
16#16861165—21no—
17C2×C22⋊Dic127—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×Dic254D411yescharacteristic, complemented
28C22×Dic127—21no—
29C22×Dic127—21no—
30C22×Dic127D411yescharacteristic, complemented
31#14523238—41no—
32#14523236—41no—
33#14523232—21no—
34C4⋊Dic127—21no—
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37Dic127⋊C4—21no—
38Dic127⋊C4—41no—
39C22⋊Dic127—21no—
40C22⋊Dic127—21no—
41C22⋊Dic127—21no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45C22⋊Dic127—41no—
46C22⋊Dic127D411yescharacteristic, complemented
47C4×Dic127—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
62Dic254—41no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—42no—
68C2×Dic127—41no—
69C2×Dic127—41no—
70C2×Dic127—81no—
71C127⋊C8—41no—
72C23×C127—21no—
73C23×C127—21no—
74C23×C127—21no—
75C23×C127—41no—
76C23×C127—42no—
77C23×C127C2×D411yescharacteristic
78C23×C127C2×D411yescharacteristic
79Q8×C127—21no—
80D4×C127—21no—
81D4×C127—41no—
82D4×C127—42no—
83D4×C127—41no—
84D4×C127—41no—
85D4×C127—41no—
86D4×C127—82no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—21no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127C2×D411yescharacteristic
93C22×C127—22no—
94C22×C127—21no—
95C22×C127—21no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127—42no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—81no—
103C22×C127—81no—
104C22×C127—81no—
105C22×C127#12011yescharacteristic
106C4×C127—21no—
107C4×C127—41no—
108C4×C127—41no—
109C4×C127—81no—
110Dic127—41no—
111Dic127—41no—
112Dic127—41no—
113Dic127—41no—
114Dic127—81no—
115C2×C127—21no—
116C2×C127—41no—
117C2×C127—41no—
118C2×C127—42no—
119C2×C127—81no—
120C2×C127B(4,𝔽2)11yescharacteristic
121#2197—1271nomaximal, Sylow
122C127#219711yescharacteristic, complemented, Sylow
123#551—1271no—
124#458—1271no—
125B(4,𝔽2)D12711yescharacteristic
126#455—1271no—
127#354—1271no—
128#352—1271no—
129AΣL(1,𝔽16)—1271no—
130H(5,𝔽2)C2×D12711yescharacteristic
131#137—2541no—
132#126—2541no—
133#124—1271no—
134#123—1271no—
135#123—2541no—
136#123—2541no—
137C22⋊Q8—2541no—
138#120—21no—
139#120C2×D12711yescharacteristic
140C4×D4—2541no—
141#117—2541no—
142C2×C22⋊C4—2541no—
143#104—2541no—
144#100—1271no—
145#99—21no—
146#99—1271no—
147#99—2541no—
148#99C2×D12711yescharacteristic
149C24C22⋊D12711yescharacteristic
150#41—21no—
151#41—41no—
152C2×Q8—1271no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4C22×D12711yescharacteristic
159C22×C4—1271no—
160C22×C4—2541no—
161C22×C4—2541no—
162Q16—5081no—
163SD16—5081no—
164M16—2541no—
165C4⋊C4—5081no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C4⋊C4—2541no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—41no—
173C22⋊C4—1271no—
174C22⋊C4—5081no—
175C22⋊C4—2541no—
176C22⋊C4—2541no—
177C22⋊C4—2541no—
178C22⋊C4—2541no—
179C22⋊C4—2541no—
180C22⋊C4—2541no—
181C22⋊C4C22⋊D12711yescharacteristic
182C42—1271no—
183C42—2541no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23D4×D12711yescharacteristic
190C23C2×C22⋊D12711yescharacteristic
191Q8—21no—
192Q8—5081no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—42no—
198D4—41no—
199D4—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—10161no—
206C2×C4—5081no—
207C2×C4—5082no—
208C2×C4—5081no—
209C2×C4—2541no—
210C2×C4—2541no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4D4×D12711yescharacteristic
214C8—5081no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22#1686117711yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—10161no—
233C4—5081no—
234C4—5081no—
235C4—5081no—
236C4—5081no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—81no—
242C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting, prime radicalB(4,𝔽2)×C127C2
socleC2×C127B(4,𝔽2)
hypercenterB(4,𝔽2)D127
hypocenterC127#2197
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. #22550557C2semidirect
2. #22550066C2semidirect
3. #22550054C2semidirect
4. #22549670C2semidirect
5. #22549665C2semidirect
6. #22549663C2semidirect
27. C2×Dic254D4semidirect
30. C22×Dic127D4semidirect
46. C22⋊Dic127D4semidirect
48. C4×Dic127D4semidirect
122. C127#2197semidirect

Maximal subgroups

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

subgroupindex
#225505572
#225500662
#225500542
#225496702
#225496652
#225496632
B(4,𝔽2)×C1272
#2197127