Finite Group Databasebeta

← #30639941

Subgroups — #30639941

Order 16256.

326 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 69 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#22550443C211yesmaximal, characteristic
2#22549918C211yesmaximal, characteristic, complemented
3C2×#16861135C211yesmaximal, characteristic, complemented
4#22549878C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861205—21no—
10#16861196—21no—
11#16861180—21no—
12#16861180—21no—
13#16861178—22no—
14#16861168C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C22⋊Dic127—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861143—21no—
19#16861141—21no—
20#16861135—22no—
21#16861134—21no—
22#16861131—21no—
23C22×D4×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25C2×M16×C127C2211yescharacteristic
26#100×C127—22no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254D411yescharacteristic
32C2×D508D411yescharacteristic
33C2×C22⋊D127—22no—
34C2×C22⋊D127—22no—
35C2×C22⋊D127—21no—
36C22×Dic127—21no—
37C22×Dic127—21no—
38C2×C4×D127—21no—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127D411yescharacteristic
42C4⋊Dic127D411yescharacteristic
43C22⋊Dic127—22no—
44C22⋊Dic127—22no—
45C22⋊Dic127—21no—
46C22⋊Dic127—44no—
47C4×Dic127—21no—
48C4×Dic127—21no—
49C4×Dic127—21no—
50C24×C127D412yes—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58M16×C127—21no—
59M16×C127—21no—
60M16×C127—21no—
61M16×C127D412yescomplemented
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C22×D127—21no—
65Dic254—21no—
66Dic254—21no—
67D508—21no—
68D508—21no—
69C22⋊D127—41no—
70C22⋊D127—41no—
71C22⋊D127—82no—
72C22⋊D127—82no—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C2×Dic127—42no—
80C2×Dic127—41no—
81C4×D127—41no—
82C4×D127—41no—
83C23×C127—22no—
84C23×C127—22no—
85C23×C127—22no—
86C23×C127—42no—
87C23×C127—42no—
88C23×C127C2×D411yescharacteristic
89D4×C127—41no—
90D4×C127—41no—
91D4×C127—41no—
92D4×C127—41no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127#4111yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C8×C127—22no—
100C8×C127—41no—
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—44no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—82no—
108C22×C127#12011yescharacteristic
109C22×C127#12111yescharacteristic
110C22×C127#12311yescharacteristic
111C2×D127—41no—
112C2×D127—41no—
113C2×D127—41no—
114C4×C127#12111yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12311yescharacteristic
117C4×C127#12411yescharacteristic
118Dic127—42no—
119Dic127—41no—
120Dic127—41no—
121Dic127—41no—
122Dic127—81no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127—82no—
126C2×C127#39411yescharacteristic
127C2×C127#45311yescharacteristic
128C2×C127#45511yescharacteristic
129D127—81no—
130#2037—1271nomaximal, Sylow
131C127#203711yescharacteristic, complemented, Sylow
132#535—1271no—
133#428—1271no—
134C2×#104—1271no—
135#417—1271no—
136C2×#100D12711yescharacteristic, complemented
137#390—1271no—
138#339—1271no—
139C2×#41—1271no—
140C22×D4C2×D12711yescharacteristic
141C2×M16C2×D12711yescharacteristic
142C2×M16C2×D12711yescharacteristic
143#127—2541no—
144#124—2541no—
145#121—2541no—
146#121—2541no—
147#120—2542no—
148#117—1271no—
149C2×C4⋊C4—1271no—
150C2×C22⋊C4—2542no—
151C2×C42—1271no—
152#107—2541no—
153#106—2541no—
154#104—2542no—
155#103—2541no—
156#102—2541no—
157#100—22no—
158C24D50812yes—
159#41—2541no—
160#41—2541no—
161#41—2541no—
162#41—2541no—
163C2×Q8—1271no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—1271no—
171C2×D4—2542no—
172C2×D4—2542no—
173C2×D4—2541no—
174C22×C4—2541no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178M16—21no—
179M16—21no—
180M16—21no—
181M16C22⋊D12712yescomplemented
182C2×C8—21no—
183C2×C8—21no—
184C4⋊C4—1271no—
185C4⋊C4—1271no—
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C22⋊C4—5084no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C22⋊C4—2541no—
192C42—2541no—
193C42—2541no—
194C42—2541no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—42no—
199C23—42no—
200C23—2541no—
201C23C2×D50811yescharacteristic
202Q8—2541no—
203Q8—2541no—
204D4—41no—
205D4—41no—
206D4—41no—
207D4—41no—
208D4—10162no—
209D4—10162no—
210D4—5081no—
211D4—5081no—
212D4—2541no—
213D4—2541no—
214C2×C4—5082no—
215C2×C4—5081no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4—5081no—
220C2×C4—2542no—
221C2×C4—2541no—
222C2×C4—2541no—
223C2×C4—2541no—
224C2×C4D4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4#1452324711yescharacteristic
228C2×C4#1452324711yescharacteristic
229C2×C4#1452324811yescharacteristic
230C8—22no—
231C8—41no—
232C22—21no—
233C22—21no—
234C22—44no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—82no—
239C22—5081no—
240C22—5081no—
241C22—5081no—
242C22#1686117911yescharacteristic
243C22#1686118311yescharacteristic
244C22#1686119311yescharacteristic
245C4—10161no—
246C4—5082no—
247C4—5081no—
248C4—5081no—
249C4—5081no—
250C4#1686118011yescharacteristic
251C4#1686118111yescharacteristic
252C4#1686119111yescharacteristic
253C4#1686119511yescharacteristic
254C2—21no—
255C2—21no—
256C2—82no—
257C2—10161no—
258C2#2254979211yescharacteristic
259C2#2255004611yescharacteristic
260C2#2255005611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127#120
prime radical#22550443C2
hypercenterC2×#100D127
hypocenterC127#2037
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #22549918C2semidirect
3. C2×#16861135C2semidirect
4. #22549878C2semidirect
5. #22549779C2semidirect
6. #22549637C2semidirect
7. C2×#100×C127C2semidirect
61. M16×C127D4semidirect
131. C127#2037semidirect
136. C2×#100D127semidirect
181. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504432
#225499182
C2×#168611352
#225498782
#225497792
#225496372
C2×#100×C1272
#2037127