Finite Group Databasebeta

← #30639938

Subgroups — #30639938

Order 16256.

326 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 67 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#22550440C211yesmaximal, characteristic
2#22549916C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic, complemented
4C2×#16861125C211yesmaximal, characteristic, complemented
5#22549777C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7#417×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861205—21no—
11#16861198—21no—
12#16861183—21no—
13#16861183—21no—
14#16861177—22no—
15#16861168C2211yescharacteristic
16C2×(C2×D127)⋊C4—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861142—21no—
19#16861140—21no—
20#16861136—22no—
21#16861125—22no—
22C2×#41×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25#103×C127—21no—
26#102×C127—21no—
27C23×D127D412yes—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×D508—21no—
34C2×C22⋊D127—22no—
35C2×C22⋊D127—22no—
36C22×Dic127—21no—
37#14523232—21no—
38#14523232D412yescomplemented
39C2×C127⋊C8—21no—
40C4⋊Dic127D411yescharacteristic
41C4⋊Dic127D411yescharacteristic
42C22⋊Dic127—21no—
43(C2×D127)⋊C4—22no—
44(C2×D127)⋊C4—22no—
45(C2×D127)⋊C4—44no—
46C4×Dic127—21no—
47C4×Dic127—21no—
48C4×Dic127—21no—
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127—21no—
53C2×Q8×C127D411yescharacteristic
54C2×D4×C127—21no—
55C2×D4×C127D411yescharacteristic
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59M16×C127—21no—
60M16×C127—21no—
61C2×C8×C127—21no—
62C4⋊C4×C127—21no—
63C4⋊C4×C127—21no—
64C22×D127—22no—
65C22×D127—22no—
66C22×D127—22no—
67C22×D127—42no—
68C22×D127—42no—
69D508—41no—
70D508—41no—
71D508—41no—
72D508—41no—
73C22⋊D127—82no—
74C22⋊D127—82no—
75C2×Dic127—22no—
76C2×Dic127—21no—
77C2×Dic127—21no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C127⋊C8—22no—
81C23×C127—21no—
82C23×C127C2×D411yescharacteristic
83Q8×C127—21no—
84Q8×C127—21no—
85D4×C127—21no—
86D4×C127—21no—
87D4×C127—41no—
88D4×C127—41no—
89C2×C4×C127—21no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—42no—
93C2×C4×C127—41no—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C2×C4×C127#4111yescharacteristic
100C8×C127—41no—
101C22×C127—21no—
102C22×C127—21no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127—41no—
106C22×C127#12011yescharacteristic
107C22×C127#12111yescharacteristic
108C22×C127#12311yescharacteristic
109C2×D127—44no—
110C2×D127—42no—
111C2×D127—42no—
112C2×D127—42no—
113C2×D127—82no—
114C4×C127—41no—
115C4×C127—81no—
116C4×C127#12111yescharacteristic
117C4×C127#12111yescharacteristic
118C4×C127#12311yescharacteristic
119C4×C127#12411yescharacteristic
120Dic127—42no—
121Dic127—41no—
122Dic127—41no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127—81no—
126C2×C127#39411yescharacteristic
127C2×C127#45311yescharacteristic
128C2×C127#45511yescharacteristic
129D127—82no—
130#2037—1271nomaximal, Sylow
131C127#203711yescharacteristic, complemented, Sylow
132#535—1271no—
133#428—1271no—
134C2×#104—1271no—
135#417D12711yescharacteristic, complemented
136C2×#100—1271no—
137#390—1271no—
138#339—1271no—
139C2×#41C2×D12711yescharacteristic
140C22×D4—1271no—
141C2×M16—1271no—
142C2×M16C2×D12711yescharacteristic
143#127—2541no—
144#124—2541no—
145#121—2541no—
146#121—2541no—
147#120—2542no—
148#117—1271no—
149C2×C4⋊C4C2×D12711yescharacteristic
150C2×C22⋊C4—2542no—
151C2×C42—1271no—
152#107—2541no—
153#106—2541no—
154#104—2542no—
155#103—21no—
156#102—21no—
157#100—2542no—
158C24—1272no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163C2×Q8C22⋊D12711yescharacteristic
164C2×D4—21no—
165C2×D4—2542no—
166C2×D4—2542no—
167C2×D4—2541no—
168C2×D4—2541no—
169C2×D4—2541no—
170C2×D4—2541no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4C22⋊D12711yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178M16—21no—
179M16—21no—
180M16—1272no—
181M16—2541no—
182C2×C8—21no—
183C2×C8—2541no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—1271no—
187C4⋊C4—1271no—
188C22⋊C4—5084no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C22⋊C4—2541no—
192C42—2541no—
193C42—2541no—
194C42—2541no—
195C23—21no—
196C23—5082no—
197C23—5082no—
198C23—2542no—
199C23—2542no—
200C23—2542no—
201C23D4×D12711yescharacteristic
202Q8—21no—
203Q8—21no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208D4—10162no—
209D4—10162no—
210D4—5081no—
211D4—5081no—
212D4—5081no—
213D4—5081no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—5081no—
220C2×C4—5081no—
221C2×C4—2542no—
222C2×C4—2541no—
223C2×C4—2541no—
224C2×C4D4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4#1452324711yescharacteristic
228C2×C4#1452324811yescharacteristic
229C2×C4#1452324811yescharacteristic
230C8—41no—
231C8—2542no—
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—10162no—
238C22—5084no—
239C22—5082no—
240C22—5082no—
241C22—5082no—
242C22#1686117711yescharacteristic
243C22#1686118211yescharacteristic
244C22#1686119111yescharacteristic
245C4—41no—
246C4—81no—
247C4—5082no—
248C4—5081no—
249C4—5081no—
250C4#1686118111yescharacteristic
251C4#1686118211yescharacteristic
252C4#1686119211yescharacteristic
253C4#1686119511yescharacteristic
254C2—21no—
255C2—21no—
256C2—81no—
257C2—10162no—
258C2#2254979111yescharacteristic
259C2#2255004211yescharacteristic
260C2#2255005211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#417×C127C2
socleC22×C127#120
prime radical#22550440C2
hypercenter#417D127
hypocenterC127#2037
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #22549916C2semidirect
3. C2×#16861136C2semidirect
4. C2×#16861125C2semidirect
5. #22549777C2semidirect
6. #22549637C2semidirect
7. #417×C127C2semidirect
38. #14523232D4semidirect
131. C127#2037semidirect
135. #417D127semidirect

Maximal subgroups

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

subgroupindex
#225504402
#225499162
C2×#168611362
C2×#168611252
#225497772
#225496372
#417×C1272
#2037127