Finite Group Databasebeta

← #30639957

Subgroups — #30639957

Order 16256.

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

normal subgroups only

This group has 290 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#22550454C211yesmaximal, characteristic
2C2×#16861136C211yesmaximal, characteristic, complemented
3C2×#16861135C211yesmaximal, characteristic, complemented
4#22549875C211yesmaximal, characteristic, complemented
5#22549741C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7#417×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861188—21no—
11#16861185—21no—
12#16861183—21no—
13#16861181—22no—
14#16861180—21no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×C4×Dic127C2211yescharacteristic
17#16861136—22no—
18#16861135—22no—
19#16861132—21no—
20#16861129—21no—
21C127⋊#95—22no—
22C2×#41×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25#103×C127—21no—
26#102×C127—21no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254D411yescharacteristic
32C2×D508D411yescharacteristic
33C2×C22⋊D127—22no—
34C2×C22⋊D127—21no—
35C22×Dic127—21no—
36C22×Dic127D412yes—
37C2×C4×D127—21no—
38#14523232—21no—
39#14523232—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43Dic127⋊C4—22no—
44C22⋊Dic127—22no—
45(C2×D127)⋊C4—22no—
46C4×Dic127—21no—
47C4×Dic127D412yescomplemented
48#41×C127—21no—
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52C2×Q8×C127D411yescharacteristic
53C2×D4×C127—21no—
54C2×D4×C127D411yescharacteristic
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58M16×C127—21no—
59M16×C127—21no—
60C2×C8×C127—21no—
61C4⋊C4×C127—21no—
62C4⋊C4×C127—21no—
63C22×D127—21no—
64Dic254—21no—
65Dic254—21no—
66D508—21no—
67D508—21no—
68C22⋊D127—41no—
69C22⋊D127—41no—
70C22⋊D127—82no—
71C2×Dic127—22no—
72C2×Dic127—21no—
73C2×Dic127—22no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—42no—
77C4×D127—41no—
78C4×D127—41no—
79C127⋊C8—41no—
80C23×C127—21no—
81C23×C127C2×D411yescharacteristic
82Q8×C127—21no—
83Q8×C127—21no—
84D4×C127—21no—
85D4×C127—21no—
86D4×C127—41no—
87D4×C127—41no—
88C2×C4×C127—21no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—42no—
92C2×C4×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—41no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—41no—
105C22×C127#12011yescharacteristic
106C22×C127#12111yescharacteristic
107C22×C127#12411yescharacteristic
108C2×D127—41no—
109C2×D127—41no—
110C2×D127—41no—
111C4×C127—41no—
112C4×C127—81no—
113C4×C127#12111yescharacteristic
114C4×C127#12111yescharacteristic
115C4×C127#12311yescharacteristic
116C4×C127#12311yescharacteristic
117Dic127—42no—
118Dic127—41no—
119Dic127—42no—
120C2×C127—21no—
121C2×C127—21no—
122C2×C127—81no—
123C2×C127#39411yescharacteristic
124C2×C127#45411yescharacteristic
125C2×C127#45511yescharacteristic
126D127—81no—
127#2040—1271nomaximal, Sylow
128C127#204011yescharacteristic, complemented, Sylow
129#537—1271no—
130C2×#104—1271no—
131C2×#104—1271no—
132#417—1271no—
133#417D12711yescharacteristic, complemented
134#382—1271no—
135#343—1271no—
136C2×#41—1271no—
137C2×#41C2×D12711yescharacteristic
138C2×M16—1271no—
139C2×M16C2×D12711yescharacteristic
140C22⋊Q8—2541no—
141C22⋊Q8—2541no—
142#121—2541no—
143#121—2542no—
144#121—2541no—
145C2×C4⋊C4—1271no—
146C2×C4⋊C4C2×D12711yescharacteristic
147C2×C42—1271no—
148#104—2542no—
149#104—2542no—
150#103—21no—
151#103—2541no—
152#102—21no—
153#102—2541no—
154#95—2542no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—2541no—
160#41—2541no—
161#41—2541no—
162#41—2541no—
163C2×Q8—1271no—
164C2×Q8C22⋊D12711yescharacteristic
165C2×D4—21no—
166C2×D4—1271no—
167C2×D4—2542no—
168C2×D4—2541no—
169C2×D4C22⋊D12711yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—1272no—
173C22×C4—2541no—
174C22×C4—2541no—
175C22×C4C22×D12711yescharacteristic
176M16—21no—
177M16—21no—
178M16—2541no—
179M16—2541no—
180C2×C8—21no—
181C2×C8—2541no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—2542no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C22⋊C4—2542no—
188C22⋊C4—2542no—
189C42—1272no—
190C42—2541no—
191C23—21no—
192C23—2541no—
193C23D4×D12711yescharacteristic
194Q8—21no—
195Q8—21no—
196Q8—2541no—
197Q8—2541no—
198D4—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—10162no—
203D4—5081no—
204D4—5081no—
205D4—2541no—
206D4—2541no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—5081no—
213C2×C4—5081no—
214C2×C4—5081no—
215C2×C4—5081no—
216C2×C4—5082no—
217C2×C4—2542no—
218C2×C4—2541no—
219C2×C4—2542no—
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4#1452324711yescharacteristic
224C2×C4#1452324811yescharacteristic
225C2×C4#1452324811yescharacteristic
226C8—41no—
227C8—5081no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—5081no—
234C22—5081no—
235C22—5081no—
236C22#1686117711yescharacteristic
237C22#1686118211yescharacteristic
238C22#1686119511yescharacteristic
239C4—41no—
240C4—81no—
241C4—5082no—
242C4—5081no—
243C4—5082no—
244C4#1686118111yescharacteristic
245C4#1686118211yescharacteristic
246C4#1686119111yescharacteristic
247C4#1686119211yescharacteristic
248C2—21no—
249C2—21no—
250C2—81no—
251C2—10161no—
252C2#2254979111yescharacteristic
253C2#2255004711yescharacteristic
254C2#2255005211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#417×C127C2
socleC22×C127#120
prime radical#22550454C2
hypercenter#417D127
hypocenterC127#2040
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. C2×#16861136C2semidirect
3. C2×#16861135C2semidirect
4. #22549875C2semidirect
5. #22549741C2semidirect
6. #22549644C2semidirect
7. #417×C127C2semidirect
47. C4×Dic127D4semidirect
128. C127#2040semidirect
133. #417D127semidirect

Maximal subgroups

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

subgroupindex
#225504542
C2×#168611362
C2×#168611352
#225498752
#225497412
#225496442
#417×C1272
#2040127