Finite Group Databasebeta

← #30639953

Subgroups — #30639953

Order 16256.

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

normal subgroups only

This group has 306 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#22550454C211yesmaximal, characteristic
2C2×#16861137C211yesmaximal, characteristic, complemented
3#22549878C211yesmaximal, characteristic, complemented
4#22549877C211yesmaximal, characteristic, complemented
5#22549746C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861188—21no—
11#16861185—21no—
12#16861183—21no—
13#16861181—22no—
14#16861180—21no—
15C2×C4⋊Dic127—22no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×C4⋊Dic127C2211yescharacteristic
18#16861137—22no—
19#16861134—21no—
20#16861133—21no—
21#16861131—21no—
22#16861130—21no—
23C2×#41×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25C2×C42×C127C2211yescharacteristic
26#104×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—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—
43C4⋊Dic127—21no—
44C4⋊Dic127—21no—
45C4⋊Dic127—44no—
46Dic127⋊C4—22no—
47C22⋊Dic127—22no—
48(C2×D127)⋊C4—22no—
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—
62C42×C127—21no—
63C42×C127D412yescomplemented
64C22×D127—21no—
65Dic254—21no—
66Dic254—21no—
67D508—21no—
68D508—21no—
69C22⋊D127—41no—
70C22⋊D127—41no—
71C22⋊D127—82no—
72C2×Dic127—22no—
73C2×Dic127—21no—
74C2×Dic127—42no—
75C2×Dic127—42no—
76C2×Dic127—41no—
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—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—21no—
92C2×C4×C127—41no—
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#12411yescharacteristic
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—41no—
112C4×C127—24no—
113C4×C127—41no—
114C4×C127#12111yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12311yescharacteristic
117C4×C127#12311yescharacteristic
118Dic127—41no—
119Dic127—42no—
120Dic127—81no—
121C2×C127—21no—
122C2×C127—21no—
123C2×C127—81no—
124C2×C127#39411yescharacteristic
125C2×C127#45311yescharacteristic
126C2×C127#45611yescharacteristic
127D127—81no—
128#2039—1271nomaximal, Sylow
129C127#203911yescharacteristic, complemented, Sylow
130#537—1271no—
131C2×#104—1271no—
132C2×#104D12711yescharacteristic, complemented
133#417—1271no—
134#417—1271no—
135#384—1271no—
136#343—1271no—
137C2×#41—1271no—
138C2×#41C2×D12711yescharacteristic
139C2×M16—1271no—
140C2×M16C2×D12711yescharacteristic
141C22⋊Q8—2541no—
142C22⋊Q8—2541no—
143#121—2541no—
144#121—2542no—
145#121—2541no—
146C2×C4⋊C4—1271no—
147C2×C4⋊C4—1271no—
148C2×C4⋊C4—2542no—
149C2×C42C2×D12711yescharacteristic
150#104—22no—
151#104—2542no—
152#103—2541no—
153#103—2541no—
154#102—2541no—
155#102—2541no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160#41—2541no—
161#41—2541no—
162#41—2541no—
163#41—2541no—
164C2×Q8—1271no—
165C2×Q8D50811yescharacteristic
166C2×D4—21no—
167C2×D4—1271no—
168C2×D4—2542no—
169C2×D4—2541no—
170C2×D4D50811yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—1272no—
174C22×C4—2541no—
175C22×C4—2541no—
176C22×C4C22×D12711yescharacteristic
177M16—21no—
178M16—21no—
179M16—2541no—
180M16—2541no—
181C2×C8—21no—
182C2×C8—2541no—
183C4⋊C4—5084no—
184C4⋊C4—2542no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C4⋊C4—2541no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C42—21no—
192C42C22⋊D12712yescomplemented
193C23—21no—
194C23—2541no—
195C23D4×D12711yescharacteristic
196Q8—21no—
197Q8—21no—
198Q8—2541no—
199Q8—2541no—
200D4—21no—
201D4—21no—
202D4—41no—
203D4—41no—
204D4—10162no—
205D4—5081no—
206D4—5081no—
207D4—2541no—
208D4—2541no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—5082no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4—5082no—
220C2×C4—2542no—
221C2×C4—2541no—
222C2×C4D4×D12711yescharacteristic
223C2×C4C2×C22⋊D12711yescharacteristic
224C2×C4C2×D50811yescharacteristic
225C2×C4#1452324711yescharacteristic
226C2×C4#1452324711yescharacteristic
227C2×C4#1452324811yescharacteristic
228C8—41no—
229C8—5081no—
230C22—21no—
231C22—21no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—5081no—
236C22—5081no—
237C22—5081no—
238C22#1686117911yescharacteristic
239C22#1686118111yescharacteristic
240C22#1686119511yescharacteristic
241C4—24no—
242C4—41no—
243C4—10161no—
244C4—5082no—
245C4—5081no—
246C4#1686118011yescharacteristic
247C4#1686118311yescharacteristic
248C4#1686119111yescharacteristic
249C4#1686119311yescharacteristic
250C2—21no—
251C2—21no—
252C2—81no—
253C2—10161no—
254C2#2254979211yescharacteristic
255C2#2255004511yescharacteristic
256C2#2255006011yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC22×C127#120
prime radical#22550454C2
hypercenterC2×#104D127
hypocenterC127#2039
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16861137C2semidirect
3. #22549878C2semidirect
4. #22549877C2semidirect
5. #22549746C2semidirect
6. #22549644C2semidirect
7. C2×#104×C127C2semidirect
63. C42×C127D4semidirect
129. C127#2039semidirect
132. C2×#104D127semidirect
192. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504542
C2×#168611372
#225498782
#225498772
#225497462
#225496442
C2×#104×C1272
#2039127