Finite Group Databasebeta

← #30639952

Subgroups — #30639952

Order 16256.

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

normal subgroups only

This group has 306 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#22549745C211yesmaximal, 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×Dic127⋊C4—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861136—22no—
19#16861135—22no—
20#16861132—21no—
21#16861129—21no—
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—
44Dic127⋊C4—44no—
45C22⋊Dic127—22no—
46(C2×D127)⋊C4—22no—
47C4×Dic127—21no—
48C4×Dic127D412yescomplemented
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—21no—
65Dic254—21no—
66Dic254—21no—
67D508—21no—
68D508—21no—
69C22⋊D127—41no—
70C22⋊D127—41no—
71C22⋊D127—82no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—21no—
77C2×Dic127—42no—
78C4×D127—41no—
79C4×D127—41no—
80C127⋊C8—41no—
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#12411yescharacteristic
109C2×D127—41no—
110C2×D127—41no—
111C2×D127—41no—
112C4×C127—41no—
113C4×C127—81no—
114C4×C127#12111yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12311yescharacteristic
117C4×C127#12311yescharacteristic
118Dic127—24no—
119Dic127—41no—
120Dic127—42no—
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×#104—1271no—
133#417—1271no—
134#417D12711yescharacteristic, complemented
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—2542no—
148C2×C4⋊C4C2×D12711yescharacteristic
149C2×C42—1271no—
150#104—2542no—
151#104—2542no—
152#103—21no—
153#103—2541no—
154#102—21no—
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×Q8C22⋊D12711yescharacteristic
166C2×D4—21no—
167C2×D4—1271no—
168C2×D4—2542no—
169C2×D4—2541no—
170C2×D4C22⋊D12711yescharacteristic
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—21no—
184C4⋊C4—21no—
185C4⋊C4—5084no—
186C4⋊C4—2542no—
187C4⋊C4—2541no—
188C4⋊C4—2541no—
189C22⋊C4—2542no—
190C22⋊C4—2542no—
191C42—1272no—
192C42—2541no—
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—21no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—5081no—
215C2×C4—5081no—
216C2×C4—5082no—
217C2×C4—2542no—
218C2×C4—2541no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4—2542no—
222C2×C4D4×D12711yescharacteristic
223C2×C4D4×D12711yescharacteristic
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4#1452324711yescharacteristic
226C2×C4#1452324811yescharacteristic
227C2×C4#1452324811yescharacteristic
228C8—41no—
229C8—5081no—
230C22—21no—
231C22—21no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—5081no—
236C22—5081no—
237C22—5081no—
238C22#1686117711yescharacteristic
239C22#1686118211yescharacteristic
240C22#1686119511yescharacteristic
241C4—41no—
242C4—81no—
243C4—5082no—
244C4—5081no—
245C4—2544no—
246C4#1686118111yescharacteristic
247C4#1686118211yescharacteristic
248C4#1686119111yescharacteristic
249C4#1686119211yescharacteristic
250C2—21no—
251C2—21no—
252C2—81no—
253C2—10161no—
254C2#2254979111yescharacteristic
255C2#2255004211yescharacteristic
256C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#417×C127C2
socleC22×C127#120
prime radical#22550454C2
hypercenter#417D127
hypocenterC127#2039
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. #22549745C2semidirect
6. #22549644C2semidirect
7. #417×C127C2semidirect
48. C4×Dic127D4semidirect
129. C127#2039semidirect
134. #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
#225497452
#225496442
#417×C1272
#2039127