Finite Group Databasebeta

← #30637700

Subgroups — #30637700

Order 16256.

282 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 51 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#22550453C211yesmaximal, characteristic
2#22550017C211yesmaximal, characteristic, complemented
3#22550014C211yesmaximal, characteristic, complemented
4#22549797C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic, complemented
8C2×#14523247—21no—
9C22×Dic254C2211yescharacteristic
10C2×#14523237—21no—
11C2×#14523236—21no—
12#16861187—21no—
13#16861186—21no—
14#16861183C2211yescharacteristic
15#16861182—21no—
16#16861181—21no—
17#16861181—21no—
18C2×C4⋊Dic127—21no—
19C2×Dic127⋊C4C2211yescharacteristic
20#16861134—21no—
21#16861132—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic
23C127⋊#95—21no—
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27#14523247—42no—
28#14523247—42no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—42no—
33C2×D508D411yescharacteristic
34C2×C22⋊D127—21no—
35C2×C22⋊D127—21no—
36C2×C22⋊D127—21no—
37C22×Dic127—21no—
38C22×Dic127—21no—
39C22×Dic127D411yescharacteristic
40C22×Dic127D411yescharacteristic
41C2×C4×D127—21no—
42#14523237—42no—
43#14523236—42no—
44C2×C127⋊C8—21no—
45C4⋊Dic127—42no—
46C4⋊Dic127D411yescharacteristic
47Dic127⋊C4—21no—
48Dic127⋊C4—21no—
49Dic127⋊C4—21no—
50C22⋊Dic127—21no—
51(C2×D127)⋊C4—21no—
52(C2×D127)⋊C4—21no—
53C2×D4×C127—21no—
54C2×D4×C127D411yescharacteristic
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57C2×C8×C127—21no—
58C4⋊C4×C127D411yescharacteristic
59C22⋊C4×C127—21no—
60C22×D127—21no—
61Dic254—22no—
62Dic254—22no—
63Dic254—42no—
64Dic254—41no—
65Dic254—41no—
66D508—41no—
67C22⋊D127—42no—
68C22⋊D127—41no—
69C22⋊D127—41no—
70C22⋊D127—81no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—42no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C2×Dic127SD1612yescomplemented
81C4×D127—42no—
82C127⋊C8—41no—
83C23×C127—21no—
84C23×C127C2×D411yescharacteristic
85D4×C127—41no—
86D4×C127—41no—
87D4×C127—41no—
88C2×C4×C127—22no—
89C2×C4×C127—21no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C8×C127—41no—
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127#12011yescharacteristic
102C22×C127C2×SD1611yescharacteristic
103C22×C127AGL(1,ℤ/8)11yescharacteristic
104C2×D127—41no—
105C2×D127—41no—
106C2×D127—41no—
107C4×C127—21no—
108C4×C127—21no—
109C4×C127—42no—
110C4×C127—41no—
111Dic127—22no—
112Dic127—41no—
113Dic127—42no—
114Dic127—41no—
115C2×C127—21no—
116C2×C127—21no—
117C2×C127—81no—
118C2×C127#45011yescharacteristic
119C2×C127#45611yescharacteristic
120C2×C127B(4,𝔽2)11yescharacteristic
121D127—81no—
122#1604—1271nomaximal, Sylow
123C127#160411yescharacteristic, complemented, Sylow
124#537—1271no—
125#448—1271no—
126#448—1271no—
127#395—1271no—
128#327—1271no—
129#327D12711yescharacteristic, complemented
130#324—1271no—
131C2×#41—2541no—
132C22×Q8—1271no—
133C2×SD16—2541no—
134C2×SD16—2541no—
135C22⋊Q8—2541no—
136C22⋊Q8—2541no—
137#121—1271no—
138#121—2541no—
139#121—2541no—
140#121—2541no—
141#121C2×D12711yescharacteristic
142C2×C4⋊C4—1271no—
143C2×C4⋊C4—2541no—
144#103—2541no—
145#103—2541no—
146C22⋊C8—1271no—
147C22⋊C8C2×D12711yescharacteristic
148#95—2541no—
149#95C2×D12711yescharacteristic
150#41—5082no—
151#41—5082no—
152C2×Q8—5082no—
153C2×Q8—2541no—
154C2×Q8—2541no—
155C2×Q8—2541no—
156C2×D4—21no—
157C2×D4—1271no—
158C2×D4—2541no—
159C2×D4—2541no—
160C2×D4—2541no—
161C2×D4D50811yescharacteristic
162C22×C4—21no—
163C22×C4—1271no—
164C22×C4—1271no—
165C22×C4—2541no—
166C22×C4—2541no—
167C22×C4—2541no—
168C22×C4C22×D12711yescharacteristic
169SD16—5082no—
170SD16—5082no—
171C2×C8—21no—
172C2×C8—2541no—
173C4⋊C4—1271no—
174C4⋊C4—5082no—
175C4⋊C4—2541no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C4⋊C4D50811yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C22⋊C4—2541no—
183C23—21no—
184C23—2541no—
185C23D4×D12711yescharacteristic
186Q8—5081no—
187Q8—5081no—
188Q8—5082no—
189Q8—2542no—
190Q8—2542no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—10161no—
195D4—5081no—
196D4—5081no—
197D4—5081no—
198D4—5082no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—1272no—
205C2×C4—5082no—
206C2×C4—5081no—
207C2×C4—5081no—
208C2×C4—5082no—
209C2×C4—5081no—
210C2×C4—2542no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4—2541no—
214C2×C4—2541no—
215C2×C4D4×D12711yescharacteristic
216C2×C4C2×D50811yescharacteristic
217C8—41no—
218C8—5081no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—5081no—
225C22—5081no—
226C22—5081no—
227C22#1686117911yescharacteristic
228C22SD16×D12711yescharacteristic
229C22#1686123311yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—5081no—
235C4—5082no—
236C4—5081no—
237C4—2542no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—10161no—
242C2#2255002811yescharacteristic
243C2#2255006011yescharacteristic
244C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
prime radical#22550453C2
hypercenter#327D127
hypocenterC127#1604
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22550017C2semidirect
3. #22550014C2semidirect
4. #22549797C2semidirect
5. #22549609C2semidirect
6. #22549601C2semidirect
7. #327×C127C2semidirect
80. C2×Dic127SD16semidirect
123. C127#1604semidirect
129. #327D127semidirect

Maximal subgroups

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

subgroupindex
#225504532
#225500172
#225500142
#225497972
#225496092
#225496012
#327×C1272
#1604127