Finite Group Databasebeta

← #30650097

Subgroups — #30650097

Order 16256.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#22550410C211yesmaximal, characteristic
2#22550183C212yesmaximal, complemented
3#22550168C211yesmaximal, characteristic, complemented
4#22550163C211yesmaximal, characteristic, complemented
5#22550132C212yesmaximal, complemented
6#22550124C212yesmaximal
7#22550098C212yesmaximal, complemented
8#22550083C211yesmaximal, characteristic, complemented
9#22550078C211yesmaximal, characteristic, complemented
10#22549900C211yesmaximal, characteristic, complemented
11#533×C127C211yesmaximal, characteristic, complemented
12C2×#14523238C2212yes—
13C2×#14523235C2212yes—
14C2×#14523232C2212yes—
15#16861203—21no—
16#16861201C2211yescharacteristic
17#16861197C2211yescharacteristic
18#16861192—21no—
19#16861188C2211yescharacteristic
20#16861183C2211yescharacteristic
21C4×Dic254C2211yescharacteristic
22C4×D508C2211yescharacteristic
23C4×C22⋊D127—21no—
24#16861166—21no—
25#16861142C2212yes—
26#16861140C2212yes—
27C4⋊(C127⋊C8)C2211yescharacteristic
28C4⋊(C127⋊C8)C2211yescharacteristic
29C4⋊(C127⋊C8)C2212yes—
30#16861133C2212yes—
31#16861133C2212yes—
32#16861132C2212yes—
33#16861129C2212yes—
34C4⋊Q8×C127C2211yescharacteristic
35C4⋊Q8×C127C2211yescharacteristic
36#125×C127—21no—
37C22⋊Q8×C127—21no—
38C22⋊Q8×C127C2212yes—
39C4×Q8×C127—21no—
40C4×Q8×C127C2212yes—
41#117×C127—21no—
42C2×C42×C127C2211yescharacteristic
43C2×Dic254C2311yescharacteristic
44C2×D508C2311yescharacteristic
45C2×C22⋊D127—21no—
46C2×C4×D127—21no—
47#14523238—42no—
48#14523235—42no—
49#14523232—42no—
50C2×C127⋊C8C2312yes—
51C2×C127⋊C8C2312yes—
52C4⋊Dic127C2311yescharacteristic
53C4⋊Dic127C2311yescharacteristic
54Dic127⋊C4—21no—
55Dic127⋊C4—21no—
56Dic127⋊C4—21no—
57Dic127⋊C4—21no—
58C22⋊Dic127—21no—
59(C2×D127)⋊C4—21no—
60(C2×D127)⋊C4—21no—
61(C2×D127)⋊C4—21no—
62(C2×D127)⋊C4—21no—
63C4×Dic127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127C2312yes—
66C22×C4×C127D411yescharacteristic
67C22×C4×C127D411yescharacteristic
68C22×C4×C127C2311yescharacteristic
69C4⋊C4×C127—21no—
70C4⋊C4×C127—21no—
71C4⋊C4×C127—21no—
72C4⋊C4×C127—22no—
73C4⋊C4×C127—22no—
74C4⋊C4×C127C2312yes—
75C22⋊C4×C127—22no—
76C42×C127—22no—
77C42×C127D411yescharacteristic
78C42×C127D411yescharacteristic
79C42×C127C2311yescharacteristic
80C42×C127C2311yescharacteristic
81C22×D127—21no—
82Dic254—21no—
83Dic254—21no—
84D508—21no—
85D508—21no—
86C22⋊D127—42no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127—21no—
90C4×D127—42no—
91C127⋊C8—22no—
92C127⋊C8—22no—
93C23×C127C2×D411yescharacteristic
94Q8×C127—22no—
95Q8×C127—22no—
96Q8×C127—42no—
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—41no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2411yescharacteristic
109C22×C127—21no—
110C22×C127—21no—
111C22×C127—21no—
112C22×C127C22×D411yescharacteristic
113C2×D127—41no—
114C2×D127—41no—
115C2×D127—41no—
116C4×C127—21no—
117C4×C127—21no—
118C4×C127—21no—
119C4×C127—21no—
120C4×C127—42no—
121C4×C127—42no—
122C4×C127#13712yes—
123C4×C127H(5,𝔽2)11yescharacteristic
124C4×C1272−1+411yescharacteristic
125Dic127—41no—
126Dic127—41no—
127Dic127—41no—
128C2×C127—41no—
129C2×C127#53711yescharacteristic
130C2×C127C2×#13711yescharacteristic
131C2×C127#57511yescharacteristic
132D127—81no—
133#3252—1271nomaximal, Sylow
134C127#325211yescharacteristic, complemented, Sylow
135#533D12711yescharacteristic, complemented
136#529—1271no—
137#479—1272no—
138#477—1271no—
139#476—1271no—
140#470—1272no—
141#468—1272no—
142#464—1272no—
143#462—1271no—
144#461—1271no—
145#423—1271no—
146C2×Q16—1272no—
147C2×SD16—1272no—
148C2×M16—1272no—
149C4⋊Q8C2×D12711yescharacteristic
150C4⋊Q8C2×D12711yescharacteristic
151#126—2541no—
152#125—21no—
153#125—1271no—
154#124—1271no—
155#123—2541no—
156C22⋊Q8—21no—
157C22⋊Q8—1271no—
158C22⋊Q8C2×D12712yes—
159#121—1271no—
160C4×Q8—21no—
161C4×Q8—1271no—
162C4×Q8C2×D12712yes—
163C4×D4—1271no—
164C4×D4—2541no—
165#117—21no—
166#117—2541no—
167C2×C42C2×D12711yescharacteristic
168#107—1272no—
169#106—1272no—
170C4⋊C8—1271no—
171C4⋊C8—1271no—
172C4⋊C8—1272no—
173#103—1272no—
174#103—1272no—
175#103—1272no—
176#102—1272no—
177C2×Q8—21no—
178C2×Q8—1271no—
179C2×Q8C22×D12712yes—
180C2×D4—1271no—
181C2×D4—2541no—
182C22×C4—2541no—
183C22×C4C22⋊D12711yescharacteristic
184C22×C4C22⋊D12711yescharacteristic
185C22×C4C22×D12711yescharacteristic
186Q16—5082no—
187SD16—5082no—
188M16—5082no—
189C2×C8—1272no—
190C2×C8—1272no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—1271no—
197C4⋊C4—1271no—
198C4⋊C4—2541no—
199C4⋊C4—2541no—
200C4⋊C4—2541no—
201C4⋊C4—2541no—
202C4⋊C4C22×D12712yes—
203C22⋊C4—22no—
204C22⋊C4—2541no—
205C22⋊C4—2541no—
206C22⋊C4—2541no—
207C22⋊C4—2541no—
208C22⋊C4—2541no—
209C42—22no—
210C42—2541no—
211C42C22⋊D12711yescharacteristic
212C42C22⋊D12711yescharacteristic
213C42C22×D12711yescharacteristic
214C42C22×D12711yescharacteristic
215C23—2541no—
216C23C2×C22⋊D12711yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—42no—
220Q8—2541no—
221Q8—2541no—
222D4—5082no—
223D4—2541no—
224D4—2541no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—5082no—
232C2×C4—2541no—
233C2×C4—2541no—
234C2×C4—2541no—
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4C2×C22⋊D12711yescharacteristic
237C2×C4C2×C22⋊D12711yescharacteristic
238C2×C4C2×C22⋊D12711yescharacteristic
239C2×C4C2×C22⋊D12711yescharacteristic
240C2×C4C23×D12711yescharacteristic
241C8—2542no—
242C8—2542no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—5081no—
247C22—5081no—
248C22—5081no—
249C22C22×C22⋊D12711yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—5081no—
257C4—5081no—
258C4—5081no—
259C4#1686124012yes—
260C4#1686125511yescharacteristic
261C4#1686125611yescharacteristic
262C2—41no—
263C2—10161no—
264C2#2255045411yescharacteristic
265C2C2×#1686124011yescharacteristic
266C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#533×C127C2
socleC22×C127C22×D4
prime radical#16861183C22
hypercenter#533D127
hypocenterC127#3252
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #22550183C2semidirect
3. #22550168C2semidirect
4. #22550163C2semidirect
5. #22550132C2semidirect
7. #22550098C2semidirect
8. #22550083C2semidirect
9. #22550078C2semidirect
10. #22549900C2semidirect
11. #533×C127C2semidirect
134. C127#3252semidirect
135. #533D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
#225501832
#225501682
#225501632
#225501322
#225501242
#225500982
#225500832
#225500782
#225499002
#533×C1272
#3252127