Finite Group Databasebeta

← #30650123

Subgroups — #30650123

Order 16256.

342 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 115 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#22550431C211yesmaximal, characteristic
2#22550430C211yesmaximal, characteristic
3#22550160C212yesmaximal, complemented
4#22550158C211yesmaximal, characteristic, complemented
5#22550135C212yesmaximal
6#22550133C212yesmaximal
7#22550087C211yesmaximal, characteristic, complemented
8#22550086C211yesmaximal, characteristic, complemented
9#22550085C212yesmaximal, complemented
10#22549898C211yesmaximal, characteristic
11#475×C127C211yesmaximal, characteristic, complemented
12C2×Dic508C2212yes—
13C2×#14523238C2212yes—
14C2×#14523231C2212yes—
15#16861209C2211yescharacteristic
16#16861208C2211yescharacteristic
17#16861207C2211yescharacteristic
18#16861200—21no—
19#16861199—21no—
20#16861189C2212yes—
21#16861189C2211yescharacteristic
22#16861187—21no—
23#16861185—21no—
24#16861185C2211yescharacteristic
25Q8×Dic127—21no—
26C4×Dic254—21no—
27C4×Dic254C2211yescharacteristic
28#16861174C2212yes—
29#16861167—21no—
30#16861166—21no—
31C42×D127C2211yescharacteristic
32#16861140C2212yes—
33C4⋊(C127⋊C8)C2211yescharacteristic
34Dic127⋊C8C2212yes—
35#16861134C2212yes—
36#16861132C2212yes—
37#16861132C2212yes—
38C4⋊Q8×C127C2211yescharacteristic
39C4×Q8×C127C2211yescharacteristic
40#106×C127C2212yes—
41C4⋊C8×C127C2211yescharacteristic
42#103×C127C2212yes—
43C2×Dic254—21no—
44C2×Dic254C2311yescharacteristic
45C2×Dic254C2312yes—
46C2×C4×D127D411yescharacteristic
47C2×C4×D127D411yescharacteristic
48C2×C4×D127C2311yescharacteristic
49Dic508—42no—
50#14523238—42no—
51#14523231—42no—
52C2×C127⋊C8C2312yes—
53C4⋊Dic127—21no—
54C4⋊Dic127—22no—
55C4⋊Dic127—21no—
56C4⋊Dic127—21no—
57C4⋊Dic127C2311yescharacteristic
58Dic127⋊C4—22no—
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62Dic127⋊C4—21no—
63Dic127⋊C4—21no—
64(C2×D127)⋊C4—22no—
65(C2×D127)⋊C4—21no—
66(C2×D127)⋊C4—21no—
67C4×Dic127—22no—
68C4×Dic127—21no—
69C4×Dic127D411yescharacteristic
70C4×Dic127D411yescharacteristic
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127—21no—
73C2×Q8×C127C2311yescharacteristic
74C2×C8×C127C2312yes—
75C4⋊C4×C127—21no—
76C4⋊C4×C127—21no—
77C4⋊C4×C127C2311yescharacteristic
78C4⋊C4×C127C2312yes—
79C42×C127—21no—
80C42×C127C2311yescharacteristic
81C22×D127C2×D411yescharacteristic
82Dic254—22no—
83Dic254—21no—
84Dic254—21no—
85Dic254—22no—
86Dic254—42no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127—22no—
90C2×Dic127C2×D411yescharacteristic
91C2×Dic127C2×D411yescharacteristic
92C2×Dic127C2×D411yescharacteristic
93C4×D127—22no—
94C4×D127—21no—
95C4×D127—21no—
96C4×D127—22no—
97C127⋊C8—22no—
98Q8×C127—21no—
99Q8×C127—21no—
100Q8×C127—42no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—22no—
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2×D411yescharacteristic
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127C22×D411yescharacteristic
109C2×D127—21no—
110C2×D127—21no—
111C2×D127—21no—
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—42no—
115C4×C127—41no—
116C4×C127#13712yes—
117C4×C127H(5,𝔽2)11yescharacteristic
118C4×C1272−1+411yescharacteristic
119Dic127—21no—
120Dic127—21no—
121Dic127—41no—
122Dic127—42no—
123Dic127—41no—
124Dic127#13712yes—
125C2×C127#53711yescharacteristic
126C2×C127C2×#13711yescharacteristic
127C2×C127C2×#13711yescharacteristic
128D127—41no—
129#3255—1271nomaximal, Sylow
130C127#325511yescharacteristic, complemented, Sylow
131#533—1271no—
132#533—1271no—
133#475—1271no—
134#475—1272no—
135#475D12711yescharacteristic, complemented
136#470—1272no—
137#470—1272no—
138#462—1271no—
139#462—1271no—
140#462—1272no—
141#423—1271no—
142C2×Q16—1272no—
143C2×Q16—1272no—
144C2×M16—1272no—
145C4⋊Q8—1271no—
146C4⋊Q8—1271no—
147C4⋊Q8—1271no—
148C4⋊Q8C2×D12711yescharacteristic
149#125—2541no—
150#125—2541no—
151C22⋊Q8—1271no—
152C22⋊Q8—1271no—
153C22⋊Q8—1272no—
154C22⋊Q8—2541no—
155C22⋊Q8—2541no—
156C4×Q8—1272no—
157C4×Q8—1271no—
158C4×Q8—2541no—
159C4×Q8—2541no—
160C4×Q8C2×D12711yescharacteristic
161#117—2541no—
162#117—2541no—
163C2×C42—1271no—
164#106—1272no—
165#106C2×D12712yes—
166C4⋊C8—1272no—
167C4⋊C8—1271no—
168C4⋊C8C2×D12711yescharacteristic
169#103—1272no—
170#103—1272no—
171#103—1272no—
172#103C2×D12712yes—
173C2×Q8—21no—
174C2×Q8—1272no—
175C2×Q8—1271no—
176C2×Q8—2541no—
177C2×Q8C22×D12711yescharacteristic
178C22×C4—1271no—
179C22×C4—1271no—
180C22×C4—1271no—
181Q16—5082no—
182Q16—5082no—
183M16—5082no—
184C2×C8—1272no—
185C2×C8C22×D12712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—1271no—
189C4⋊C4—2541no—
190C4⋊C4—2541no—
191C4⋊C4—2542no—
192C4⋊C4—2541no—
193C4⋊C4—2541no—
194C4⋊C4—2541no—
195C4⋊C4—2542no—
196C4⋊C4—2541no—
197C4⋊C4—2541no—
198C4⋊C4—2541no—
199C4⋊C4C22×D12711yescharacteristic
200C4⋊C4C22×D12712yes—
201C22⋊C4—2541no—
202C22⋊C4—2541no—
203C22⋊C4—2542no—
204C42—21no—
205C42—1271no—
206C42—1271no—
207C42—1271no—
208C42—2542no—
209C42—2541no—
210C42C22×D12711yescharacteristic
211C23—1271no—
212Q8—21no—
213Q8—21no—
214Q8—42no—
215Q8—5082no—
216Q8—2541no—
217Q8—2542no—
218Q8—2542no—
219Q8—2541no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—1271no—
224C2×C4—1271no—
225C2×C4—1271no—
226C2×C4—2542no—
227C2×C4—2541no—
228C2×C4—2542no—
229C2×C4—2541no—
230C2×C4—2542no—
231C2×C4—2541no—
232C2×C4—2541no—
233C2×C4D4×D12711yescharacteristic
234C2×C4D4×D12711yescharacteristic
235C2×C4C23×D12711yescharacteristic
236C8—22no—
237C8—2542no—
238C22—2541no—
239C22—2541no—
240C22—2541no—
241C22C2×D4×D12711yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—41no—
246C4—1272no—
247C4—5081no—
248C4—5082no—
249C4—5081no—
250C4—2541no—
251C4—2541no—
252C4#1686123812yes—
253C4#1686125511yescharacteristic
254C4#1686125711yescharacteristic
255C2—5081no—
256C2#2255045611yescharacteristic
257C2#137×D12711yescharacteristic
258C2C2×#1686123811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#475×C127C2
socleC22×C127C22×D4
prime radicalC22×D127C2×D4
hypercenter#475D127
hypocenterC127#3255
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #22550160C2semidirect
4. #22550158C2semidirect
7. #22550087C2semidirect
8. #22550086C2semidirect
9. #22550085C2semidirect
11. #475×C127C2semidirect
130. C127#3255semidirect
135. #475D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#225504302
#225501602
#225501582
#225501352
#225501332
#225500872
#225500862
#225500852
#225498982
#475×C1272
#3255127