Finite Group Databasebeta

← #30642003

Subgroups — #30642003

Order 16256.

528 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#16861174C211yesmaximal, characteristic, complemented
2C2×#16861167C211yesmaximal, characteristic
3#22549811C211yesmaximal, characteristic
4#22549810C212yesmaximal, complemented
5#22549800C212yesmaximal
6#22549796C211yesmaximal, characteristic, complemented
7#22549789C211yesmaximal, characteristic
8#22549727C211yesmaximal, characteristic, complemented
9#22549727C212yesmaximal, complemented
10#22549723C212yesmaximal
11#400×C127C211yesmaximal, characteristic, complemented
12C22×Dic254C2211yescharacteristic
13C22×C4×D127C2211yescharacteristic
14#16861174—24no—
15#16861167—24no—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×Dic127⋊C4C2212yes—
18C2×Dic127⋊C4C2212yes—
19C2×Dic127⋊C4C2212yes—
20C2×Dic127⋊C4C2212yes—
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2211yescharacteristic
23C2×(C2×D127)⋊C4C2211yescharacteristic
24C2×(C2×D127)⋊C4C2212yes—
25C2×C4×Dic127C2211yescharacteristic
26C2×C4×Dic127C2212yes—
27C2×C4×Dic127C2212yes—
28C2×C4×Dic127C2211yescharacteristic
29C127⋊#95C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2211yescharacteristic
32C2×C4⋊C4×C127C2211yescharacteristic
33C2×C4⋊C4×C127C2211yescharacteristic
34#95×C127C2211yescharacteristic
35#95×C127C2212yes—
36#95×C127C2212yes—
37C23×D127C2311yescharacteristic
38C2×Dic254—24no—
39C2×Dic254—22no—
40C22×Dic127C2311yescharacteristic
41C22×Dic127C2311yescharacteristic
42C22×Dic127C2312yes—
43C22×Dic127C2312yes—
44C22×Dic127C2311yescharacteristic
45C2×C4×D127—22no—
46C2×C4×D127—22no—
47C2×C4×D127Q814yescomplemented
48C4⋊Dic127—22no—
49Dic127⋊C4—24no—
50Dic127⋊C4—24no—
51Dic127⋊C4—24no—
52Dic127⋊C4—24no—
53(C2×D127)⋊C4—22no—
54(C2×D127)⋊C4—24no—
55(C2×D127)⋊C4—24no—
56(C2×D127)⋊C4—22no—
57C4×Dic127—22no—
58C4×Dic127—24no—
59C4×Dic127—24no—
60C4×Dic127D414yes—
61C22×C4×C127C2312yes—
62C22×C4×C127C2311yescharacteristic
63C22×C4×C127C2312yes—
64C22×C4×C127C2311yescharacteristic
65C22×C4×C127C2311yescharacteristic
66C4⋊C4×C127—22no—
67C4⋊C4×C127—22no—
68C22×D127—21no—
69C22×D127—22no—
70C22×D127—22no—
71C22×D127—21no—
72C22×D127C2×Q812yes—
73Dic254—28no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C2×Dic127C2×D412yes—
82C2×Dic127C2×D412yes—
83C2×Dic127C2×Q812yes—
84C2×Dic127#4114yes—
85C2×Dic127#4114yes—
86C2×Dic127#4112yes—
87C4×D127—28no—
88C23×C127C2411yescharacteristic
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—22no—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127C2×D412yes—
101C2×C4×C127C2×Q812yes—
102C22×C127C22×D411yescharacteristic
103C22×C127C22×Q811yescharacteristic
104C22×C127C2×#4111yescharacteristic
105C22×C127C2×#4112yes—
106C22×C127C2×#4112yes—
107C2×D127—22no—
108C2×D127—24no—
109C2×D127—41no—
110C2×D127—42no—
111C2×D127—41no—
112C4×C127—22no—
113C4×C127—22no—
114C4×C127—41no—
115C4×C127—42no—
116C4×C127—42no—
117Dic127—24no—
118Dic127—24no—
119Dic127—22no—
120Dic127—41no—
121Dic127C22⋊Q814yes—
122C2×C127C2×C22⋊Q811yescharacteristic
123C2×C127#52511yescharacteristic
124C2×C127#52912yes—
125C2×C127#53211yescharacteristic
126C2×C127#53312yes—
127D127—42no—
128#2396—1271nomaximal, Sylow
129C127#239611yescharacteristic, complemented, Sylow
130C2×C4×Q8—1271no—
131C2×#117—1271no—
132#400D12711yescharacteristic, complemented
133#399—1271no—
134#398—1272no—
135#396—1272no—
136#395—1271no—
137#393—1271no—
138C4×C4⋊C4—1272no—
139C4×C4⋊C4—1271no—
140C4×C22⋊C4—1272no—
141C22×Q8—1271no—
142C23×C4—1271no—
143C4×Q8—2544no—
144#117—2544no—
145C2×C4⋊C4—1271no—
146C2×C4⋊C4—1272no—
147C2×C4⋊C4—1272no—
148C2×C4⋊C4—1272no—
149C2×C4⋊C4—1272no—
150C2×C4⋊C4C2×D12711yescharacteristic
151C2×C4⋊C4C2×D12711yescharacteristic
152C2×C22⋊C4—1271no—
153C2×C22⋊C4—1271no—
154C2×C22⋊C4—1272no—
155C2×C22⋊C4—1272no—
156C2×C42—1271no—
157C2×C42—1271no—
158C2×C42—1272no—
159C2×C42—1272no—
160#95—1272no—
161#95—1272no—
162#95—1271no—
163#95C2×D12712yes—
164#95C2×D12711yescharacteristic
165#95C2×D12712yes—
166C24—1271no—
167C2×Q8—2544no—
168C2×Q8—2542no—
169C22×C4—1271no—
170C22×C4—1272no—
171C22×C4—1272no—
172C22×C4—1271no—
173C22×C4—1274no—
174C22×C4—1271no—
175C22×C4—2542no—
176C22×C4—2542no—
177C22×C4C22×D12712yes—
178C22×C4C22×D12711yescharacteristic
179C22×C4C22×D12712yes—
180C22×C4C22×D12711yescharacteristic
181C22×C4C22×D12711yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—2544no—
185C4⋊C4—2544no—
186C4⋊C4—2544no—
187C4⋊C4—2544no—
188C4⋊C4—2542no—
189C22⋊C4—2544no—
190C22⋊C4—2544no—
191C22⋊C4—2542no—
192C22⋊C4—2542no—
193C42—1274no—
194C42—2544no—
195C42—2542no—
196C42—2544no—
197C23—1272no—
198C23—2541no—
199C23—2542no—
200C23—2542no—
201C23—2541no—
202C23C23×D12711yescharacteristic
203Q8—2548no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—1272no—
216C2×C4—1272no—
217C2×C4—1272no—
218C2×C4—1272no—
219C2×C4—1274no—
220C2×C4—1274no—
221C2×C4—2548no—
222C2×C4—2541no—
223C2×C4—2542no—
224C2×C4—2542no—
225C2×C4—2542no—
226C2×C4—2542no—
227C2×C4—2542no—
228C2×C4—2542no—
229C2×C4D4×D12712yes—
230C2×C4Q8×D12712yes—
231C22—5081no—
232C22—5082no—
233C22—5081no—
234C22—2542no—
235C22—2544no—
236C22C2×D4×D12711yescharacteristic
237C22C2×Q8×D12711yescharacteristic
238C22#41×D12711yescharacteristic
239C22#41×D12712yes—
240C22#41×D12712yes—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—1274no—
247C4—5081no—
248C4—2544no—
249C4—2544no—
250C4—2542no—
251C2—5082no—
252C2C22⋊Q8×D12711yescharacteristic
253C2#2255038411yescharacteristic
254C2#2255040912yes—
255C2#2255042511yescharacteristic
256C2#2255043212yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#400×C127C2
prime radicalC23×D127C23
hypercenter#400D127
hypocenterC127#2396
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#16861174C2semidirect
4. #22549810C2semidirect
6. #22549796C2semidirect
8. #22549727C2semidirect
9. #22549727C2semidirect
11. #400×C127C2semidirect
47. C2×C4×D127Q8semidirect
129. C127#2396semidirect
132. #400D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611742
C2×#168611672
#225498112
#225498102
#225498002
#225497962
#225497892
#225497272
#225497232
#400×C1272
#2396127