Finite Group Databasebeta

← #30642015

Subgroups — #30642015

Order 16256.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×(C22⋊D127)⋊C4C211yesmaximal, characteristic
3#22549817C212yesmaximal, complemented
4#22549802C211yesmaximal, characteristic, complemented
5#22549791C212yesmaximal
6#22549789C211yesmaximal, characteristic, complemented
7#22549788C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic127C212yesmaximal, complemented
9#22549723C212yesmaximal, complemented
10#22549721C211yesmaximal, characteristic, complemented
11#397×C127C211yesmaximal, characteristic, complemented
12C22×Dic254C2211yescharacteristic
13C22×C22⋊D127C2211yescharacteristic
14C23×Dic127C2211yescharacteristic
15C22×C4×D127C2211yescharacteristic
16#16861174—24no—
17(C22⋊D127)⋊C4—24no—
18C2×Dic127⋊C4C2212yes—
19C2×Dic127⋊C4C2211yescharacteristic
20C2×C22⋊Dic127C2212yes—
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2211yescharacteristic
23C2×(C2×D127)⋊C4C2212yes—
24C2×C4×Dic127C2212yes—
25C2×C4×Dic127C2212yes—
26C2×C4×Dic127C2211yescharacteristic
27C127⋊#95C2211yescharacteristic
28C127⋊#95C2212yes—
29C127⋊#95C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C2×C4⋊C4×C127C2211yescharacteristic
33C2×C22⋊C4×C127C2211yescharacteristic
34C2×C22⋊C4×C127C2212yes—
35#95×C127C2211yescharacteristic
36#95×C127C2212yes—
37C23×D127C2311yescharacteristic
38C2×Dic254—24no—
39C2×Dic254—22no—
40C2×C22⋊D127—22no—
41C2×C22⋊D127—22no—
42C2×C22⋊D127—22no—
43C22×Dic127—22no—
44C22×Dic127—24no—
45C22×Dic127C2311yescharacteristic
46C22×Dic127C2312yes—
47C22×Dic127C2311yescharacteristic
48C22×Dic127C2312yes—
49C22×Dic127C2311yescharacteristic
50C2×C4×D127—22no—
51C2×C4×D127—24no—
52Dic127⋊C4—22no—
53Dic127⋊C4—24no—
54C22⋊Dic127—24no—
55(C2×D127)⋊C4—24no—
56(C2×D127)⋊C4—24no—
57(C2×D127)⋊C4—22no—
58C4×Dic127—24no—
59C4×Dic127—24no—
60C4×Dic127D414yes—
61C24×C127C2311yescharacteristic
62C22×C4×C127C2311yescharacteristic
63C22×C4×C127C2312yes—
64C22×C4×C127C2312yes—
65C22×C4×C127C2311yescharacteristic
66C4⋊C4×C127—22no—
67C22⋊C4×C127—24no—
68C22⋊C4×C127—22no—
69C22×D127—21no—
70C22×D127—22no—
71C22×D127—22no—
72C22×D127—21no—
73C22×D127#4112yes—
74Dic254—28no—
75C22⋊D127—28no—
76C2×Dic127—22no—
77C2×Dic127—21no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C2×Dic127—21no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127—44no—
86C2×Dic127C2×D412yes—
87C2×Dic127C2×D412yes—
88C2×Dic127#4114yes—
89C2×Dic127#4114yes—
90C4×D127—44no—
91C23×C127—21no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—21no—
95C23×C127#4112yes—
96C23×C127C2411yescharacteristic
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127C2×D412yes—
107C22×C127—22no—
108C22×C127—24no—
109C22×C127—41no—
110C22×C127—41no—
111C22×C127—42no—
112C22×C127C22×D411yescharacteristic
113C22×C127C2×#4111yescharacteristic
114C22×C127C2×#4112yes—
115C22×C127C2×#4112yes—
116C22×C127C2×#4111yescharacteristic
117C2×D127—22no—
118C2×D127—24no—
119C2×D127—41no—
120C2×D127—41no—
121C2×D127—42no—
122C4×C127—22no—
123C4×C127—42no—
124C4×C127—41no—
125C4×C127—42no—
126Dic127—22no—
127Dic127—24no—
128Dic127—41no—
129Dic127—42no—
130Dic127#12414yes—
131C2×C127—42no—
132C2×C127#52511yescharacteristic
133C2×C127#52511yescharacteristic
134C2×C127C2×#12411yescharacteristic
135C2×C127#52912yes—
136C2×C127#52912yes—
137D127—42no—
138#2398—1271nomaximal, Sylow
139C127#239811yescharacteristic, complemented, Sylow
140C2×C4×Q8—1271no—
141C2×C4×D4—1271no—
142#400—1272no—
143#397—1271no—
144#397D12711yescharacteristic, complemented
145#394—1272no—
146#393—1271no—
147#393—1271no—
148C4×C22⋊C4—1272no—
149C4×C22⋊C4—1272no—
150#376—1271no—
151C22×Q8—1271no—
152C22×D4—1271no—
153C23×C4—1271no—
154C23×C4—1271no—
155C4×Q8—2544no—
156C4×D4—2544no—
157C2×C4⋊C4—1271no—
158C2×C4⋊C4—1272no—
159C2×C4⋊C4C2×D12711yescharacteristic
160C2×C22⋊C4—1272no—
161C2×C22⋊C4—1271no—
162C2×C22⋊C4—1272no—
163C2×C22⋊C4—1272no—
164C2×C22⋊C4C2×D12711yescharacteristic
165C2×C22⋊C4C2×D12712yes—
166C2×C42—1271no—
167C2×C42—1272no—
168C2×C42—1272no—
169#95—1271no—
170#95—1272no—
171#95—1272no—
172#95—1272no—
173#95—1272no—
174#95C2×D12711yescharacteristic
175#95C2×D12712yes—
176C24—1271no—
177C24C22×D12711yescharacteristic
178C2×Q8—2544no—
179C2×Q8—2542no—
180C2×D4—2542no—
181C2×D4—2542no—
182C2×D4—2542no—
183C22×C4—1271no—
184C22×C4—1272no—
185C22×C4—1271no—
186C22×C4—1272no—
187C22×C4—1271no—
188C22×C4—2542no—
189C22×C4—2544no—
190C22×C4—2542no—
191C22×C4—2544no—
192C22×C4C22×D12712yes—
193C22×C4C22×D12711yescharacteristic
194C22×C4C22×D12712yes—
195C22×C4C22×D12711yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—2542no—
198C4⋊C4—2544no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—2544no—
202C22⋊C4—2544no—
203C22⋊C4—2544no—
204C22⋊C4—2542no—
205C42—1274no—
206C42—2544no—
207C42—2544no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—1272no—
213C23—2541no—
214C23—2542no—
215C23—2542no—
216C23—2541no—
217C23#1452324712yes—
218C23C23×D12711yescharacteristic
219Q8—2548no—
220D4—2548no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—1272no—
231C2×C4—1274no—
232C2×C4—1272no—
233C2×C4—1274no—
234C2×C4—5084no—
235C2×C4—5084no—
236C2×C4—2541no—
237C2×C4—2542no—
238C2×C4—2542no—
239C2×C4—2542no—
240C2×C4—2542no—
241C2×C4—2542no—
242C2×C4—2541no—
243C2×C4—2542no—
244C2×C4—2541no—
245C2×C4D4×D12712yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—5081no—
252C22—5081no—
253C22—5082no—
254C22—2542no—
255C22—2544no—
256C22C2×D4×D12711yescharacteristic
257C22#41×D12712yes—
258C22#41×D12711yescharacteristic
259C22#41×D12712yes—
260C22C2×#1452324711yescharacteristic
261C4—22no—
262C4—42no—
263C4—41no—
264C4—42no—
265C4—1274no—
266C4—5081no—
267C4—5082no—
268C4—2544no—
269C4—2542no—
270C2—42no—
271C2—5082no—
272C2#2255038411yescharacteristic
273C2#2255038611yescharacteristic
274C2#124×D12711yescharacteristic
275C2#2255040312yes—
276C2#2255040912yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#397×C127C2
prime radicalC22×C22⋊D127C22
hypercenter#397D127
hypocenterC127#2398
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#16861174C2semidirect
3. #22549817C2semidirect
4. #22549802C2semidirect
6. #22549789C2semidirect
7. #22549788C2semidirect
8. C22⋊C4×Dic127C2semidirect
9. #22549723C2semidirect
10. #22549721C2semidirect
11. #397×C127C2semidirect
139. C127#2398semidirect
144. #397D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611742
C2×(C22⋊D127)⋊C42
#225498172
#225498022
#225497912
#225497892
#225497882
C22⋊C4×Dic1272
#225497232
#225497212
#397×C1272
#2398127