Finite Group Databasebeta

← #30642012

Subgroups — #30642012

Order 16256.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×Q8×Dic127C211yesmaximal, characteristic, complemented
2C2×(C22⋊D127)⋊C4C211yesmaximal, characteristic
3#22549818C212yesmaximal, complemented
4#22549803C211yesmaximal, characteristic, complemented
5#22549801C211yesmaximal, characteristic, complemented
6#22549792C212yesmaximal
7#22549787C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic127C212yesmaximal, complemented
9#22549723C212yesmaximal, complemented
10#22549721C211yesmaximal, characteristic, complemented
11#393×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2211yescharacteristic
13C23×Dic127C2211yescharacteristic
14C22×C4×D127C2211yescharacteristic
15Q8×Dic127—24no—
16(C22⋊D127)⋊C4—24no—
17C2×C4⋊Dic127C2212yes—
18C2×C4⋊Dic127C2211yescharacteristic
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—
32C22×Q8×C127C2211yescharacteristic
33C2×C22⋊C4×C127C2212yes—
34C2×C22⋊C4×C127C2211yescharacteristic
35#95×C127C2211yescharacteristic
36#95×C127C2212yes—
37C23×D127C2311yescharacteristic
38C2×C22⋊D127—22no—
39C2×C22⋊D127—22no—
40C2×C22⋊D127—22no—
41C22×Dic127—22no—
42C22×Dic127—24no—
43C22×Dic127C2311yescharacteristic
44C22×Dic127C2312yes—
45C22×Dic127C2312yes—
46C22×Dic127C2311yescharacteristic
47C22×Dic127C2311yescharacteristic
48C2×C4×D127—22no—
49C2×C4×D127—24no—
50C4⋊Dic127—24no—
51C4⋊Dic127—22no—
52Dic127⋊C4—22no—
53C22⋊Dic127—24no—
54(C2×D127)⋊C4—24no—
55(C2×D127)⋊C4—22no—
56(C2×D127)⋊C4—24no—
57C4×Dic127—24no—
58C4×Dic127—24no—
59C4×Dic127D414yes—
60C24×C127C2311yescharacteristic
61C2×Q8×C127—22no—
62C2×Q8×C127—24no—
63C22×C4×C127C2311yescharacteristic
64C22×C4×C127C2312yes—
65C22×C4×C127C2311yescharacteristic
66C22×C4×C127C2312yes—
67C22⋊C4×C127—22no—
68C22⋊C4×C127—24no—
69C22×D127—21no—
70C22×D127—22no—
71C22×D127—22no—
72C22×D127—21no—
73C22×D127#4112yes—
74C22⋊D127—28no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—22no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127—44no—
86C2×Dic127C2×D412yes—
87C2×Dic127C2×D412yes—
88C2×Dic127#4114yes—
89C4×D127—44no—
90C23×C127—21no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—21no—
94C23×C127#4112yes—
95C23×C127C2411yescharacteristic
96Q8×C127—28no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—21no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127C2×D412yes—
106C2×C4×C127#4114yes—
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—24no—
124C4×C127—41no—
125C4×C127—42no—
126Dic127—22no—
127Dic127—42no—
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#397—1271no—
145#394—1272no—
146#393—1271no—
147#393D12711yescharacteristic, complemented
148C4×C22⋊C4—1272no—
149C4×C22⋊C4—1272no—
150#376—1271no—
151C22×Q8C2×D12711yescharacteristic
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⋊C4—1271no—
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—22no—
179C2×Q8—24no—
180C2×D4—2542no—
181C2×D4—2542no—
182C2×D4—2542no—
183C22×C4—1271no—
184C22×C4—1271no—
185C22×C4—1272no—
186C22×C4—1271no—
187C22×C4—1272no—
188C22×C4—2542no—
189C22×C4—2544no—
190C22×C4—2542no—
191C22×C4—2544no—
192C22×C4C22×D12712yes—
193C22×C4C22×D12711yescharacteristic
194C22×C4C22×D12711yescharacteristic
195C22×C4C22×D12712yes—
196C4⋊C4—2542no—
197C4⋊C4—2544no—
198C4⋊C4—2542no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—2544no—
202C22⋊C4—2544no—
203C22⋊C4—2542no—
204C22⋊C4—2544no—
205C42—1274no—
206C42—2544no—
207C42—2544no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—1272no—
213C23—2541no—
214C23—2542no—
215C23—2542no—
216C23—2541no—
217C23#1452324912yes—
218C23C23×D12711yescharacteristic
219Q8—28no—
220D4—2548no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—1272no—
230C2×C4—1274no—
231C2×C4—1272no—
232C2×C4—5084no—
233C2×C4—5084no—
234C2×C4—2542no—
235C2×C4—2541no—
236C2×C4—2542no—
237C2×C4—2542no—
238C2×C4—2542no—
239C2×C4—2542no—
240C2×C4—2542no—
241C2×C4—2542no—
242C2×C4—2541no—
243C2×C4—2541no—
244C2×C4D4×D12712yes—
245C2×C4#1452324714yes—
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×D12711yescharacteristic
258C22#41×D12712yes—
259C22C2×#1452324712yes—
260C22C2×#1452324911yescharacteristic
261C4—22no—
262C4—24no—
263C4—41no—
264C4—42no—
265C4—1274no—
266C4—5082no—
267C4—5081no—
268C4—5082no—
269C4—2542no—
270C2—42no—
271C2—5082no—
272C2#2255038211yescharacteristic
273C2#2255038911yescharacteristic
274C2#124×D12711yescharacteristic
275C2#2255040512yes—
276C2#2255041212yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#393×C127C2
prime radicalC22×C22⋊D127C22
hypercenter#393D127
hypocenterC127#2398
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic127C2semidirect
3. #22549818C2semidirect
4. #22549803C2semidirect
5. #22549801C2semidirect
7. #22549787C2semidirect
8. C22⋊C4×Dic127C2semidirect
9. #22549723C2semidirect
10. #22549721C2semidirect
11. #393×C127C2semidirect
139. C127#2398semidirect
147. #393D127semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1272
C2×(C22⋊D127)⋊C42
#225498182
#225498032
#225498012
#225497922
#225497872
C22⋊C4×Dic1272
#225497232
#225497212
#393×C1272
#2398127