Finite Group Databasebeta

← #30647059

Subgroups — #30647059

Order 16256.

688 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 169 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×#16861192C211yesmaximal, characteristic, complemented
2C2×#16861181C211yesmaximal, characteristic
3C22×C4×Dic127C211yesmaximal, characteristic, complemented
4#22549776C212yesmaximal, complemented
5#22549771C212yesmaximal, complemented
6#22549770C212yesmaximal, complemented
7#22549766C212yesmaximal, complemented
8#22549756C211yesmaximal, characteristic, complemented
9#22549753C211yesmaximal, characteristic, complemented
10#22549734C211yesmaximal, characteristic, complemented
11C2×#123×C127C211yesmaximal, characteristic, complemented
12C22×C22⋊D127C2211yescharacteristic
13C22×C22⋊D127C2211yescharacteristic
14C23×Dic127C2211yescharacteristic
15C23×Dic127C2211yescharacteristic
16#16861192—24no—
17#16861181—24no—
18C2×Dic127⋊C4C2212yes—
19C2×Dic127⋊C4C2211yescharacteristic
20C2×Dic127⋊C4C2212yes—
21C2×C22⋊Dic127C2211yescharacteristic
22C2×C22⋊Dic127C2212yes—
23C2×C22⋊Dic127C2211yescharacteristic
24C2×(C2×D127)⋊C4C2211yescharacteristic
25C2×(C2×D127)⋊C4C2212yes—
26C2×(C2×D127)⋊C4C2212yes—
27C2×C4×Dic127—24no—
28C2×C4×Dic127C2212yes—
29C2×C4×Dic127C2212yes—
30C127⋊#95C2212yes—
31C127⋊#95C2212yes—
32C127⋊#95C2212yes—
33C22×D4×C127C2211yescharacteristic
34C23×C4×C127C2211yescharacteristic
35#123×C127—24no—
36C2×C4⋊C4×C127C2212yes—
37C2×C22⋊C4×C127C2212yes—
38C2×C22⋊C4×C127C2211yescharacteristic
39C23×D127C2311yescharacteristic
40C2×C22⋊D127—22no—
41C2×C22⋊D127—24no—
42C2×C22⋊D127—22no—
43C2×C22⋊D127—22no—
44C2×C22⋊D127—22no—
45C22×Dic127—24no—
46C22×Dic127—24no—
47C22×Dic127—22no—
48C22×Dic127D414yes—
49C22×Dic127C2311yescharacteristic
50C22×Dic127C2312yes—
51C22×Dic127C2311yescharacteristic
52C22×Dic127C2312yes—
53C22×Dic127C2311yescharacteristic
54Dic127⋊C4—22no—
55Dic127⋊C4—24no—
56Dic127⋊C4—24no—
57C22⋊Dic127—22no—
58C22⋊Dic127—24no—
59C22⋊Dic127—22no—
60(C2×D127)⋊C4—22no—
61(C2×D127)⋊C4—24no—
62(C2×D127)⋊C4—24no—
63C4×Dic127—24no—
64C4×Dic127—24no—
65C24×C127C2311yescharacteristic
66C24×C127C2311yescharacteristic
67C2×D4×C127—22no—
68C2×D4×C127—22no—
69C2×D4×C127—22no—
70C22×C4×C127—22no—
71C22×C4×C127—22no—
72C22×C4×C127D414yescomplemented
73C22×C4×C127C2312yes—
74C22×C4×C127C2311yescharacteristic
75C22×C4×C127C2312yes—
76C4⋊C4×C127—24no—
77C22⋊C4×C127—22no—
78C22⋊C4×C127—24no—
79C22×D127—21no—
80C22×D127—21no—
81C22×D127—22no—
82C22×D127—22no—
83C22×D127—21no—
84C22⋊D127—44no—
85C22⋊D127—44no—
86C2×Dic127—28no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127—22no—
90C2×Dic127—28no—
91C2×Dic127—22no—
92C2×Dic127—21no—
93C2×Dic127—21no—
94C2×Dic127—22no—
95C2×Dic127—22no—
96C2×Dic127—21no—
97C2×Dic127C2×D412yes—
98C2×Dic127C2×D412yes—
99C2×Dic127#4114yes—
100C2×Dic127#4114yes—
101C23×C127—22no—
102C23×C127—22no—
103C23×C127—21no—
104C23×C127—21no—
105C23×C127—21no—
106C23×C127—22no—
107C23×C127—22no—
108C23×C127C2×D412yes—
109C23×C127C2×D412yes—
110C23×C127#4112yes—
111C23×C127C2411yescharacteristic
112D4×C127—44no—
113C2×C4×C127—28no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127C2×D414yes—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—24no—
126C22×C127—24no—
127C22×C127—41no—
128C22×C127—41no—
129C22×C127—41no—
130C22×C127—42no—
131C22×C127—42no—
132C22×C127#12114yes—
133C22×C127C22×D411yescharacteristic
134C22×C127C22×D411yescharacteristic
135C22×C127C2×#4112yes—
136C22×C127C2×#4112yes—
137C22×C127C2×#4111yescharacteristic
138C2×D127—41no—
139C2×D127—41no—
140C2×D127—41no—
141C2×D127—42no—
142C2×D127—42no—
143C4×C127—24no—
144C4×C127—41no—
145C4×C127—42no—
146Dic127—22no—
147Dic127—24no—
148Dic127—41no—
149Dic127—42no—
150Dic127#12314yescomplemented
151C2×C127—24no—
152C2×C127—81no—
153C2×C127C2×#12111yescharacteristic
154C2×C127C2×#12311yescharacteristic
155C2×C127#52511yescharacteristic
156C2×C127#52912yes—
157C2×C127#53212yes—
158D127—81no—
159#2854—1271nomaximal, Sylow
160C127#285411yescharacteristic, complemented, Sylow
161C2×#123—1271no—
162C2×#123D12711yescharacteristic, complemented
163C2×#121—1271no—
164C22×C42—1271no—
165#390—1272no—
166#389—1272no—
167#388—1272no—
168#388—1272no—
169#386—1271no—
170#386—1271no—
171#380—1271no—
172C22×D4—1271no—
173C22×D4—1271no—
174C22×D4C2×D12711yescharacteristic
175C23×C4—1271no—
176C23×C4—1271no—
177C23×C4C2×D12711yescharacteristic
178#123—24no—
179#123—2544no—
180#121—2544no—
181C2×C4⋊C4—1271no—
182C2×C4⋊C4—1272no—
183C2×C4⋊C4—1272no—
184C2×C4⋊C4C2×D12712yes—
185C2×C22⋊C4—1271no—
186C2×C22⋊C4—1271no—
187C2×C22⋊C4—1272no—
188C2×C22⋊C4—1271no—
189C2×C22⋊C4—1272no—
190C2×C22⋊C4—1272no—
191C2×C22⋊C4C2×D12711yescharacteristic
192C2×C22⋊C4C2×D12712yes—
193C2×C42—1272no—
194C2×C42—1272no—
195C2×C42—2544no—
196#95—1272no—
197#95—1272no—
198#95—1272no—
199C24—1271no—
200C24C22×D12711yescharacteristic
201C24C22×D12711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—2542no—
206C2×D4—2544no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4—2542no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—1271no—
213C22×C4—1274no—
214C22×C4—1271no—
215C22×C4—1272no—
216C22×C4—1272no—
217C22×C4—1271no—
218C22×C4—2544no—
219C22×C4—2544no—
220C22×C4—2542no—
221C22×C4C22⋊D12714yescomplemented
222C22×C4C22×D12712yes—
223C22×C4C22×D12711yescharacteristic
224C22×C4C22×D12712yes—
225C4⋊C4—24no—
226C4⋊C4—2544no—
227C4⋊C4—2542no—
228C4⋊C4—2544no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—2544no—
232C22⋊C4—2542no—
233C22⋊C4—2544no—
234C22⋊C4—2542no—
235C22⋊C4—2542no—
236C22⋊C4—2544no—
237C42—2544no—
238C42—2544no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—2541no—
247C23—2541no—
248C23—2542no—
249C23—2542no—
250C23—2541no—
251C23D4×D12712yes—
252C23C2×C22⋊D12712yes—
253C23#1452324712yes—
254C23C23×D12711yescharacteristic
255D4—44no—
256D4—5084no—
257D4—5084no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—28no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—1272no—
268C2×C4—1274no—
269C2×C4—1272no—
270C2×C4—1274no—
271C2×C4—2548no—
272C2×C4—2541no—
273C2×C4—2542no—
274C2×C4—2541no—
275C2×C4—2542no—
276C2×C4—2542no—
277C2×C4—2541no—
278C2×C4—2548no—
279C2×C4—2542no—
280C2×C4—2542no—
281C2×C4—2542no—
282C2×C4C2×C22⋊D12714yes—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—5081no—
293C22—5081no—
294C22—5081no—
295C22—5082no—
296C22—5082no—
297C22#1686118114yes—
298C22C2×D4×D12711yescharacteristic
299C22C22×C22⋊D12711yescharacteristic
300C22#41×D12712yes—
301C22#41×D12712yes—
302C22C2×#1452324711yescharacteristic
303C4—24no—
304C4—41no—
305C4—42no—
306C4—1274no—
307C4—5081no—
308C4—5082no—
309C4—2542no—
310C4—2544no—
311C2—24no—
312C2—81no—
313C2—10161no—
314C2C2×#1686118111yescharacteristic
315C2#123×D12711yescharacteristic
316C2#2255038411yescharacteristic
317C2#2255040312yes—
318C2#2255042212yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×#123×C127C2
prime radicalC2×#16861181C2
hypercenterC2×#123D127
hypocenterC127#2854
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16861192C2semidirect
3. C22×C4×Dic127C2semidirect
4. #22549776C2semidirect
5. #22549771C2semidirect
6. #22549770C2semidirect
7. #22549766C2semidirect
8. #22549756C2semidirect
9. #22549753C2semidirect
10. #22549734C2semidirect
11. C2×#123×C127C2semidirect
72. C22×C4×C127D4semidirect
150. Dic127#123semidirect
160. C127#2854semidirect
162. C2×#123D127semidirect
221. C22×C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611922
C2×#168611812
C22×C4×Dic1272
#225497762
#225497712
#225497702
#225497662
#225497562
#225497532
#225497342
C2×#123×C1272
#2854127