Finite Group Databasebeta

← #30650533

Subgroups — #30650533

Order 16256.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550693C211yesmaximal, characteristic, complemented
2#22550692C212yesmaximal, complemented
3#22550689C211yesmaximal, characteristic, complemented
4#22550254C211yesmaximal, characteristic, complemented
5#22550152C211yesmaximal, characteristic, complemented
6#22550150C212yesmaximal, complemented
7#22550061C212yesmaximal, complemented
8#22550057C212yesmaximal, complemented
9#22549992C211yesmaximal, characteristic, complemented
10#22549991C211yesmaximal, characteristic, complemented
11#473×C127C211yesmaximal, characteristic, complemented
12#16861257C2212yes—
13#16861256C2212yes—
14#16861242—22no—
15#16861242C2212yes—
16#16861242C2212yes—
17#16861241—22no—
18#16861241—22no—
19#16861240C2212yes—
20#16861238—22no—
21#16861232—21no—
22#16861231—21no—
23#16861231C2211yescharacteristic
24#16861230—22no—
25#16861229C2211yescharacteristic
26C2×Dic508—21no—
27C2×Dic508C2211yescharacteristic
28C2×#14523238C2212yes—
29Q16×D127—21no—
30Q16×D127—22no—
31#16861217C2212yes—
32#16861215C2211yescharacteristic
33#16861207C2212yes—
34#16861144C2211yescharacteristic
35#16861136C2211yescharacteristic
36#16861135C2212yes—
37#16861135C2211yescharacteristic
38#16861127C2212yes—
39C8×Dic127C2211yescharacteristic
40#137×C127C2212yes—
41C2×Q16×C127C2211yescharacteristic
42#131×C127C2211yescharacteristic
43#108×C127C2211yescharacteristic
44#101×C127C2212yes—
45#14523249—22no—
46#14523249—22no—
47#14523248—22no—
48#14523248—42no—
49#14523248C2311yescharacteristic
50#14523248C2312yes—
51#14523247—22no—
52#14523247—42no—
53C2×Dic254—22no—
54C2×Dic254C2312yes—
55Q8×D127—22no—
56Q8×D127—22no—
57Q8×D127—22no—
58Dic508—22no—
59Dic508—22no—
60Dic508—21no—
61Dic508—21no—
62Dic508D411yescharacteristic
63#14523238—22no—
64#14523238—22no—
65#14523238—22no—
66#14523238D411yescharacteristic
67#14523237—22no—
68#14523237—22no—
69#14523237D412yes—
70#14523236D411yescharacteristic
71#14523235—22no—
72#14523235D411yescharacteristic
73D1016D411yescharacteristic
74#14523233D411yescharacteristic
75#14523232C2311yescharacteristic
76#14523231—22no—
77#14523231—21no—
78C2×C127⋊C8C2311yescharacteristic
79C8×D127—22no—
80C8×D127—21no—
81Dic127⋊C4—42no—
82C4×Dic127C2311yescharacteristic
83#41×C127C2311yescharacteristic
84C2×Q8×C127C2312yes—
85Q16×C127—22no—
86Q16×C127—21no—
87Q16×C127—21no—
88SD16×C127—22no—
89M16×C127—21no—
90M16×C127C2312yes—
91M16×C127C2311yescharacteristic
92C2×C8×C127—21no—
93C2×C8×C127C2311yescharacteristic
94Dic254—22no—
95Dic254—22no—
96Dic254—42no—
97Dic254—42no—
98Dic254C2×D411yescharacteristic
99Dic254C2×D412yes—
100D508—42no—
101D508C2×D411yescharacteristic
102D508C2×D412yes—
103C22⋊D127—22no—
104C22⋊D127—21no—
105C22⋊D127—42no—
106C2×Dic127—21no—
107C2×Dic127—22no—
108C2×Dic127—42no—
109C4×D127—22no—
110C4×D127—21no—
111C4×D127—42no—
112C4×D127—42no—
113C4×D127—42no—
114C127⋊C8—21no—
115C127⋊C8C2×D411yescharacteristic
116C127⋊C8C2×D411yescharacteristic
117Q8×C127—22no—
118Q8×C127—22no—
119Q8×C127C2×D411yescharacteristic
120D4×C127—21no—
121D4×C127C2×D411yescharacteristic
122C2×C4×C127—21no—
123C2×C4×C127—22no—
124C2×C4×C127C2411yescharacteristic
125C8×C127—21no—
126C8×C127—22no—
127C8×C127C2×D411yescharacteristic
128C8×C127C2×D411yescharacteristic
129C22×C127—21no—
130C22×C127H(5,𝔽2)11yescharacteristic
131C2×D127—22no—
132C2×D127—21no—
133C4×C127—21no—
134C4×C127—42no—
135C4×C127C22×D411yescharacteristic
136C4×C127C22×D411yescharacteristic
137Dic127—22no—
138Dic127—21no—
139Dic127—22no—
140Dic127—42no—
141C2×C127—21no—
142C2×C127—41no—
143C2×C127D4211yescharacteristic
144D127—42no—
145D127—41no—
146#3293—1271nomaximal, Sylow
147C127#329311yescharacteristic, complemented, Sylow
148#578—1272no—
149#578—1271no—
150#578—1271no—
151#494—1271no—
152#473—1271no—
153#473—1272no—
154#473D12711yescharacteristic, complemented
155#456—1272no—
156#456—1272no—
157#443—1271no—
158#443—1271no—
1592−1+4—1272no—
1602−1+4—1272no—
161#137—1272no—
162#137—1272no—
163#137—1272no—
164#137—2542no—
165#137—2542no—
166#137—2542no—
167#137—2542no—
168#137C2×D12712yes—
169#135—1271no—
170#135—1271no—
171#135—2541no—
172#135—2542no—
173#135—2541no—
174C2×Q16—1272no—
175C2×Q16—1271no—
176C2×Q16—2541no—
177C2×Q16—2542no—
178C2×Q16—2541no—
179C2×Q16C2×D12711yescharacteristic
180#131—1272no—
181#131—1271no—
182#131C2×D12711yescharacteristic
183C4⋊Q8—1272no—
184#108—1271no—
185#108C2×D12711yescharacteristic
186#104—1271no—
187#104—1272no—
188#104—1271no—
189#101—1272no—
190#101C2×D12712yes—
191C4×C8—1271no—
192#41—1271no—
193#41—1272no—
194#41—5082no—
195#41—5082no—
196#41—2542no—
197#41—2542no—
198#41—2542no—
199#41—2542no—
200#41C22×D12711yescharacteristic
201C2×Q8—1272no—
202C2×Q8—2542no—
203C2×Q8—2542no—
204C2×Q8—2542no—
205C2×Q8—2542no—
206C2×Q8C22×D12712yes—
207Q16—22no—
208Q16—21no—
209Q16—21no—
210Q16—1271no—
211Q16—1271no—
212Q16—2542no—
213Q16—2542no—
214Q16—2542no—
215Q16—2542no—
216Q16—2542no—
217Q16—2541no—
218Q16—2541no—
219SD16—22no—
220SD16—1271no—
221SD16—1271no—
222SD16—1272no—
223SD16—2542no—
224SD16—2542no—
225SD16—2542no—
226D8—1271no—
227D8—1271no—
228M16—21no—
229M16—1271no—
230M16—2542no—
231M16—2541no—
232M16C22×D12712yes—
233M16C22×D12711yescharacteristic
234C2×C8—21no—
235C2×C8—1271no—
236C2×C8—2542no—
237C2×C8—2541no—
238C2×C8C22×D12711yescharacteristic
239C4⋊C4—5082no—
240C42—1271no—
241Q8—22no—
242Q8—22no—
243Q8—1271no—
244Q8—1272no—
245Q8—5082no—
246Q8—5082no—
247Q8—2542no—
248Q8—2542no—
249Q8D4×D12711yescharacteristic
250D4—21no—
251D4—1271no—
252D4—1272no—
253D4—5082no—
254D4—5082no—
255D4—2541no—
256D4—2542no—
257D4D4×D12711yescharacteristic
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—5082no—
261C2×C4—5082no—
262C2×C4—5082no—
263C2×C4—5082no—
264C2×C4—2542no—
265C2×C4—2541no—
266C2×C4—2542no—
267C2×C4—2541no—
268C2×C4C23×D12711yescharacteristic
269C8—21no—
270C8—22no—
271C8—1271no—
272C8—1271no—
273C8—2541no—
274C8D4×D12711yescharacteristic
275C8D4×D12711yescharacteristic
276C22—21no—
277C22—2542no—
278C22—2541no—
279C22#1686125511yescharacteristic
280C4—21no—
281C4—42no—
282C4—5082no—
283C4—2542no—
284C4—2541no—
285C4—2542no—
286C4C2×D4×D12711yescharacteristic
287C4C2×D4×D12711yescharacteristic
288C2—21no—
289C2—41no—
290C2—5082no—
291C2—5081no—
292C2#2255051011yescharacteristic
subgroupG/H
centerC2#22550510
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#473×C127C2
socleC2×C127D42
hypercenter#473D127
hypocenterC127#3293
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #22550693C2semidirect
2. #22550692C2semidirect
3. #22550689C2semidirect
4. #22550254C2semidirect
5. #22550152C2semidirect
6. #22550150C2semidirect
7. #22550061C2semidirect
8. #22550057C2semidirect
9. #22549992C2semidirect
10. #22549991C2semidirect
11. #473×C127C2semidirect
147. C127#3293semidirect
154. #473D127semidirect

Maximal subgroups

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

subgroupindex
#225506932
#225506922
#225506892
#225502542
#225501522
#225501502
#225500612
#225500572
#225499922
#225499912
#473×C1272
#3293127