Subgroups — #30653498

Order 16256.

1424 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550711	C₂	1	2	yes	maximal, complemented
2	#22550693	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550690	C₂	1	3	yes	maximal, complemented
4	#22550687	C₂	1	2	yes	maximal, complemented
5	#22550686	C₂	1	6	yes	maximal, complemented
6	#22550677	C₂	1	3	yes	maximal, complemented
7	#22550672	C₂	1	6	yes	maximal, complemented
8	C₂×#16861228	C₂	1	3	yes	maximal, complemented
9	#135×D₁₂₇	C₂	1	3	yes	maximal, complemented
10	#131×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#576×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16861256	—	2	2	no	—
13	#16861256	C₂²	1	2	yes	complemented
14	#16861256	C₂²	1	6	yes	complemented
15	#16861254	—	2	6	no	—
16	#16861254	C₂²	1	6	yes	complemented
17	C₂×#14523247	—	2	6	no	—
18	C₂×#14523247	C₂²	1	6	yes	complemented
19	#41×D₁₂₇	—	2	2	no	—
20	#41×D₁₂₇	C₂²	1	1	yes	characteristic
21	#41×D₁₂₇	C₂²	1	6	yes	complemented
22	#16861242	C₂²	1	6	yes	complemented
23	#16861239	C₂²	1	6	yes	complemented
24	#16861238	C₂²	1	12	yes	complemented
25	#16861235	C₂²	1	6	yes	complemented
26	#16861233	C₂²	1	12	yes	complemented
27	#16861232	C₂²	1	1	yes	characteristic, complemented
28	#16861231	C₂²	1	3	yes	complemented
29	#16861230	C₂²	1	6	yes	complemented
30	#16861229	C₂²	1	6	yes	complemented
31	#16861228	C₂²	1	3	yes	complemented
32	#16861228	C₂²	1	6	yes	complemented
33	#16861228	C₂²	1	6	yes	complemented
34	#16861228	C₂²	1	6	yes	complemented
35	C₂×Dic₅₀₈	C₂²	1	3	yes	complemented
36	Q₁₆×D₁₂₇	C₂²	1	3	yes	complemented
37	C₂×#14523236	C₂²	1	6	yes	complemented
38	SD₁₆×D₁₂₇	C₂²	1	6	yes	complemented
39	D₈×D₁₂₇	C₂²	1	3	yes	complemented
40	#16861217	C₂²	1	3	yes	complemented
41	#16861216	C₂²	1	1	yes	characteristic, complemented
42	#16861215	C₂²	1	3	yes	complemented
43	M₁₆×D₁₂₇	C₂²	1	3	yes	complemented
44	C₂×C₈×D₁₂₇	C₂²	1	3	yes	complemented
45	H(5,𝔽₂)×C₁₂₇	C₂²	1	2	yes	complemented
46	AGL(1,ℤ/8)×C₁₂₇	C₂²	1	6	yes	complemented
47	#135×C₁₂₇	C₂²	1	3	yes	complemented
48	C₂×D₈×C₁₂₇	C₂²	1	3	yes	complemented
49	#131×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
50	#14523249	D₄	1	1	yes	characteristic
51	#14523249	C₂³	1	2	yes	—
52	#14523248	—	2	6	no	—
53	#14523248	—	2	6	no	—
54	#14523248	C₂³	1	6	yes	—
55	#14523248	C₂³	1	3	yes	—
56	#14523247	—	2	6	no	—
57	#14523247	—	2	12	no	—
58	#14523247	—	2	6	no	—
59	#14523247	—	2	6	no	—
60	#14523247	—	2	6	no	—
61	#14523247	—	2	6	no	—
62	#14523247	D₄	1	3	yes	—
63	#14523247	C₂³	1	6	yes	—
64	#14523247	C₂³	1	12	yes	—
65	C₂×Dic₂₅₄	—	2	6	no	—
66	C₂×Dic₂₅₄	C₂³	1	6	yes	—
67	Q₈×D₁₂₇	—	2	2	no	—
68	Q₈×D₁₂₇	D₄	1	1	yes	characteristic
69	Q₈×D₁₂₇	C₂³	1	6	yes	—
70	C₂×C₂²⋊D₁₂₇	—	2	6	no	—
71	C₂×C₂²⋊D₁₂₇	—	2	12	no	—
72	D₄×D₁₂₇	—	2	6	no	—
73	D₄×D₁₂₇	—	2	6	no	—
74	D₄×D₁₂₇	D₄	1	3	yes	—
75	D₄×D₁₂₇	C₂³	1	6	yes	—
76	C₂²×Dic₁₂₇	—	2	6	no	—
77	C₂×C₄×D₁₂₇	—	2	6	no	—
78	C₂×C₄×D₁₂₇	C₂³	1	3	yes	—
79	Dic₅₀₈	C₂³	1	3	yes	—
80	Dic₅₀₈	C₂³	1	6	yes	—
81	#14523238	C₂³	1	6	yes	—
82	#14523237	C₂³	1	6	yes	—
83	#14523236	C₂³	1	6	yes	—
84	#14523236	C₂³	1	12	yes	—
85	#14523235	C₂³	1	2	yes	—
86	D₁₀₁₆	C₂³	1	1	yes	characteristic
87	#14523233	C₂³	1	6	yes	—
88	#14523232	C₂³	1	3	yes	—
89	#14523231	C₂³	1	6	yes	—
90	C₂×C₁₂₇⋊C₈	C₂³	1	3	yes	—
91	C₈×D₁₂₇	C₂³	1	1	yes	characteristic
92	C₈×D₁₂₇	C₂³	1	3	yes	—
93	C₈×D₁₂₇	C₂³	1	3	yes	—
94	C₈×D₁₂₇	C₂³	1	3	yes	—
95	#41×C₁₂₇	—	2	2	no	—
96	#41×C₁₂₇	C₂³	1	6	yes	—
97	#41×C₁₂₇	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₂₇	—	2	6	no	—
99	C₂×D₄×C₁₂₇	C₂³	1	6	yes	—
100	Q₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
101	SD₁₆×C₁₂₇	C₂³	1	6	yes	—
102	D₈×C₁₂₇	C₂³	1	3	yes	—
103	D₈×C₁₂₇	C₂³	1	6	yes	—
104	M₁₆×C₁₂₇	C₂³	1	3	yes	—
105	C₂×C₈×C₁₂₇	C₂³	1	3	yes	—
106	C₂²×D₁₂₇	—	2	6	no	—
107	C₂²×D₁₂₇	C₂×D₄	1	3	yes	—
108	Dic₂₅₄	—	2	6	no	—
109	Dic₂₅₄	—	2	6	no	—
110	Dic₂₅₄	C₂×D₄	1	3	yes	—
111	Dic₂₅₄	C₂⁴	1	6	yes	—
112	D₅₀₈	—	2	2	no	—
113	D₅₀₈	C₂×D₄	1	3	yes	—
114	D₅₀₈	C₂⁴	1	2	yes	—
115	C₂²⋊D₁₂₇	—	2	6	no	—
116	C₂²⋊D₁₂₇	—	2	12	no	—
117	C₂²⋊D₁₂₇	—	2	6	no	—
118	C₂²⋊D₁₂₇	—	2	6	no	—
119	C₂²⋊D₁₂₇	C₂×D₄	1	6	yes	—
120	C₂×Dic₁₂₇	—	2	6	no	—
121	C₂×Dic₁₂₇	—	2	6	no	—
122	C₂×Dic₁₂₇	—	2	12	no	—
123	C₂×Dic₁₂₇	C₂×D₄	1	3	yes	—
124	C₄×D₁₂₇	—	2	6	no	—
125	C₄×D₁₂₇	—	2	6	no	—
126	C₄×D₁₂₇	—	2	2	no	—
127	C₄×D₁₂₇	C₂×D₄	1	3	yes	—
128	C₄×D₁₂₇	C₂×D₄	1	3	yes	—
129	C₄×D₁₂₇	C₂⁴	1	1	yes	characteristic
130	C₄×D₁₂₇	C₂⁴	1	3	yes	—
131	C₁₂₇⋊C₈	C₂⁴	1	1	yes	characteristic
132	C₁₂₇⋊C₈	C₂⁴	1	3	yes	—
133	C₂³×C₁₂₇	—	2	6	no	—
134	Q₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
135	Q₈×C₁₂₇	C₂⁴	1	2	yes	—
136	D₄×C₁₂₇	—	2	6	no	—
137	D₄×C₁₂₇	—	2	6	no	—
138	D₄×C₁₂₇	C₂×D₄	1	3	yes	—
139	D₄×C₁₂₇	C₂⁴	1	6	yes	—
140	C₂×C₄×C₁₂₇	—	2	6	no	—
141	C₂×C₄×C₁₂₇	C₂⁴	1	3	yes	—
142	C₈×C₁₂₇	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₂₇	C₂⁴	1	3	yes	—
144	C₂²×C₁₂₇	—	2	6	no	—
145	C₂²×C₁₂₇	—	8	6	no	—
146	C₂²×C₁₂₇	C₂²×D₄	1	3	yes	—
147	C₂×D₁₂₇	—	2	2	no	—
148	C₂×D₁₂₇	—	4	3	no	—
149	C₂×D₁₂₇	—	8	6	no	—
150	C₂×D₁₂₇	C₂²×D₄	1	3	yes	—
151	C₂×D₁₂₇	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₁₂₇	—	2	2	no	—
153	C₄×C₁₂₇	C₂²×D₄	1	3	yes	—
154	C₄×C₁₂₇	C₂⁵	1	1	yes	characteristic
155	Dic₁₂₇	—	2	6	no	—
156	Dic₁₂₇	C₂²×D₄	1	1	yes	characteristic
157	Dic₁₂₇	C₂²×D₄	1	3	yes	—
158	C₂×C₁₂₇	—	2	3	no	—
159	C₂×C₁₂₇	—	4	6	no	—
160	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
161	D₁₂₇	—	2	1	no	—
162	D₁₂₇	—	2	3	no	—
163	D₁₂₇	—	4	2	no	—
164	#3584	—	127	1	no	maximal, Sylow
165	C₁₂₇	#3584	1	1	yes	characteristic, complemented, Sylow
166	#585	—	127	2	no	—
167	#578	—	127	3	no	—
168	#578	—	127	1	no	—
169	#577	—	127	2	no	—
170	#577	—	127	6	no	—
171	#576	—	127	3	no	—
172	#576	D₁₂₇	1	1	yes	characteristic, complemented
173	#575	—	127	6	no	—
174	C₂×#135	—	127	3	no	—
175	C₂×#135	—	127	3	no	—
176	C₂×#131	—	127	1	no	—
177	2₋¹⁺⁴	—	127	6	no	—
178	2₋¹⁺⁴	—	127	2	no	—
179	2₋¹⁺⁴	—	254	2	no	—
180	H(5,𝔽₂)	—	127	6	no	—
181	H(5,𝔽₂)	—	254	6	no	—
182	H(5,𝔽₂)	C₂×D₁₂₇	1	2	yes	complemented
183	C₂×#41	—	127	1	no	—
184	C₂×#41	—	127	6	no	—
185	C₂×#41	—	127	6	no	—
186	C₂×#41	—	254	6	no	—
187	C₂×#41	—	254	2	no	—
188	#137	—	127	6	no	—
189	#137	—	127	12	no	—
190	#137	—	127	6	no	—
191	AGL(1,ℤ/8)	—	127	6	no	—
192	AGL(1,ℤ/8)	—	127	12	no	—
193	AGL(1,ℤ/8)	C₂×D₁₂₇	1	6	yes	complemented
194	#135	—	127	6	no	—
195	#135	—	127	1	no	—
196	#135	—	127	6	no	—
197	#135	—	127	3	no	—
198	#135	—	127	6	no	—
199	#135	—	127	6	no	—
200	#135	—	127	6	no	—
201	#135	—	127	3	no	—
202	#135	C₂×D₁₂₇	1	3	yes	complemented
203	C₂×Q₁₆	—	127	3	no	—
204	C₂×Q₁₆	—	127	3	no	—
205	C₂×SD₁₆	—	127	6	no	—
206	C₂×SD₁₆	—	127	6	no	—
207	C₂×D₈	—	127	3	no	—
208	C₂×D₈	C₂×D₁₂₇	1	3	yes	complemented
209	#131	—	127	1	no	—
210	#131	—	127	3	no	—
211	#131	—	127	3	no	—
212	#131	C₂×D₁₂₇	1	1	yes	characteristic, complemented
213	C₂×M₁₆	—	127	3	no	—
214	C₂²×C₈	—	127	3	no	—
215	#41	—	2	2	no	—
216	#41	—	127	1	no	—
217	#41	—	127	3	no	—
218	#41	—	127	3	no	—
219	#41	—	127	2	no	—
220	#41	—	127	6	no	—
221	#41	—	127	12	no	—
222	#41	—	127	6	no	—
223	#41	—	254	6	no	—
224	#41	—	254	6	no	—
225	#41	—	254	6	no	—
226	#41	—	254	6	no	—
227	#41	—	254	6	no	—
228	#41	—	254	12	no	—
229	#41	—	254	6	no	—
230	#41	—	254	6	no	—
231	#41	C₂²×D₁₂₇	1	6	yes	—
232	#41	C₂²×D₁₂₇	1	1	yes	characteristic
233	C₂×Q₈	—	127	1	no	—
234	C₂×Q₈	—	127	6	no	—
235	C₂×Q₈	—	127	6	no	—
236	C₂×Q₈	—	254	6	no	—
237	C₂×Q₈	—	254	2	no	—
238	C₂×D₄	—	2	6	no	—
239	C₂×D₄	—	127	3	no	—
240	C₂×D₄	—	127	6	no	—
241	C₂×D₄	—	254	6	no	—
242	C₂×D₄	—	254	6	no	—
243	C₂×D₄	—	254	12	no	—
244	C₂×D₄	—	254	6	no	—
245	C₂×D₄	C₂²×D₁₂₇	1	6	yes	—
246	C₂²×C₄	—	127	3	no	—
247	C₂²×C₄	—	254	6	no	—
248	C₂²×C₄	—	254	6	no	—
249	Q₁₆	—	127	6	no	—
250	Q₁₆	—	127	3	no	—
251	Q₁₆	—	127	6	no	—
252	Q₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
253	SD₁₆	—	127	2	no	—
254	SD₁₆	—	127	6	no	—
255	SD₁₆	—	127	12	no	—
256	SD₁₆	—	127	6	no	—
257	SD₁₆	C₂²×D₁₂₇	1	6	yes	—
258	D₈	—	127	6	no	—
259	D₈	—	127	1	no	—
260	D₈	C₂²×D₁₂₇	1	3	yes	—
261	D₈	C₂²×D₁₂₇	1	6	yes	—
262	M₁₆	—	127	3	no	—
263	M₁₆	—	127	6	no	—
264	M₁₆	C₂²×D₁₂₇	1	3	yes	—
265	C₂×C₈	—	127	3	no	—
266	C₂×C₈	—	127	1	no	—
267	C₂×C₈	—	127	3	no	—
268	C₂×C₈	—	127	3	no	—
269	C₂×C₈	—	127	3	no	—
270	C₂×C₈	C₂²×D₁₂₇	1	3	yes	—
271	C₂³	—	2	6	no	—
272	C₂³	—	127	3	no	—
273	C₂³	—	254	6	no	—
274	Q₈	—	127	3	no	—
275	Q₈	—	127	6	no	—
276	Q₈	—	254	6	no	—
277	Q₈	—	254	6	no	—
278	Q₈	D₄×D₁₂₇	1	1	yes	characteristic
279	Q₈	C₂³×D₁₂₇	1	2	yes	—
280	D₄	—	2	6	no	—
281	D₄	—	2	6	no	—
282	D₄	—	127	6	no	—
283	D₄	—	127	3	no	—
284	D₄	—	127	2	no	—
285	D₄	—	254	2	no	—
286	D₄	—	254	6	no	—
287	D₄	—	254	6	no	—
288	D₄	—	254	12	no	—
289	D₄	—	254	6	no	—
290	D₄	D₄×D₁₂₇	1	3	yes	—
291	D₄	C₂³×D₁₂₇	1	6	yes	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	127	3	no	—
294	C₂×C₄	—	127	3	no	—
295	C₂×C₄	—	127	3	no	—
296	C₂×C₄	—	127	1	no	—
297	C₂×C₄	—	127	3	no	—
298	C₂×C₄	—	254	2	no	—
299	C₂×C₄	—	254	6	no	—
300	C₂×C₄	—	254	6	no	—
301	C₂×C₄	—	254	12	no	—
302	C₂×C₄	—	254	6	no	—
303	C₂×C₄	—	254	6	no	—
304	C₂×C₄	C₂³×D₁₂₇	1	3	yes	—
305	C₈	—	127	1	no	—
306	C₈	—	127	3	no	—
307	C₈	C₂³×D₁₂₇	1	1	yes	characteristic
308	C₈	C₂³×D₁₂₇	1	3	yes	—
309	C₂²	—	2	6	no	—
310	C₂²	—	8	6	no	—
311	C₂²	—	127	1	no	—
312	C₂²	—	127	3	no	—
313	C₂²	—	1016	6	no	—
314	C₂²	—	508	3	no	—
315	C₂²	—	254	2	no	—
316	C₂²	C₂×D₄×D₁₂₇	1	3	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	127	1	no	—
319	C₄	—	127	3	no	—
320	C₄	—	254	6	no	—
321	C₄	C₂×D₄×D₁₂₇	1	3	yes	—
322	C₄	C₂⁴×D₁₂₇	1	1	yes	characteristic
323	C₂	—	2	3	no	—
324	C₂	—	4	6	no	—
325	C₂	—	508	2	no	—
326	C₂	—	254	3	no	—
327	C₂	—	254	1	no	—
328	C₂	C₂²×D₄×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₂₇
commutator, cosocle	C₄×C₁₂₇	C₂⁵
Frattini	C₄	C₂⁴×D₁₂₇
Fitting	#576×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂³×D₄
hypercenter	#576	D₁₂₇
hypocenter	C₁₂₇	#3584
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. #22550711	C₂	semidirect
2. #22550693	C₂	semidirect
3. #22550690	C₂	semidirect
4. #22550687	C₂	semidirect
5. #22550686	C₂	semidirect
6. #22550677	C₂	semidirect
7. #22550672	C₂	semidirect
8. C₂×#16861228	C₂	semidirect
9. #135×D₁₂₇	C₂	semidirect
10. #131×D₁₂₇	C₂	semidirect
11. #576×C₁₂₇	C₂	semidirect
13. #16861256	C₂²	semidirect
14. #16861256	C₂²	semidirect
16. #16861254	C₂²	semidirect
18. C₂×#14523247	C₂²	semidirect
21. #41×D₁₂₇	C₂²	semidirect
22. #16861242	C₂²	semidirect
23. #16861239	C₂²	semidirect
24. #16861238	C₂²	semidirect
25. #16861235	C₂²	semidirect
26. #16861233	C₂²	semidirect
27. #16861232	C₂²	semidirect
28. #16861231	C₂²	semidirect
29. #16861230	C₂²	semidirect
30. #16861229	C₂²	semidirect
31. #16861228	C₂²	semidirect
32. #16861228	C₂²	semidirect
33. #16861228	C₂²	semidirect
34. #16861228	C₂²	semidirect
35. C₂×Dic₅₀₈	C₂²	semidirect
36. Q₁₆×D₁₂₇	C₂²	semidirect
37. C₂×#14523236	C₂²	semidirect
38. SD₁₆×D₁₂₇	C₂²	semidirect
39. D₈×D₁₂₇	C₂²	semidirect
40. #16861217	C₂²	semidirect
41. #16861216	C₂²	semidirect
42. #16861215	C₂²	semidirect
43. M₁₆×D₁₂₇	C₂²	semidirect
44. C₂×C₈×D₁₂₇	C₂²	semidirect
45. H(5,𝔽₂)×C₁₂₇	C₂²	semidirect
46. AGL(1,ℤ/8)×C₁₂₇	C₂²	semidirect
47. #135×C₁₂₇	C₂²	semidirect
48. C₂×D₈×C₁₂₇	C₂²	semidirect
49. #131×C₁₂₇	C₂²	semidirect
165. C₁₂₇	#3584	semidirect
172. #576	D₁₂₇	semidirect
182. H(5,𝔽₂)	C₂×D₁₂₇	semidirect
193. AGL(1,ℤ/8)	C₂×D₁₂₇	semidirect
202. #135	C₂×D₁₂₇	semidirect
208. C₂×D₈	C₂×D₁₂₇	semidirect
212. #131	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550711	2
#22550693	2
#22550690	2
#22550687	2
#22550686	2
#22550677	2
#22550672	2
C₂×#16861228	2
#135×D₁₂₇	2
#131×D₁₂₇	2
#576×C₁₂₇	2
#3584	127