Subgroups — D₄×#14523231 #30647461

Order 16256.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523231	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549955	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549953	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549947	C₂	1	2	yes	maximal, complemented
6	#22549944	C₂	1	2	yes	maximal, complemented
7	D₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549898	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549838	C₂	1	2	yes	maximal, complemented
10	C₄×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₁₂₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₂₇	C₂²	1	2	yes	—
14	C₂×#14523231	—	2	8	no	—
15	C₂×#14523231	C₂²	1	1	yes	characteristic
16	C₂×#14523231	C₂²	1	2	yes	—
17	C₂×#14523231	C₂²	1	1	yes	characteristic
18	C₂×#14523231	C₂²	1	2	yes	complemented
19	C₂×#14523231	C₂²	1	2	yes	complemented
20	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
21	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
22	D₄×Dic₁₂₇	C₄	1	1	yes	characteristic
23	#16861171	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
25	(C₂²⋊D₁₂₇)⋊C₄	C₄	1	2	yes	—
26	C₄⋊C₄×D₁₂₇	C₄	1	1	yes	characteristic
27	C₂²⋊C₄×D₁₂₇	C₄	1	2	yes	—
28	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
29	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
30	Dic₁₂₇⋊C₈	C₂²	1	2	yes	complemented
31	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	complemented
32	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	complemented
33	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	complemented
34	#16861119	C₂²	1	1	yes	characteristic, complemented
35	#16861119	C₂²	1	1	yes	characteristic, complemented
36	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	complemented
38	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	complemented
41	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₁₂₇	C₂×C₄	1	2	yes	—
43	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
45	D₄×D₁₂₇	—	2	4	no	—
46	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₂₇	—	2	4	no	—
48	C₂×C₄×D₁₂₇	—	2	4	no	—
49	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
54	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
55	#14523231	—	2	4	no	—
56	#14523231	—	2	2	no	—
57	#14523231	—	2	4	no	—
58	#14523231	—	2	4	no	—
59	#14523231	D₄	1	4	yes	direct product factor
60	C₂×C₁₂₇⋊C₈	—	2	4	no	—
61	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
62	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
63	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
65	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
67	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
68	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
74	C₂×C₈×C₁₂₇	—	2	4	no	—
75	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
76	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
80	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
81	C₂²×D₁₂₇	—	2	4	no	—
82	C₂²×D₁₂₇	—	2	4	no	—
83	C₂²×D₁₂₇	—	4	2	no	—
84	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
85	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
86	D₅₀₈	—	2	2	no	—
87	C₂²⋊D₁₂₇	—	2	4	no	—
88	C₂×Dic₁₂₇	—	2	4	no	—
89	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₁₂₇	—	2	4	no	—
94	C₄×D₁₂₇	—	2	2	no	—
95	C₄×D₁₂₇	—	2	2	no	—
96	C₄×D₁₂₇	—	2	2	no	—
97	C₄×D₁₂₇	—	2	4	no	—
98	C₄×D₁₂₇	—	4	1	no	—
99	C₄×D₁₂₇	C₂×D₄	1	2	yes	—
100	C₄×D₁₂₇	#41	1	2	yes	—
101	C₁₂₇⋊C₈	—	2	2	no	—
102	C₁₂₇⋊C₈	—	2	1	no	—
103	C₁₂₇⋊C₈	C₂×D₄	1	2	yes	complemented
104	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
105	D₄×C₁₂₇	M₁₆	1	4	yes	complemented
106	C₂×C₄×C₁₂₇	—	2	2	no	—
107	C₂×C₄×C₁₂₇	—	2	2	no	—
108	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
112	C₈×C₁₂₇	—	2	2	no	—
113	C₈×C₁₂₇	—	2	1	no	—
114	C₈×C₁₂₇	C₂×D₄	1	2	yes	complemented
115	C₂²×C₁₂₇	—	2	2	no	—
116	C₂²×C₁₂₇	—	2	2	no	—
117	C₂²×C₁₂₇	C₂×M₁₆	1	4	yes	complemented
118	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₁₂₇	—	2	1	no	—
120	C₂×D₁₂₇	—	2	1	no	—
121	C₂×D₁₂₇	—	2	2	no	—
122	C₂×D₁₂₇	—	2	2	no	—
123	C₂×D₁₂₇	—	2	2	no	—
124	C₂×D₁₂₇	—	4	8	no	—
125	C₂×D₁₂₇	C₄×D₄	1	2	yes	—
126	C₄×C₁₂₇	—	2	2	no	—
127	C₄×C₁₂₇	—	2	1	no	—
128	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	complemented
129	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
131	Dic₁₂₇	—	2	2	no	—
132	Dic₁₂₇	—	2	1	no	—
133	Dic₁₂₇	—	2	1	no	—
134	Dic₁₂₇	C₄×D₄	1	2	yes	—
135	C₂×C₁₂₇	—	2	4	no	—
136	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
138	C₂×C₁₂₇	#568	1	1	yes	characteristic
139	D₁₂₇	—	2	2	no	—
140	D₁₂₇	—	4	2	no	—
141	D₄×M₁₆	—	127	1	no	maximal, Sylow
142	C₁₂₇	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
143	C₂²×M₁₆	—	127	2	no	—
144	C₂×C₄×D₄	—	127	1	no	—
145	#436	—	127	1	no	—
146	#436	—	127	1	no	—
147	#435	—	127	2	no	—
148	#435	—	127	2	no	—
149	C₈×D₄	—	127	1	no	—
150	C₈×D₄	D₁₂₇	1	1	yes	characteristic, complemented
151	#423	—	127	1	no	—
152	#407	—	127	2	no	—
153	C₄×M₁₆	—	127	1	no	—
154	C₂²×D₄	—	127	1	no	—
155	C₂³×C₄	—	127	2	no	—
156	C₂×M₁₆	—	127	1	no	—
157	C₂×M₁₆	—	127	2	no	—
158	C₂×M₁₆	—	127	1	no	—
159	C₂×M₁₆	—	127	2	no	—
160	C₂×M₁₆	—	127	2	no	—
161	C₂×M₁₆	—	254	8	no	—
162	C₂²×C₈	—	127	2	no	—
163	C₂²×C₈	C₂×D₁₂₇	1	2	yes	complemented
164	C₄×D₄	—	127	1	no	—
165	C₄×D₄	—	127	2	no	—
166	C₄×D₄	—	127	1	no	—
167	C₄×D₄	—	127	2	no	—
168	C₄×D₄	—	127	1	no	—
169	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	127	1	no	—
171	C₂×C₂²⋊C₄	—	127	2	no	—
172	C₂×C₄²	—	127	1	no	—
173	C₄⋊C₈	—	127	2	no	—
174	C₄⋊C₈	—	127	1	no	—
175	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
176	C₂²⋊C₈	—	127	2	no	—
177	C₂²⋊C₈	—	127	2	no	—
178	C₂²⋊C₈	—	127	2	no	—
179	C₂²⋊C₈	C₂×D₁₂₇	1	2	yes	complemented
180	#97	—	127	1	no	—
181	#97	—	127	1	no	—
182	C₄×C₈	—	127	1	no	—
183	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
184	C₂⁴	—	127	2	no	—
185	C₂×D₄	—	127	2	no	—
186	C₂×D₄	—	127	1	no	—
187	C₂×D₄	—	254	4	no	—
188	C₂×D₄	C₄×D₁₂₇	1	1	yes	characteristic
189	C₂²×C₄	—	127	1	no	—
190	C₂²×C₄	—	127	2	no	—
191	C₂²×C₄	—	127	1	no	—
192	C₂²×C₄	—	127	1	no	—
193	C₂²×C₄	—	127	2	no	—
194	C₂²×C₄	—	127	2	no	—
195	C₂²×C₄	—	127	2	no	—
196	C₂²×C₄	—	254	4	no	—
197	C₂²×C₄	—	254	4	no	—
198	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
199	M₁₆	—	127	4	no	—
200	M₁₆	—	254	4	no	—
201	M₁₆	—	254	2	no	—
202	M₁₆	—	254	4	no	—
203	M₁₆	—	254	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	127	1	no	—
206	C₂×C₈	—	127	1	no	—
207	C₂×C₈	—	127	2	no	—
208	C₂×C₈	—	254	4	no	—
209	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
211	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
212	C₄⋊C₄	—	127	2	no	—
213	C₄⋊C₄	—	127	1	no	—
214	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
215	C₂²⋊C₄	—	127	2	no	—
216	C₂²⋊C₄	—	127	2	no	—
217	C₂²⋊C₄	—	127	2	no	—
218	C₂²⋊C₄	C₄×D₁₂₇	1	2	yes	—
219	C₄²	—	127	1	no	—
220	C₄²	—	127	1	no	—
221	C₄²	—	127	1	no	—
222	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
223	C₂³	—	127	1	no	—
224	C₂³	—	127	2	no	—
225	C₂³	—	508	2	no	—
226	C₂³	—	254	4	no	—
227	C₂³	—	254	4	no	—
228	C₂³	C₂×C₄×D₁₂₇	1	2	yes	—
229	D₄	—	254	4	no	—
230	D₄	—	254	2	no	—
231	D₄	#14523231	1	4	yes	direct product factor
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	127	1	no	—
235	C₂×C₄	—	127	2	no	—
236	C₂×C₄	—	127	2	no	—
237	C₂×C₄	—	127	1	no	—
238	C₂×C₄	—	127	1	no	—
239	C₂×C₄	—	127	2	no	—
240	C₂×C₄	—	508	1	no	—
241	C₂×C₄	—	254	4	no	—
242	C₂×C₄	—	254	4	no	—
243	C₂×C₄	—	254	4	no	—
244	C₂×C₄	—	254	2	no	—
245	C₂×C₄	—	254	2	no	—
246	C₂×C₄	—	254	2	no	—
247	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	127	2	no	—
254	C₈	—	254	2	no	—
255	C₈	—	254	1	no	—
256	C₈	D₄×D₁₂₇	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	127	2	no	—
260	C₂²	—	508	8	no	—
261	C₂²	—	254	2	no	—
262	C₂²	—	254	2	no	—
263	C₂²	—	254	2	no	—
264	C₂²	—	254	1	no	—
265	C₂²	—	254	1	no	—
266	C₂²	C₂×#14523231	1	4	yes	complemented
267	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	127	2	no	—
271	C₄	—	254	2	no	—
272	C₄	—	254	1	no	—
273	C₄	—	254	1	no	—
274	C₄	C₂×#14523231	1	2	yes	complemented
275	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
276	C₄	#41×D₁₂₇	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	508	2	no	—
279	C₂	—	254	2	no	—
280	C₂	C₄×D₄×D₁₂₇	1	1	yes	characteristic
281	C₂	C₂²×#14523231	1	1	yes	characteristic
282	C₂	#22550641	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₈×D₄×C₁₂₇	C₂
prime radical	C₂×D₄×D₁₂₇	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₈×D₄	D₁₂₇
hypocenter	C₁₂₇	D₄×M₁₆
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523231	C₂	semidirect
3. #22549955	C₂	semidirect
4. #22549953	C₂	semidirect
5. #22549947	C₂	semidirect
6. #22549944	C₂	semidirect
7. D₄×C₁₂₇⋊C₈	C₂	semidirect
8. #22549898	C₂	semidirect
9. #22549838	C₂	semidirect
10. C₄×#14523231	C₂	semidirect
11. C₈×D₄×C₁₂₇	C₂	semidirect
18. C₂×#14523231	C₂²	semidirect
19. C₂×#14523231	C₂²	semidirect
20. C₂²×C₁₂₇⋊C₈	C₂²	semidirect
29. C₄⋊(C₁₂₇⋊C₈)	C₂²	semidirect
30. Dic₁₂₇⋊C₈	C₂²	semidirect
31. C₂²⋊(C₁₂₇⋊C₈)	C₂²	semidirect
32. (C₂×D₁₂₇)⋊C₈	C₂²	semidirect
33. (C₂×D₁₂₇)⋊C₈	C₂²	semidirect
34. #16861119	C₂²	semidirect
35. #16861119	C₂²	semidirect
36. C₄×C₁₂₇⋊C₈	C₂²	semidirect
37. C₂²×C₈×C₁₂₇	C₂²	semidirect
39. C₄⋊C₈×C₁₂₇	C₂²	semidirect
40. C₂²⋊C₈×C₁₂₇	C₂²	semidirect
41. C₄×C₈×C₁₂₇	C₂²	semidirect
59. #14523231	D₄	direct (N × H)
103. C₁₂₇⋊C₈	C₂×D₄	semidirect
105. D₄×C₁₂₇	M₁₆	semidirect
114. C₈×C₁₂₇	C₂×D₄	semidirect
117. C₂²×C₁₂₇	C₂×M₁₆	semidirect
128. C₄×C₁₂₇	C₂×M₁₆	semidirect
142. C₁₂₇	D₄×M₁₆	semidirect
150. C₈×D₄	D₁₂₇	semidirect
163. C₂²×C₈	C₂×D₁₂₇	semidirect
175. C₄⋊C₈	C₂×D₁₂₇	semidirect
179. C₂²⋊C₈	C₂×D₁₂₇	semidirect
183. C₄×C₈	C₂×D₁₂₇	semidirect
231. D₄	#14523231	direct (N × H)
256. C₈	D₄×D₁₂₇	semidirect
266. C₂²	C₂×#14523231	semidirect
274. C₄	C₂×#14523231	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523231	2
C₄×D₄×D₁₂₇	2
#22549955	2
#22549953	2
#22549947	2
#22549944	2
D₄×C₁₂₇⋊C₈	2
#22549898	2
#22549838	2
C₄×#14523231	2
C₈×D₄×C₁₂₇	2
D₄×M₁₆	127

Subgroups — D4×#14523231 #30647461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#14523231 #30647461