Subgroups — #518×D₂₇₁ #46849547

Order 34688.

1920 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₂₇₁	C₂	1	1	yes	maximal, characteristic
2	#31774372	C₂	1	1	yes	maximal, characteristic, complemented
3	#31774371	C₂	1	2	yes	maximal, complemented
4	#31774370	C₂	1	4	yes	maximal, complemented
5	#31774369	C₂	1	8	yes	maximal, complemented
6	C₄×D₄×D₂₇₁	C₂	1	8	yes	maximal, complemented
7	#117×D₂₇₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₄×D₂₇₁	C₂	1	4	yes	maximal, complemented
9	#518×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₂₇₁	C₂²	1	2	yes	—
11	C₂²×D₁₀₈₄	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₂₇₁	C₂²	1	2	yes	—
13	C₂×D₄×D₂₇₁	C₄	1	16	yes	complemented
14	C₂×D₄×D₂₇₁	C₂²	1	4	yes	—
15	C₂×D₄×D₂₇₁	C₂²	1	4	yes	—
16	C₂²×C₄×D₂₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₇₁	C₂²	1	8	yes	complemented
18	C₄×D₁₀₈₄	C₂²	1	8	yes	complemented
19	D₄×Dic₂₇₁	C₂²	1	8	yes	complemented
20	#22576122	C₂²	1	8	yes	complemented
21	C₄×C₂²⋊D₂₇₁	C₂²	1	16	yes	complemented
22	(C₂²⋊D₂₇₁)⋊C₄	C₂²	1	16	yes	complemented
23	#22576119	C₂²	1	2	yes	complemented
24	#22576118	C₂²	1	4	yes	complemented
25	#22576117	C₂²	1	4	yes	complemented
26	#22576116	C₂²	1	4	yes	complemented
27	C₄⋊C₄×D₂₇₁	C₂²	1	4	yes	complemented
28	C₂×C₂²⋊Dic₂₇₁	C₂²	1	4	yes	complemented
29	C₂×(C₂×D₂₇₁)⋊C₄	C₂²	1	8	yes	complemented
30	C₂²⋊C₄×D₂₇₁	C₂²	1	4	yes	—
31	C₂²⋊C₄×D₂₇₁	C₂²	1	8	yes	complemented
32	C₂²⋊C₄×D₂₇₁	C₂²	1	8	yes	complemented
33	C₂²⋊C₄×D₂₇₁	C₂²	1	8	yes	complemented
34	C₄²×D₂₇₁	C₂²	1	4	yes	complemented
35	C₂²×D₄×C₂₇₁	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₂₇₁	C₂²	1	8	yes	complemented
37	#117×C₂₇₁	C₂²	1	2	yes	complemented
38	C₂×C₂²⋊C₄×C₂₇₁	C₂²	1	4	yes	complemented
39	C₂³×D₂₇₁	—	4	8	no	—
40	C₂³×D₂₇₁	C₂×C₄	1	16	yes	complemented
41	C₂³×D₂₇₁	C₂³	1	1	yes	characteristic
42	C₂³×D₂₇₁	C₂³	1	4	yes	—
43	C₂³×D₂₇₁	C₂³	1	4	yes	—
44	C₂³×D₂₇₁	C₂³	1	2	yes	—
45	C₂×D₁₀₈₄	C₂×C₄	1	8	yes	complemented
46	C₂×D₁₀₈₄	C₂³	1	4	yes	—
47	C₂×C₂²⋊D₂₇₁	C₂×C₄	1	16	yes	complemented
48	C₂×C₂²⋊D₂₇₁	C₂³	1	8	yes	—
49	D₄×D₂₇₁	C₂×C₄	1	32	yes	complemented
50	D₄×D₂₇₁	C₂×C₄	1	32	yes	complemented
51	C₂²×Dic₂₇₁	C₂³	1	1	yes	characteristic
52	C₂²×Dic₂₇₁	C₂³	1	8	yes	complemented
53	C₂×C₄×D₂₇₁	—	4	16	no	—
54	C₂×C₄×D₂₇₁	C₂×C₄	1	8	yes	complemented
55	C₂×C₄×D₂₇₁	C₂³	1	8	yes	—
56	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
57	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
58	C₂×C₄×D₂₇₁	C₂³	1	16	yes	complemented
59	C₄⋊Dic₂₇₁	C₂³	1	4	yes	complemented
60	Dic₂₇₁⋊C₄	C₂³	1	8	yes	complemented
61	C₂²⋊Dic₂₇₁	C₂³	1	4	yes	—
62	C₂²⋊Dic₂₇₁	C₂³	1	8	yes	complemented
63	(C₂×D₂₇₁)⋊C₄	C₂³	1	8	yes	—
64	(C₂×D₂₇₁)⋊C₄	C₂³	1	16	yes	complemented
65	C₄×Dic₂₇₁	C₂³	1	8	yes	complemented
66	C₄×Dic₂₇₁	C₂³	1	4	yes	complemented
67	C₂⁴×C₂₇₁	C₂³	1	2	yes	—
68	C₂×D₄×C₂₇₁	C₂×C₄	1	8	yes	complemented
69	C₂×D₄×C₂₇₁	C₂³	1	4	yes	—
70	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₂₇₁	C₂³	1	8	yes	complemented
72	C₄⋊C₄×C₂₇₁	C₂³	1	4	yes	complemented
73	C₂²⋊C₄×C₂₇₁	C₂³	1	4	yes	—
74	C₂²⋊C₄×C₂₇₁	C₂³	1	8	yes	complemented
75	C₄²×C₂₇₁	C₂³	1	4	yes	complemented
76	C₂²×D₂₇₁	—	2	16	no	—
77	C₂²×D₂₇₁	—	2	4	no	—
78	C₂²×D₂₇₁	—	4	32	no	—
79	C₂²×D₂₇₁	—	4	4	no	—
80	C₂²×D₂₇₁	—	4	8	no	—
81	C₂²×D₂₇₁	C₂²×C₄	1	4	yes	—
82	C₂²×D₂₇₁	C₂²×C₄	1	16	yes	—
83	C₂²×D₂₇₁	C₂²×C₄	1	16	yes	complemented
84	C₂²×D₂₇₁	C₂²×C₄	1	8	yes	complemented
85	C₂²×D₂₇₁	C₂⁴	1	1	yes	characteristic
86	C₂²×D₂₇₁	C₂⁴	1	4	yes	—
87	C₂²×D₂₇₁	C₂⁴	1	1	yes	characteristic
88	D₁₀₈₄	C₂²×C₄	1	16	yes	complemented
89	C₂²⋊D₂₇₁	C₂²×C₄	1	32	yes	complemented
90	C₂×Dic₂₇₁	—	4	8	no	—
91	C₂×Dic₂₇₁	C₂²×C₄	1	4	yes	complemented
92	C₂×Dic₂₇₁	C₂⁴	1	8	yes	—
93	C₂×Dic₂₇₁	C₂⁴	1	2	yes	—
94	C₄×D₂₇₁	—	2	16	no	—
95	C₄×D₂₇₁	—	4	16	no	—
96	C₄×D₂₇₁	C₂²×C₄	1	8	yes	—
97	C₄×D₂₇₁	C₂²×C₄	1	8	yes	complemented
98	C₂³×C₂₇₁	—	4	4	no	—
99	C₂³×C₂₇₁	C₂²×C₄	1	8	yes	complemented
100	C₂³×C₂₇₁	C₂⁴	1	1	yes	characteristic
101	C₂³×C₂₇₁	C₂⁴	1	4	yes	—
102	D₄×C₂₇₁	C₂²×C₄	1	16	yes	complemented
103	C₂×C₄×C₂₇₁	—	4	8	no	—
104	C₂×C₄×C₂₇₁	C₂²×C₄	1	4	yes	complemented
105	C₂×C₄×C₂₇₁	C₂⁴	1	8	yes	—
106	C₂×C₄×C₂₇₁	C₂⁴	1	2	yes	—
107	C₂²×C₂₇₁	—	2	8	no	—
108	C₂²×C₂₇₁	—	2	2	no	—
109	C₂²×C₂₇₁	—	4	8	no	—
110	C₂²×C₂₇₁	C₂³×C₄	1	2	yes	—
111	C₂²×C₂₇₁	C₂³×C₄	1	8	yes	—
112	C₂²×C₂₇₁	C₂⁵	1	1	yes	characteristic
113	C₂×D₂₇₁	—	2	8	no	—
114	C₂×D₂₇₁	—	2	2	no	—
115	C₂×D₂₇₁	—	2	8	no	—
116	C₂×D₂₇₁	—	2	32	no	—
117	C₂×D₂₇₁	—	4	16	no	—
118	C₂×D₂₇₁	—	4	8	no	—
119	C₂×D₂₇₁	C₂³×C₄	1	2	yes	—
120	C₂×D₂₇₁	C₂³×C₄	1	8	yes	—
121	C₂×D₂₇₁	C₂³×C₄	1	2	yes	—
122	C₂×D₂₇₁	H(5,𝔽₂)	1	4	yes	—
123	C₄×C₂₇₁	—	2	8	no	—
124	C₄×C₂₇₁	C₂³×C₄	1	4	yes	—
125	Dic₂₇₁	—	2	8	no	—
126	Dic₂₇₁	C₂³×C₄	1	4	yes	—
127	C₂×C₂₇₁	—	2	2	no	—
128	C₂×C₂₇₁	—	2	8	no	—
129	C₂×C₂₇₁	C₂⁴×C₄	1	1	yes	characteristic
130	C₂×C₂₇₁	C₂×H(5,𝔽₂)	1	2	yes	—
131	D₂₇₁	—	2	2	no	—
132	D₂₇₁	—	2	8	no	—
133	D₂₇₁	#518	1	4	yes	direct product factor
134	C₂₇₁	C₂×#518	1	1	yes	characteristic, complemented, Sylow
135	C₂×#518	—	271	1	no	maximal, Sylow
136	C₂³×D₄	—	271	1	no	—
137	#518	—	271	1	no	—
138	#518	—	271	8	no	—
139	#518	—	271	2	no	—
140	#518	—	271	4	no	—
141	#518	D₂₇₁	1	1	yes	characteristic, direct product factor
142	C₂×C₄×D₄	—	271	8	no	—
143	C₂×#117	—	271	2	no	—
144	C₂²×C₂²⋊C₄	—	271	4	no	—
145	C₂⁵	—	271	2	no	—
146	C₂²×D₄	—	271	2	no	—
147	C₂²×D₄	—	271	16	no	—
148	C₂²×D₄	—	271	4	no	—
149	C₂²×D₄	—	271	4	no	—
150	C₂²×D₄	—	271	1	no	—
151	C₂²×D₄	C₂×D₂₇₁	1	1	yes	characteristic
152	C₂³×C₄	—	271	1	no	—
153	C₂³×C₄	—	271	8	no	—
154	C₄×D₄	—	271	8	no	—
155	C₄×D₄	—	271	16	no	—
156	C₄×D₄	—	271	8	no	—
157	C₄×D₄	—	271	16	no	—
158	C₄×D₄	—	271	8	no	—
159	C₄×D₄	C₂×D₂₇₁	1	8	yes	complemented
160	#117	—	271	2	no	—
161	#117	—	271	4	no	—
162	#117	—	271	4	no	—
163	#117	—	271	4	no	—
164	#117	C₂×D₂₇₁	1	2	yes	complemented
165	C₂×C₄⋊C₄	—	271	4	no	—
166	C₂×C₂²⋊C₄	—	271	4	no	—
167	C₂×C₂²⋊C₄	—	271	8	no	—
168	C₂×C₂²⋊C₄	—	271	4	no	—
169	C₂×C₂²⋊C₄	—	271	8	no	—
170	C₂×C₂²⋊C₄	—	271	8	no	—
171	C₂×C₂²⋊C₄	—	271	8	no	—
172	C₂×C₂²⋊C₄	C₂×D₂₇₁	1	4	yes	complemented
173	C₂×C₄²	—	271	4	no	—
174	C₂⁴	—	271	1	no	—
175	C₂⁴	—	271	4	no	—
176	C₂⁴	—	271	16	no	—
177	C₂⁴	—	271	4	no	—
178	C₂⁴	—	271	2	no	—
179	C₂⁴	—	1084	8	no	—
180	C₂⁴	C₂²×D₂₇₁	1	2	yes	—
181	C₂×D₄	—	271	16	no	—
182	C₂×D₄	—	271	8	no	—
183	C₂×D₄	—	271	32	no	—
184	C₂×D₄	—	271	32	no	—
185	C₂×D₄	—	271	8	no	—
186	C₂×D₄	—	271	4	no	—
187	C₂×D₄	C₄×D₂₇₁	1	8	yes	complemented
188	C₂×D₄	C₂²×D₂₇₁	1	4	yes	—
189	C₂²×C₄	—	271	1	no	—
190	C₂²×C₄	—	271	8	no	—
191	C₂²×C₄	—	271	2	no	—
192	C₂²×C₄	—	271	2	no	—
193	C₂²×C₄	—	271	8	no	—
194	C₂²×C₄	—	271	8	no	—
195	C₂²×C₄	—	271	16	no	—
196	C₂²×C₄	—	1084	16	no	—
197	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₂₇₁	1	8	yes	complemented
199	C₄⋊C₄	—	271	8	no	—
200	C₄⋊C₄	—	271	4	no	—
201	C₄⋊C₄	C₂²×D₂₇₁	1	4	yes	complemented
202	C₂²⋊C₄	—	271	4	no	—
203	C₂²⋊C₄	—	271	8	no	—
204	C₂²⋊C₄	—	271	8	no	—
205	C₂²⋊C₄	—	271	16	no	—
206	C₂²⋊C₄	C₂²×D₂₇₁	1	4	yes	—
207	C₂²⋊C₄	C₂²×D₂₇₁	1	8	yes	complemented
208	C₄²	—	271	8	no	—
209	C₄²	—	271	4	no	—
210	C₄²	C₂²×D₂₇₁	1	4	yes	complemented
211	C₂³	—	4	4	no	—
212	C₂³	—	271	1	no	—
213	C₂³	—	271	4	no	—
214	C₂³	—	271	16	no	—
215	C₂³	—	271	1	no	—
216	C₂³	—	271	4	no	—
217	C₂³	—	271	16	no	—
218	C₂³	—	271	8	no	—
219	C₂³	—	542	16	no	—
220	C₂³	—	542	4	no	—
221	C₂³	—	1084	8	no	—
222	C₂³	—	1084	4	no	—
223	C₂³	—	1084	32	no	—
224	C₂³	C₂×C₄×D₂₇₁	1	8	yes	complemented
225	C₂³	C₂³×D₂₇₁	1	1	yes	characteristic
226	C₂³	C₂³×D₂₇₁	1	4	yes	—
227	D₄	—	271	32	no	—
228	D₄	—	271	16	no	—
229	D₄	C₂×C₄×D₂₇₁	1	16	yes	complemented
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	271	4	no	—
232	C₂×C₄	—	271	2	no	—
233	C₂×C₄	—	271	8	no	—
234	C₂×C₄	—	271	8	no	—
235	C₂×C₄	—	271	8	no	—
236	C₂×C₄	—	542	16	no	—
237	C₂×C₄	—	1084	16	no	—
238	C₂×C₄	—	1084	8	no	—
239	C₂×C₄	C₂×C₄×D₂₇₁	1	4	yes	complemented
240	C₂×C₄	C₂³×D₂₇₁	1	8	yes	—
241	C₂×C₄	C₂³×D₂₇₁	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	271	4	no	—
246	C₂²	—	271	2	no	—
247	C₂²	—	271	2	no	—
248	C₂²	—	271	8	no	—
249	C₂²	—	542	8	no	—
250	C₂²	—	542	32	no	—
251	C₂²	—	542	8	no	—
252	C₂²	—	542	2	no	—
253	C₂²	—	1084	8	no	—
254	C₂²	—	1084	16	no	—
255	C₂²	C₂²×C₄×D₂₇₁	1	2	yes	—
256	C₂²	C₂²×C₄×D₂₇₁	1	8	yes	—
257	C₂²	C₂⁴×D₂₇₁	1	1	yes	characteristic
258	C₄	—	2	8	no	—
259	C₄	—	271	4	no	—
260	C₄	—	542	8	no	—
261	C₄	C₂²×C₄×D₂₇₁	1	4	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	2	8	no	—
264	C₂	—	271	4	no	—
265	C₂	—	542	2	no	—
266	C₂	—	542	8	no	—
267	C₂	C₂³×C₄×D₂₇₁	1	1	yes	characteristic
268	C₂	H(5,𝔽₂)×D₂₇₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₇₁
commutator	C₂×C₂₇₁	C₂⁴×C₄
Fitting	#518×C₂₇₁	C₂
socle, cosocle	C₂²×C₂₇₁	C₂⁵
prime radical	C₂²×D₄×D₂₇₁	C₂
hypercenter	#518	D₂₇₁
hypocenter	C₂₇₁	C₂×#518
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
2. #31774372	C₂	semidirect
3. #31774371	C₂	semidirect
4. #31774370	C₂	semidirect
5. #31774369	C₂	semidirect
6. C₄×D₄×D₂₇₁	C₂	semidirect
7. #117×D₂₇₁	C₂	semidirect
8. C₂×C₂²⋊C₄×D₂₇₁	C₂	semidirect
9. #518×C₂₇₁	C₂	semidirect
13. C₂×D₄×D₂₇₁	C₄	semidirect
17. C₂²×C₄×D₂₇₁	C₂²	semidirect
18. C₄×D₁₀₈₄	C₂²	semidirect
19. D₄×Dic₂₇₁	C₂²	semidirect
20. #22576122	C₂²	semidirect
21. C₄×C₂²⋊D₂₇₁	C₂²	semidirect
22. (C₂²⋊D₂₇₁)⋊C₄	C₂²	semidirect
23. #22576119	C₂²	semidirect
24. #22576118	C₂²	semidirect
25. #22576117	C₂²	semidirect
26. #22576116	C₂²	semidirect
27. C₄⋊C₄×D₂₇₁	C₂²	semidirect
28. C₂×C₂²⋊Dic₂₇₁	C₂²	semidirect
29. C₂×(C₂×D₂₇₁)⋊C₄	C₂²	semidirect
31. C₂²⋊C₄×D₂₇₁	C₂²	semidirect
32. C₂²⋊C₄×D₂₇₁	C₂²	semidirect
33. C₂²⋊C₄×D₂₇₁	C₂²	semidirect
34. C₄²×D₂₇₁	C₂²	semidirect
36. C₄×D₄×C₂₇₁	C₂²	semidirect
37. #117×C₂₇₁	C₂²	semidirect
38. C₂×C₂²⋊C₄×C₂₇₁	C₂²	semidirect
40. C₂³×D₂₇₁	C₂×C₄	semidirect
45. C₂×D₁₀₈₄	C₂×C₄	semidirect
47. C₂×C₂²⋊D₂₇₁	C₂×C₄	semidirect
49. D₄×D₂₇₁	C₂×C₄	semidirect
50. D₄×D₂₇₁	C₂×C₄	semidirect
52. C₂²×Dic₂₇₁	C₂³	semidirect
54. C₂×C₄×D₂₇₁	C₂×C₄	semidirect
58. C₂×C₄×D₂₇₁	C₂³	semidirect
59. C₄⋊Dic₂₇₁	C₂³	semidirect
60. Dic₂₇₁⋊C₄	C₂³	semidirect
62. C₂²⋊Dic₂₇₁	C₂³	semidirect
64. (C₂×D₂₇₁)⋊C₄	C₂³	semidirect
65. C₄×Dic₂₇₁	C₂³	semidirect
66. C₄×Dic₂₇₁	C₂³	semidirect
68. C₂×D₄×C₂₇₁	C₂×C₄	semidirect
71. C₂²×C₄×C₂₇₁	C₂³	semidirect
72. C₄⋊C₄×C₂₇₁	C₂³	semidirect
74. C₂²⋊C₄×C₂₇₁	C₂³	semidirect
75. C₄²×C₂₇₁	C₂³	semidirect
83. C₂²×D₂₇₁	C₂²×C₄	semidirect
84. C₂²×D₂₇₁	C₂²×C₄	semidirect
88. D₁₀₈₄	C₂²×C₄	semidirect
89. C₂²⋊D₂₇₁	C₂²×C₄	semidirect
91. C₂×Dic₂₇₁	C₂²×C₄	semidirect
97. C₄×D₂₇₁	C₂²×C₄	semidirect
99. C₂³×C₂₇₁	C₂²×C₄	semidirect
102. D₄×C₂₇₁	C₂²×C₄	semidirect
104. C₂×C₄×C₂₇₁	C₂²×C₄	semidirect
133. D₂₇₁	#518	direct (N × H)
134. C₂₇₁	C₂×#518	semidirect
141. #518	D₂₇₁	direct (N × H)
159. C₄×D₄	C₂×D₂₇₁	semidirect
164. #117	C₂×D₂₇₁	semidirect
172. C₂×C₂²⋊C₄	C₂×D₂₇₁	semidirect
187. C₂×D₄	C₄×D₂₇₁	semidirect
198. C₂²×C₄	C₂²×D₂₇₁	semidirect
201. C₄⋊C₄	C₂²×D₂₇₁	semidirect
207. C₂²⋊C₄	C₂²×D₂₇₁	semidirect
210. C₄²	C₂²×D₂₇₁	semidirect
224. C₂³	C₂×C₄×D₂₇₁	semidirect
229. D₄	C₂×C₄×D₂₇₁	semidirect
239. C₂×C₄	C₂×C₄×D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×D₂₇₁	2
#31774372	2
#31774371	2
#31774370	2
#31774369	2
C₄×D₄×D₂₇₁	2
#117×D₂₇₁	2
C₂×C₂²⋊C₄×D₂₇₁	2
#518×C₂₇₁	2
C₂×#518	271

Subgroups — #518×D271 #46849547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #518×D₂₇₁ #46849547