Subgroups — #46845211

Order 34688.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×D₂₇₁	C₂	1	1	yes	maximal, characteristic
2	#31774369	C₂	1	2	yes	maximal, complemented
3	#31774007	C₂	1	2	yes	maximal, complemented
4	#31774006	C₂	1	2	yes	maximal, complemented
5	C₄×D₂₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#31773979	C₂	1	1	yes	maximal, characteristic, complemented
8	#31773934	C₂	1	1	yes	maximal, characteristic, complemented
9	#31773900	C₂	1	2	yes	maximal, complemented
10	#31773851	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₂₇₁	C₂²	1	2	yes	—
13	C₂×D₂₁₆₈	C₂²	1	1	yes	characteristic
14	C₂×#16864639	C₂²	1	2	yes	—
15	D₈×D₂₇₁	C₄	1	8	yes	complemented
16	C₂×#16864637	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₇₁	C₂²	1	1	yes	characteristic
18	C₄×D₁₀₈₄	C₂²	1	2	yes	complemented
19	D₄×Dic₂₇₁	C₂²	1	2	yes	complemented
20	#22576122	C₂²	1	2	yes	complemented
21	C₄×C₂²⋊D₂₇₁	—	2	2	no	—
22	(C₂²⋊D₂₇₁)⋊C₄	—	2	2	no	—
23	#22576118	C₂²	1	2	yes	—
24	#22576117	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₂₇₁)⋊C₄	—	2	2	no	—
26	C₂²⋊C₄×D₂₇₁	—	2	2	no	—
27	#22576094	C₂²	1	1	yes	characteristic, complemented
28	#22576091	C₂²	1	2	yes	complemented
29	#22576082	C₂²	1	2	yes	complemented
30	#22576081	C₂²	1	2	yes	complemented
31	#22576080	C₂²	1	2	yes	complemented
32	#22576071	C₂²	1	1	yes	characteristic, complemented
33	#22576070	C₂²	1	1	yes	characteristic, complemented
34	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₈×C₂₇₁	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₂₇₁	C₂²	1	2	yes	complemented
37	#107×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
38	#102×C₂₇₁	C₂²	1	2	yes	complemented
39	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
40	C₂³×D₂₇₁	—	2	2	no	—
41	C₂×D₁₀₈₄	C₂³	1	2	yes	—
42	C₂×C₂²⋊D₂₇₁	—	2	2	no	—
43	D₄×D₂₇₁	—	2	4	no	—
44	D₄×D₂₇₁	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₂₇₁	—	2	2	no	—
46	C₂×C₄×D₂₇₁	—	2	2	no	—
47	C₂×C₄×D₂₇₁	—	2	2	no	—
48	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
49	D₂₁₆₈	C₂×C₄	1	4	yes	complemented
50	#16864639	C₂×C₄	1	8	yes	complemented
51	#16864637	—	2	2	no	—
52	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
54	C₈×D₂₇₁	C₂×C₄	1	4	yes	complemented
55	C₄⋊Dic₂₇₁	C₂³	1	2	yes	—
56	Dic₂₇₁⋊C₄	D₄	1	1	yes	characteristic
57	Dic₂₇₁⋊C₄	D₄	1	1	yes	characteristic
58	C₂²⋊Dic₂₇₁	—	2	2	no	—
59	(C₂×D₂₇₁)⋊C₄	—	2	2	no	—
60	(C₂×D₂₇₁)⋊C₄	—	2	2	no	—
61	(C₂×D₂₇₁)⋊C₄	—	2	2	no	—
62	(C₂×D₂₇₁)⋊C₄	D₄	1	1	yes	characteristic
63	(C₂×D₂₇₁)⋊C₄	D₄	1	1	yes	characteristic
64	C₄×Dic₂₇₁	—	2	2	no	—
65	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₂₇₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₂₇₁	—	2	2	no	—
68	D₈×C₂₇₁	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₂₇₁	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₂₇₁	—	2	2	no	—
73	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
74	C₂²×D₂₇₁	—	2	2	no	—
75	C₂²×D₂₇₁	—	2	8	no	—
76	C₂²×D₂₇₁	—	8	2	no	—
77	C₂²×D₂₇₁	—	8	2	no	—
78	C₂²×D₂₇₁	C₂×D₄	1	1	yes	characteristic
79	D₁₀₈₄	—	2	2	no	—
80	D₁₀₈₄	C₂²×C₄	1	4	yes	—
81	C₂²⋊D₂₇₁	—	2	8	no	—
82	C₂×Dic₂₇₁	—	2	2	no	—
83	C₂×Dic₂₇₁	—	8	2	no	—
84	C₂×Dic₂₇₁	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₂₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₂₇₁	C₂×D₄	1	1	yes	characteristic
87	C₄×D₂₇₁	—	8	2	no	—
88	C₄×D₂₇₁	—	8	2	no	—
89	C₄×D₂₇₁	C₂²×C₄	1	2	yes	—
90	C₄×D₂₇₁	#41	1	2	yes	—
91	C₂₇₁⋊C₈	—	2	1	no	—
92	C₂₇₁⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂³×C₂₇₁	—	2	2	no	—
94	D₄×C₂₇₁	—	2	2	no	—
95	D₄×C₂₇₁	C₂²×C₄	1	4	yes	—
96	C₂×C₄×C₂₇₁	—	2	2	no	—
97	C₂×C₄×C₂₇₁	—	8	2	no	—
98	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
101	C₈×C₂₇₁	—	2	1	no	—
102	C₈×C₂₇₁	C₂²×C₄	1	2	yes	—
103	C₂²×C₂₇₁	—	2	4	no	—
104	C₂²×C₂₇₁	—	4	2	no	—
105	C₂²×C₂₇₁	—	4	2	no	—
106	C₂²×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
107	C₂×D₂₇₁	—	2	1	no	—
108	C₂×D₂₇₁	—	2	1	no	—
109	C₂×D₂₇₁	—	2	4	no	—
110	C₂×D₂₇₁	—	4	2	no	—
111	C₂×D₂₇₁	—	4	2	no	—
112	C₂×D₂₇₁	—	8	8	no	—
113	C₂×D₂₇₁	C₄×D₄	1	2	yes	—
114	C₄×C₂₇₁	—	2	1	no	—
115	C₄×C₂₇₁	—	2	1	no	—
116	C₄×C₂₇₁	—	4	2	no	—
117	C₄×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₂₇₁	C₂×#41	1	1	yes	characteristic
119	Dic₂₇₁	—	2	1	no	—
120	Dic₂₇₁	—	2	1	no	—
121	Dic₂₇₁	—	4	2	no	—
122	Dic₂₇₁	C₄×D₄	1	2	yes	—
123	C₂×C₂₇₁	—	4	4	no	—
124	C₂×C₂₇₁	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂×C₂₇₁	#576	1	1	yes	characteristic
126	C₂×C₂₇₁	#576	1	1	yes	characteristic
127	D₂₇₁	—	2	2	no	—
128	D₂₇₁	—	4	4	no	—
129	C₂₇₁	#2948	1	1	yes	characteristic, complemented, Sylow
130	#2948	—	271	1	no	maximal, Sylow
131	C₂²×D₈	—	271	1	no	—
132	#518	—	271	2	no	—
133	#442	—	271	2	no	—
134	#442	—	271	2	no	—
135	C₄×D₈	—	271	1	no	—
136	C₄×D₈	—	271	1	no	—
137	C₄×D₈	—	271	1	no	—
138	C₄×D₈	D₂₇₁	1	1	yes	characteristic, complemented
139	#427	—	271	1	no	—
140	#418	—	271	2	no	—
141	#405	—	271	1	no	—
142	C₂²×D₄	—	271	2	no	—
143	C₂×D₈	—	271	2	no	—
144	C₂×D₈	—	271	8	no	—
145	C₂×D₈	—	271	1	no	—
146	C₂×D₈	C₂×D₂₇₁	1	1	yes	characteristic
147	C₂×M₁₆	—	271	1	no	—
148	C₂²×C₈	—	271	1	no	—
149	C₄×D₄	—	271	2	no	—
150	C₄×D₄	—	271	2	no	—
151	C₄×D₄	—	271	2	no	—
152	C₄×D₄	—	542	2	no	—
153	C₄×D₄	—	542	2	no	—
154	C₄×D₄	C₂×D₂₇₁	1	2	yes	complemented
155	#117	—	271	1	no	—
156	#117	—	271	2	no	—
157	C₂×C₂²⋊C₄	—	542	2	no	—
158	C₂×C₂²⋊C₄	—	542	2	no	—
159	#107	—	271	1	no	—
160	#107	C₂×D₂₇₁	1	1	yes	characteristic, complemented
161	#106	—	271	2	no	—
162	#102	—	271	2	no	—
163	#102	—	271	2	no	—
164	#102	—	271	2	no	—
165	#102	C₂×D₂₇₁	1	2	yes	complemented
166	#97	—	271	1	no	—
167	#97	—	271	1	no	—
168	C₄×C₈	—	271	1	no	—
169	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic, complemented
170	C₂⁴	—	542	2	no	—
171	C₂×D₄	—	271	8	no	—
172	C₂×D₄	—	271	2	no	—
173	C₂×D₄	—	542	4	no	—
174	C₂×D₄	—	542	2	no	—
175	C₂×D₄	C₂²×D₂₇₁	1	2	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	271	1	no	—
178	C₂²×C₄	—	542	2	no	—
179	C₂²×C₄	—	542	2	no	—
180	C₂²×C₄	—	542	2	no	—
181	D₈	—	271	8	no	—
182	D₈	—	271	4	no	—
183	D₈	C₄×D₂₇₁	1	4	yes	complemented
184	M₁₆	—	542	2	no	—
185	C₂×C₈	—	271	1	no	—
186	C₂×C₈	—	271	4	no	—
187	C₂×C₈	—	271	1	no	—
188	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
190	C₄⋊C₄	—	271	1	no	—
191	C₄⋊C₄	—	271	1	no	—
192	C₄⋊C₄	—	271	2	no	—
193	C₄⋊C₄	C₂²×D₂₇₁	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	271	1	no	—
196	C₂²⋊C₄	—	271	1	no	—
197	C₂²⋊C₄	—	542	2	no	—
198	C₂²⋊C₄	—	542	2	no	—
199	C₂²⋊C₄	—	542	2	no	—
200	C₂²⋊C₄	—	542	2	no	—
201	C₄²	—	271	1	no	—
202	C₄²	—	542	2	no	—
203	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	—	271	1	no	—
206	C₂³	—	542	8	no	—
207	C₂³	—	542	2	no	—
208	C₂³	—	2168	2	no	—
209	C₂³	—	2168	2	no	—
210	D₄	—	2	2	no	—
211	D₄	—	271	4	no	—
212	D₄	—	542	8	no	—
213	D₄	—	542	2	no	—
214	D₄	C₂×C₄×D₂₇₁	1	4	yes	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	271	1	no	—
218	C₂×C₄	—	271	2	no	—
219	C₂×C₄	—	271	2	no	—
220	C₂×C₄	—	271	1	no	—
221	C₂×C₄	—	271	1	no	—
222	C₂×C₄	—	542	2	no	—
223	C₂×C₄	—	2168	2	no	—
224	C₂×C₄	—	2168	2	no	—
225	C₂×C₄	—	2168	2	no	—
226	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
227	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
228	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
229	C₈	—	2	1	no	—
230	C₈	—	271	2	no	—
231	C₈	—	542	1	no	—
232	C₈	C₂×C₄×D₂₇₁	1	2	yes	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	271	2	no	—
237	C₂²	—	542	1	no	—
238	C₂²	—	542	1	no	—
239	C₂²	—	542	4	no	—
240	C₂²	—	1084	2	no	—
241	C₂²	—	1084	2	no	—
242	C₂²	—	2168	8	no	—
243	C₂²	C₂×D₄×D₂₇₁	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	271	2	no	—
248	C₄	—	542	1	no	—
249	C₄	—	542	1	no	—
250	C₄	—	1084	2	no	—
251	C₄	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
252	C₄	#41×D₂₇₁	1	1	yes	characteristic
253	C₂	—	4	4	no	—
254	C₂	—	542	2	no	—
255	C₂	—	1084	4	no	—
256	C₂	C₄×D₄×D₂₇₁	1	1	yes	characteristic
257	C₂	#31774698	1	1	yes	characteristic
258	C₂	#31774699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₇₁
commutator	C₄×C₂₇₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	C₄×D₈×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂²×D₄
prime radical	C₂×D₈×D₂₇₁	C₂
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	C₄×D₈	D₂₇₁
hypocenter	C₂₇₁	#2948
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #31774369	C₂	semidirect
3. #31774007	C₂	semidirect
4. #31774006	C₂	semidirect
5. C₄×D₂₁₆₈	C₂	semidirect
6. D₈×Dic₂₇₁	C₂	semidirect
7. #31773979	C₂	semidirect
8. #31773934	C₂	semidirect
9. #31773900	C₂	semidirect
10. #31773851	C₂	semidirect
11. C₄×D₈×C₂₇₁	C₂	semidirect
15. D₈×D₂₇₁	C₄	semidirect
18. C₄×D₁₀₈₄	C₂²	semidirect
19. D₄×Dic₂₇₁	C₂²	semidirect
20. #22576122	C₂²	semidirect
27. #22576094	C₂²	semidirect
28. #22576091	C₂²	semidirect
29. #22576082	C₂²	semidirect
30. #22576081	C₂²	semidirect
31. #22576080	C₂²	semidirect
32. #22576071	C₂²	semidirect
33. #22576070	C₂²	semidirect
34. C₈×Dic₂₇₁	C₂²	semidirect
36. C₄×D₄×C₂₇₁	C₂²	semidirect
37. #107×C₂₇₁	C₂²	semidirect
38. #102×C₂₇₁	C₂²	semidirect
39. C₄×C₈×C₂₇₁	C₂²	semidirect
44. D₄×D₂₇₁	C₂×C₄	semidirect
49. D₂₁₆₈	C₂×C₄	semidirect
50. #16864639	C₂×C₄	semidirect
54. C₈×D₂₇₁	C₂×C₄	semidirect
68. D₈×C₂₇₁	C₂×C₄	semidirect
129. C₂₇₁	#2948	semidirect
138. C₄×D₈	D₂₇₁	semidirect
154. C₄×D₄	C₂×D₂₇₁	semidirect
160. #107	C₂×D₂₇₁	semidirect
165. #102	C₂×D₂₇₁	semidirect
169. C₄×C₈	C₂×D₂₇₁	semidirect
183. D₈	C₄×D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×D₂₇₁	2
#31774369	2
#31774007	2
#31774006	2
C₄×D₂₁₆₈	2
D₈×Dic₂₇₁	2
#31773979	2
#31773934	2
#31773900	2
#31773851	2
C₄×D₈×C₂₇₁	2
#2948	271