Subgroups — #46845276

Order 34688.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₂₇₁	C₂	1	1	yes	maximal, characteristic
2	#31774662	C₂	1	2	yes	maximal, complemented
3	#31774017	C₂	1	2	yes	maximal, complemented
4	#31774016	C₂	1	2	yes	maximal, complemented
5	#31774015	C₂	1	1	yes	maximal, characteristic, complemented
6	#31774014	C₂	1	1	yes	maximal, characteristic, complemented
7	#31774013	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#31773912	C₂	1	2	yes	maximal, complemented
10	#31773851	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₂₇₁	C₂²	1	2	yes	—
13	#22576183	C₄	1	1	yes	characteristic, complemented
14	#22576182	C₂²	1	1	yes	characteristic
15	#22576181	C₄	1	2	yes	complemented
16	#22576180	C₂²	1	2	yes	—
17	#22576179	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₂₇₁	C₄	1	1	yes	characteristic, complemented
19	SD₁₆×D₂₇₁	C₄	1	2	yes	complemented
20	D₈×D₂₇₁	C₄	1	1	yes	characteristic, complemented
21	#22576168	—	2	2	no	—
22	#22576168	C₂²	1	2	yes	complemented
23	#22576167	C₂²	1	2	yes	complemented
24	#22576166	—	2	2	no	—
25	#22576166	C₂²	1	2	yes	complemented
26	C₂×#16864637	—	2	2	no	—
27	C₂×#16864637	C₂²	1	1	yes	characteristic
28	M₁₆×D₂₇₁	—	2	2	no	—
29	M₁₆×D₂₇₁	C₂²	1	2	yes	—
30	C₂×C₈×D₂₇₁	C₂²	1	1	yes	characteristic
31	#22576117	C₂²	1	1	yes	characteristic
32	#22576096	C₂²	1	1	yes	characteristic, complemented
33	#22576095	C₂²	1	2	yes	complemented
34	#22576088	C₂²	1	2	yes	complemented
35	#22576087	C₂²	1	2	yes	complemented
36	#22576086	C₂²	1	2	yes	complemented
37	#22576071	C₂²	1	1	yes	characteristic, complemented
38	#22576070	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic, complemented
40	#135×C₂₇₁	C₂²	1	1	yes	characteristic
41	#131×C₂₇₁	C₂²	1	2	yes	complemented
42	#108×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
43	#104×C₂₇₁	C₂²	1	2	yes	complemented
44	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic, complemented
45	#16864655	C₂×C₄	1	2	yes	—
46	#16864654	—	2	2	no	—
47	#16864654	C₂³	1	2	yes	—
48	#16864653	—	2	2	no	—
49	#16864653	C₂×C₄	1	2	yes	—
50	Q₈×D₂₇₁	C₂×C₄	1	2	yes	—
51	D₄×D₂₇₁	—	2	2	no	—
52	D₄×D₂₇₁	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₂₇₁	—	2	2	no	—
54	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
55	Dic₁₀₈₄	C₂×C₄	1	1	yes	characteristic
56	#16864644	C₂×C₄	1	2	yes	—
57	#16864643	C₂×C₄	1	2	yes	—
58	#16864642	C₂×C₄	1	2	yes	—
59	#16864641	C₂×C₄	1	2	yes	—
60	D₂₁₆₈	C₂×C₄	1	1	yes	characteristic
61	#16864639	C₂×C₄	1	2	yes	—
62	#16864638	—	2	2	no	—
63	#16864638	C₂³	1	2	yes	—
64	#16864637	—	2	2	no	—
65	#16864637	—	2	2	no	—
66	#16864637	—	2	2	no	—
67	#16864637	D₄	1	1	yes	characteristic
68	#16864637	D₄	1	1	yes	characteristic
69	#16864637	D₄	1	1	yes	characteristic
70	#16864637	D₄	1	1	yes	characteristic
71	C₂×C₂₇₁⋊C₈	—	2	2	no	—
72	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×D₂₇₁	—	2	2	no	—
75	C₈×D₂₇₁	—	2	2	no	—
76	C₈×D₂₇₁	—	2	2	no	—
77	C₈×D₂₇₁	C₂×C₄	1	1	yes	characteristic
78	C₈×D₂₇₁	C₂×C₄	1	1	yes	characteristic
79	C₈×D₂₇₁	C₂×C₄	1	1	yes	characteristic
80	C₈×D₂₇₁	C₂×C₄	1	1	yes	characteristic
81	Dic₂₇₁⋊C₄	—	2	1	no	—
82	(C₂×D₂₇₁)⋊C₄	—	2	1	no	—
83	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
84	#41×C₂₇₁	C₂³	1	2	yes	—
85	Q₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
86	SD₁₆×C₂₇₁	C₂×C₄	1	2	yes	—
87	D₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
88	M₁₆×C₂₇₁	—	2	2	no	—
89	M₁₆×C₂₇₁	C₂³	1	2	yes	—
90	C₂×C₈×C₂₇₁	—	2	2	no	—
91	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
93	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
94	C₂²×D₂₇₁	—	2	2	no	—
95	C₂²×D₂₇₁	#41	1	1	yes	characteristic
96	Dic₅₄₂	—	2	2	no	—
97	Dic₅₄₂	C₂²×C₄	1	2	yes	—
98	D₁₀₈₄	—	2	2	no	—
99	D₁₀₈₄	C₂²×C₄	1	2	yes	—
100	C₂²⋊D₂₇₁	—	2	2	no	—
101	C₂²⋊D₂₇₁	—	2	2	no	—
102	C₂²⋊D₂₇₁	—	2	2	no	—
103	C₂×Dic₂₇₁	—	2	2	no	—
104	C₂×Dic₂₇₁	—	2	1	no	—
105	C₂×Dic₂₇₁	#41	1	1	yes	characteristic
106	C₄×D₂₇₁	—	2	2	no	—
107	C₄×D₂₇₁	—	2	2	no	—
108	C₄×D₂₇₁	—	2	2	no	—
109	C₄×D₂₇₁	C₂²×C₄	1	1	yes	characteristic
110	C₄×D₂₇₁	C₂²×C₄	1	1	yes	characteristic
111	C₄×D₂₇₁	C₂×D₄	1	1	yes	characteristic
112	C₄×D₂₇₁	C₂×D₄	1	1	yes	characteristic
113	C₂₇₁⋊C₈	—	2	2	no	—
114	C₂₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
117	C₂₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₂₇₁	C₂²×C₄	1	2	yes	—
119	D₄×C₂₇₁	—	2	2	no	—
120	D₄×C₂₇₁	C₂²×C₄	1	2	yes	—
121	C₂×C₄×C₂₇₁	—	2	2	no	—
122	C₂×C₄×C₂₇₁	—	2	1	no	—
123	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
124	C₈×C₂₇₁	—	2	2	no	—
125	C₈×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
126	C₈×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
127	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
128	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₂₇₁	—	2	2	no	—
130	C₂²×C₂₇₁	C₂×#41	1	1	yes	characteristic
131	C₂×D₂₇₁	—	2	2	no	—
132	C₂×D₂₇₁	—	4	1	no	—
133	C₂×D₂₇₁	—	8	2	no	—
134	C₂×D₂₇₁	C₄×D₄	1	1	yes	characteristic
135	C₂×D₂₇₁	C₄×D₄	1	1	yes	characteristic
136	C₄×C₂₇₁	—	2	2	no	—
137	C₄×C₂₇₁	—	4	1	no	—
138	C₄×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
140	Dic₂₇₁	—	2	2	no	—
141	Dic₂₇₁	—	4	1	no	—
142	Dic₂₇₁	C₄×D₄	1	1	yes	characteristic
143	Dic₂₇₁	C₄×D₄	1	1	yes	characteristic
144	C₂×C₂₇₁	—	2	1	no	—
145	C₂×C₂₇₁	—	4	2	no	—
146	C₂×C₂₇₁	C₂×C₄×D₄	1	1	yes	characteristic
147	D₂₇₁	—	2	1	no	—
148	D₂₇₁	—	2	1	no	—
149	D₂₇₁	—	4	2	no	—
150	C₂₇₁	#2957	1	1	yes	characteristic, complemented, Sylow
151	#2957	—	271	1	no	maximal, Sylow
152	C₂×#135	—	271	1	no	—
153	#568	—	271	2	no	—
154	#444	—	271	2	no	—
155	#444	—	271	2	no	—
156	#443	—	271	1	no	—
157	#443	—	271	1	no	—
158	#443	—	271	1	no	—
159	#443	D₂₇₁	1	1	yes	characteristic, complemented
160	C₂×#108	—	271	1	no	—
161	#421	—	271	2	no	—
162	#405	—	271	1	no	—
163	C₂×#41	—	271	2	no	—
164	#135	—	271	2	no	—
165	#135	—	271	1	no	—
166	#135	—	271	2	no	—
167	#135	—	271	1	no	—
168	#135	—	271	1	no	—
169	#135	C₂×D₂₇₁	1	1	yes	characteristic
170	C₂×Q₁₆	—	271	1	no	—
171	C₂×SD₁₆	—	271	2	no	—
172	C₂×D₈	—	271	1	no	—
173	#131	—	271	2	no	—
174	#131	—	271	2	no	—
175	#131	—	271	2	no	—
176	#131	—	542	2	no	—
177	#131	—	542	2	no	—
178	#131	C₂×D₂₇₁	1	2	yes	complemented
179	C₂×M₁₆	—	271	1	no	—
180	C₂×M₁₆	—	271	2	no	—
181	C₂×M₁₆	—	542	2	no	—
182	C₂×M₁₆	—	542	2	no	—
183	C₂²×C₈	—	271	1	no	—
184	#117	—	271	1	no	—
185	#108	—	271	2	no	—
186	#108	—	271	1	no	—
187	#108	C₂×D₂₇₁	1	1	yes	characteristic, complemented
188	#104	—	271	2	no	—
189	#104	—	271	2	no	—
190	#104	—	271	2	no	—
191	#104	C₂×D₂₇₁	1	2	yes	complemented
192	#97	—	271	1	no	—
193	#97	—	271	1	no	—
194	C₄×C₈	—	271	1	no	—
195	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic, complemented
196	#41	—	271	2	no	—
197	#41	—	271	2	no	—
198	#41	—	271	2	no	—
199	#41	—	542	2	no	—
200	#41	—	542	2	no	—
201	#41	C₂²×D₂₇₁	1	2	yes	—
202	C₂×Q₈	—	271	2	no	—
203	C₂×D₄	—	271	2	no	—
204	C₂×D₄	—	542	2	no	—
205	C₂²×C₄	—	271	1	no	—
206	C₂²×C₄	—	542	2	no	—
207	Q₁₆	—	271	2	no	—
208	Q₁₆	—	271	1	no	—
209	Q₁₆	C₄×D₂₇₁	1	1	yes	characteristic
210	SD₁₆	—	271	2	no	—
211	SD₁₆	—	271	2	no	—
212	SD₁₆	—	271	2	no	—
213	SD₁₆	C₄×D₂₇₁	1	2	yes	—
214	D₈	—	271	2	no	—
215	D₈	—	271	1	no	—
216	D₈	C₄×D₂₇₁	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	271	1	no	—
219	M₁₆	—	271	1	no	—
220	M₁₆	—	271	1	no	—
221	M₁₆	—	271	1	no	—
222	M₁₆	—	271	2	no	—
223	M₁₆	—	542	2	no	—
224	M₁₆	—	542	2	no	—
225	M₁₆	—	542	2	no	—
226	M₁₆	—	542	2	no	—
227	M₁₆	C₂²×D₂₇₁	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	271	1	no	—
230	C₂×C₈	—	271	1	no	—
231	C₂×C₈	—	271	1	no	—
232	C₂×C₈	—	271	1	no	—
233	C₂×C₈	—	271	1	no	—
234	C₂×C₈	—	271	1	no	—
235	C₂×C₈	—	542	2	no	—
236	C₂×C₈	—	542	2	no	—
237	C₂×C₈	—	542	2	no	—
238	C₂×C₈	—	542	2	no	—
239	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
241	C₄⋊C₄	—	542	1	no	—
242	C₂²⋊C₄	—	542	1	no	—
243	C₄²	—	271	1	no	—
244	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
245	C₂³	—	271	1	no	—
246	C₂³	—	542	2	no	—
247	Q₈	—	271	2	no	—
248	Q₈	—	542	2	no	—
249	Q₈	C₂×C₄×D₂₇₁	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	271	2	no	—
252	D₄	—	542	2	no	—
253	D₄	—	542	2	no	—
254	D₄	—	542	2	no	—
255	D₄	—	542	2	no	—
256	D₄	C₂×C₄×D₂₇₁	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	271	1	no	—
260	C₂×C₄	—	271	1	no	—
261	C₂×C₄	—	271	1	no	—
262	C₂×C₄	—	271	1	no	—
263	C₂×C₄	—	271	1	no	—
264	C₂×C₄	—	542	2	no	—
265	C₂×C₄	—	542	2	no	—
266	C₂×C₄	—	542	2	no	—
267	C₂×C₄	—	542	2	no	—
268	C₂×C₄	—	542	1	no	—
269	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	271	1	no	—
272	C₈	—	271	1	no	—
273	C₈	—	271	1	no	—
274	C₈	—	271	1	no	—
275	C₈	—	542	2	no	—
276	C₈	C₂×C₄×D₂₇₁	1	1	yes	characteristic
277	C₈	C₂×C₄×D₂₇₁	1	1	yes	characteristic
278	C₈	D₄×D₂₇₁	1	1	yes	characteristic
279	C₈	D₄×D₂₇₁	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	271	1	no	—
282	C₂²	—	271	1	no	—
283	C₂²	—	542	2	no	—
284	C₂²	—	1084	1	no	—
285	C₂²	—	2168	2	no	—
286	C₂²	#41×D₂₇₁	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	271	1	no	—
290	C₄	—	271	1	no	—
291	C₄	—	542	2	no	—
292	C₄	—	1084	1	no	—
293	C₄	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
294	C₄	C₂×D₄×D₂₇₁	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	542	1	no	—
298	C₂	—	542	1	no	—
299	C₂	—	1084	2	no	—
300	C₂	C₄×D₄×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₇₁
commutator	C₄×C₂₇₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	#443×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₂×C₄×D₄
prime radical	#135×D₂₇₁	C₂
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	#443	D₂₇₁
hypocenter	C₂₇₁	#2957
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #31774662	C₂	semidirect
3. #31774017	C₂	semidirect
4. #31774016	C₂	semidirect
5. #31774015	C₂	semidirect
6. #31774014	C₂	semidirect
7. #31774013	C₂	semidirect
8. #108×D₂₇₁	C₂	semidirect
9. #31773912	C₂	semidirect
10. #31773851	C₂	semidirect
11. #443×C₂₇₁	C₂	semidirect
13. #22576183	C₄	semidirect
15. #22576181	C₄	semidirect
17. #22576179	C₄	semidirect
18. Q₁₆×D₂₇₁	C₄	semidirect
19. SD₁₆×D₂₇₁	C₄	semidirect
20. D₈×D₂₇₁	C₄	semidirect
22. #22576168	C₂²	semidirect
23. #22576167	C₂²	semidirect
25. #22576166	C₂²	semidirect
32. #22576096	C₂²	semidirect
33. #22576095	C₂²	semidirect
34. #22576088	C₂²	semidirect
35. #22576087	C₂²	semidirect
36. #22576086	C₂²	semidirect
37. #22576071	C₂²	semidirect
38. #22576070	C₂²	semidirect
39. C₈×Dic₂₇₁	C₂²	semidirect
41. #131×C₂₇₁	C₂²	semidirect
42. #108×C₂₇₁	C₂²	semidirect
43. #104×C₂₇₁	C₂²	semidirect
44. C₄×C₈×C₂₇₁	C₂²	semidirect
150. C₂₇₁	#2957	semidirect
159. #443	D₂₇₁	semidirect
178. #131	C₂×D₂₇₁	semidirect
187. #108	C₂×D₂₇₁	semidirect
191. #104	C₂×D₂₇₁	semidirect
195. C₄×C₈	C₂×D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#135×D₂₇₁	2
#31774662	2
#31774017	2
#31774016	2
#31774015	2
#31774014	2
#31774013	2
#108×D₂₇₁	2
#31773912	2
#31773851	2
#443×C₂₇₁	2
#2957	271