Subgroups — C₂×C₃₁×#3057 #21517204

Order 7936.

2870 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#535	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#3057	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃₁×#468	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×#465	C₂	1	4	yes	maximal, complemented
8	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₃₁	—	2	8	no	—
15	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
17	C₂²×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
18	C₂×C₃₁×#131	—	2	8	no	—
19	C₂×C₃₁×#131	C₂²	1	8	yes	complemented
20	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	complemented
22	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₃₁	C₂²	1	2	yes	complemented
24	C₃₁×#536	—	2	4	no	—
25	C₃₁×#536	C₂²	1	8	yes	complemented
26	C₃₁×#535	—	2	4	no	—
27	C₃₁×#535	C₂²	1	8	yes	—
28	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
29	C₂×C₃₁×#127	—	2	1	no	—
30	C₂×C₃₁×#124	—	2	1	no	—
31	C₂×C₃₁×#124	—	2	1	no	—
32	C₂×C₃₁×#123	—	2	2	no	—
33	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
34	C₂×C₃₁×#121	C₂²	1	4	yes	—
35	C₂×C₃₁×#120	—	2	2	no	—
36	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
38	C₃₁×#468	C₂²	1	32	yes	complemented
39	C₃₁×#465	C₂²	1	32	yes	complemented
40	C₃₁×#427	C₂²	1	8	yes	complemented
41	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
42	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
43	C₃₁×#419	C₂²	1	8	yes	complemented
44	C₃₁×#418	C₂²	1	8	yes	complemented
45	C₂×C₃₁×#103	C₂²	1	4	yes	complemented
46	C₂×C₃₁×#102	C₂²	1	4	yes	—
47	C₂⁵×C₃₁	—	2	1	no	—
48	C₂×#41×C₃₁	—	2	4	no	—
49	C₂×#41×C₃₁	—	2	4	no	—
50	C₂×#41×C₃₁	D₄	1	4	yes	—
51	C₂×#41×C₃₁	C₂³	1	4	yes	—
52	C₂²×Q₈×C₃₁	—	2	1	no	—
53	C₂²×Q₈×C₃₁	—	2	2	no	—
54	C₂²×Q₈×C₃₁	—	2	4	no	—
55	C₂²×Q₈×C₃₁	—	2	4	no	—
56	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
58	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
59	C₂²×D₄×C₃₁	—	2	4	no	—
60	C₂²×D₄×C₃₁	—	2	4	no	—
61	C₂²×D₄×C₃₁	—	2	1	no	—
62	C₂²×D₄×C₃₁	—	2	2	no	—
63	C₂²×D₄×C₃₁	—	2	4	no	—
64	C₂²×D₄×C₃₁	D₄	1	2	yes	—
65	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
66	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
68	C₂³×C₄×C₃₁	—	2	1	no	—
69	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×SD₁₆×C₃₁	—	2	8	no	—
72	C₂×SD₁₆×C₃₁	—	2	8	no	—
73	C₂×SD₁₆×C₃₁	—	2	32	no	—
74	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
76	C₃₁×#131	—	2	32	no	—
77	C₂×M₁₆×C₃₁	—	2	4	no	—
78	C₂×M₁₆×C₃₁	—	2	8	no	—
79	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
80	C₂×M₁₆×C₃₁	C₂³	1	8	yes	complemented
81	C₂²×C₈×C₃₁	—	2	8	no	—
82	C₂²×C₈×C₃₁	D₄	1	8	yes	complemented
83	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
84	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
87	C₂²×C₈×C₃₁	C₂³	1	8	yes	complemented
88	C₃₁×C₄⋊Q₈	—	2	8	no	—
89	C₃₁×#127	—	2	8	no	—
90	C₃₁×#124	—	2	8	no	—
91	C₃₁×#124	—	2	8	no	—
92	C₃₁×#123	—	2	16	no	—
93	C₃₁×C₂²⋊Q₈	—	2	8	no	—
94	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	complemented
95	C₃₁×#121	—	2	8	no	—
96	C₃₁×#121	C₂³	1	16	yes	—
97	C₃₁×#120	—	2	16	no	—
98	C₃₁×#117	—	2	2	no	—
99	C₃₁×#117	—	2	2	no	—
100	C₃₁×#117	C₂³	1	4	yes	—
101	C₃₁×#117	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
108	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
109	C₂×C₄²×C₃₁	—	2	1	no	—
110	C₂×C₄²×C₃₁	—	2	1	no	—
111	C₃₁×#107	C₂³	1	8	yes	complemented
112	C₃₁×#106	C₂³	1	8	yes	complemented
113	C₃₁×#103	C₂³	1	16	yes	complemented
114	C₃₁×#102	C₂³	1	16	yes	—
115	C₂⁴×C₃₁	—	2	2	no	—
116	C₂⁴×C₃₁	—	2	4	no	—
117	C₂⁴×C₃₁	—	4	4	no	—
118	C₂⁴×C₃₁	—	8	4	no	—
119	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
120	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
121	#41×C₃₁	—	2	16	no	—
122	#41×C₃₁	—	2	8	no	—
123	#41×C₃₁	—	2	8	no	—
124	C₂×Q₈×C₃₁	—	2	4	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	2	4	no	—
127	C₂×Q₈×C₃₁	—	2	4	no	—
128	C₂×Q₈×C₃₁	—	2	8	no	—
129	C₂×Q₈×C₃₁	—	2	4	no	—
130	C₂×Q₈×C₃₁	—	2	8	no	—
131	C₂×Q₈×C₃₁	—	4	4	no	—
132	C₂×Q₈×C₃₁	—	4	8	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
134	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
135	C₂×D₄×C₃₁	—	2	8	no	—
136	C₂×D₄×C₃₁	—	2	16	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	2	8	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	4	no	—
142	C₂×D₄×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	2	8	no	—
144	C₂×D₄×C₃₁	—	4	4	no	—
145	C₂×D₄×C₃₁	—	4	8	no	—
146	C₂×D₄×C₃₁	—	8	8	no	—
147	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
148	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
149	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₃₁	—	2	4	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	4	4	no	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₃₁	#41	1	4	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
166	SD₁₆×C₃₁	—	2	16	no	—
167	SD₁₆×C₃₁	—	2	16	no	—
168	M₁₆×C₃₁	—	2	8	no	—
169	M₁₆×C₃₁	—	2	16	no	—
170	C₂×C₈×C₃₁	—	2	4	no	—
171	C₂×C₈×C₃₁	—	2	16	no	—
172	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
173	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
174	C₂×C₈×C₃₁	C₂×D₄	1	16	yes	complemented
175	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
176	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
177	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₃₁	—	2	16	no	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
182	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
183	C₂²⋊C₄×C₃₁	—	2	4	no	—
184	C₂²⋊C₄×C₃₁	—	2	4	no	—
185	C₂²⋊C₄×C₃₁	—	2	16	no	—
186	C₂²⋊C₄×C₃₁	—	2	16	no	—
187	C₄²×C₃₁	—	2	4	no	—
188	C₄²×C₃₁	—	2	4	no	—
189	C₂×#3057	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	4	no	—
191	C₂³×C₃₁	—	2	4	no	—
192	C₂³×C₃₁	—	2	8	no	—
193	C₂³×C₃₁	—	2	4	no	—
194	C₂³×C₃₁	—	4	8	no	—
195	C₂³×C₃₁	—	4	4	no	—
196	C₂³×C₃₁	—	4	8	no	—
197	C₂³×C₃₁	—	8	8	no	—
198	C₂³×C₃₁	—	8	16	no	—
199	C₂³×C₃₁	#121	1	4	yes	—
200	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
201	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
202	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₃₁	—	2	8	no	—
204	Q₈×C₃₁	—	2	4	no	—
205	Q₈×C₃₁	—	2	4	no	—
206	Q₈×C₃₁	—	4	4	no	—
207	Q₈×C₃₁	—	4	8	no	—
208	D₄×C₃₁	—	2	16	no	—
209	D₄×C₃₁	—	2	8	no	—
210	D₄×C₃₁	—	2	4	no	—
211	D₄×C₃₁	—	2	4	no	—
212	D₄×C₃₁	—	4	4	no	—
213	D₄×C₃₁	—	4	8	no	—
214	D₄×C₃₁	—	8	16	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	8	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	4	8	no	—
223	C₂×C₄×C₃₁	—	4	4	no	—
224	C₂×C₄×C₃₁	—	4	4	no	—
225	C₂×C₄×C₃₁	—	4	4	no	—
226	C₂×C₄×C₃₁	#121	1	4	yes	—
227	C₂×C₄×C₃₁	#121	1	8	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
229	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
234	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
235	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
236	C₈×C₃₁	—	2	4	no	—
237	C₈×C₃₁	C₂²×D₄	1	4	yes	—
238	C₈×C₃₁	C₂²×D₄	1	4	yes	—
239	C₂³×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂×#536	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂×#535	C₂×C₃₁	1	1	yes	characteristic
243	#3057	C₂×C₃₁	1	16	yes	direct product factor
244	C₂×#468	C₂×C₃₁	1	4	yes	complemented
245	C₂×#465	C₂×C₃₁	1	4	yes	complemented
246	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂×#419	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂×#418	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂²×C₃₁	—	2	8	no	—
250	C₂²×C₃₁	—	2	4	no	—
251	C₂²×C₃₁	—	2	4	no	—
252	C₂²×C₃₁	—	2	8	no	—
253	C₂²×C₃₁	—	4	8	no	—
254	C₂²×C₃₁	—	4	4	no	—
255	C₂²×C₃₁	—	4	8	no	—
256	C₂²×C₃₁	—	8	16	no	—
257	C₂²×C₃₁	—	8	8	no	—
258	C₂²×C₃₁	C₂×#121	1	4	yes	—
259	C₂²×C₃₁	C₂×#121	1	2	yes	—
260	C₂²×C₃₁	#577	1	4	yes	—
261	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
262	C₄×C₃₁	—	2	4	no	—
263	C₄×C₃₁	—	4	4	no	—
264	C₄×C₃₁	—	4	4	no	—
265	C₄×C₃₁	—	4	4	no	—
266	C₄×C₃₁	C₂×#121	1	2	yes	—
267	C₄×C₃₁	C₂×#121	1	4	yes	—
268	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
269	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
270	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
271	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
273	C₂²×SD₁₆	—	2	8	no	—
274	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
275	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
276	C₂²×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
277	C₂×#131	—	2	8	no	—
278	C₂×#131	C₂²×C₃₁	1	8	yes	complemented
279	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
280	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	complemented
281	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
282	C₂³×C₈	C₂²×C₃₁	1	2	yes	complemented
283	#536	—	2	4	no	—
284	#536	C₂²×C₃₁	1	8	yes	complemented
285	#535	—	2	4	no	—
286	#535	C₂²×C₃₁	1	8	yes	—
287	C₂×C₄⋊Q₈	—	2	1	no	—
288	C₂×#127	—	2	1	no	—
289	C₂×#124	—	2	1	no	—
290	C₂×#124	—	2	1	no	—
291	C₂×#123	—	2	2	no	—
292	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
293	C₂×#121	C₂²×C₃₁	1	4	yes	—
294	C₂×#120	—	2	2	no	—
295	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
297	#468	C₂²×C₃₁	1	32	yes	complemented
298	#465	C₂²×C₃₁	1	32	yes	complemented
299	#427	C₂²×C₃₁	1	8	yes	complemented
300	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
301	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
302	#419	C₂²×C₃₁	1	8	yes	complemented
303	#418	C₂²×C₃₁	1	8	yes	complemented
304	C₂×#103	C₂²×C₃₁	1	4	yes	complemented
305	C₂×#102	C₂²×C₃₁	1	4	yes	—
306	C₂×C₃₁	—	2	4	no	—
307	C₂×C₃₁	—	4	4	no	—
308	C₂×C₃₁	—	8	4	no	—
309	C₂×C₃₁	#3057	1	4	yes	direct product factor
310	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
311	C₂×C₃₁	C₂×#577	1	2	yes	—
312	C₂⁵	—	2	1	no	—
313	C₂×#41	—	2	4	no	—
314	C₂×#41	—	2	4	no	—
315	C₂×#41	D₄×C₃₁	1	4	yes	—
316	C₂×#41	C₂³×C₃₁	1	4	yes	—
317	C₂²×Q₈	—	2	1	no	—
318	C₂²×Q₈	—	2	2	no	—
319	C₂²×Q₈	—	2	4	no	—
320	C₂²×Q₈	—	2	4	no	—
321	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
322	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
323	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
324	C₂²×D₄	—	2	4	no	—
325	C₂²×D₄	—	2	4	no	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	—	2	2	no	—
328	C₂²×D₄	—	2	4	no	—
329	C₂²×D₄	D₄×C₃₁	1	2	yes	—
330	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
331	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
332	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
333	C₂³×C₄	—	2	1	no	—
334	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
335	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×SD₁₆	—	2	8	no	—
337	C₂×SD₁₆	—	2	8	no	—
338	C₂×SD₁₆	—	2	32	no	—
339	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
340	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
341	#131	—	2	32	no	—
342	C₂×M₁₆	—	2	4	no	—
343	C₂×M₁₆	—	2	8	no	—
344	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
345	C₂×M₁₆	C₂³×C₃₁	1	8	yes	complemented
346	C₂²×C₈	—	2	8	no	—
347	C₂²×C₈	D₄×C₃₁	1	8	yes	complemented
348	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
349	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
350	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
352	C₂²×C₈	C₂³×C₃₁	1	8	yes	complemented
353	C₄⋊Q₈	—	2	8	no	—
354	#127	—	2	8	no	—
355	#124	—	2	8	no	—
356	#124	—	2	8	no	—
357	#123	—	2	16	no	—
358	C₂²⋊Q₈	—	2	8	no	—
359	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	complemented
360	#121	—	2	8	no	—
361	#121	C₂³×C₃₁	1	16	yes	—
362	#120	—	2	16	no	—
363	#117	—	2	2	no	—
364	#117	—	2	2	no	—
365	#117	C₂³×C₃₁	1	4	yes	—
366	#117	C₂³×C₃₁	1	4	yes	—
367	C₂×C₄⋊C₄	—	2	4	no	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
370	C₂×C₂²⋊C₄	—	2	1	no	—
371	C₂×C₂²⋊C₄	—	2	1	no	—
372	C₂×C₂²⋊C₄	—	2	4	no	—
373	C₂×C₂²⋊C₄	—	2	4	no	—
374	C₂×C₄²	—	2	1	no	—
375	C₂×C₄²	—	2	1	no	—
376	#107	C₂³×C₃₁	1	8	yes	complemented
377	#106	C₂³×C₃₁	1	8	yes	complemented
378	#103	C₂³×C₃₁	1	16	yes	complemented
379	#102	C₂³×C₃₁	1	16	yes	—
380	C₃₁	C₂×#3057	1	1	yes	characteristic, direct product factor, Sylow
381	C₂⁴	—	2	2	no	—
382	C₂⁴	—	2	4	no	—
383	C₂⁴	—	4	4	no	—
384	C₂⁴	—	8	4	no	—
385	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
386	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
387	#41	—	2	16	no	—
388	#41	—	2	8	no	—
389	#41	—	2	8	no	—
390	C₂×Q₈	—	2	4	no	—
391	C₂×Q₈	—	2	2	no	—
392	C₂×Q₈	—	2	4	no	—
393	C₂×Q₈	—	2	4	no	—
394	C₂×Q₈	—	2	8	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	2	8	no	—
397	C₂×Q₈	—	4	4	no	—
398	C₂×Q₈	—	4	8	no	—
399	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
400	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	16	no	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	8	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	8	no	—
412	C₂×D₄	—	8	8	no	—
413	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
414	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
415	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
424	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
425	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
428	C₂²×C₄	#41×C₃₁	1	4	yes	—
429	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
430	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
432	SD₁₆	—	2	16	no	—
433	SD₁₆	—	2	16	no	—
434	M₁₆	—	2	8	no	—
435	M₁₆	—	2	16	no	—
436	C₂×C₈	—	2	4	no	—
437	C₂×C₈	—	2	16	no	—
438	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
439	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
440	C₂×C₈	C₂×D₄×C₃₁	1	16	yes	complemented
441	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
442	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
443	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
444	C₄⋊C₄	—	2	16	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
448	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	16	no	—
452	C₂²⋊C₄	—	2	16	no	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	4	no	—
455	C₂³	—	2	4	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	8	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	4	8	no	—
460	C₂³	—	4	4	no	—
461	C₂³	—	4	8	no	—
462	C₂³	—	8	8	no	—
463	C₂³	—	8	16	no	—
464	C₂³	C₃₁×#121	1	4	yes	—
465	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
466	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
467	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
468	Q₈	—	2	8	no	—
469	Q₈	—	2	4	no	—
470	Q₈	—	2	4	no	—
471	Q₈	—	4	4	no	—
472	Q₈	—	4	8	no	—
473	D₄	—	2	16	no	—
474	D₄	—	2	8	no	—
475	D₄	—	2	4	no	—
476	D₄	—	2	4	no	—
477	D₄	—	4	4	no	—
478	D₄	—	4	8	no	—
479	D₄	—	8	16	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	8	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	4	8	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	4	4	no	—
490	C₂×C₄	—	4	4	no	—
491	C₂×C₄	C₃₁×#121	1	4	yes	—
492	C₂×C₄	C₃₁×#121	1	8	yes	—
493	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
494	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
495	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
497	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
498	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
499	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
500	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
501	C₈	—	2	4	no	—
502	C₈	C₂²×D₄×C₃₁	1	4	yes	—
503	C₈	C₂²×D₄×C₃₁	1	4	yes	—
504	C₂²	—	2	8	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	2	4	no	—
507	C₂²	—	2	8	no	—
508	C₂²	—	4	8	no	—
509	C₂²	—	4	4	no	—
510	C₂²	—	4	8	no	—
511	C₂²	—	8	16	no	—
512	C₂²	—	8	8	no	—
513	C₂²	C₂×C₃₁×#121	1	4	yes	—
514	C₂²	C₂×C₃₁×#121	1	2	yes	—
515	C₂²	C₃₁×#577	1	4	yes	—
516	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
517	C₄	—	2	4	no	—
518	C₄	—	4	4	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	4	no	—
521	C₄	C₂×C₃₁×#121	1	2	yes	—
522	C₄	C₂×C₃₁×#121	1	4	yes	—
523	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
524	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
525	C₂	—	2	4	no	—
526	C₂	—	4	4	no	—
527	C₂	—	8	4	no	—
528	C₂	C₃₁×#3057	1	4	yes	direct product factor
529	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
530	C₂	C₂×C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×SD₁₆×C₃₁	C₂	semidirect
2. C₂²×C₃₁×#131	C₂	semidirect
3. C₂×C₃₁×#536	C₂	semidirect
5. C₃₁×#3057	C₂	direct (N × H)
6. C₂×C₃₁×#468	C₂	semidirect
7. C₂×C₃₁×#465	C₂	semidirect
8. C₂×C₃₁×#427	C₂	semidirect
9. C₂×C₃₁×#419	C₂	semidirect
10. C₂×C₃₁×#418	C₂	semidirect
17. C₂²×SD₁₆×C₃₁	C₂²	semidirect
19. C₂×C₃₁×#131	C₂²	semidirect
21. C₂²×M₁₆×C₃₁	C₂²	semidirect
23. C₂³×C₈×C₃₁	C₂²	semidirect
25. C₃₁×#536	C₂²	semidirect
33. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
38. C₃₁×#468	C₂²	semidirect
39. C₃₁×#465	C₂²	semidirect
40. C₃₁×#427	C₂²	semidirect
41. C₂×C₃₁×#107	C₂²	semidirect
42. C₂×C₃₁×#106	C₂²	semidirect
43. C₃₁×#419	C₂²	semidirect
44. C₃₁×#418	C₂²	semidirect
45. C₂×C₃₁×#103	C₂²	semidirect
80. C₂×M₁₆×C₃₁	C₂³	semidirect
82. C₂²×C₈×C₃₁	D₄	semidirect
87. C₂²×C₈×C₃₁	C₂³	semidirect
94. C₃₁×C₂²⋊Q₈	C₂³	semidirect
111. C₃₁×#107	C₂³	semidirect
112. C₃₁×#106	C₂³	semidirect
113. C₃₁×#103	C₂³	semidirect
174. C₂×C₈×C₃₁	C₂×D₄	semidirect
189. C₂×#3057	C₃₁	direct (N × H)
239. C₂³×SD₁₆	C₂×C₃₁	semidirect
240. C₂²×#131	C₂×C₃₁	semidirect
241. C₂×#536	C₂×C₃₁	semidirect
243. #3057	C₂×C₃₁	direct (N × H)
244. C₂×#468	C₂×C₃₁	semidirect
245. C₂×#465	C₂×C₃₁	semidirect
246. C₂×#427	C₂×C₃₁	semidirect
247. C₂×#419	C₂×C₃₁	semidirect
248. C₂×#418	C₂×C₃₁	semidirect
276. C₂²×SD₁₆	C₂²×C₃₁	semidirect
278. C₂×#131	C₂²×C₃₁	semidirect
280. C₂²×M₁₆	C₂²×C₃₁	semidirect
282. C₂³×C₈	C₂²×C₃₁	semidirect
284. #536	C₂²×C₃₁	semidirect
292. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
297. #468	C₂²×C₃₁	semidirect
298. #465	C₂²×C₃₁	semidirect
299. #427	C₂²×C₃₁	semidirect
300. C₂×#107	C₂²×C₃₁	semidirect
301. C₂×#106	C₂²×C₃₁	semidirect
302. #419	C₂²×C₃₁	semidirect
303. #418	C₂²×C₃₁	semidirect
304. C₂×#103	C₂²×C₃₁	semidirect
309. C₂×C₃₁	#3057	direct (N × H)
345. C₂×M₁₆	C₂³×C₃₁	semidirect
347. C₂²×C₈	D₄×C₃₁	semidirect
352. C₂²×C₈	C₂³×C₃₁	semidirect
359. C₂²⋊Q₈	C₂³×C₃₁	semidirect
376. #107	C₂³×C₃₁	semidirect
377. #106	C₂³×C₃₁	semidirect
378. #103	C₂³×C₃₁	semidirect
380. C₃₁	C₂×#3057	direct (N × H)
440. C₂×C₈	C₂×D₄×C₃₁	semidirect
528. C₂	C₃₁×#3057	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×SD₁₆×C₃₁	2
C₂²×C₃₁×#131	2
C₂×C₃₁×#536	2
C₂×C₃₁×#535	2
C₃₁×#3057	2
C₂×C₃₁×#468	2
C₂×C₃₁×#465	2
C₂×C₃₁×#427	2
C₂×C₃₁×#419	2
C₂×C₃₁×#418	2
C₂×#3057	31

Subgroups — C2×C31×#3057 #21517204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3057 #21517204