Subgroups — C₂×C₃₁×#2957 #21516940

Order 7936.

2642 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#568	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2957	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#444	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#443	C₂	1	4	yes	maximal, complemented
6	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#421	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×#135	C₄	1	2	yes	complemented
11	C₂×C₃₁×#135	C₄	1	2	yes	complemented
12	C₂×C₃₁×#135	C₄	1	8	yes	complemented
13	C₂×C₃₁×#135	C₂²	1	2	yes	—
14	C₂×C₃₁×#135	C₂²	1	2	yes	—
15	C₂×C₃₁×#135	C₂²	1	8	yes	—
16	C₂²×Q₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
17	C₂²×SD₁₆×C₃₁	C₄	1	2	yes	complemented
18	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#568	—	2	8	no	—
20	C₃₁×#568	C₂²	1	16	yes	complemented
21	C₂×C₃₁×#131	—	2	4	no	—
22	C₂×C₃₁×#131	C₂²	1	8	yes	complemented
23	C₂²×M₁₆×C₃₁	—	2	4	no	—
24	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
25	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
28	C₃₁×#444	C₂²	1	32	yes	complemented
29	C₃₁×#443	C₂²	1	32	yes	complemented
30	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
31	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
32	C₂×C₃₁×#108	C₂²	1	8	yes	complemented
33	C₃₁×#421	C₂²	1	16	yes	complemented
34	C₂×C₃₁×#104	C₂²	1	8	yes	complemented
35	C₃₁×#405	C₂²	1	8	yes	complemented
36	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
37	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
38	C₂×#41×C₃₁	—	2	2	no	—
39	C₂×#41×C₃₁	—	2	4	no	—
40	C₂×#41×C₃₁	—	2	2	no	—
41	C₂×#41×C₃₁	—	2	4	no	—
42	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
44	C₂×#41×C₃₁	C₂³	1	4	yes	—
45	C₂×#41×C₃₁	C₂³	1	8	yes	—
46	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#135	C₂×C₄	1	8	yes	—
52	C₃₁×#135	C₂×C₄	1	8	yes	—
53	C₃₁×#135	C₂×C₄	1	16	yes	complemented
54	C₃₁×#135	C₂×C₄	1	16	yes	complemented
55	C₃₁×#135	C₂³	1	8	yes	—
56	C₃₁×#135	C₂³	1	8	yes	—
57	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
59	C₂×Q₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
60	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
61	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
62	C₂×SD₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
63	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×D₈×C₃₁	C₂×C₄	1	8	yes	complemented
66	C₃₁×#131	—	2	32	no	—
67	C₃₁×#131	—	2	16	no	—
68	C₃₁×#131	C₂³	1	32	yes	complemented
69	C₂×M₁₆×C₃₁	—	2	2	no	—
70	C₂×M₁₆×C₃₁	—	2	4	no	—
71	C₂×M₁₆×C₃₁	—	2	32	no	—
72	C₂×M₁₆×C₃₁	—	2	8	no	—
73	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
74	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
75	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
76	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
77	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
78	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
79	C₂²×C₈×C₃₁	—	2	2	no	—
80	C₂²×C₈×C₃₁	—	2	8	no	—
81	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
86	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
89	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
90	C₃₁×#117	—	2	2	no	—
91	C₃₁×#117	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
95	C₃₁×#108	C₂³	1	8	yes	complemented
96	C₃₁×#108	C₂³	1	8	yes	complemented
97	C₃₁×#104	C₂³	1	32	yes	complemented
98	C₃₁×#97	C₂³	1	8	yes	complemented
99	C₄×C₈×C₃₁	C₂³	1	8	yes	complemented
100	C₂⁴×C₃₁	—	2	2	no	—
101	C₂⁴×C₃₁	#41	1	1	yes	characteristic
102	#41×C₃₁	—	2	8	no	—
103	#41×C₃₁	—	2	8	no	—
104	#41×C₃₁	—	2	8	no	—
105	#41×C₃₁	—	2	8	no	—
106	#41×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	2	4	no	—
108	#41×C₃₁	—	2	8	no	—
109	#41×C₃₁	C₂²×C₄	1	8	yes	—
110	#41×C₃₁	C₂²×C₄	1	16	yes	—
111	#41×C₃₁	C₂⁴	1	8	yes	—
112	C₂×Q₈×C₃₁	—	2	2	no	—
113	C₂×Q₈×C₃₁	—	2	4	no	—
114	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
115	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
116	C₂×D₄×C₃₁	—	2	4	no	—
117	C₂×D₄×C₃₁	—	2	8	no	—
118	C₂×D₄×C₃₁	—	2	4	no	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
123	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	4	no	—
126	C₂²×C₄×C₃₁	—	2	8	no	—
127	C₂²×C₄×C₃₁	—	2	8	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	#41	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₃₁	C₂²×C₄	1	8	yes	—
141	Q₁₆×C₃₁	C₂²×C₄	1	8	yes	—
142	SD₁₆×C₃₁	C₂²×C₄	1	16	yes	—
143	SD₁₆×C₃₁	C₂²×C₄	1	16	yes	—
144	D₈×C₃₁	C₂²×C₄	1	8	yes	—
145	D₈×C₃₁	C₂²×C₄	1	8	yes	—
146	M₁₆×C₃₁	—	2	8	no	—
147	M₁₆×C₃₁	—	2	4	no	—
148	M₁₆×C₃₁	—	2	32	no	—
149	M₁₆×C₃₁	C₂×D₄	1	8	yes	—
150	M₁₆×C₃₁	C₂×D₄	1	8	yes	—
151	M₁₆×C₃₁	C₂⁴	1	8	yes	—
152	C₂×C₈×C₃₁	—	2	8	no	—
153	C₂×C₈×C₃₁	—	2	4	no	—
154	C₂×C₈×C₃₁	—	2	32	no	—
155	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
158	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
161	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
162	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
163	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
164	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
165	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
166	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
167	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
168	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
169	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
170	C₄⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	—	2	4	no	—
172	C₄²×C₃₁	—	2	2	no	—
173	C₄²×C₃₁	C₂⁴	1	4	yes	—
174	C₂×#2957	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂³×C₃₁	—	2	4	no	—
176	C₂³×C₃₁	—	2	8	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	8	4	no	—
179	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
180	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
181	C₂³×C₃₁	C₄×D₄	1	2	yes	—
182	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
183	C₂³×C₃₁	C₂×#41	1	2	yes	—
184	Q₈×C₃₁	—	2	4	no	—
185	Q₈×C₃₁	—	2	8	no	—
186	Q₈×C₃₁	C₂³×C₄	1	8	yes	—
187	D₄×C₃₁	—	2	8	no	—
188	D₄×C₃₁	—	2	16	no	—
189	D₄×C₃₁	—	2	8	no	—
190	D₄×C₃₁	—	2	4	no	—
191	D₄×C₃₁	—	2	8	no	—
192	D₄×C₃₁	C₂³×C₄	1	8	yes	—
193	C₂×C₄×C₃₁	—	2	8	no	—
194	C₂×C₄×C₃₁	—	2	4	no	—
195	C₂×C₄×C₃₁	—	2	16	no	—
196	C₂×C₄×C₃₁	—	2	8	no	—
197	C₂×C₄×C₃₁	—	2	8	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	4	4	no	—
200	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
202	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
203	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
205	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
207	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
208	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
210	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
217	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
218	C₈×C₃₁	—	2	8	no	—
219	C₈×C₃₁	C₂³×C₄	1	2	yes	—
220	C₈×C₃₁	C₂³×C₄	1	2	yes	—
221	C₈×C₃₁	C₂³×C₄	1	2	yes	—
222	C₈×C₃₁	C₂³×C₄	1	2	yes	—
223	C₈×C₃₁	C₂²×D₄	1	8	yes	—
224	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic
225	C₂×#568	C₂×C₃₁	1	2	yes	complemented
226	#2957	C₂×C₃₁	1	16	yes	direct product factor
227	C₂×#444	C₂×C₃₁	1	4	yes	complemented
228	C₂×#443	C₂×C₃₁	1	4	yes	complemented
229	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
230	C₂×#421	C₂×C₃₁	1	2	yes	complemented
231	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	2	8	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	8	no	—
239	C₂²×C₃₁	—	8	8	no	—
240	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
241	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
242	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
243	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
245	C₄×C₃₁	—	2	8	no	—
246	C₄×C₃₁	—	4	4	no	—
247	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
248	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
249	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
250	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
251	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
252	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
253	C₂²×#41	C₂²×C₃₁	1	2	yes	—
254	C₂×#135	C₄×C₃₁	1	2	yes	complemented
255	C₂×#135	C₄×C₃₁	1	2	yes	complemented
256	C₂×#135	C₄×C₃₁	1	8	yes	complemented
257	C₂×#135	C₂²×C₃₁	1	2	yes	—
258	C₂×#135	C₂²×C₃₁	1	2	yes	—
259	C₂×#135	C₂²×C₃₁	1	8	yes	—
260	C₂²×Q₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
261	C₂²×SD₁₆	C₄×C₃₁	1	2	yes	complemented
262	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic, complemented
263	#568	—	2	8	no	—
264	#568	C₂²×C₃₁	1	16	yes	complemented
265	C₂×#131	—	2	4	no	—
266	C₂×#131	C₂²×C₃₁	1	8	yes	complemented
267	C₂²×M₁₆	—	2	4	no	—
268	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
269	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
270	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
272	#444	C₂²×C₃₁	1	32	yes	complemented
273	#443	C₂²×C₃₁	1	32	yes	complemented
274	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
275	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
276	C₂×#108	C₂²×C₃₁	1	8	yes	complemented
277	#421	C₂²×C₃₁	1	16	yes	complemented
278	C₂×#104	C₂²×C₃₁	1	8	yes	complemented
279	#405	C₂²×C₃₁	1	8	yes	complemented
280	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
281	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
282	C₂×C₃₁	—	2	1	no	—
283	C₂×C₃₁	—	2	2	no	—
284	C₂×C₃₁	—	2	1	no	—
285	C₂×C₃₁	—	2	2	no	—
286	C₂×C₃₁	—	4	8	no	—
287	C₂×C₃₁	#2957	1	2	yes	direct product factor
288	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
289	C₂×#41	—	2	2	no	—
290	C₂×#41	—	2	4	no	—
291	C₂×#41	—	2	2	no	—
292	C₂×#41	—	2	4	no	—
293	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
294	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
295	C₂×#41	C₂³×C₃₁	1	4	yes	—
296	C₂×#41	C₂³×C₃₁	1	8	yes	—
297	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
298	C₂²×D₄	—	2	2	no	—
299	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
300	C₂³×C₄	—	2	2	no	—
301	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
302	#135	C₂×C₄×C₃₁	1	8	yes	—
303	#135	C₂×C₄×C₃₁	1	8	yes	—
304	#135	C₂×C₄×C₃₁	1	16	yes	complemented
305	#135	C₂×C₄×C₃₁	1	16	yes	complemented
306	#135	C₂³×C₃₁	1	8	yes	—
307	#135	C₂³×C₃₁	1	8	yes	—
308	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
309	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
310	C₂×Q₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
311	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
312	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
313	C₂×SD₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
314	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
315	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
316	C₂×D₈	C₂×C₄×C₃₁	1	8	yes	complemented
317	#131	—	2	32	no	—
318	#131	—	2	16	no	—
319	#131	C₂³×C₃₁	1	32	yes	complemented
320	C₂×M₁₆	—	2	2	no	—
321	C₂×M₁₆	—	2	4	no	—
322	C₂×M₁₆	—	2	32	no	—
323	C₂×M₁₆	—	2	8	no	—
324	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
325	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
326	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
327	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
328	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
329	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
330	C₂²×C₈	—	2	2	no	—
331	C₂²×C₈	—	2	8	no	—
332	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
335	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
336	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
337	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
339	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
340	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
341	#117	—	2	2	no	—
342	#117	C₂³×C₃₁	1	4	yes	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₂²⋊C₄	—	2	1	no	—
345	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
346	#108	C₂³×C₃₁	1	8	yes	complemented
347	#108	C₂³×C₃₁	1	8	yes	complemented
348	#104	C₂³×C₃₁	1	32	yes	complemented
349	#97	C₂³×C₃₁	1	8	yes	complemented
350	C₄×C₈	C₂³×C₃₁	1	8	yes	complemented
351	C₃₁	C₂×#2957	1	1	yes	characteristic, direct product factor, Sylow
352	C₂⁴	—	2	2	no	—
353	C₂⁴	#41×C₃₁	1	1	yes	characteristic
354	#41	—	2	8	no	—
355	#41	—	2	8	no	—
356	#41	—	2	8	no	—
357	#41	—	2	8	no	—
358	#41	—	2	4	no	—
359	#41	—	2	4	no	—
360	#41	—	2	8	no	—
361	#41	C₂²×C₄×C₃₁	1	8	yes	—
362	#41	C₂²×C₄×C₃₁	1	16	yes	—
363	#41	C₂⁴×C₃₁	1	8	yes	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	4	no	—
366	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
367	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	8	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	8	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
375	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	8	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	#41×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
392	Q₁₆	C₂²×C₄×C₃₁	1	8	yes	—
393	Q₁₆	C₂²×C₄×C₃₁	1	8	yes	—
394	SD₁₆	C₂²×C₄×C₃₁	1	16	yes	—
395	SD₁₆	C₂²×C₄×C₃₁	1	16	yes	—
396	D₈	C₂²×C₄×C₃₁	1	8	yes	—
397	D₈	C₂²×C₄×C₃₁	1	8	yes	—
398	M₁₆	—	2	8	no	—
399	M₁₆	—	2	4	no	—
400	M₁₆	—	2	32	no	—
401	M₁₆	C₂×D₄×C₃₁	1	8	yes	—
402	M₁₆	C₂×D₄×C₃₁	1	8	yes	—
403	M₁₆	C₂⁴×C₃₁	1	8	yes	—
404	C₂×C₈	—	2	4	no	—
405	C₂×C₈	—	2	8	no	—
406	C₂×C₈	—	2	32	no	—
407	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
408	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
409	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
410	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
411	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
412	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
413	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
414	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
415	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
416	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
417	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
418	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
419	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
420	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
421	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
422	C₄⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₄²	—	2	2	no	—
425	C₄²	C₂⁴×C₃₁	1	4	yes	—
426	C₂³	—	2	4	no	—
427	C₂³	—	2	8	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	8	4	no	—
430	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
431	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
432	C₂³	C₄×D₄×C₃₁	1	2	yes	—
433	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
434	C₂³	C₂×#41×C₃₁	1	2	yes	—
435	Q₈	—	2	4	no	—
436	Q₈	—	2	8	no	—
437	Q₈	C₂³×C₄×C₃₁	1	8	yes	—
438	D₄	—	2	8	no	—
439	D₄	—	2	16	no	—
440	D₄	—	2	8	no	—
441	D₄	—	2	4	no	—
442	D₄	—	2	8	no	—
443	D₄	C₂³×C₄×C₃₁	1	8	yes	—
444	C₂×C₄	—	2	8	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	16	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
452	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
454	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
456	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
457	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
458	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
459	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
468	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
469	C₈	—	2	8	no	—
470	C₈	C₂³×C₄×C₃₁	1	2	yes	—
471	C₈	C₂³×C₄×C₃₁	1	2	yes	—
472	C₈	C₂³×C₄×C₃₁	1	2	yes	—
473	C₈	C₂³×C₄×C₃₁	1	2	yes	—
474	C₈	C₂²×D₄×C₃₁	1	8	yes	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	2	8	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	8	no	—
482	C₂²	—	8	8	no	—
483	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
484	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
485	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
486	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
487	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
488	C₄	—	2	8	no	—
489	C₄	—	4	4	no	—
490	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
491	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
492	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
493	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
494	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
495	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
496	C₂	—	2	1	no	—
497	C₂	—	2	2	no	—
498	C₂	—	2	1	no	—
499	C₂	—	2	2	no	—
500	C₂	—	4	8	no	—
501	C₂	C₃₁×#2957	1	2	yes	direct product factor
502	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂²×C₃₁×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#568	C₂	semidirect
3. C₃₁×#2957	C₂	direct (N × H)
4. C₂×C₃₁×#444	C₂	semidirect
5. C₂×C₃₁×#443	C₂	semidirect
6. C₂²×C₃₁×#108	C₂	semidirect
7. C₂×C₃₁×#421	C₂	semidirect
8. C₂×C₃₁×#405	C₂	semidirect
10. C₂×C₃₁×#135	C₄	semidirect
11. C₂×C₃₁×#135	C₄	semidirect
12. C₂×C₃₁×#135	C₄	semidirect
16. C₂²×Q₁₆×C₃₁	C₄	semidirect
17. C₂²×SD₁₆×C₃₁	C₄	semidirect
18. C₂²×D₈×C₃₁	C₄	semidirect
20. C₃₁×#568	C₂²	semidirect
22. C₂×C₃₁×#131	C₂²	semidirect
28. C₃₁×#444	C₂²	semidirect
29. C₃₁×#443	C₂²	semidirect
30. C₂×C₃₁×#108	C₂²	semidirect
31. C₂×C₃₁×#108	C₂²	semidirect
32. C₂×C₃₁×#108	C₂²	semidirect
33. C₃₁×#421	C₂²	semidirect
34. C₂×C₃₁×#104	C₂²	semidirect
35. C₃₁×#405	C₂²	semidirect
36. C₂×C₃₁×#97	C₂²	semidirect
37. C₂×C₄×C₈×C₃₁	C₂²	semidirect
53. C₃₁×#135	C₂×C₄	semidirect
54. C₃₁×#135	C₂×C₄	semidirect
59. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
62. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
65. C₂×D₈×C₃₁	C₂×C₄	semidirect
68. C₃₁×#131	C₂³	semidirect
95. C₃₁×#108	C₂³	semidirect
96. C₃₁×#108	C₂³	semidirect
97. C₃₁×#104	C₂³	semidirect
98. C₃₁×#97	C₂³	semidirect
99. C₄×C₈×C₃₁	C₂³	semidirect
174. C₂×#2957	C₃₁	direct (N × H)
225. C₂×#568	C₂×C₃₁	semidirect
226. #2957	C₂×C₃₁	direct (N × H)
227. C₂×#444	C₂×C₃₁	semidirect
228. C₂×#443	C₂×C₃₁	semidirect
229. C₂²×#108	C₂×C₃₁	semidirect
230. C₂×#421	C₂×C₃₁	semidirect
231. C₂×#405	C₂×C₃₁	semidirect
254. C₂×#135	C₄×C₃₁	semidirect
255. C₂×#135	C₄×C₃₁	semidirect
256. C₂×#135	C₄×C₃₁	semidirect
260. C₂²×Q₁₆	C₄×C₃₁	semidirect
261. C₂²×SD₁₆	C₄×C₃₁	semidirect
262. C₂²×D₈	C₄×C₃₁	semidirect
264. #568	C₂²×C₃₁	semidirect
266. C₂×#131	C₂²×C₃₁	semidirect
272. #444	C₂²×C₃₁	semidirect
273. #443	C₂²×C₃₁	semidirect
274. C₂×#108	C₂²×C₃₁	semidirect
275. C₂×#108	C₂²×C₃₁	semidirect
276. C₂×#108	C₂²×C₃₁	semidirect
277. #421	C₂²×C₃₁	semidirect
278. C₂×#104	C₂²×C₃₁	semidirect
279. #405	C₂²×C₃₁	semidirect
280. C₂×#97	C₂²×C₃₁	semidirect
281. C₂×C₄×C₈	C₂²×C₃₁	semidirect
287. C₂×C₃₁	#2957	direct (N × H)
304. #135	C₂×C₄×C₃₁	semidirect
305. #135	C₂×C₄×C₃₁	semidirect
310. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
313. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
316. C₂×D₈	C₂×C₄×C₃₁	semidirect
319. #131	C₂³×C₃₁	semidirect
346. #108	C₂³×C₃₁	semidirect
347. #108	C₂³×C₃₁	semidirect
348. #104	C₂³×C₃₁	semidirect
349. #97	C₂³×C₃₁	semidirect
350. C₄×C₈	C₂³×C₃₁	semidirect
351. C₃₁	C₂×#2957	direct (N × H)
501. C₂	C₃₁×#2957	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#135	2
C₂×C₃₁×#568	2
C₃₁×#2957	2
C₂×C₃₁×#444	2
C₂×C₃₁×#443	2
C₂²×C₃₁×#108	2
C₂×C₃₁×#421	2
C₂×C₃₁×#405	2
C₂×#2957	31

Subgroups — C2×C31×#2957 #21516940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2957 #21516940