Subgroups — C₂×C₃₁×#3365 #21518061

Order 7936.

1894 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#559	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3365	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#488	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#478	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#464	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic
10	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#559	—	2	4	no	—
16	C₃₁×#559	C₂²	1	8	yes	—
17	C₃₁×#557	—	2	4	no	—
18	C₃₁×#557	C₂²	1	8	yes	complemented
19	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
20	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#125	—	2	2	no	—
23	C₂×C₃₁×#125	C₂²	1	2	yes	—
24	C₂×C₃₁×#124	—	2	1	no	—
25	C₂×C₃₁×#124	C₂²	1	2	yes	—
26	C₂×C₃₁×#123	—	2	2	no	—
27	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₃₁	—	2	1	no	—
33	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
34	C₃₁×#488	C₂²	1	16	yes	complemented
35	C₃₁×#486	C₂²	1	8	yes	—
36	C₃₁×#478	C₂²	1	16	yes	complemented
37	C₃₁×#475	C₂²	1	8	yes	complemented
38	C₃₁×#464	C₂²	1	16	yes	complemented
39	C₃₁×#460	C₂²	1	8	yes	—
40	C₈×Q₈×C₃₁	C₂²	1	8	yes	complemented
41	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
42	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
43	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#106	C₂²	1	2	yes	—
45	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
47	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#103	C₂²	1	2	yes	—
49	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#102	C₂²	1	2	yes	—
51	C₂×C₃₁×#102	C₂²	1	2	yes	—
52	C₂×C₃₁×#102	C₂²	1	2	yes	—
53	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
54	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
55	C₂²×Q₈×C₃₁	—	2	1	no	—
56	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₃₁	—	2	1	no	—
60	C₂²×D₄×C₃₁	—	2	1	no	—
61	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₃₁	—	2	2	no	—
63	C₂×Q₁₆×C₃₁	—	2	4	no	—
64	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
65	C₂×SD₁₆×C₃₁	—	2	8	no	—
66	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
70	C₃₁×C₄⋊Q₈	—	2	8	no	—
71	C₃₁×C₄⋊Q₈	—	2	2	no	—
72	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
73	C₃₁×#127	—	2	2	no	—
74	C₃₁×#127	C₂³	1	4	yes	—
75	C₃₁×#125	—	2	16	no	—
76	C₃₁×#125	—	2	4	no	—
77	C₃₁×#125	C₂³	1	8	yes	—
78	C₃₁×#124	—	2	8	no	—
79	C₃₁×#124	—	2	4	no	—
80	C₃₁×#124	C₂³	1	8	yes	—
81	C₃₁×#123	—	2	16	no	—
82	C₄×Q₈×C₃₁	—	2	8	no	—
83	C₄×Q₈×C₃₁	—	2	4	no	—
84	C₄×Q₈×C₃₁	—	2	2	no	—
85	C₄×Q₈×C₃₁	—	2	2	no	—
86	C₄×Q₈×C₃₁	D₄	1	4	yes	—
87	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
88	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
89	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
90	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
91	C₄×D₄×C₃₁	—	2	8	no	—
92	C₄×D₄×C₃₁	—	2	4	no	—
93	C₄×D₄×C₃₁	C₂³	1	8	yes	—
94	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
96	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
97	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
108	C₂×C₄²×C₃₁	—	2	1	no	—
109	C₂×C₄²×C₃₁	—	2	2	no	—
110	C₂×C₄²×C₃₁	—	2	1	no	—
111	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
113	C₃₁×#107	C₂³	1	4	yes	—
114	C₃₁×#106	C₂³	1	8	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
117	C₃₁×#103	C₂³	1	4	yes	—
118	C₃₁×#103	C₂³	1	4	yes	—
119	C₃₁×#103	C₂³	1	8	yes	—
120	C₃₁×#102	C₂³	1	8	yes	—
121	C₃₁×#102	C₂³	1	8	yes	—
122	C₃₁×#102	C₂³	1	8	yes	—
123	C₄×C₈×C₃₁	C₂³	1	8	yes	—
124	C₄×C₈×C₃₁	C₂³	1	4	yes	—
125	C₂⁴×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	2	4	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	4	no	—
130	C₂×Q₈×C₃₁	—	2	2	no	—
131	C₂×Q₈×C₃₁	—	2	2	no	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	4	4	no	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
135	C₂×Q₈×C₃₁	#41	1	4	yes	—
136	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
137	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	—	2	4	no	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	4	no	—
142	C₂×D₄×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	4	4	no	—
144	C₂×D₄×C₃₁	—	8	2	no	—
145	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₃₁	—	2	1	no	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	1	no	—
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	8	no	—
152	C₂²×C₄×C₃₁	—	8	4	no	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₃₁	—	2	8	no	—
159	SD₁₆×C₃₁	—	2	16	no	—
160	C₂×C₈×C₃₁	—	2	4	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	2	2	no	—
163	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
164	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
165	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
166	C₄⋊C₄×C₃₁	—	2	8	no	—
167	C₄⋊C₄×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	8	no	—
170	C₄⋊C₄×C₃₁	—	2	4	no	—
171	C₄⋊C₄×C₃₁	—	2	8	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
177	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
178	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
179	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂²⋊C₄×C₃₁	—	2	8	no	—
181	C₂²⋊C₄×C₃₁	—	2	8	no	—
182	C₂²⋊C₄×C₃₁	—	2	8	no	—
183	C₄²×C₃₁	—	2	4	no	—
184	C₄²×C₃₁	—	2	8	no	—
185	C₄²×C₃₁	—	2	4	no	—
186	C₄²×C₃₁	C₂×D₄	1	4	yes	—
187	C₄²×C₃₁	C₂×D₄	1	2	yes	—
188	C₄²×C₃₁	C₂⁴	1	4	yes	—
189	C₄²×C₃₁	C₂⁴	1	2	yes	—
190	C₂×#3365	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₃₁	—	2	4	no	—
192	C₂³×C₃₁	—	4	4	no	—
193	C₂³×C₃₁	—	4	4	no	—
194	C₂³×C₃₁	—	4	4	no	—
195	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	Q₈×C₃₁	—	2	4	no	—
197	Q₈×C₃₁	—	2	8	no	—
198	Q₈×C₃₁	—	2	2	no	—
199	Q₈×C₃₁	—	2	2	no	—
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	4	8	no	—
203	Q₈×C₃₁	C₂×#41	1	4	yes	—
204	D₄×C₃₁	—	2	4	no	—
205	D₄×C₃₁	—	2	4	no	—
206	D₄×C₃₁	—	4	4	no	—
207	D₄×C₃₁	—	4	8	no	—
208	D₄×C₃₁	—	8	4	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	4	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	—	8	8	no	—
223	C₂×C₄×C₃₁	#135	1	4	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
230	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
231	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
232	C₈×C₃₁	—	2	4	no	—
233	C₈×C₃₁	—	2	2	no	—
234	C₈×C₃₁	—	2	2	no	—
235	C₂×#559	C₂×C₃₁	1	1	yes	characteristic
236	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#3365	C₂×C₃₁	1	16	yes	direct product factor
238	C₂×#488	C₂×C₃₁	1	2	yes	complemented
239	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
240	C₂×#475	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂×#478	C₂×C₃₁	1	2	yes	complemented
242	C₂×#464	C₂×C₃₁	1	2	yes	complemented
243	C₂×#460	C₂×C₃₁	1	1	yes	characteristic
244	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
246	C₂×C₄×SD₁₆	C₂×C₃₁	1	2	yes	complemented
247	C₂²×C₃₁	—	4	4	no	—
248	C₂²×C₃₁	—	4	4	no	—
249	C₂²×C₃₁	—	4	4	no	—
250	C₂²×C₃₁	—	8	8	no	—
251	C₂²×C₃₁	#548	1	2	yes	—
252	C₂²×C₃₁	C₂×#135	1	2	yes	—
253	C₂²×C₃₁	#577	1	2	yes	—
254	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
255	C₄×C₃₁	—	2	4	no	—
256	C₄×C₃₁	—	2	4	no	—
257	C₄×C₃₁	—	2	4	no	—
258	C₄×C₃₁	—	2	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	4	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	#548	1	2	yes	—
264	C₄×C₃₁	C₂×#135	1	4	yes	—
265	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
266	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
267	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
269	#559	—	2	4	no	—
270	#559	C₂²×C₃₁	1	8	yes	—
271	#557	—	2	4	no	—
272	#557	C₂²×C₃₁	1	8	yes	complemented
273	C₂×C₄⋊Q₈	—	2	1	no	—
274	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×#125	—	2	2	no	—
277	C₂×#125	C₂²×C₃₁	1	2	yes	—
278	C₂×#124	—	2	1	no	—
279	C₂×#124	C₂²×C₃₁	1	2	yes	—
280	C₂×#123	—	2	2	no	—
281	C₂×C₄×Q₈	—	2	1	no	—
282	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
284	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×C₄×D₄	—	2	1	no	—
287	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
288	#488	C₂²×C₃₁	1	16	yes	complemented
289	#486	C₂²×C₃₁	1	8	yes	—
290	#478	C₂²×C₃₁	1	16	yes	complemented
291	#475	C₂²×C₃₁	1	8	yes	complemented
292	#464	C₂²×C₃₁	1	16	yes	complemented
293	#460	C₂²×C₃₁	1	8	yes	—
294	C₈×Q₈	C₂²×C₃₁	1	8	yes	complemented
295	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
296	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
297	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×#106	C₂²×C₃₁	1	2	yes	—
299	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
300	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
301	C₂×#103	C₂²×C₃₁	1	2	yes	—
302	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#102	C₂²×C₃₁	1	2	yes	—
305	C₂×#102	C₂²×C₃₁	1	2	yes	—
306	C₂×#102	C₂²×C₃₁	1	2	yes	—
307	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
308	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
309	C₂×C₃₁	—	8	4	no	—
310	C₂×C₃₁	#3365	1	4	yes	direct product factor
311	C₂×C₃₁	C₂×#548	1	1	yes	characteristic
312	C₂×C₃₁	C₂²×#135	1	1	yes	characteristic
313	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
316	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₂²×D₄	—	2	1	no	—
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
321	C₂³×C₄	—	2	2	no	—
322	C₂×Q₁₆	—	2	4	no	—
323	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
324	C₂×SD₁₆	—	2	8	no	—
325	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
326	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
327	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
328	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
329	C₄⋊Q₈	—	2	8	no	—
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
332	#127	—	2	2	no	—
333	#127	C₂³×C₃₁	1	4	yes	—
334	#125	—	2	16	no	—
335	#125	—	2	4	no	—
336	#125	C₂³×C₃₁	1	8	yes	—
337	#124	—	2	8	no	—
338	#124	—	2	4	no	—
339	#124	C₂³×C₃₁	1	8	yes	—
340	#123	—	2	16	no	—
341	C₄×Q₈	—	2	8	no	—
342	C₄×Q₈	—	2	4	no	—
343	C₄×Q₈	—	2	2	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	D₄×C₃₁	1	4	yes	—
346	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
347	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
348	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
349	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
350	C₄×D₄	—	2	8	no	—
351	C₄×D₄	—	2	4	no	—
352	C₄×D₄	C₂³×C₃₁	1	8	yes	—
353	C₂×C₄⋊C₄	—	2	2	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	—	2	2	no	—
357	C₂×C₄⋊C₄	—	2	1	no	—
358	C₂×C₄⋊C₄	—	2	2	no	—
359	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
360	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
361	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
364	C₂×C₂²⋊C₄	—	2	2	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₂²⋊C₄	—	2	2	no	—
367	C₂×C₄²	—	2	1	no	—
368	C₂×C₄²	—	2	2	no	—
369	C₂×C₄²	—	2	1	no	—
370	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
371	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
372	#107	C₂³×C₃₁	1	4	yes	—
373	#106	C₂³×C₃₁	1	8	yes	—
374	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
375	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
376	#103	C₂³×C₃₁	1	4	yes	—
377	#103	C₂³×C₃₁	1	4	yes	—
378	#103	C₂³×C₃₁	1	8	yes	—
379	#102	C₂³×C₃₁	1	8	yes	—
380	#102	C₂³×C₃₁	1	8	yes	—
381	#102	C₂³×C₃₁	1	8	yes	—
382	C₄×C₈	C₂³×C₃₁	1	8	yes	—
383	C₄×C₈	C₂³×C₃₁	1	4	yes	—
384	C₃₁	C₂×#3365	1	1	yes	characteristic, direct product factor, Sylow
385	C₂⁴	—	2	2	no	—
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	4	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	2	no	—
392	C₂×Q₈	—	2	2	no	—
393	C₂×Q₈	—	4	4	no	—
394	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
395	C₂×Q₈	#41×C₃₁	1	4	yes	—
396	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
397	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	4	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	4	4	no	—
404	C₂×D₄	—	8	2	no	—
405	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	8	no	—
412	C₂²×C₄	—	8	4	no	—
413	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
416	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
418	Q₁₆	—	2	8	no	—
419	SD₁₆	—	2	16	no	—
420	C₂×C₈	—	2	4	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
424	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
425	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	8	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	8	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
436	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
437	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
438	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
439	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
440	C₂²⋊C₄	—	2	8	no	—
441	C₂²⋊C₄	—	2	8	no	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₄²	—	2	4	no	—
444	C₄²	—	2	8	no	—
445	C₄²	—	2	4	no	—
446	C₄²	C₂×D₄×C₃₁	1	4	yes	—
447	C₄²	C₂×D₄×C₃₁	1	2	yes	—
448	C₄²	C₂⁴×C₃₁	1	4	yes	—
449	C₄²	C₂⁴×C₃₁	1	2	yes	—
450	C₂³	—	2	4	no	—
451	C₂³	—	4	4	no	—
452	C₂³	—	4	4	no	—
453	C₂³	—	4	4	no	—
454	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
455	Q₈	—	2	4	no	—
456	Q₈	—	2	8	no	—
457	Q₈	—	2	2	no	—
458	Q₈	—	2	2	no	—
459	Q₈	—	2	2	no	—
460	Q₈	—	4	2	no	—
461	Q₈	—	4	8	no	—
462	Q₈	C₂×#41×C₃₁	1	4	yes	—
463	D₄	—	2	4	no	—
464	D₄	—	2	4	no	—
465	D₄	—	4	4	no	—
466	D₄	—	4	8	no	—
467	D₄	—	8	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	8	8	no	—
482	C₂×C₄	C₃₁×#135	1	4	yes	—
483	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
484	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
485	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
488	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
489	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
490	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
491	C₈	—	2	4	no	—
492	C₈	—	2	2	no	—
493	C₈	—	2	2	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	4	no	—
496	C₂²	—	4	4	no	—
497	C₂²	—	8	8	no	—
498	C₂²	C₃₁×#548	1	2	yes	—
499	C₂²	C₂×C₃₁×#135	1	2	yes	—
500	C₂²	C₃₁×#577	1	2	yes	—
501	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
502	C₄	—	2	4	no	—
503	C₄	—	2	4	no	—
504	C₄	—	2	4	no	—
505	C₄	—	2	2	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	4	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	2	no	—
510	C₄	C₃₁×#548	1	2	yes	—
511	C₄	C₂×C₃₁×#135	1	4	yes	—
512	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
513	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
514	C₂	—	8	4	no	—
515	C₂	C₃₁×#3365	1	4	yes	direct product factor
516	C₂	C₂×C₃₁×#548	1	1	yes	characteristic
517	C₂	C₂²×C₃₁×#135	1	1	yes	characteristic
518	C₂	C₂×C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#557	C₂	semidirect
3. C₃₁×#3365	C₂	direct (N × H)
4. C₂×C₃₁×#488	C₂	semidirect
6. C₂×C₃₁×#475	C₂	semidirect
7. C₂×C₃₁×#478	C₂	semidirect
8. C₂×C₃₁×#464	C₂	semidirect
10. C₂×C₈×Q₈×C₃₁	C₂	semidirect
11. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
12. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
18. C₃₁×#557	C₂²	semidirect
34. C₃₁×#488	C₂²	semidirect
36. C₃₁×#478	C₂²	semidirect
37. C₃₁×#475	C₂²	semidirect
38. C₃₁×#464	C₂²	semidirect
40. C₈×Q₈×C₃₁	C₂²	semidirect
41. C₄×Q₁₆×C₃₁	C₂²	semidirect
42. C₄×SD₁₆×C₃₁	C₂²	semidirect
190. C₂×#3365	C₃₁	direct (N × H)
236. C₂×#557	C₂×C₃₁	semidirect
237. #3365	C₂×C₃₁	direct (N × H)
238. C₂×#488	C₂×C₃₁	semidirect
240. C₂×#475	C₂×C₃₁	semidirect
241. C₂×#478	C₂×C₃₁	semidirect
242. C₂×#464	C₂×C₃₁	semidirect
244. C₂×C₈×Q₈	C₂×C₃₁	semidirect
245. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
246. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
272. #557	C₂²×C₃₁	semidirect
288. #488	C₂²×C₃₁	semidirect
290. #478	C₂²×C₃₁	semidirect
291. #475	C₂²×C₃₁	semidirect
292. #464	C₂²×C₃₁	semidirect
294. C₈×Q₈	C₂²×C₃₁	semidirect
295. C₄×Q₁₆	C₂²×C₃₁	semidirect
296. C₄×SD₁₆	C₂²×C₃₁	semidirect
310. C₂×C₃₁	#3365	direct (N × H)
384. C₃₁	C₂×#3365	direct (N × H)
515. C₂	C₃₁×#3365	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#559	2
C₂×C₃₁×#557	2
C₃₁×#3365	2
C₂×C₃₁×#488	2
C₂×C₃₁×#486	2
C₂×C₃₁×#475	2
C₂×C₃₁×#478	2
C₂×C₃₁×#464	2
C₂×C₃₁×#460	2
C₂×C₈×Q₈×C₃₁	2
C₂×C₄×Q₁₆×C₃₁	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×#3365	31

Subgroups — C2×C31×#3365 #21518061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3365 #21518061