Subgroups — C₃₁×#60167 #21516926

Order 7936.

2706 conjugacy classes of subgroups.

The subgroup lattice has 530 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	C₃₁×#3428	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2952	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2951	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2950	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2949	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2948	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2943	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×#135	C₂	1	4	yes	maximal, complemented
10	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2896	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#135	C₄	1	16	yes	complemented
15	C₂×C₃₁×#135	C₂²	1	2	yes	—
16	C₂×C₃₁×#135	C₂²	1	2	yes	—
17	C₂×C₃₁×#135	C₂²	1	2	yes	—
18	C₂×C₃₁×#135	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
21	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#520	—	2	4	no	—
25	C₃₁×#520	C₂²	1	8	yes	complemented
26	C₃₁×#519	C₂²	1	4	yes	—
27	C₃₁×#518	—	2	4	no	—
28	C₃₁×#518	C₂²	1	4	yes	—
29	C₄×#41×C₃₁	—	2	4	no	—
30	C₄×#41×C₃₁	C₂²	1	8	yes	complemented
31	C₂×C₃₁×#117	—	2	4	no	—
32	C₂×C₃₁×#117	C₂²	1	2	yes	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₃₁×#442	C₂²	1	8	yes	complemented
35	C₃₁×#441	C₂²	1	8	yes	complemented
36	C₃₁×#440	C₂²	1	16	yes	complemented
37	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
38	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
39	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
40	C₃₁×#427	C₂²	1	2	yes	—
41	C₃₁×#427	C₂²	1	2	yes	—
42	C₃₁×#427	C₂²	1	2	yes	complemented
43	C₃₁×#427	C₂²	1	2	yes	complemented
44	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
45	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
46	C₃₁×#419	C₂²	1	4	yes	—
47	C₃₁×#418	C₂²	1	4	yes	—
48	C₃₁×#417	C₂²	1	8	yes	complemented
49	C₃₁×#416	C₂²	1	8	yes	complemented
50	C₃₁×#405	C₂²	1	2	yes	—
51	C₃₁×#405	C₂²	1	2	yes	—
52	C₃₁×#405	C₂²	1	2	yes	complemented
53	C₃₁×#405	C₂²	1	2	yes	complemented
54	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
55	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
56	C₂×#41×C₃₁	—	2	2	no	—
57	C₂×#41×C₃₁	—	2	2	no	—
58	C₂×#41×C₃₁	—	2	8	no	—
59	C₂×#41×C₃₁	C₂×C₄	1	16	yes	complemented
60	C₂×#41×C₃₁	C₂³	1	4	yes	—
61	C₂×#41×C₃₁	C₂³	1	4	yes	—
62	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
63	C₂²×D₄×C₃₁	—	2	2	no	—
64	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
65	C₂³×C₄×C₃₁	—	2	2	no	—
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#135	C₂×C₄	1	16	yes	complemented
68	C₃₁×#135	C₂×C₄	1	16	yes	complemented
69	C₃₁×#135	C₂×C₄	1	16	yes	complemented
70	C₃₁×#135	C₂×C₄	1	16	yes	complemented
71	C₂×Q₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
72	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
74	C₂×SD₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
75	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
77	C₂×D₈×C₃₁	C₂×C₄	1	8	yes	complemented
78	C₂×D₈×C₃₁	C₂³	1	2	yes	—
79	C₂×D₈×C₃₁	C₂³	1	2	yes	—
80	C₂×M₁₆×C₃₁	—	2	4	no	—
81	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
82	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
83	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
84	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
85	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₄×Q₈×C₃₁	—	2	8	no	—
92	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
93	C₄×D₄×C₃₁	—	2	16	no	—
94	C₄×D₄×C₃₁	—	2	8	no	—
95	C₄×D₄×C₃₁	—	2	16	no	—
96	C₄×D₄×C₃₁	C₂³	1	16	yes	—
97	C₃₁×#117	—	2	2	no	—
98	C₃₁×#117	—	2	2	no	—
99	C₃₁×#117	—	2	8	no	—
100	C₃₁×#117	—	2	8	no	—
101	C₃₁×#117	—	2	8	no	—
102	C₃₁×#117	D₄	1	2	yes	—
103	C₃₁×#117	D₄	1	2	yes	—
104	C₃₁×#117	C₂³	1	4	yes	—
105	C₃₁×#117	C₂³	1	4	yes	—
106	C₃₁×#117	C₂³	1	2	yes	—
107	C₃₁×#117	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
109	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
113	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
114	C₂×C₄²×C₃₁	—	2	2	no	—
115	C₂×C₄²×C₃₁	—	2	8	no	—
116	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
117	C₃₁×#107	C₂³	1	4	yes	—
118	C₃₁×#107	C₂³	1	4	yes	—
119	C₃₁×#106	C₂³	1	4	yes	—
120	C₃₁×#106	C₂³	1	4	yes	—
121	C₃₁×#103	C₂³	1	16	yes	—
122	C₃₁×#102	C₂³	1	16	yes	—
123	C₃₁×#97	C₂³	1	4	yes	—
124	C₃₁×#97	C₂³	1	4	yes	—
125	C₄×C₈×C₃₁	C₂³	1	4	yes	—
126	C₄×C₈×C₃₁	C₂³	1	4	yes	—
127	C₂⁴×C₃₁	—	2	2	no	—
128	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
129	#41×C₃₁	—	2	16	no	—
130	#41×C₃₁	—	2	16	no	—
131	#41×C₃₁	—	2	8	no	—
132	#41×C₃₁	—	2	8	no	—
133	#41×C₃₁	C₂²×C₄	1	16	yes	—
134	#41×C₃₁	C₂²×C₄	1	16	yes	—
135	C₂×Q₈×C₃₁	—	2	2	no	—
136	C₂×Q₈×C₃₁	—	2	4	no	—
137	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
138	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
139	C₂×D₄×C₃₁	—	2	16	no	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	4	no	—
142	C₂×D₄×C₃₁	—	2	2	no	—
143	C₂×D₄×C₃₁	—	2	4	no	—
144	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
145	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	16	no	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	4	no	—
151	C₂²×C₄×C₃₁	—	2	16	no	—
152	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	#41	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
161	Q₁₆×C₃₁	C₂²×C₄	1	8	yes	—
162	Q₁₆×C₃₁	C₂²×C₄	1	8	yes	—
163	SD₁₆×C₃₁	C₂²×C₄	1	16	yes	—
164	SD₁₆×C₃₁	C₂²×C₄	1	16	yes	—
165	D₈×C₃₁	C₂²×C₄	1	8	yes	—
166	D₈×C₃₁	C₂²×C₄	1	8	yes	—
167	M₁₆×C₃₁	—	2	4	no	—
168	M₁₆×C₃₁	—	2	4	no	—
169	C₂×C₈×C₃₁	—	2	4	no	—
170	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
171	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
172	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
173	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
174	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
175	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
176	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
177	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	16	no	—
183	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
185	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
186	C₂²⋊C₄×C₃₁	—	2	4	no	—
187	C₂²⋊C₄×C₃₁	—	2	4	no	—
188	C₂²⋊C₄×C₃₁	—	2	16	no	—
189	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
190	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
191	C₄²×C₃₁	—	2	16	no	—
192	C₄²×C₃₁	—	2	4	no	—
193	C₄²×C₃₁	—	2	4	no	—
194	C₄²×C₃₁	C₂×D₄	1	4	yes	—
195	C₄²×C₃₁	C₂⁴	1	4	yes	—
196	#60167	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	4	no	—
198	C₂³×C₃₁	—	2	8	no	—
199	C₂³×C₃₁	—	4	2	no	—
200	C₂³×C₃₁	—	8	4	no	—
201	C₂³×C₃₁	C₄×D₄	1	4	yes	—
202	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₃₁	—	2	8	no	—
206	Q₈×C₃₁	—	2	4	no	—
207	Q₈×C₃₁	C₂³×C₄	1	8	yes	—
208	D₄×C₃₁	—	2	16	no	—
209	D₄×C₃₁	—	2	16	no	—
210	D₄×C₃₁	—	2	8	no	—
211	D₄×C₃₁	—	2	4	no	—
212	D₄×C₃₁	C₂³×C₄	1	8	yes	—
213	C₂×C₄×C₃₁	—	2	8	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	16	no	—
216	C₂×C₄×C₃₁	—	2	16	no	—
217	C₂×C₄×C₃₁	—	2	8	no	—
218	C₂×C₄×C₃₁	—	4	4	no	—
219	C₂×C₄×C₃₁	—	8	16	no	—
220	C₂×C₄×C₃₁	—	8	4	no	—
221	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
222	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
223	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
224	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
225	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
226	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
232	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
233	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
234	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
235	C₈×C₃₁	—	2	4	no	—
236	C₈×C₃₁	C₂³×C₄	1	2	yes	—
237	C₈×C₃₁	C₂³×C₄	1	2	yes	—
238	C₈×C₃₁	C₂³×C₄	1	2	yes	—
239	C₈×C₃₁	C₂³×C₄	1	2	yes	—
240	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic
241	#3428	C₂×C₃₁	1	2	yes	complemented
242	#2952	C₂×C₃₁	1	4	yes	complemented
243	#2951	C₂×C₃₁	1	4	yes	complemented
244	#2950	C₂×C₃₁	1	2	yes	complemented
245	#2949	C₂×C₃₁	1	4	yes	complemented
246	#2948	C₂×C₃₁	1	2	yes	complemented
247	#2943	C₂×C₃₁	1	4	yes	complemented
248	C₄×#135	C₂×C₃₁	1	4	yes	complemented
249	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#2896	C₂×C₃₁	1	2	yes	complemented
251	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	8	no	—
256	C₂²×C₃₁	—	4	4	no	—
257	C₂²×C₃₁	—	4	8	no	—
258	C₂²×C₃₁	—	8	8	no	—
259	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
260	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
261	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
262	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
263	C₄×C₃₁	—	2	8	no	—
264	C₄×C₃₁	—	2	8	no	—
265	C₄×C₃₁	—	4	8	no	—
266	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
267	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
268	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
269	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
270	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
271	C₂²×#41	C₂²×C₃₁	1	2	yes	—
272	C₂×#135	C₄×C₃₁	1	16	yes	complemented
273	C₂×#135	C₂²×C₃₁	1	2	yes	—
274	C₂×#135	C₂²×C₃₁	1	2	yes	—
275	C₂×#135	C₂²×C₃₁	1	2	yes	—
276	C₂×#135	C₂²×C₃₁	1	2	yes	—
277	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
278	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
279	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
281	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
282	#520	—	2	4	no	—
283	#520	C₂²×C₃₁	1	8	yes	complemented
284	#519	C₂²×C₃₁	1	4	yes	—
285	#518	—	2	4	no	—
286	#518	C₂²×C₃₁	1	4	yes	—
287	C₄×#41	—	2	4	no	—
288	C₄×#41	C₂²×C₃₁	1	8	yes	complemented
289	C₂×#117	—	2	4	no	—
290	C₂×#117	C₂²×C₃₁	1	2	yes	—
291	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
292	#442	C₂²×C₃₁	1	8	yes	complemented
293	#441	C₂²×C₃₁	1	8	yes	complemented
294	#440	C₂²×C₃₁	1	16	yes	complemented
295	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
296	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
297	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
298	#427	C₂²×C₃₁	1	2	yes	—
299	#427	C₂²×C₃₁	1	2	yes	—
300	#427	C₂²×C₃₁	1	2	yes	complemented
301	#427	C₂²×C₃₁	1	2	yes	complemented
302	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
303	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
304	#419	C₂²×C₃₁	1	4	yes	—
305	#418	C₂²×C₃₁	1	4	yes	—
306	#417	C₂²×C₃₁	1	8	yes	complemented
307	#416	C₂²×C₃₁	1	8	yes	complemented
308	#405	C₂²×C₃₁	1	2	yes	—
309	#405	C₂²×C₃₁	1	2	yes	—
310	#405	C₂²×C₃₁	1	2	yes	complemented
311	#405	C₂²×C₃₁	1	2	yes	complemented
312	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
313	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
314	C₂×C₃₁	—	2	2	no	—
315	C₂×C₃₁	—	2	2	no	—
316	C₂×C₃₁	—	2	2	no	—
317	C₂×C₃₁	—	4	8	no	—
318	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
319	C₂×C₃₁	#3584	1	2	yes	—
320	C₂×#41	—	2	2	no	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	8	no	—
323	C₂×#41	C₂×C₄×C₃₁	1	16	yes	complemented
324	C₂×#41	C₂³×C₃₁	1	4	yes	—
325	C₂×#41	C₂³×C₃₁	1	4	yes	—
326	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
327	C₂²×D₄	—	2	2	no	—
328	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
329	C₂³×C₄	—	2	2	no	—
330	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
331	#135	C₂×C₄×C₃₁	1	16	yes	complemented
332	#135	C₂×C₄×C₃₁	1	16	yes	complemented
333	#135	C₂×C₄×C₃₁	1	16	yes	complemented
334	#135	C₂×C₄×C₃₁	1	16	yes	complemented
335	C₂×Q₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
336	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
337	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
338	C₂×SD₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
339	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
340	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
341	C₂×D₈	C₂×C₄×C₃₁	1	8	yes	complemented
342	C₂×D₈	C₂³×C₃₁	1	2	yes	—
343	C₂×D₈	C₂³×C₃₁	1	2	yes	—
344	C₂×M₁₆	—	2	4	no	—
345	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
346	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
347	C₂²×C₈	C₂×C₄×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	1	yes	characteristic
352	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
354	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
355	C₄×Q₈	—	2	8	no	—
356	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
357	C₄×D₄	—	2	16	no	—
358	C₄×D₄	—	2	8	no	—
359	C₄×D₄	—	2	16	no	—
360	C₄×D₄	C₂³×C₃₁	1	16	yes	—
361	#117	—	2	2	no	—
362	#117	—	2	2	no	—
363	#117	—	2	8	no	—
364	#117	—	2	8	no	—
365	#117	—	2	8	no	—
366	#117	D₄×C₃₁	1	2	yes	—
367	#117	D₄×C₃₁	1	2	yes	—
368	#117	C₂³×C₃₁	1	4	yes	—
369	#117	C₂³×C₃₁	1	4	yes	—
370	#117	C₂³×C₃₁	1	2	yes	—
371	#117	C₂³×C₃₁	1	2	yes	—
372	C₂×C₄⋊C₄	—	2	8	no	—
373	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	—	2	8	no	—
377	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
378	C₂×C₄²	—	2	2	no	—
379	C₂×C₄²	—	2	8	no	—
380	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
381	#107	C₂³×C₃₁	1	4	yes	—
382	#107	C₂³×C₃₁	1	4	yes	—
383	#106	C₂³×C₃₁	1	4	yes	—
384	#106	C₂³×C₃₁	1	4	yes	—
385	#103	C₂³×C₃₁	1	16	yes	—
386	#102	C₂³×C₃₁	1	16	yes	—
387	#97	C₂³×C₃₁	1	4	yes	—
388	#97	C₂³×C₃₁	1	4	yes	—
389	C₄×C₈	C₂³×C₃₁	1	4	yes	—
390	C₄×C₈	C₂³×C₃₁	1	4	yes	—
391	C₃₁	#60167	1	1	yes	characteristic, direct product factor, Sylow
392	C₂⁴	—	2	2	no	—
393	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
394	#41	—	2	16	no	—
395	#41	—	2	16	no	—
396	#41	—	2	8	no	—
397	#41	—	2	8	no	—
398	#41	C₂²×C₄×C₃₁	1	16	yes	—
399	#41	C₂²×C₄×C₃₁	1	16	yes	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	4	no	—
402	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
403	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
404	C₂×D₄	—	2	16	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
410	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	16	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	16	no	—
417	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
418	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	#41×C₃₁	1	4	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
425	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
426	Q₁₆	C₂²×C₄×C₃₁	1	8	yes	—
427	Q₁₆	C₂²×C₄×C₃₁	1	8	yes	—
428	SD₁₆	C₂²×C₄×C₃₁	1	16	yes	—
429	SD₁₆	C₂²×C₄×C₃₁	1	16	yes	—
430	D₈	C₂²×C₄×C₃₁	1	8	yes	—
431	D₈	C₂²×C₄×C₃₁	1	8	yes	—
432	M₁₆	—	2	4	no	—
433	M₁₆	—	2	4	no	—
434	C₂×C₈	—	2	4	no	—
435	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
436	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
437	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
438	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
439	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
440	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
441	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
442	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
443	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
444	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
445	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	16	no	—
448	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
449	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
450	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	16	no	—
454	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
455	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
456	C₄²	—	2	4	no	—
457	C₄²	—	2	16	no	—
458	C₄²	—	2	4	no	—
459	C₄²	C₂×D₄×C₃₁	1	4	yes	—
460	C₄²	C₂⁴×C₃₁	1	4	yes	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	8	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	8	4	no	—
465	C₂³	C₄×D₄×C₃₁	1	4	yes	—
466	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
467	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
468	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	Q₈	—	2	8	no	—
470	Q₈	—	2	4	no	—
471	Q₈	C₂³×C₄×C₃₁	1	8	yes	—
472	D₄	—	2	16	no	—
473	D₄	—	2	16	no	—
474	D₄	—	2	8	no	—
475	D₄	—	2	4	no	—
476	D₄	C₂³×C₄×C₃₁	1	8	yes	—
477	C₂×C₄	—	2	8	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	16	no	—
480	C₂×C₄	—	2	16	no	—
481	C₂×C₄	—	2	8	no	—
482	C₂×C₄	—	4	4	no	—
483	C₂×C₄	—	8	16	no	—
484	C₂×C₄	—	8	4	no	—
485	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
486	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
487	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
488	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
489	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
490	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
492	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
496	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
497	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
498	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
499	C₈	—	2	4	no	—
500	C₈	C₂³×C₄×C₃₁	1	2	yes	—
501	C₈	C₂³×C₄×C₃₁	1	2	yes	—
502	C₈	C₂³×C₄×C₃₁	1	2	yes	—
503	C₈	C₂³×C₄×C₃₁	1	2	yes	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	2	8	no	—
508	C₂²	—	4	4	no	—
509	C₂²	—	4	8	no	—
510	C₂²	—	8	8	no	—
511	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
512	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
513	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
514	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
515	C₄	—	2	8	no	—
516	C₄	—	2	8	no	—
517	C₄	—	4	8	no	—
518	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
519	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
520	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
521	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
522	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
523	C₂	—	2	2	no	—
524	C₂	—	2	2	no	—
525	C₂	—	2	2	no	—
526	C₂	—	4	8	no	—
527	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
528	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₂²×C₃₁×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3428	C₂	semidirect
3. C₃₁×#2952	C₂	semidirect
4. C₃₁×#2951	C₂	semidirect
5. C₃₁×#2950	C₂	semidirect
6. C₃₁×#2949	C₂	semidirect
7. C₃₁×#2948	C₂	semidirect
8. C₃₁×#2943	C₂	semidirect
9. C₄×C₃₁×#135	C₂	semidirect
10. C₂×C₃₁×#427	C₂	semidirect
11. C₃₁×#2896	C₂	semidirect
12. C₂×C₃₁×#405	C₂	semidirect
14. C₂×C₃₁×#135	C₄	semidirect
25. C₃₁×#520	C₂²	semidirect
30. C₄×#41×C₃₁	C₂²	semidirect
34. C₃₁×#442	C₂²	semidirect
35. C₃₁×#441	C₂²	semidirect
36. C₃₁×#440	C₂²	semidirect
37. C₄×Q₁₆×C₃₁	C₂²	semidirect
38. C₄×SD₁₆×C₃₁	C₂²	semidirect
39. C₄×D₈×C₃₁	C₂²	semidirect
42. C₃₁×#427	C₂²	semidirect
43. C₃₁×#427	C₂²	semidirect
44. C₂×C₃₁×#107	C₂²	semidirect
45. C₂×C₃₁×#106	C₂²	semidirect
48. C₃₁×#417	C₂²	semidirect
49. C₃₁×#416	C₂²	semidirect
52. C₃₁×#405	C₂²	semidirect
53. C₃₁×#405	C₂²	semidirect
54. C₂×C₃₁×#97	C₂²	semidirect
55. C₂×C₄×C₈×C₃₁	C₂²	semidirect
59. C₂×#41×C₃₁	C₂×C₄	semidirect
67. C₃₁×#135	C₂×C₄	semidirect
68. C₃₁×#135	C₂×C₄	semidirect
69. C₃₁×#135	C₂×C₄	semidirect
70. C₃₁×#135	C₂×C₄	semidirect
71. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
74. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
77. C₂×D₈×C₃₁	C₂×C₄	semidirect
83. C₂²×C₈×C₃₁	C₂×C₄	semidirect
196. #60167	C₃₁	direct (N × H)
241. #3428	C₂×C₃₁	semidirect
242. #2952	C₂×C₃₁	semidirect
243. #2951	C₂×C₃₁	semidirect
244. #2950	C₂×C₃₁	semidirect
245. #2949	C₂×C₃₁	semidirect
246. #2948	C₂×C₃₁	semidirect
247. #2943	C₂×C₃₁	semidirect
248. C₄×#135	C₂×C₃₁	semidirect
249. C₂×#427	C₂×C₃₁	semidirect
250. #2896	C₂×C₃₁	semidirect
251. C₂×#405	C₂×C₃₁	semidirect
272. C₂×#135	C₄×C₃₁	semidirect
283. #520	C₂²×C₃₁	semidirect
288. C₄×#41	C₂²×C₃₁	semidirect
292. #442	C₂²×C₃₁	semidirect
293. #441	C₂²×C₃₁	semidirect
294. #440	C₂²×C₃₁	semidirect
295. C₄×Q₁₆	C₂²×C₃₁	semidirect
296. C₄×SD₁₆	C₂²×C₃₁	semidirect
297. C₄×D₈	C₂²×C₃₁	semidirect
300. #427	C₂²×C₃₁	semidirect
301. #427	C₂²×C₃₁	semidirect
302. C₂×#107	C₂²×C₃₁	semidirect
303. C₂×#106	C₂²×C₃₁	semidirect
306. #417	C₂²×C₃₁	semidirect
307. #416	C₂²×C₃₁	semidirect
310. #405	C₂²×C₃₁	semidirect
311. #405	C₂²×C₃₁	semidirect
312. C₂×#97	C₂²×C₃₁	semidirect
313. C₂×C₄×C₈	C₂²×C₃₁	semidirect
323. C₂×#41	C₂×C₄×C₃₁	semidirect
331. #135	C₂×C₄×C₃₁	semidirect
332. #135	C₂×C₄×C₃₁	semidirect
333. #135	C₂×C₄×C₃₁	semidirect
334. #135	C₂×C₄×C₃₁	semidirect
335. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
338. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
341. C₂×D₈	C₂×C₄×C₃₁	semidirect
347. C₂²×C₈	C₂×C₄×C₃₁	semidirect
391. C₃₁	#60167	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
C₃₁×#3428	2
C₃₁×#2952	2
C₃₁×#2951	2
C₃₁×#2950	2
C₃₁×#2949	2
C₃₁×#2948	2
C₃₁×#2943	2
C₄×C₃₁×#135	2
C₂×C₃₁×#427	2
C₃₁×#2896	2
C₂×C₃₁×#405	2
#60167	31

Subgroups — C31×#60167 #21516926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60167 #21516926