Subgroups — C₃₁×#60169 #21516928

Order 7936.

2666 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#137	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#519	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2950	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2949	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2947	C₂	1	8	yes	maximal, complemented
7	C₄×C₃₁×#137	C₂	1	8	yes	maximal, complemented
8	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2896	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#137	C₄	1	16	yes	complemented
15	C₂×C₃₁×#137	C₂²	1	4	yes	—
16	C₂×C₃₁×#137	C₂²	1	4	yes	—
17	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#520	—	2	2	no	—
22	C₃₁×#520	C₂²	1	4	yes	—
23	C₃₁×#519	—	2	2	no	—
24	C₃₁×#519	—	2	2	no	—
25	C₃₁×#519	C₂²	1	4	yes	—
26	C₃₁×#519	C₂²	1	2	yes	—
27	C₃₁×#519	C₂²	1	4	yes	complemented
28	C₃₁×#518	—	2	2	no	—
29	C₃₁×#518	C₂²	1	2	yes	—
30	C₄×#41×C₃₁	—	2	2	no	—
31	C₄×#41×C₃₁	C₂²	1	4	yes	—
32	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
33	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
34	C₂×C₃₁×#117	—	2	2	no	—
35	C₂×C₃₁×#117	—	2	2	no	—
36	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
39	C₃₁×#441	C₂²	1	16	yes	complemented
40	C₃₁×#440	C₂²	1	16	yes	complemented
41	C₄×Q₁₆×C₃₁	C₂²	1	16	yes	complemented
42	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
43	C₃₁×#428	C₂²	1	4	yes	complemented
44	C₃₁×#428	C₂²	1	4	yes	complemented
45	C₃₁×#427	C₂²	1	4	yes	—
46	C₃₁×#419	C₂²	1	4	yes	—
47	C₃₁×#419	C₂²	1	2	yes	—
48	C₃₁×#419	C₂²	1	4	yes	complemented
49	C₃₁×#418	C₂²	1	2	yes	—
50	C₃₁×#417	C₂²	1	4	yes	—
51	C₃₁×#416	C₂²	1	4	yes	—
52	C₂×C₃₁×#103	C₂²	1	4	yes	complemented
53	C₃₁×#405	C₂²	1	4	yes	—
54	C₃₁×#405	C₂²	1	4	yes	complemented
55	C₄×M₁₆×C₃₁	C₂²	1	4	yes	complemented
56	C₂×#41×C₃₁	—	2	1	no	—
57	C₂×#41×C₃₁	—	2	1	no	—
58	C₂×#41×C₃₁	—	2	4	no	—
59	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
60	C₂×#41×C₃₁	C₂³	1	2	yes	—
61	C₂×#41×C₃₁	C₂³	1	2	yes	—
62	C₂²×Q₈×C₃₁	—	2	1	no	—
63	C₂²×Q₈×C₃₁	—	2	1	no	—
64	C₂²×Q₈×C₃₁	—	2	1	no	—
65	C₂²×Q₈×C₃₁	—	2	4	no	—
66	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
67	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
68	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
69	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
71	C₂²×D₄×C₃₁	—	2	1	no	—
72	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	—	2	1	no	—
74	C₂³×C₄×C₃₁	—	2	1	no	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#137	C₂×C₄	1	32	yes	complemented
77	C₃₁×#137	C₂×C₄	1	32	yes	complemented
78	C₂×Q₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
79	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
80	C₂×SD₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
81	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
82	C₂×M₁₆×C₃₁	—	2	4	no	—
83	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
84	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
85	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
87	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
88	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
89	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
90	C₄×Q₈×C₃₁	—	2	8	no	—
91	C₄×Q₈×C₃₁	—	2	8	no	—
92	C₄×Q₈×C₃₁	—	2	4	no	—
93	C₄×Q₈×C₃₁	—	2	8	no	—
94	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
95	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
96	C₄×D₄×C₃₁	—	2	8	no	—
97	C₄×D₄×C₃₁	—	2	4	no	—
98	C₄×D₄×C₃₁	—	2	8	no	—
99	C₄×D₄×C₃₁	C₂³	1	8	yes	—
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₃₁×#117	—	2	1	no	—
103	C₃₁×#117	—	2	1	no	—
104	C₃₁×#117	—	2	4	no	—
105	C₃₁×#117	—	2	4	no	—
106	C₃₁×#117	—	2	8	no	—
107	C₃₁×#117	—	2	8	no	—
108	C₃₁×#117	—	2	4	no	—
109	C₃₁×#117	D₄	1	2	yes	—
110	C₃₁×#117	D₄	1	2	yes	—
111	C₃₁×#117	C₂³	1	2	yes	—
112	C₃₁×#117	C₂³	1	2	yes	—
113	C₃₁×#117	C₂³	1	2	yes	—
114	C₃₁×#117	C₂³	1	2	yes	—
115	C₃₁×#117	C₂³	1	2	yes	—
116	C₃₁×#117	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
119	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
123	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
125	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
126	C₂×C₄²×C₃₁	—	2	4	no	—
127	C₂×C₄²×C₃₁	—	2	1	no	—
128	C₂×C₄²×C₃₁	—	2	1	no	—
129	C₂×C₄²×C₃₁	—	2	4	no	—
130	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
131	C₃₁×#107	C₂³	1	8	yes	—
132	C₃₁×#106	C₂³	1	8	yes	—
133	C₃₁×#103	C₂³	1	16	yes	—
134	C₃₁×#103	C₂³	1	8	yes	—
135	C₃₁×#102	C₂³	1	8	yes	—
136	C₃₁×#97	C₂³	1	8	yes	—
137	C₄×C₈×C₃₁	C₂³	1	8	yes	—
138	C₂⁴×C₃₁	—	2	1	no	—
139	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
140	#41×C₃₁	—	2	8	no	—
141	#41×C₃₁	—	2	8	no	—
142	#41×C₃₁	—	2	4	no	—
143	#41×C₃₁	—	2	4	no	—
144	#41×C₃₁	C₂²×C₄	1	8	yes	—
145	#41×C₃₁	C₂²×C₄	1	8	yes	—
146	C₂×Q₈×C₃₁	—	2	8	no	—
147	C₂×Q₈×C₃₁	—	2	8	no	—
148	C₂×Q₈×C₃₁	—	2	8	no	—
149	C₂×Q₈×C₃₁	—	2	2	no	—
150	C₂×Q₈×C₃₁	—	2	2	no	—
151	C₂×Q₈×C₃₁	—	2	2	no	—
152	C₂×Q₈×C₃₁	—	2	1	no	—
153	C₂×Q₈×C₃₁	—	2	4	no	—
154	C₂×Q₈×C₃₁	—	2	4	no	—
155	C₂×Q₈×C₃₁	—	2	4	no	—
156	C₂×Q₈×C₃₁	—	2	2	no	—
157	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
158	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
159	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
160	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
161	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
162	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
163	C₂×D₄×C₃₁	—	2	8	no	—
164	C₂×D₄×C₃₁	—	2	2	no	—
165	C₂×D₄×C₃₁	—	2	2	no	—
166	C₂×D₄×C₃₁	—	2	1	no	—
167	C₂×D₄×C₃₁	—	2	2	no	—
168	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
169	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₃₁	—	2	8	no	—
171	C₂²×C₄×C₃₁	—	2	2	no	—
172	C₂²×C₄×C₃₁	—	2	1	no	—
173	C₂²×C₄×C₃₁	—	2	2	no	—
174	C₂²×C₄×C₃₁	—	2	2	no	—
175	C₂²×C₄×C₃₁	—	2	8	no	—
176	C₂²×C₄×C₃₁	—	2	2	no	—
177	C₂²×C₄×C₃₁	—	2	2	no	—
178	C₂²×C₄×C₃₁	—	2	2	no	—
179	C₂²×C₄×C₃₁	—	2	2	no	—
180	C₂²×C₄×C₃₁	—	2	8	no	—
181	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₃₁	#41	1	4	yes	—
187	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
188	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
190	Q₁₆×C₃₁	C₂²×C₄	1	32	yes	—
191	SD₁₆×C₃₁	C₂²×C₄	1	32	yes	—
192	M₁₆×C₃₁	—	2	4	no	—
193	M₁₆×C₃₁	—	2	4	no	—
194	M₁₆×C₃₁	C₂²×C₄	1	8	yes	—
195	M₁₆×C₃₁	C₂²×C₄	1	8	yes	—
196	C₂×C₈×C₃₁	—	2	4	no	—
197	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
198	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
199	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
200	C₄⋊C₄×C₃₁	—	2	2	no	—
201	C₄⋊C₄×C₃₁	—	2	8	no	—
202	C₄⋊C₄×C₃₁	—	2	16	no	—
203	C₄⋊C₄×C₃₁	—	2	2	no	—
204	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
205	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
206	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
207	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
208	C₂²⋊C₄×C₃₁	—	2	2	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	—	2	2	no	—
211	C₂²⋊C₄×C₃₁	—	2	2	no	—
212	C₂²⋊C₄×C₃₁	—	2	8	no	—
213	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
214	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
215	C₄²×C₃₁	—	2	2	no	—
216	C₄²×C₃₁	—	2	8	no	—
217	C₄²×C₃₁	—	2	16	no	—
218	C₄²×C₃₁	—	2	2	no	—
219	C₄²×C₃₁	—	2	2	no	—
220	C₄²×C₃₁	—	2	2	no	—
221	C₄²×C₃₁	C₂×D₄	1	4	yes	—
222	C₄²×C₃₁	C₂⁴	1	4	yes	—
223	#60169	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂³×C₃₁	—	2	2	no	—
225	C₂³×C₃₁	—	2	4	no	—
226	C₂³×C₃₁	—	4	2	no	—
227	C₂³×C₃₁	—	8	2	no	—
228	C₂³×C₃₁	C₄×D₄	1	4	yes	—
229	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
230	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₃₁	—	2	8	no	—
233	Q₈×C₃₁	—	2	8	no	—
234	Q₈×C₃₁	—	2	8	no	—
235	Q₈×C₃₁	—	2	4	no	—
236	Q₈×C₃₁	—	2	4	no	—
237	Q₈×C₃₁	—	2	2	no	—
238	Q₈×C₃₁	C₂³×C₄	1	8	yes	—
239	Q₈×C₃₁	C₂³×C₄	1	4	yes	—
240	D₄×C₃₁	—	2	8	no	—
241	D₄×C₃₁	—	2	8	no	—
242	D₄×C₃₁	—	2	4	no	—
243	D₄×C₃₁	—	2	2	no	—
244	D₄×C₃₁	C₂³×C₄	1	4	yes	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	2	8	no	—
247	C₂×C₄×C₃₁	—	2	8	no	—
248	C₂×C₄×C₃₁	—	2	8	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	2	8	no	—
252	C₂×C₄×C₃₁	—	2	8	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	4	4	no	—
256	C₂×C₄×C₃₁	—	4	4	no	—
257	C₂×C₄×C₃₁	—	8	8	no	—
258	C₂×C₄×C₃₁	—	8	2	no	—
259	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
260	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
261	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
262	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
263	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
264	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
266	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
270	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
271	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
272	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
273	C₈×C₃₁	—	2	4	no	—
274	C₈×C₃₁	C₂³×C₄	1	8	yes	—
275	C₂²×#137	C₂×C₃₁	1	1	yes	characteristic
276	#3428	C₂×C₃₁	1	1	yes	characteristic
277	C₂×#519	C₂×C₃₁	1	1	yes	characteristic, complemented
278	#2950	C₂×C₃₁	1	4	yes	complemented
279	#2949	C₂×C₃₁	1	4	yes	complemented
280	#2947	C₂×C₃₁	1	8	yes	complemented
281	C₄×#137	C₂×C₃₁	1	8	yes	complemented
282	#2911	C₂×C₃₁	1	1	yes	characteristic, complemented
283	#2896	C₂×C₃₁	1	1	yes	characteristic
284	C₂×#419	C₂×C₃₁	1	1	yes	characteristic, complemented
285	#2873	C₂×C₃₁	1	1	yes	characteristic, complemented
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	2	4	no	—
290	C₂²×C₃₁	—	4	4	no	—
291	C₂²×C₃₁	—	4	4	no	—
292	C₂²×C₃₁	—	8	4	no	—
293	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
294	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
295	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
296	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
297	C₄×C₃₁	—	2	8	no	—
298	C₄×C₃₁	—	2	4	no	—
299	C₄×C₃₁	—	2	8	no	—
300	C₄×C₃₁	—	4	4	no	—
301	C₄×C₃₁	—	4	4	no	—
302	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
303	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
304	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
305	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
306	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
307	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
308	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×#137	C₄×C₃₁	1	16	yes	complemented
310	C₂×#137	C₂²×C₃₁	1	4	yes	—
311	C₂×#137	C₂²×C₃₁	1	4	yes	—
312	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
313	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
314	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
315	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
316	#520	—	2	2	no	—
317	#520	C₂²×C₃₁	1	4	yes	—
318	#519	—	2	2	no	—
319	#519	—	2	2	no	—
320	#519	C₂²×C₃₁	1	4	yes	—
321	#519	C₂²×C₃₁	1	2	yes	—
322	#519	C₂²×C₃₁	1	4	yes	complemented
323	#518	—	2	2	no	—
324	#518	C₂²×C₃₁	1	2	yes	—
325	C₄×#41	—	2	2	no	—
326	C₄×#41	C₂²×C₃₁	1	4	yes	—
327	C₂×C₄×Q₈	—	2	2	no	—
328	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
329	C₂×#117	—	2	2	no	—
330	C₂×#117	—	2	2	no	—
331	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
334	#441	C₂²×C₃₁	1	16	yes	complemented
335	#440	C₂²×C₃₁	1	16	yes	complemented
336	C₄×Q₁₆	C₂²×C₃₁	1	16	yes	complemented
337	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
338	#428	C₂²×C₃₁	1	4	yes	complemented
339	#428	C₂²×C₃₁	1	4	yes	complemented
340	#427	C₂²×C₃₁	1	4	yes	—
341	#419	C₂²×C₃₁	1	4	yes	—
342	#419	C₂²×C₃₁	1	2	yes	—
343	#419	C₂²×C₃₁	1	4	yes	complemented
344	#418	C₂²×C₃₁	1	2	yes	—
345	#417	C₂²×C₃₁	1	4	yes	—
346	#416	C₂²×C₃₁	1	4	yes	—
347	C₂×#103	C₂²×C₃₁	1	4	yes	complemented
348	#405	C₂²×C₃₁	1	4	yes	—
349	#405	C₂²×C₃₁	1	4	yes	complemented
350	C₄×M₁₆	C₂²×C₃₁	1	4	yes	complemented
351	C₂×C₃₁	—	2	2	no	—
352	C₂×C₃₁	—	2	2	no	—
353	C₂×C₃₁	—	2	2	no	—
354	C₂×C₃₁	—	4	4	no	—
355	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
356	C₂×C₃₁	#3586	1	2	yes	—
357	C₂×#41	—	2	1	no	—
358	C₂×#41	—	2	1	no	—
359	C₂×#41	—	2	4	no	—
360	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
361	C₂×#41	C₂³×C₃₁	1	2	yes	—
362	C₂×#41	C₂³×C₃₁	1	2	yes	—
363	C₂²×Q₈	—	2	1	no	—
364	C₂²×Q₈	—	2	1	no	—
365	C₂²×Q₈	—	2	1	no	—
366	C₂²×Q₈	—	2	4	no	—
367	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
368	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
369	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
370	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
372	C₂²×D₄	—	2	1	no	—
373	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
374	C₂³×C₄	—	2	1	no	—
375	C₂³×C₄	—	2	1	no	—
376	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
377	#137	C₂×C₄×C₃₁	1	32	yes	complemented
378	#137	C₂×C₄×C₃₁	1	32	yes	complemented
379	C₂×Q₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
380	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
381	C₂×SD₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
382	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
383	C₂×M₁₆	—	2	4	no	—
384	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
385	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
386	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
387	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
388	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
389	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
390	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
391	C₄×Q₈	—	2	8	no	—
392	C₄×Q₈	—	2	8	no	—
393	C₄×Q₈	—	2	4	no	—
394	C₄×Q₈	—	2	8	no	—
395	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
396	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
397	C₄×D₄	—	2	8	no	—
398	C₄×D₄	—	2	4	no	—
399	C₄×D₄	—	2	8	no	—
400	C₄×D₄	C₂³×C₃₁	1	8	yes	—
401	#117	—	2	1	no	—
402	#117	—	2	1	no	—
403	#117	—	2	1	no	—
404	#117	—	2	1	no	—
405	#117	—	2	4	no	—
406	#117	—	2	4	no	—
407	#117	—	2	8	no	—
408	#117	—	2	8	no	—
409	#117	—	2	4	no	—
410	#117	D₄×C₃₁	1	2	yes	—
411	#117	D₄×C₃₁	1	2	yes	—
412	#117	C₂³×C₃₁	1	2	yes	—
413	#117	C₂³×C₃₁	1	2	yes	—
414	#117	C₂³×C₃₁	1	2	yes	—
415	#117	C₂³×C₃₁	1	2	yes	—
416	#117	C₂³×C₃₁	1	2	yes	—
417	#117	C₂³×C₃₁	1	2	yes	—
418	C₂×C₄⋊C₄	—	2	4	no	—
419	C₂×C₄⋊C₄	—	2	4	no	—
420	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₂²⋊C₄	—	2	4	no	—
426	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
427	C₂×C₄²	—	2	4	no	—
428	C₂×C₄²	—	2	1	no	—
429	C₂×C₄²	—	2	1	no	—
430	C₂×C₄²	—	2	4	no	—
431	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
432	#107	C₂³×C₃₁	1	8	yes	—
433	#106	C₂³×C₃₁	1	8	yes	—
434	#103	C₂³×C₃₁	1	16	yes	—
435	#103	C₂³×C₃₁	1	8	yes	—
436	#102	C₂³×C₃₁	1	8	yes	—
437	#97	C₂³×C₃₁	1	8	yes	—
438	C₄×C₈	C₂³×C₃₁	1	8	yes	—
439	C₃₁	#60169	1	1	yes	characteristic, direct product factor, Sylow
440	C₂⁴	—	2	1	no	—
441	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
442	#41	—	2	8	no	—
443	#41	—	2	8	no	—
444	#41	—	2	4	no	—
445	#41	—	2	4	no	—
446	#41	C₂²×C₄×C₃₁	1	8	yes	—
447	#41	C₂²×C₄×C₃₁	1	8	yes	—
448	C₂×Q₈	—	2	8	no	—
449	C₂×Q₈	—	2	8	no	—
450	C₂×Q₈	—	2	8	no	—
451	C₂×Q₈	—	2	2	no	—
452	C₂×Q₈	—	2	2	no	—
453	C₂×Q₈	—	2	2	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	4	no	—
456	C₂×Q₈	—	2	4	no	—
457	C₂×Q₈	—	2	4	no	—
458	C₂×Q₈	—	2	2	no	—
459	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
460	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
461	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
462	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
463	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
464	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
465	C₂×D₄	—	2	8	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
471	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
472	C₂²×C₄	—	2	8	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	8	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	8	no	—
483	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
484	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
485	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₂²×C₄	#41×C₃₁	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
492	Q₁₆	C₂²×C₄×C₃₁	1	32	yes	—
493	SD₁₆	C₂²×C₄×C₃₁	1	32	yes	—
494	M₁₆	—	2	4	no	—
495	M₁₆	—	2	4	no	—
496	M₁₆	C₂²×C₄×C₃₁	1	8	yes	—
497	M₁₆	C₂²×C₄×C₃₁	1	8	yes	—
498	C₂×C₈	—	2	4	no	—
499	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
500	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
501	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	—	2	16	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
507	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
508	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
509	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	8	no	—
515	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
516	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
517	C₄²	—	2	2	no	—
518	C₄²	—	2	8	no	—
519	C₄²	—	2	16	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	2	no	—
522	C₄²	—	2	2	no	—
523	C₄²	C₂×D₄×C₃₁	1	4	yes	—
524	C₄²	C₂⁴×C₃₁	1	4	yes	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	4	2	no	—
528	C₂³	—	8	2	no	—
529	C₂³	C₄×D₄×C₃₁	1	4	yes	—
530	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
531	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
532	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
533	Q₈	—	2	8	no	—
534	Q₈	—	2	8	no	—
535	Q₈	—	2	8	no	—
536	Q₈	—	2	4	no	—
537	Q₈	—	2	4	no	—
538	Q₈	—	2	2	no	—
539	Q₈	C₂³×C₄×C₃₁	1	8	yes	—
540	Q₈	C₂³×C₄×C₃₁	1	4	yes	—
541	D₄	—	2	8	no	—
542	D₄	—	2	8	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	2	no	—
545	D₄	C₂³×C₄×C₃₁	1	4	yes	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	8	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	8	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	8	8	no	—
559	C₂×C₄	—	8	2	no	—
560	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
561	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
562	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
563	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
564	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
565	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
567	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
568	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
569	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
570	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
571	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
572	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
573	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
574	C₈	—	2	4	no	—
575	C₈	C₂³×C₄×C₃₁	1	8	yes	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	8	4	no	—
583	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
584	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
585	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
586	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
587	C₄	—	2	4	no	—
588	C₄	—	2	8	no	—
589	C₄	—	2	8	no	—
590	C₄	—	4	4	no	—
591	C₄	—	4	4	no	—
592	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
593	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
594	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
595	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
596	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
597	C₂	—	2	2	no	—
598	C₂	—	2	2	no	—
599	C₂	—	2	2	no	—
600	C₂	—	4	4	no	—
601	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
602	C₂	C₃₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#519	C₂	semidirect
4. C₃₁×#2950	C₂	semidirect
5. C₃₁×#2949	C₂	semidirect
6. C₃₁×#2947	C₂	semidirect
7. C₄×C₃₁×#137	C₂	semidirect
8. C₃₁×#2911	C₂	semidirect
10. C₂×C₃₁×#419	C₂	semidirect
11. C₃₁×#2873	C₂	semidirect
14. C₂×C₃₁×#137	C₄	semidirect
27. C₃₁×#519	C₂²	semidirect
33. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
39. C₃₁×#441	C₂²	semidirect
40. C₃₁×#440	C₂²	semidirect
41. C₄×Q₁₆×C₃₁	C₂²	semidirect
42. C₄×SD₁₆×C₃₁	C₂²	semidirect
43. C₃₁×#428	C₂²	semidirect
44. C₃₁×#428	C₂²	semidirect
48. C₃₁×#419	C₂²	semidirect
52. C₂×C₃₁×#103	C₂²	semidirect
54. C₃₁×#405	C₂²	semidirect
55. C₄×M₁₆×C₃₁	C₂²	semidirect
66. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
76. C₃₁×#137	C₂×C₄	semidirect
77. C₃₁×#137	C₂×C₄	semidirect
78. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
80. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
83. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
223. #60169	C₃₁	direct (N × H)
277. C₂×#519	C₂×C₃₁	semidirect
278. #2950	C₂×C₃₁	semidirect
279. #2949	C₂×C₃₁	semidirect
280. #2947	C₂×C₃₁	semidirect
281. C₄×#137	C₂×C₃₁	semidirect
282. #2911	C₂×C₃₁	semidirect
284. C₂×#419	C₂×C₃₁	semidirect
285. #2873	C₂×C₃₁	semidirect
309. C₂×#137	C₄×C₃₁	semidirect
322. #519	C₂²×C₃₁	semidirect
328. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
334. #441	C₂²×C₃₁	semidirect
335. #440	C₂²×C₃₁	semidirect
336. C₄×Q₁₆	C₂²×C₃₁	semidirect
337. C₄×SD₁₆	C₂²×C₃₁	semidirect
338. #428	C₂²×C₃₁	semidirect
339. #428	C₂²×C₃₁	semidirect
343. #419	C₂²×C₃₁	semidirect
347. C₂×#103	C₂²×C₃₁	semidirect
349. #405	C₂²×C₃₁	semidirect
350. C₄×M₁₆	C₂²×C₃₁	semidirect
367. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
377. #137	C₂×C₄×C₃₁	semidirect
378. #137	C₂×C₄×C₃₁	semidirect
379. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
381. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
384. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
439. C₃₁	#60169	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×#137	2
C₃₁×#3428	2
C₂×C₃₁×#519	2
C₃₁×#2950	2
C₃₁×#2949	2
C₃₁×#2947	2
C₄×C₃₁×#137	2
C₃₁×#2911	2
C₃₁×#2896	2
C₂×C₃₁×#419	2
C₃₁×#2873	2
#60169	31

Subgroups — C31×#60169 #21516928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60169 #21516928