Subgroups — C₃₁×#60470 #21517229

Order 7936.

2604 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3586	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3585	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#577	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3573	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3069	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3068	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3067	C₂	1	6	yes	maximal, complemented
8	C₂×C₃₁×#471	C₂	1	6	yes	maximal, complemented
9	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2889	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2888	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#585	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
15	C₃₁×#578	—	2	4	no	—
16	C₃₁×#577	—	2	4	no	—
17	C₃₁×#577	—	2	4	no	—
18	C₃₁×#577	—	2	4	no	—
19	C₃₁×#577	—	2	4	no	—
20	C₃₁×#576	—	2	4	no	—
21	C₃₁×#575	C₂²	1	6	yes	—
22	C₃₁×#575	C₂²	1	6	yes	complemented
23	C₂×C₃₁×#137	C₂²	1	6	yes	complemented
24	C₂×C₃₁×#137	C₂²	1	3	yes	complemented
25	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	6	yes	—
26	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	3	yes	complemented
27	C₂×C₃₁×#135	C₂²	1	3	yes	complemented
28	C₂²×SD₁₆×C₃₁	C₂²	1	3	yes	complemented
29	C₃₁×#568	—	2	4	no	—
30	C₃₁×#568	C₂²	1	1	yes	characteristic
31	C₃₁×#568	C₂²	1	1	yes	characteristic
32	C₃₁×#568	C₂²	1	1	yes	characteristic
33	C₃₁×#568	C₂²	1	6	yes	complemented
34	C₂×C₃₁×#131	—	2	4	no	—
35	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#131	C₂²	1	6	yes	complemented
37	C₃₁×#473	C₂²	1	12	yes	complemented
38	C₃₁×#472	C₂²	1	12	yes	complemented
39	C₃₁×#471	C₂²	1	12	yes	complemented
40	C₃₁×#471	C₂²	1	24	yes	complemented
41	C₃₁×#471	C₂²	1	4	yes	complemented
42	C₃₁×#430	C₂²	1	6	yes	complemented
43	C₂×C₃₁×#108	C₂²	1	6	yes	complemented
44	C₃₁×#413	C₂²	1	6	yes	complemented
45	C₃₁×#413	C₂²	1	6	yes	complemented
46	C₂×C₃₁×#101	C₂²	1	6	yes	complemented
47	C₂×C₃₁×#100	C₂²	1	6	yes	complemented
48	C₃₁×2₋¹⁺⁴	—	2	1	no	—
49	C₃₁×2₋¹⁺⁴	—	2	4	no	—
50	C₃₁×2₋¹⁺⁴	—	2	4	no	—
51	C₃₁×2₋¹⁺⁴	D₄	1	4	yes	—
52	C₃₁×H(5,𝔽₂)	—	2	3	no	—
53	C₃₁×H(5,𝔽₂)	—	2	4	no	—
54	C₃₁×H(5,𝔽₂)	—	2	4	no	—
55	C₃₁×H(5,𝔽₂)	D₄	1	4	yes	—
56	C₂×#41×C₃₁	—	2	4	no	—
57	C₂×#41×C₃₁	—	2	1	no	—
58	C₂×#41×C₃₁	—	2	3	no	—
59	C₂×#41×C₃₁	C₂³	1	6	yes	—
60	C₂×#41×C₃₁	C₂³	1	3	yes	—
61	C₂×#41×C₃₁	C₂³	1	3	yes	—
62	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₃₁	—	2	1	no	—
64	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
65	C₂²×D₄×C₃₁	—	2	3	no	—
66	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
67	C₃₁×#137	—	2	12	no	—
68	C₃₁×#137	—	2	6	no	—
69	C₃₁×#137	—	2	6	no	—
70	C₃₁×#137	—	2	12	no	—
71	C₃₁×#137	C₂³	1	12	yes	—
72	C₃₁×#137	C₂³	1	4	yes	—
73	C₃₁×AGL(1,ℤ/8)	—	2	12	no	—
74	C₃₁×AGL(1,ℤ/8)	—	2	6	no	—
75	C₃₁×AGL(1,ℤ/8)	—	2	6	no	—
76	C₃₁×AGL(1,ℤ/8)	—	2	12	no	—
77	C₃₁×AGL(1,ℤ/8)	C₂³	1	12	yes	—
78	C₃₁×AGL(1,ℤ/8)	C₂³	1	4	yes	—
79	C₃₁×#135	—	2	12	no	—
80	C₃₁×#135	—	2	12	no	—
81	C₃₁×#135	—	2	6	no	—
82	C₃₁×#135	—	2	6	no	—
83	C₂×Q₁₆×C₃₁	—	2	6	no	—
84	C₂×Q₁₆×C₃₁	C₂³	1	3	yes	—
85	C₂×SD₁₆×C₃₁	—	2	6	no	—
86	C₂×SD₁₆×C₃₁	—	2	6	no	—
87	C₂×SD₁₆×C₃₁	—	2	6	no	—
88	C₂×SD₁₆×C₃₁	—	2	6	no	—
89	C₂×SD₁₆×C₃₁	C₂³	1	3	yes	—
90	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
91	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
92	C₂×D₈×C₃₁	—	2	6	no	—
93	C₂×D₈×C₃₁	C₂³	1	3	yes	—
94	C₃₁×#131	—	2	4	no	—
95	C₃₁×#131	—	2	4	no	—
96	C₃₁×#131	—	2	4	no	—
97	C₃₁×#131	—	2	12	no	—
98	C₃₁×#131	—	2	6	no	—
99	C₃₁×#131	—	2	6	no	—
100	C₃₁×#131	D₄	1	4	yes	—
101	C₃₁×#131	D₄	1	4	yes	—
102	C₃₁×#131	C₂³	1	4	yes	—
103	C₃₁×#131	C₂³	1	12	yes	—
104	C₂×M₁₆×C₃₁	—	2	12	no	—
105	C₂×M₁₆×C₃₁	—	2	6	no	—
106	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
107	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
108	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
109	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
110	C₂²×C₈×C₃₁	—	2	6	no	—
111	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
112	C₃₁×#108	C₂³	1	4	yes	—
113	C₃₁×#108	C₂³	1	12	yes	—
114	C₃₁×#101	C₂³	1	12	yes	—
115	C₃₁×#101	C₂³	1	4	yes	—
116	C₃₁×#100	C₂³	1	12	yes	—
117	C₃₁×#100	C₂³	1	4	yes	—
118	C₂⁴×C₃₁	—	2	3	no	—
119	#41×C₃₁	—	2	4	no	—
120	#41×C₃₁	—	2	12	no	—
121	#41×C₃₁	—	2	6	no	—
122	#41×C₃₁	—	2	6	no	—
123	#41×C₃₁	—	2	6	no	—
124	#41×C₃₁	—	2	6	no	—
125	#41×C₃₁	—	2	6	no	—
126	#41×C₃₁	—	2	6	no	—
127	#41×C₃₁	—	4	4	no	—
128	#41×C₃₁	—	4	12	no	—
129	#41×C₃₁	C₂×D₄	1	12	yes	—
130	#41×C₃₁	C₂×D₄	1	4	yes	—
131	#41×C₃₁	C₂×D₄	1	4	yes	—
132	#41×C₃₁	C₂⁴	1	4	yes	—
133	C₂×Q₈×C₃₁	—	2	4	no	—
134	C₂×Q₈×C₃₁	—	2	3	no	—
135	C₂×Q₈×C₃₁	—	2	6	no	—
136	C₂×Q₈×C₃₁	—	2	6	no	—
137	C₂×Q₈×C₃₁	—	4	4	no	—
138	C₂×Q₈×C₃₁	C₂×D₄	1	6	yes	—
139	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
140	C₂×Q₈×C₃₁	C₂⁴	1	3	yes	—
141	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
142	C₂×D₄×C₃₁	—	2	6	no	—
143	C₂×D₄×C₃₁	—	2	12	no	—
144	C₂×D₄×C₃₁	—	2	3	no	—
145	C₂×D₄×C₃₁	—	2	3	no	—
146	C₂×D₄×C₃₁	—	2	6	no	—
147	C₂×D₄×C₃₁	—	2	6	no	—
148	C₂×D₄×C₃₁	—	2	3	no	—
149	C₂×D₄×C₃₁	—	4	12	no	—
150	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
151	C₂×D₄×C₃₁	#41	1	3	yes	—
152	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
153	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
154	C₂²×C₄×C₃₁	—	2	6	no	—
155	C₂²×C₄×C₃₁	—	2	3	no	—
156	C₂²×C₄×C₃₁	—	2	3	no	—
157	C₂²×C₄×C₃₁	—	2	3	no	—
158	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
159	Q₁₆×C₃₁	—	2	12	no	—
160	Q₁₆×C₃₁	—	2	4	no	—
161	Q₁₆×C₃₁	—	2	6	no	—
162	SD₁₆×C₃₁	—	2	12	no	—
163	SD₁₆×C₃₁	—	2	4	no	—
164	SD₁₆×C₃₁	—	2	12	no	—
165	SD₁₆×C₃₁	—	2	4	no	—
166	SD₁₆×C₃₁	—	2	3	no	—
167	SD₁₆×C₃₁	—	2	3	no	—
168	SD₁₆×C₃₁	—	2	12	no	—
169	SD₁₆×C₃₁	—	2	1	no	—
170	SD₁₆×C₃₁	—	2	1	no	—
171	D₈×C₃₁	—	2	12	no	—
172	D₈×C₃₁	—	2	4	no	—
173	D₈×C₃₁	—	2	6	no	—
174	M₁₆×C₃₁	—	2	6	no	—
175	M₁₆×C₃₁	—	2	6	no	—
176	M₁₆×C₃₁	—	2	6	no	—
177	M₁₆×C₃₁	—	2	12	no	—
178	M₁₆×C₃₁	—	2	4	no	—
179	M₁₆×C₃₁	C₂×D₄	1	12	yes	—
180	M₁₆×C₃₁	C₂⁴	1	4	yes	—
181	M₁₆×C₃₁	C₂⁴	1	4	yes	—
182	M₁₆×C₃₁	C₂⁴	1	4	yes	—
183	C₂×C₈×C₃₁	—	2	6	no	—
184	C₂×C₈×C₃₁	—	2	6	no	—
185	C₂×C₈×C₃₁	—	2	6	no	—
186	C₂×C₈×C₃₁	—	2	4	no	—
187	C₂×C₈×C₃₁	—	2	12	no	—
188	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
189	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
190	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
191	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
192	#60470	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₃₁	—	2	6	no	—
194	C₂³×C₃₁	—	2	1	no	—
195	C₂³×C₃₁	—	2	3	no	—
196	C₂³×C₃₁	—	4	6	no	—
197	C₂³×C₃₁	—	16	3	no	—
198	C₂³×C₃₁	C₂×#41	1	3	yes	—
199	Q₈×C₃₁	—	2	6	no	—
200	Q₈×C₃₁	—	2	3	no	—
201	Q₈×C₃₁	—	2	3	no	—
202	Q₈×C₃₁	—	2	1	no	—
203	Q₈×C₃₁	—	2	1	no	—
204	Q₈×C₃₁	—	4	6	no	—
205	Q₈×C₃₁	#121	1	4	yes	—
206	Q₈×C₃₁	C₂²×D₄	1	4	yes	—
207	D₄×C₃₁	—	2	12	no	—
208	D₄×C₃₁	—	2	4	no	—
209	D₄×C₃₁	—	2	6	no	—
210	D₄×C₃₁	—	2	1	no	—
211	D₄×C₃₁	—	2	3	no	—
212	D₄×C₃₁	—	2	1	no	—
213	D₄×C₃₁	—	2	3	no	—
214	D₄×C₃₁	—	4	6	no	—
215	D₄×C₃₁	—	4	6	no	—
216	D₄×C₃₁	—	4	6	no	—
217	D₄×C₃₁	#121	1	12	yes	—
218	D₄×C₃₁	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₃₁	—	2	6	no	—
220	C₂×C₄×C₃₁	—	2	12	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	3	no	—
224	C₂×C₄×C₃₁	—	4	6	no	—
225	C₂×C₄×C₃₁	—	4	6	no	—
226	C₂×C₄×C₃₁	—	4	6	no	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
229	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
230	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
231	C₈×C₃₁	—	2	4	no	—
232	C₈×C₃₁	—	2	4	no	—
233	C₈×C₃₁	—	2	4	no	—
234	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₈×C₃₁	C₂²×D₄	1	6	yes	—
236	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	#3586	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#3585	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂×#577	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#3573	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#3069	C₂×C₃₁	1	6	yes	complemented
242	#3068	C₂×C₃₁	1	6	yes	complemented
243	#3067	C₂×C₃₁	1	6	yes	complemented
244	C₂×#471	C₂×C₃₁	1	6	yes	complemented
245	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#2889	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#2888	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂²×C₃₁	—	2	4	no	—
249	C₂²×C₃₁	—	4	3	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	4	3	no	—
252	C₂²×C₃₁	—	8	6	no	—
253	C₂²×C₃₁	—	8	1	no	—
254	C₂²×C₃₁	—	8	3	no	—
255	C₂²×C₃₁	—	16	6	no	—
256	C₂²×C₃₁	C₂×#121	1	6	yes	—
257	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
258	C₄×C₃₁	—	2	4	no	—
259	C₄×C₃₁	—	4	1	no	—
260	C₄×C₃₁	—	4	3	no	—
261	C₄×C₃₁	C₂×#121	1	6	yes	—
262	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
263	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
264	#585	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
267	#578	—	2	4	no	—
268	#577	—	2	4	no	—
269	#577	—	2	4	no	—
270	#577	—	2	4	no	—
271	#577	—	2	4	no	—
272	#576	—	2	4	no	—
273	#575	C₂²×C₃₁	1	6	yes	—
274	#575	C₂²×C₃₁	1	6	yes	complemented
275	C₂×#137	C₂²×C₃₁	1	6	yes	complemented
276	C₂×#137	C₂²×C₃₁	1	3	yes	complemented
277	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	6	yes	—
278	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	3	yes	complemented
279	C₂×#135	C₂²×C₃₁	1	3	yes	complemented
280	C₂²×SD₁₆	C₂²×C₃₁	1	3	yes	complemented
281	#568	—	2	4	no	—
282	#568	C₂²×C₃₁	1	1	yes	characteristic
283	#568	C₂²×C₃₁	1	1	yes	characteristic
284	#568	C₂²×C₃₁	1	1	yes	characteristic
285	#568	C₂²×C₃₁	1	6	yes	complemented
286	C₂×#131	—	2	4	no	—
287	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#131	C₂²×C₃₁	1	6	yes	complemented
289	#473	C₂²×C₃₁	1	12	yes	complemented
290	#472	C₂²×C₃₁	1	12	yes	complemented
291	#471	C₂²×C₃₁	1	12	yes	complemented
292	#471	C₂²×C₃₁	1	24	yes	complemented
293	#471	C₂²×C₃₁	1	4	yes	complemented
294	#430	C₂²×C₃₁	1	6	yes	complemented
295	C₂×#108	C₂²×C₃₁	1	6	yes	complemented
296	#413	C₂²×C₃₁	1	6	yes	complemented
297	#413	C₂²×C₃₁	1	6	yes	complemented
298	C₂×#101	C₂²×C₃₁	1	6	yes	complemented
299	C₂×#100	C₂²×C₃₁	1	6	yes	complemented
300	C₂×C₃₁	—	2	1	no	—
301	C₂×C₃₁	—	2	6	no	—
302	C₂×C₃₁	—	4	4	no	—
303	C₂×C₃₁	—	8	1	no	—
304	C₂×C₃₁	—	8	3	no	—
305	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
306	2₋¹⁺⁴	—	2	1	no	—
307	2₋¹⁺⁴	—	2	4	no	—
308	2₋¹⁺⁴	—	2	4	no	—
309	2₋¹⁺⁴	D₄×C₃₁	1	4	yes	—
310	H(5,𝔽₂)	—	2	3	no	—
311	H(5,𝔽₂)	—	2	4	no	—
312	H(5,𝔽₂)	—	2	4	no	—
313	H(5,𝔽₂)	D₄×C₃₁	1	4	yes	—
314	C₂×#41	—	2	4	no	—
315	C₂×#41	—	2	1	no	—
316	C₂×#41	—	2	3	no	—
317	C₂×#41	C₂³×C₃₁	1	6	yes	—
318	C₂×#41	C₂³×C₃₁	1	3	yes	—
319	C₂×#41	C₂³×C₃₁	1	3	yes	—
320	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
321	C₂²×Q₈	—	2	1	no	—
322	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
323	C₂²×D₄	—	2	3	no	—
324	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
325	#137	—	2	12	no	—
326	#137	—	2	6	no	—
327	#137	—	2	6	no	—
328	#137	—	2	12	no	—
329	#137	C₂³×C₃₁	1	12	yes	—
330	#137	C₂³×C₃₁	1	4	yes	—
331	AGL(1,ℤ/8)	—	2	12	no	—
332	AGL(1,ℤ/8)	—	2	6	no	—
333	AGL(1,ℤ/8)	—	2	6	no	—
334	AGL(1,ℤ/8)	—	2	12	no	—
335	AGL(1,ℤ/8)	C₂³×C₃₁	1	12	yes	—
336	AGL(1,ℤ/8)	C₂³×C₃₁	1	4	yes	—
337	#135	—	2	12	no	—
338	#135	—	2	12	no	—
339	#135	—	2	6	no	—
340	#135	—	2	6	no	—
341	C₂×Q₁₆	—	2	6	no	—
342	C₂×Q₁₆	C₂³×C₃₁	1	3	yes	—
343	C₂×SD₁₆	—	2	6	no	—
344	C₂×SD₁₆	—	2	6	no	—
345	C₂×SD₁₆	—	2	6	no	—
346	C₂×SD₁₆	—	2	6	no	—
347	C₂×SD₁₆	C₂³×C₃₁	1	3	yes	—
348	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
349	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
350	C₂×D₈	—	2	6	no	—
351	C₂×D₈	C₂³×C₃₁	1	3	yes	—
352	#131	—	2	4	no	—
353	#131	—	2	4	no	—
354	#131	—	2	4	no	—
355	#131	—	2	12	no	—
356	#131	—	2	6	no	—
357	#131	—	2	6	no	—
358	#131	D₄×C₃₁	1	4	yes	—
359	#131	D₄×C₃₁	1	4	yes	—
360	#131	C₂³×C₃₁	1	4	yes	—
361	#131	C₂³×C₃₁	1	12	yes	—
362	C₂×M₁₆	—	2	12	no	—
363	C₂×M₁₆	—	2	6	no	—
364	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
365	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
366	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
367	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
368	C₂²×C₈	—	2	6	no	—
369	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
370	#108	C₂³×C₃₁	1	4	yes	—
371	#108	C₂³×C₃₁	1	12	yes	—
372	#101	C₂³×C₃₁	1	12	yes	—
373	#101	C₂³×C₃₁	1	4	yes	—
374	#100	C₂³×C₃₁	1	12	yes	—
375	#100	C₂³×C₃₁	1	4	yes	—
376	C₃₁	#60470	1	1	yes	characteristic, direct product factor, Sylow
377	C₂⁴	—	2	3	no	—
378	#41	—	2	4	no	—
379	#41	—	2	12	no	—
380	#41	—	2	6	no	—
381	#41	—	2	6	no	—
382	#41	—	2	6	no	—
383	#41	—	2	6	no	—
384	#41	—	2	6	no	—
385	#41	—	2	6	no	—
386	#41	—	4	4	no	—
387	#41	—	4	12	no	—
388	#41	C₂×D₄×C₃₁	1	12	yes	—
389	#41	C₂×D₄×C₃₁	1	4	yes	—
390	#41	C₂×D₄×C₃₁	1	4	yes	—
391	#41	C₂⁴×C₃₁	1	4	yes	—
392	C₂×Q₈	—	2	4	no	—
393	C₂×Q₈	—	2	3	no	—
394	C₂×Q₈	—	2	6	no	—
395	C₂×Q₈	—	2	6	no	—
396	C₂×Q₈	—	4	4	no	—
397	C₂×Q₈	C₂×D₄×C₃₁	1	6	yes	—
398	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
399	C₂×Q₈	C₂⁴×C₃₁	1	3	yes	—
400	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
401	C₂×D₄	—	2	6	no	—
402	C₂×D₄	—	2	12	no	—
403	C₂×D₄	—	2	3	no	—
404	C₂×D₄	—	2	3	no	—
405	C₂×D₄	—	2	6	no	—
406	C₂×D₄	—	2	6	no	—
407	C₂×D₄	—	2	3	no	—
408	C₂×D₄	—	4	12	no	—
409	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
410	C₂×D₄	#41×C₃₁	1	3	yes	—
411	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
412	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
413	C₂²×C₄	—	2	6	no	—
414	C₂²×C₄	—	2	3	no	—
415	C₂²×C₄	—	2	3	no	—
416	C₂²×C₄	—	2	3	no	—
417	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
418	Q₁₆	—	2	12	no	—
419	Q₁₆	—	2	4	no	—
420	Q₁₆	—	2	6	no	—
421	SD₁₆	—	2	12	no	—
422	SD₁₆	—	2	4	no	—
423	SD₁₆	—	2	12	no	—
424	SD₁₆	—	2	4	no	—
425	SD₁₆	—	2	3	no	—
426	SD₁₆	—	2	3	no	—
427	SD₁₆	—	2	12	no	—
428	SD₁₆	—	2	1	no	—
429	SD₁₆	—	2	1	no	—
430	D₈	—	2	12	no	—
431	D₈	—	2	4	no	—
432	D₈	—	2	6	no	—
433	M₁₆	—	2	6	no	—
434	M₁₆	—	2	6	no	—
435	M₁₆	—	2	6	no	—
436	M₁₆	—	2	12	no	—
437	M₁₆	—	2	4	no	—
438	M₁₆	C₂×D₄×C₃₁	1	12	yes	—
439	M₁₆	C₂⁴×C₃₁	1	4	yes	—
440	M₁₆	C₂⁴×C₃₁	1	4	yes	—
441	M₁₆	C₂⁴×C₃₁	1	4	yes	—
442	C₂×C₈	—	2	6	no	—
443	C₂×C₈	—	2	6	no	—
444	C₂×C₈	—	2	6	no	—
445	C₂×C₈	—	2	4	no	—
446	C₂×C₈	—	2	12	no	—
447	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
448	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
449	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
450	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
451	C₂³	—	2	6	no	—
452	C₂³	—	2	1	no	—
453	C₂³	—	2	3	no	—
454	C₂³	—	4	6	no	—
455	C₂³	—	16	3	no	—
456	C₂³	C₂×#41×C₃₁	1	3	yes	—
457	Q₈	—	2	6	no	—
458	Q₈	—	2	3	no	—
459	Q₈	—	2	3	no	—
460	Q₈	—	2	1	no	—
461	Q₈	—	2	1	no	—
462	Q₈	—	4	6	no	—
463	Q₈	C₃₁×#121	1	4	yes	—
464	Q₈	C₂²×D₄×C₃₁	1	4	yes	—
465	D₄	—	2	12	no	—
466	D₄	—	2	4	no	—
467	D₄	—	2	6	no	—
468	D₄	—	2	1	no	—
469	D₄	—	2	3	no	—
470	D₄	—	2	1	no	—
471	D₄	—	2	3	no	—
472	D₄	—	4	6	no	—
473	D₄	—	4	6	no	—
474	D₄	—	4	6	no	—
475	D₄	C₃₁×#121	1	12	yes	—
476	D₄	C₂²×D₄×C₃₁	1	4	yes	—
477	C₂×C₄	—	2	6	no	—
478	C₂×C₄	—	2	12	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	3	no	—
482	C₂×C₄	—	4	6	no	—
483	C₂×C₄	—	4	6	no	—
484	C₂×C₄	—	4	6	no	—
485	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
486	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
487	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
488	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
489	C₈	—	2	4	no	—
490	C₈	—	2	4	no	—
491	C₈	—	2	4	no	—
492	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₈	C₂²×D₄×C₃₁	1	6	yes	—
494	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
495	C₂²	—	2	4	no	—
496	C₂²	—	4	3	no	—
497	C₂²	—	4	1	no	—
498	C₂²	—	4	3	no	—
499	C₂²	—	8	6	no	—
500	C₂²	—	8	3	no	—
501	C₂²	—	8	1	no	—
502	C₂²	—	16	6	no	—
503	C₂²	C₂×C₃₁×#121	1	6	yes	—
504	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
505	C₄	—	2	4	no	—
506	C₄	—	4	1	no	—
507	C₄	—	4	3	no	—
508	C₄	C₂×C₃₁×#121	1	6	yes	—
509	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
510	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
511	C₂	—	2	1	no	—
512	C₂	—	2	6	no	—
513	C₂	—	4	4	no	—
514	C₂	—	8	3	no	—
515	C₂	—	8	1	no	—
516	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂²×#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3586	C₂	semidirect
2. C₃₁×#3585	C₂	semidirect
3. C₂×C₃₁×#577	C₂	semidirect
4. C₃₁×#3573	C₂	semidirect
5. C₃₁×#3069	C₂	semidirect
6. C₃₁×#3068	C₂	semidirect
7. C₃₁×#3067	C₂	semidirect
8. C₂×C₃₁×#471	C₂	semidirect
9. C₃₁×#2916	C₂	semidirect
10. C₃₁×#2889	C₂	semidirect
11. C₃₁×#2888	C₂	semidirect
22. C₃₁×#575	C₂²	semidirect
23. C₂×C₃₁×#137	C₂²	semidirect
24. C₂×C₃₁×#137	C₂²	semidirect
26. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
27. C₂×C₃₁×#135	C₂²	semidirect
28. C₂²×SD₁₆×C₃₁	C₂²	semidirect
33. C₃₁×#568	C₂²	semidirect
36. C₂×C₃₁×#131	C₂²	semidirect
37. C₃₁×#473	C₂²	semidirect
38. C₃₁×#472	C₂²	semidirect
39. C₃₁×#471	C₂²	semidirect
40. C₃₁×#471	C₂²	semidirect
41. C₃₁×#471	C₂²	semidirect
42. C₃₁×#430	C₂²	semidirect
43. C₂×C₃₁×#108	C₂²	semidirect
44. C₃₁×#413	C₂²	semidirect
45. C₃₁×#413	C₂²	semidirect
46. C₂×C₃₁×#101	C₂²	semidirect
47. C₂×C₃₁×#100	C₂²	semidirect
192. #60470	C₃₁	direct (N × H)
237. #3586	C₂×C₃₁	semidirect
238. #3585	C₂×C₃₁	semidirect
239. C₂×#577	C₂×C₃₁	semidirect
240. #3573	C₂×C₃₁	semidirect
241. #3069	C₂×C₃₁	semidirect
242. #3068	C₂×C₃₁	semidirect
243. #3067	C₂×C₃₁	semidirect
244. C₂×#471	C₂×C₃₁	semidirect
245. #2916	C₂×C₃₁	semidirect
246. #2889	C₂×C₃₁	semidirect
247. #2888	C₂×C₃₁	semidirect
274. #575	C₂²×C₃₁	semidirect
275. C₂×#137	C₂²×C₃₁	semidirect
276. C₂×#137	C₂²×C₃₁	semidirect
278. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
279. C₂×#135	C₂²×C₃₁	semidirect
280. C₂²×SD₁₆	C₂²×C₃₁	semidirect
285. #568	C₂²×C₃₁	semidirect
288. C₂×#131	C₂²×C₃₁	semidirect
289. #473	C₂²×C₃₁	semidirect
290. #472	C₂²×C₃₁	semidirect
291. #471	C₂²×C₃₁	semidirect
292. #471	C₂²×C₃₁	semidirect
293. #471	C₂²×C₃₁	semidirect
294. #430	C₂²×C₃₁	semidirect
295. C₂×#108	C₂²×C₃₁	semidirect
296. #413	C₂²×C₃₁	semidirect
297. #413	C₂²×C₃₁	semidirect
298. C₂×#101	C₂²×C₃₁	semidirect
299. C₂×#100	C₂²×C₃₁	semidirect
376. C₃₁	#60470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3586	2
C₃₁×#3585	2
C₂×C₃₁×#577	2
C₃₁×#3573	2
C₃₁×#3069	2
C₃₁×#3068	2
C₃₁×#3067	2
C₂×C₃₁×#471	2
C₃₁×#2916	2
C₃₁×#2889	2
C₃₁×#2888	2
#60470	31

Subgroups — C31×#60470 #21517229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60470 #21517229