Subgroups — C₃₁×#61110 #21517869

Order 7936.

2684 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3585	C₂	1	4	yes	maximal, complemented
2	C₃₁×#3292	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3291	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3288	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3068	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3018	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#453	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2956	C₂	1	2	yes	maximal, complemented
10	C₃₁×#585	C₂²	1	2	yes	complemented
11	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	complemented
12	C₃₁×#577	—	2	8	no	—
13	C₃₁×#577	C₂²	1	16	yes	complemented
14	C₃₁×#576	—	2	8	no	—
15	C₃₁×#576	C₂²	1	16	yes	complemented
16	C₃₁×#575	—	2	8	no	—
17	C₃₁×#575	C₂²	1	2	yes	—
18	C₃₁×#575	C₂²	1	4	yes	complemented
19	C₂×C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
20	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
21	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
22	C₃₁×#568	C₂²	1	4	yes	complemented
23	C₃₁×#532	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#496	C₂²	1	4	yes	complemented
26	C₃₁×#493	C₂²	1	2	yes	complemented
27	C₃₁×#492	C₂²	1	2	yes	complemented
28	C₃₁×#472	C₂²	1	16	yes	complemented
29	C₃₁×#471	C₂²	1	16	yes	complemented
30	C₃₁×#456	C₂²	1	4	yes	complemented
31	C₃₁×#455	C₂²	1	4	yes	complemented
32	C₃₁×#454	C₂²	1	4	yes	complemented
33	C₃₁×#453	C₂²	1	8	yes	complemented
34	C₃₁×#453	C₂²	1	8	yes	complemented
35	C₃₁×#453	C₂²	1	4	yes	complemented
36	C₃₁×#444	C₂²	1	8	yes	complemented
37	C₃₁×#443	C₂²	1	8	yes	complemented
38	C₃₁×#430	C₂²	1	2	yes	complemented
39	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
40	C₃₁×#413	C₂²	1	2	yes	complemented
41	C₂×C₃₁×#100	C₂²	1	2	yes	complemented
42	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×2₋¹⁺⁴	—	2	4	no	—
44	C₃₁×2₋¹⁺⁴	C₂³	1	4	yes	—
45	C₃₁×H(5,𝔽₂)	—	2	8	no	—
46	C₃₁×H(5,𝔽₂)	—	2	4	no	—
47	C₃₁×H(5,𝔽₂)	—	4	2	no	—
48	C₃₁×H(5,𝔽₂)	—	4	2	no	—
49	C₃₁×H(5,𝔽₂)	C₂³	1	4	yes	—
50	C₃₁×H(5,𝔽₂)	C₂³	1	8	yes	complemented
51	C₂×#41×C₃₁	—	2	8	no	—
52	C₂×#41×C₃₁	—	2	4	no	—
53	C₂×#41×C₃₁	—	4	2	no	—
54	C₂×#41×C₃₁	C₂³	1	4	yes	—
55	C₂×#41×C₃₁	C₂³	1	2	yes	—
56	C₂²×D₄×C₃₁	—	2	4	no	—
57	C₂²×D₄×C₃₁	—	4	1	no	—
58	C₂²×D₄×C₃₁	—	4	2	no	—
59	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
60	C₃₁×#137	—	2	4	no	—
61	C₃₁×#137	—	2	16	no	—
62	C₃₁×#137	D₄	1	8	yes	complemented
63	C₃₁×#137	C₂³	1	8	yes	complemented
64	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
65	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
66	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
67	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
68	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
69	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
70	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
71	C₃₁×AGL(1,ℤ/8)	D₄	1	8	yes	complemented
72	C₃₁×AGL(1,ℤ/8)	C₂³	1	16	yes	—
73	C₃₁×AGL(1,ℤ/8)	C₂³	1	8	yes	complemented
74	C₃₁×#135	—	2	8	no	—
75	C₃₁×#135	—	2	16	no	—
76	C₃₁×#135	—	2	16	no	—
77	C₃₁×#135	C₂³	1	8	yes	—
78	C₂×SD₁₆×C₃₁	—	2	4	no	—
79	C₂×SD₁₆×C₃₁	—	2	16	no	—
80	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	complemented
81	C₂×D₈×C₃₁	—	2	4	no	—
82	C₂×D₈×C₃₁	—	2	16	no	—
83	C₂×D₈×C₃₁	C₂³	1	8	yes	complemented
84	C₃₁×#131	—	2	8	no	—
85	C₃₁×#131	C₂³	1	16	yes	complemented
86	C₂×M₁₆×C₃₁	—	2	8	no	—
87	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
88	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
89	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic, complemented
90	C₃₁×#127	—	4	1	no	—
91	C₃₁×#127	C₂³	1	2	yes	—
92	C₃₁×#127	C₂³	1	2	yes	—
93	C₃₁×#127	C₂³	1	1	yes	characteristic, complemented
94	C₃₁×#124	C₂³	1	2	yes	complemented
95	C₃₁×#121	—	4	1	no	—
96	C₄×D₄×C₃₁	—	4	1	no	—
97	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
98	C₃₁×#108	C₂³	1	8	yes	complemented
99	C₃₁×#104	C₂³	1	8	yes	complemented
100	C₃₁×#104	C₂³	1	8	yes	complemented
101	C₃₁×#101	C₂³	1	4	yes	complemented
102	C₃₁×#100	C₂³	1	8	yes	complemented
103	C₃₁×#100	C₂³	1	4	yes	complemented
104	C₃₁×#97	C₂³	1	2	yes	complemented
105	C₄×C₈×C₃₁	C₂³	1	2	yes	complemented
106	C₂⁴×C₃₁	—	2	2	no	—
107	C₂⁴×C₃₁	—	4	2	no	—
108	#41×C₃₁	—	2	4	no	—
109	#41×C₃₁	—	2	4	no	—
110	#41×C₃₁	—	2	8	no	—
111	#41×C₃₁	—	2	16	no	—
112	#41×C₃₁	—	2	8	no	—
113	#41×C₃₁	—	2	4	no	—
114	#41×C₃₁	—	2	4	no	—
115	#41×C₃₁	—	4	4	no	—
116	#41×C₃₁	—	4	4	no	—
117	#41×C₃₁	—	4	8	no	—
118	#41×C₃₁	—	4	8	no	—
119	#41×C₃₁	—	4	4	no	—
120	#41×C₃₁	C₂×D₄	1	8	yes	—
121	#41×C₃₁	C₂⁴	1	8	yes	—
122	C₂×Q₈×C₃₁	—	2	8	no	—
123	C₂×Q₈×C₃₁	—	4	2	no	—
124	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
125	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
126	C₂×D₄×C₃₁	—	2	4	no	—
127	C₂×D₄×C₃₁	—	2	8	no	—
128	C₂×D₄×C₃₁	—	2	16	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	2	4	no	—
131	C₂×D₄×C₃₁	—	2	4	no	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	—	4	8	no	—
134	C₂×D₄×C₃₁	—	4	4	no	—
135	C₂×D₄×C₃₁	—	4	1	no	—
136	C₂×D₄×C₃₁	—	4	2	no	—
137	C₂×D₄×C₃₁	—	4	2	no	—
138	C₂×D₄×C₃₁	—	4	8	no	—
139	C₂×D₄×C₃₁	—	4	4	no	—
140	C₂×D₄×C₃₁	—	4	8	no	—
141	C₂×D₄×C₃₁	—	4	2	no	—
142	C₂×D₄×C₃₁	—	4	8	no	—
143	C₂×D₄×C₃₁	—	8	4	no	—
144	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
145	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
146	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₃₁	—	2	4	no	—
148	C₂²×C₄×C₃₁	—	2	1	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	4	2	no	—
151	C₂²×C₄×C₃₁	—	4	4	no	—
152	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
153	Q₁₆×C₃₁	—	2	8	no	—
154	Q₁₆×C₃₁	C₂×D₄	1	8	yes	complemented
155	SD₁₆×C₃₁	—	2	16	no	—
156	SD₁₆×C₃₁	—	2	8	no	—
157	SD₁₆×C₃₁	—	2	8	no	—
158	SD₁₆×C₃₁	—	2	16	no	—
159	SD₁₆×C₃₁	C₂×D₄	1	16	yes	complemented
160	D₈×C₃₁	—	2	16	no	—
161	D₈×C₃₁	—	2	8	no	—
162	D₈×C₃₁	—	2	16	no	—
163	D₈×C₃₁	C₂×D₄	1	8	yes	complemented
164	M₁₆×C₃₁	—	2	4	no	—
165	M₁₆×C₃₁	—	2	16	no	—
166	M₁₆×C₃₁	C₂×D₄	1	8	yes	—
167	M₁₆×C₃₁	C₂⁴	1	8	yes	—
168	C₂×C₈×C₃₁	—	2	4	no	—
169	C₂×C₈×C₃₁	—	2	16	no	—
170	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
171	C₄⋊C₄×C₃₁	—	4	1	no	—
172	C₂²⋊C₄×C₃₁	—	4	2	no	—
173	C₄²×C₃₁	—	2	1	no	—
174	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
176	#61110	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₃₁	—	2	4	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	4	no	—
180	C₂³×C₃₁	—	4	4	no	—
181	C₂³×C₃₁	—	4	8	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	8	4	no	—
185	C₂³×C₃₁	—	16	4	no	—
186	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
187	Q₈×C₃₁	—	2	4	no	—
188	Q₈×C₃₁	—	2	4	no	—
189	Q₈×C₃₁	—	4	4	no	—
190	Q₈×C₃₁	C₂²×D₄	1	8	yes	—
191	D₄×C₃₁	—	2	8	no	—
192	D₄×C₃₁	—	2	8	no	—
193	D₄×C₃₁	—	2	4	no	—
194	D₄×C₃₁	—	2	4	no	—
195	D₄×C₃₁	—	2	8	no	—
196	D₄×C₃₁	—	4	16	no	—
197	D₄×C₃₁	—	4	8	no	—
198	D₄×C₃₁	—	4	4	no	—
199	D₄×C₃₁	—	4	4	no	—
200	D₄×C₃₁	—	4	4	no	—
201	D₄×C₃₁	—	8	2	no	—
202	D₄×C₃₁	—	8	4	no	—
203	D₄×C₃₁	—	8	4	no	—
204	D₄×C₃₁	C₂²×D₄	1	8	yes	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	8	no	—
208	C₂×C₄×C₃₁	—	2	8	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	4	8	no	—
212	C₂×C₄×C₃₁	—	4	4	no	—
213	C₂×C₄×C₃₁	—	4	8	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
220	C₈×C₃₁	—	2	8	no	—
221	C₈×C₃₁	C₂²×D₄	1	8	yes	—
222	#3585	C₂×C₃₁	1	4	yes	complemented
223	#3292	C₂×C₃₁	1	4	yes	complemented
224	#3291	C₂×C₃₁	1	4	yes	complemented
225	#3288	C₂×C₃₁	1	8	yes	complemented
226	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#3068	C₂×C₃₁	1	4	yes	complemented
228	#3018	C₂×C₃₁	1	2	yes	complemented
229	C₂×#453	C₂×C₃₁	1	2	yes	complemented
230	#2956	C₂×C₃₁	1	2	yes	complemented
231	C₂²×C₃₁	—	2	8	no	—
232	C₂²×C₃₁	—	4	1	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	4	no	—
235	C₂²×C₃₁	—	8	4	no	—
236	C₂²×C₃₁	—	8	4	no	—
237	C₂²×C₃₁	—	8	2	no	—
238	C₂²×C₃₁	—	8	4	no	—
239	C₂²×C₃₁	—	16	2	no	—
240	C₂²×C₃₁	—	16	4	no	—
241	C₂²×C₃₁	—	16	8	no	—
242	C₂²×C₃₁	D₄²	1	2	yes	—
243	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	8	no	—
246	C₄×C₃₁	—	4	1	no	—
247	C₄×C₃₁	—	4	2	no	—
248	C₄×C₃₁	D₄²	1	2	yes	—
249	C₄×C₃₁	C₂³×D₄	1	2	yes	—
250	#585	C₂²×C₃₁	1	2	yes	complemented
251	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	complemented
252	#577	—	2	8	no	—
253	#577	C₂²×C₃₁	1	16	yes	complemented
254	#576	—	2	8	no	—
255	#576	C₂²×C₃₁	1	16	yes	complemented
256	#575	—	2	8	no	—
257	#575	C₂²×C₃₁	1	2	yes	—
258	#575	C₂²×C₃₁	1	4	yes	complemented
259	C₂×AGL(1,ℤ/8)	—	2	8	no	—
260	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
261	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
262	#568	C₂²×C₃₁	1	4	yes	complemented
263	#532	C₂²×C₃₁	1	1	yes	characteristic, complemented
264	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
265	#496	C₂²×C₃₁	1	4	yes	complemented
266	#493	C₂²×C₃₁	1	2	yes	complemented
267	#492	C₂²×C₃₁	1	2	yes	complemented
268	#472	C₂²×C₃₁	1	16	yes	complemented
269	#471	C₂²×C₃₁	1	16	yes	complemented
270	#456	C₂²×C₃₁	1	4	yes	complemented
271	#455	C₂²×C₃₁	1	4	yes	complemented
272	#454	C₂²×C₃₁	1	4	yes	complemented
273	#453	C₂²×C₃₁	1	8	yes	complemented
274	#453	C₂²×C₃₁	1	8	yes	complemented
275	#453	C₂²×C₃₁	1	4	yes	complemented
276	#444	C₂²×C₃₁	1	8	yes	complemented
277	#443	C₂²×C₃₁	1	8	yes	complemented
278	#430	C₂²×C₃₁	1	2	yes	complemented
279	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
280	#413	C₂²×C₃₁	1	2	yes	complemented
281	C₂×#100	C₂²×C₃₁	1	2	yes	complemented
282	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
283	C₂×C₃₁	—	2	1	no	—
284	C₂×C₃₁	—	2	2	no	—
285	C₂×C₃₁	—	4	8	no	—
286	C₂×C₃₁	—	8	4	no	—
287	C₂×C₃₁	—	8	2	no	—
288	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
289	2₋¹⁺⁴	—	2	4	no	—
290	2₋¹⁺⁴	C₂³×C₃₁	1	4	yes	—
291	H(5,𝔽₂)	—	2	8	no	—
292	H(5,𝔽₂)	—	2	4	no	—
293	H(5,𝔽₂)	—	4	2	no	—
294	H(5,𝔽₂)	—	4	2	no	—
295	H(5,𝔽₂)	C₂³×C₃₁	1	4	yes	—
296	H(5,𝔽₂)	C₂³×C₃₁	1	8	yes	complemented
297	C₂×#41	—	2	8	no	—
298	C₂×#41	—	2	4	no	—
299	C₂×#41	—	4	2	no	—
300	C₂×#41	C₂³×C₃₁	1	4	yes	—
301	C₂×#41	C₂³×C₃₁	1	2	yes	—
302	C₂²×D₄	—	2	4	no	—
303	C₂²×D₄	—	4	1	no	—
304	C₂²×D₄	—	4	2	no	—
305	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
306	#137	—	2	4	no	—
307	#137	—	2	16	no	—
308	#137	D₄×C₃₁	1	8	yes	complemented
309	#137	C₂³×C₃₁	1	8	yes	complemented
310	AGL(1,ℤ/8)	—	2	8	no	—
311	AGL(1,ℤ/8)	—	2	4	no	—
312	AGL(1,ℤ/8)	—	2	16	no	—
313	AGL(1,ℤ/8)	—	2	16	no	—
314	AGL(1,ℤ/8)	—	2	16	no	—
315	AGL(1,ℤ/8)	—	2	8	no	—
316	AGL(1,ℤ/8)	—	2	8	no	—
317	AGL(1,ℤ/8)	D₄×C₃₁	1	8	yes	complemented
318	AGL(1,ℤ/8)	C₂³×C₃₁	1	16	yes	—
319	AGL(1,ℤ/8)	C₂³×C₃₁	1	8	yes	complemented
320	#135	—	2	8	no	—
321	#135	—	2	16	no	—
322	#135	—	2	16	no	—
323	#135	C₂³×C₃₁	1	8	yes	—
324	C₂×SD₁₆	—	2	4	no	—
325	C₂×SD₁₆	—	2	16	no	—
326	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	complemented
327	C₂×D₈	—	2	4	no	—
328	C₂×D₈	—	2	16	no	—
329	C₂×D₈	C₂³×C₃₁	1	8	yes	complemented
330	#131	—	2	8	no	—
331	#131	C₂³×C₃₁	1	16	yes	complemented
332	C₂×M₁₆	—	2	8	no	—
333	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
334	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
335	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic, complemented
336	#127	—	4	1	no	—
337	#127	C₂³×C₃₁	1	2	yes	—
338	#127	C₂³×C₃₁	1	2	yes	—
339	#127	C₂³×C₃₁	1	1	yes	characteristic, complemented
340	#124	C₂³×C₃₁	1	2	yes	complemented
341	#121	—	4	1	no	—
342	C₄×D₄	—	4	1	no	—
343	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
344	#108	C₂³×C₃₁	1	8	yes	complemented
345	#104	C₂³×C₃₁	1	8	yes	complemented
346	#104	C₂³×C₃₁	1	8	yes	complemented
347	#101	C₂³×C₃₁	1	4	yes	complemented
348	#100	C₂³×C₃₁	1	8	yes	complemented
349	#100	C₂³×C₃₁	1	4	yes	complemented
350	#97	C₂³×C₃₁	1	2	yes	complemented
351	C₄×C₈	C₂³×C₃₁	1	2	yes	complemented
352	C₃₁	#61110	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁴	—	2	2	no	—
354	C₂⁴	—	4	2	no	—
355	#41	—	2	4	no	—
356	#41	—	2	4	no	—
357	#41	—	2	8	no	—
358	#41	—	2	16	no	—
359	#41	—	2	8	no	—
360	#41	—	2	4	no	—
361	#41	—	2	4	no	—
362	#41	—	4	4	no	—
363	#41	—	4	4	no	—
364	#41	—	4	8	no	—
365	#41	—	4	8	no	—
366	#41	—	4	4	no	—
367	#41	C₂×D₄×C₃₁	1	8	yes	—
368	#41	C₂⁴×C₃₁	1	8	yes	—
369	C₂×Q₈	—	2	8	no	—
370	C₂×Q₈	—	4	2	no	—
371	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
372	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	—	2	8	no	—
375	C₂×D₄	—	2	16	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	4	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	4	8	no	—
381	C₂×D₄	—	4	4	no	—
382	C₂×D₄	—	4	1	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	8	no	—
386	C₂×D₄	—	4	4	no	—
387	C₂×D₄	—	4	8	no	—
388	C₂×D₄	—	4	2	no	—
389	C₂×D₄	—	4	8	no	—
390	C₂×D₄	—	8	4	no	—
391	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
392	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
393	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	4	no	—
399	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
400	Q₁₆	—	2	8	no	—
401	Q₁₆	C₂×D₄×C₃₁	1	8	yes	complemented
402	SD₁₆	—	2	16	no	—
403	SD₁₆	—	2	8	no	—
404	SD₁₆	—	2	8	no	—
405	SD₁₆	—	2	16	no	—
406	SD₁₆	C₂×D₄×C₃₁	1	16	yes	complemented
407	D₈	—	2	16	no	—
408	D₈	—	2	8	no	—
409	D₈	—	2	16	no	—
410	D₈	C₂×D₄×C₃₁	1	8	yes	complemented
411	M₁₆	—	2	4	no	—
412	M₁₆	—	2	16	no	—
413	M₁₆	C₂×D₄×C₃₁	1	8	yes	—
414	M₁₆	C₂⁴×C₃₁	1	8	yes	—
415	C₂×C₈	—	2	4	no	—
416	C₂×C₈	—	2	16	no	—
417	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
418	C₄⋊C₄	—	4	1	no	—
419	C₂²⋊C₄	—	4	2	no	—
420	C₄²	—	2	1	no	—
421	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
422	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
423	C₂³	—	2	4	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	—	4	8	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	8	4	no	—
431	C₂³	—	16	4	no	—
432	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
433	Q₈	—	2	4	no	—
434	Q₈	—	2	4	no	—
435	Q₈	—	4	4	no	—
436	Q₈	C₂²×D₄×C₃₁	1	8	yes	—
437	D₄	—	2	8	no	—
438	D₄	—	2	8	no	—
439	D₄	—	2	4	no	—
440	D₄	—	2	4	no	—
441	D₄	—	2	8	no	—
442	D₄	—	4	16	no	—
443	D₄	—	4	8	no	—
444	D₄	—	4	4	no	—
445	D₄	—	4	4	no	—
446	D₄	—	4	4	no	—
447	D₄	—	8	2	no	—
448	D₄	—	8	4	no	—
449	D₄	—	8	4	no	—
450	D₄	C₂²×D₄×C₃₁	1	8	yes	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	8	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	4	8	no	—
458	C₂×C₄	—	4	4	no	—
459	C₂×C₄	—	4	8	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
464	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
465	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
466	C₈	—	2	8	no	—
467	C₈	C₂²×D₄×C₃₁	1	8	yes	—
468	C₂²	—	2	8	no	—
469	C₂²	—	4	1	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	8	4	no	—
473	C₂²	—	8	4	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	4	no	—
476	C₂²	—	16	2	no	—
477	C₂²	—	16	8	no	—
478	C₂²	—	16	4	no	—
479	C₂²	D₄²×C₃₁	1	2	yes	—
480	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
481	C₄	—	2	2	no	—
482	C₄	—	2	8	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	2	no	—
485	C₄	D₄²×C₃₁	1	2	yes	—
486	C₄	C₂³×D₄×C₃₁	1	2	yes	—
487	C₂	—	2	1	no	—
488	C₂	—	2	2	no	—
489	C₂	—	4	8	no	—
490	C₂	—	8	4	no	—
491	C₂	—	8	2	no	—
492	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂×D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3585	C₂	semidirect
2. C₃₁×#3292	C₂	semidirect
3. C₃₁×#3291	C₂	semidirect
4. C₃₁×#3288	C₂	semidirect
5. C₃₁×#3151	C₂	semidirect
6. C₃₁×#3068	C₂	semidirect
7. C₃₁×#3018	C₂	semidirect
8. C₂×C₃₁×#453	C₂	semidirect
9. C₃₁×#2956	C₂	semidirect
10. C₃₁×#585	C₂²	semidirect
11. C₂×C₃₁×H(5,𝔽₂)	C₂²	semidirect
13. C₃₁×#577	C₂²	semidirect
15. C₃₁×#576	C₂²	semidirect
18. C₃₁×#575	C₂²	semidirect
20. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
21. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
22. C₃₁×#568	C₂²	semidirect
23. C₃₁×#532	C₂²	semidirect
24. C₂×C₃₁×#127	C₂²	semidirect
25. C₃₁×#496	C₂²	semidirect
26. C₃₁×#493	C₂²	semidirect
27. C₃₁×#492	C₂²	semidirect
28. C₃₁×#472	C₂²	semidirect
29. C₃₁×#471	C₂²	semidirect
30. C₃₁×#456	C₂²	semidirect
31. C₃₁×#455	C₂²	semidirect
32. C₃₁×#454	C₂²	semidirect
33. C₃₁×#453	C₂²	semidirect
34. C₃₁×#453	C₂²	semidirect
35. C₃₁×#453	C₂²	semidirect
36. C₃₁×#444	C₂²	semidirect
37. C₃₁×#443	C₂²	semidirect
38. C₃₁×#430	C₂²	semidirect
39. C₂×C₃₁×#104	C₂²	semidirect
40. C₃₁×#413	C₂²	semidirect
41. C₂×C₃₁×#100	C₂²	semidirect
42. C₄×M₁₆×C₃₁	C₂²	semidirect
50. C₃₁×H(5,𝔽₂)	C₂³	semidirect
62. C₃₁×#137	D₄	semidirect
63. C₃₁×#137	C₂³	semidirect
71. C₃₁×AGL(1,ℤ/8)	D₄	semidirect
73. C₃₁×AGL(1,ℤ/8)	C₂³	semidirect
80. C₂×SD₁₆×C₃₁	C₂³	semidirect
83. C₂×D₈×C₃₁	C₂³	semidirect
85. C₃₁×#131	C₂³	semidirect
89. C₃₁×C₄⋊Q₈	C₂³	semidirect
93. C₃₁×#127	C₂³	semidirect
94. C₃₁×#124	C₂³	semidirect
98. C₃₁×#108	C₂³	semidirect
99. C₃₁×#104	C₂³	semidirect
100. C₃₁×#104	C₂³	semidirect
101. C₃₁×#101	C₂³	semidirect
102. C₃₁×#100	C₂³	semidirect
103. C₃₁×#100	C₂³	semidirect
104. C₃₁×#97	C₂³	semidirect
105. C₄×C₈×C₃₁	C₂³	semidirect
154. Q₁₆×C₃₁	C₂×D₄	semidirect
159. SD₁₆×C₃₁	C₂×D₄	semidirect
163. D₈×C₃₁	C₂×D₄	semidirect
176. #61110	C₃₁	direct (N × H)
222. #3585	C₂×C₃₁	semidirect
223. #3292	C₂×C₃₁	semidirect
224. #3291	C₂×C₃₁	semidirect
225. #3288	C₂×C₃₁	semidirect
226. #3151	C₂×C₃₁	semidirect
227. #3068	C₂×C₃₁	semidirect
228. #3018	C₂×C₃₁	semidirect
229. C₂×#453	C₂×C₃₁	semidirect
230. #2956	C₂×C₃₁	semidirect
250. #585	C₂²×C₃₁	semidirect
251. C₂×H(5,𝔽₂)	C₂²×C₃₁	semidirect
253. #577	C₂²×C₃₁	semidirect
255. #576	C₂²×C₃₁	semidirect
258. #575	C₂²×C₃₁	semidirect
260. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
261. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
262. #568	C₂²×C₃₁	semidirect
263. #532	C₂²×C₃₁	semidirect
264. C₂×#127	C₂²×C₃₁	semidirect
265. #496	C₂²×C₃₁	semidirect
266. #493	C₂²×C₃₁	semidirect
267. #492	C₂²×C₃₁	semidirect
268. #472	C₂²×C₃₁	semidirect
269. #471	C₂²×C₃₁	semidirect
270. #456	C₂²×C₃₁	semidirect
271. #455	C₂²×C₃₁	semidirect
272. #454	C₂²×C₃₁	semidirect
273. #453	C₂²×C₃₁	semidirect
274. #453	C₂²×C₃₁	semidirect
275. #453	C₂²×C₃₁	semidirect
276. #444	C₂²×C₃₁	semidirect
277. #443	C₂²×C₃₁	semidirect
278. #430	C₂²×C₃₁	semidirect
279. C₂×#104	C₂²×C₃₁	semidirect
280. #413	C₂²×C₃₁	semidirect
281. C₂×#100	C₂²×C₃₁	semidirect
282. C₄×M₁₆	C₂²×C₃₁	semidirect
296. H(5,𝔽₂)	C₂³×C₃₁	semidirect
308. #137	D₄×C₃₁	semidirect
309. #137	C₂³×C₃₁	semidirect
317. AGL(1,ℤ/8)	D₄×C₃₁	semidirect
319. AGL(1,ℤ/8)	C₂³×C₃₁	semidirect
326. C₂×SD₁₆	C₂³×C₃₁	semidirect
329. C₂×D₈	C₂³×C₃₁	semidirect
331. #131	C₂³×C₃₁	semidirect
335. C₄⋊Q₈	C₂³×C₃₁	semidirect
339. #127	C₂³×C₃₁	semidirect
340. #124	C₂³×C₃₁	semidirect
344. #108	C₂³×C₃₁	semidirect
345. #104	C₂³×C₃₁	semidirect
346. #104	C₂³×C₃₁	semidirect
347. #101	C₂³×C₃₁	semidirect
348. #100	C₂³×C₃₁	semidirect
349. #100	C₂³×C₃₁	semidirect
350. #97	C₂³×C₃₁	semidirect
351. C₄×C₈	C₂³×C₃₁	semidirect
352. C₃₁	#61110	direct (N × H)
401. Q₁₆	C₂×D₄×C₃₁	semidirect
406. SD₁₆	C₂×D₄×C₃₁	semidirect
410. D₈	C₂×D₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3585	2
C₃₁×#3292	2
C₃₁×#3291	2
C₃₁×#3288	2
C₃₁×#3151	2
C₃₁×#3068	2
C₃₁×#3018	2
C₂×C₃₁×#453	2
C₃₁×#2956	2
#61110	31

Subgroups — C31×#61110 #21517869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61110 #21517869