Subgroups — C₃₁×#62243 #21519002

Order 7936.

2142 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3496	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3335	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3298	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3295	C₂	1	4	yes	maximal, complemented
6	D₄×SD₁₆×C₃₁	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3246	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3187	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3167	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#465	C₂	1	4	yes	maximal, complemented
11	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	—	2	1	no	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
14	C₂³×C₈×C₃₁	C₂²	1	2	yes	complemented
15	C₃₁×#554	—	2	4	no	—
16	C₃₁×#554	C₂²	1	4	yes	complemented
17	D₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
18	C₃₁×#547	C₂²	1	4	yes	—
19	C₃₁×#546	—	2	4	no	—
20	D₄²×C₃₁	C₂²	1	4	yes	—
21	C₃₁×#544	—	2	1	no	—
22	C₃₁×#543	—	2	2	no	—
23	C₃₁×#537	—	2	1	no	—
24	C₃₁×#534	—	2	2	no	—
25	C₃₁×#533	C₂²	1	2	yes	complemented
26	C₃₁×#532	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
28	C₂×C₃₁×#121	C₂²	1	4	yes	—
29	C₃₁×#518	—	2	1	no	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#498	C₂²	1	2	yes	complemented
32	C₃₁×#492	C₂²	1	2	yes	complemented
33	C₃₁×#487	C₂²	1	2	yes	complemented
34	C₃₁×#486	C₂²	1	2	yes	complemented
35	C₃₁×#483	C₂²	1	8	yes	complemented
36	C₃₁×#481	C₂²	1	8	yes	complemented
37	C₃₁×#476	C₂²	1	4	yes	complemented
38	C₃₁×#475	C₂²	1	4	yes	complemented
39	C₃₁×#465	C₂²	1	8	yes	complemented
40	C₃₁×#465	C₂²	1	4	yes	complemented
41	C₃₁×#465	C₂²	1	4	yes	complemented
42	C₃₁×#465	C₂²	1	8	yes	complemented
43	C₃₁×#465	C₂²	1	8	yes	complemented
44	C₃₁×#461	C₂²	1	4	yes	complemented
45	C₃₁×#460	C₂²	1	4	yes	complemented
46	C₃₁×#450	C₂²	1	8	yes	complemented
47	C₃₁×#448	C₂²	1	8	yes	complemented
48	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
49	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
50	C₈×D₄×C₃₁	C₂²	1	4	yes	complemented
51	C₈×D₄×C₃₁	C₂²	1	4	yes	complemented
52	C₂×C₃₁×#106	C₂²	1	4	yes	complemented
53	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
54	C₂×C₃₁×#103	C₂²	1	4	yes	complemented
55	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
56	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
57	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
58	C₃₁×H(5,𝔽₂)	—	4	8	no	—
59	C₂×#41×C₃₁	—	2	4	no	—
60	C₂×#41×C₃₁	—	2	2	no	—
61	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
62	C₂²×D₄×C₃₁	—	2	4	no	—
63	C₂²×D₄×C₃₁	—	2	4	no	—
64	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×SD₁₆×C₃₁	—	4	8	no	—
68	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
69	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
70	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
71	C₂²×C₈×C₃₁	—	4	4	no	—
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
74	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
76	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
77	C₃₁×C₄⋊Q₈	—	2	4	no	—
78	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
79	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
80	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
81	C₃₁×#127	C₂³	1	2	yes	—
82	C₃₁×#125	—	2	4	no	—
83	C₃₁×#125	—	2	2	no	—
84	C₃₁×#124	—	2	2	no	—
85	C₃₁×#123	—	2	8	no	—
86	C₃₁×C₂²⋊Q₈	—	2	4	no	—
87	C₃₁×C₂²⋊Q₈	—	2	4	no	—
88	C₃₁×C₂²⋊Q₈	—	2	2	no	—
89	C₃₁×C₂²⋊Q₈	—	2	4	no	—
90	C₃₁×C₂²⋊Q₈	—	2	8	no	—
91	C₃₁×C₂²⋊Q₈	—	2	8	no	—
92	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
93	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
94	C₃₁×#121	—	2	4	no	—
95	C₃₁×#121	—	2	4	no	—
96	C₃₁×#121	—	2	2	no	—
97	C₃₁×#121	—	2	4	no	—
98	C₃₁×#121	C₂³	1	8	yes	—
99	C₃₁×#121	C₂³	1	4	yes	—
100	C₃₁×#120	—	2	8	no	—
101	C₄×Q₈×C₃₁	—	2	2	no	—
102	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
103	C₄×D₄×C₃₁	—	2	4	no	—
104	C₄×D₄×C₃₁	—	2	4	no	—
105	C₄×D₄×C₃₁	—	2	2	no	—
106	C₄×D₄×C₃₁	D₄	1	4	yes	—
107	C₄×D₄×C₃₁	C₂³	1	4	yes	—
108	C₄×D₄×C₃₁	C₂³	1	4	yes	—
109	C₃₁×#117	—	2	2	no	—
110	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
111	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
116	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₃₁×#106	C₂³	1	4	yes	—
119	C₃₁×#106	C₂³	1	8	yes	—
120	C₃₁×#106	C₂³	1	4	yes	—
121	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
122	C₃₁×#103	C₂³	1	4	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#103	C₂³	1	4	yes	—
125	C₃₁×#102	C₂³	1	4	yes	—
126	C₃₁×#102	C₂³	1	8	yes	—
127	C₃₁×#102	C₂³	1	4	yes	—
128	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
129	C₄×C₈×C₃₁	C₂³	1	2	yes	—
130	C₄×C₈×C₃₁	C₂³	1	2	yes	—
131	C₂⁴×C₃₁	—	2	4	no	—
132	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
133	#41×C₃₁	—	4	16	no	—
134	#41×C₃₁	—	4	8	no	—
135	C₂×Q₈×C₃₁	—	2	2	no	—
136	C₂×Q₈×C₃₁	—	2	2	no	—
137	C₂×Q₈×C₃₁	—	2	4	no	—
138	C₂×Q₈×C₃₁	—	4	4	no	—
139	C₂×Q₈×C₃₁	—	4	4	no	—
140	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
141	C₂×D₄×C₃₁	—	2	8	no	—
142	C₂×D₄×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	2	2	no	—
144	C₂×D₄×C₃₁	—	2	2	no	—
145	C₂×D₄×C₃₁	—	2	4	no	—
146	C₂×D₄×C₃₁	—	4	4	no	—
147	C₂×D₄×C₃₁	—	4	4	no	—
148	C₂×D₄×C₃₁	—	4	16	no	—
149	C₂×D₄×C₃₁	—	4	16	no	—
150	C₂×D₄×C₃₁	SD₁₆	1	8	yes	complemented
151	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
152	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₃₁	—	2	4	no	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	—	2	4	no	—
157	C₂²×C₄×C₃₁	—	4	2	no	—
158	C₂²×C₄×C₃₁	—	4	2	no	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
165	SD₁₆×C₃₁	—	2	8	no	—
166	SD₁₆×C₃₁	—	4	8	no	—
167	SD₁₆×C₃₁	—	4	4	no	—
168	C₂×C₈×C₃₁	—	2	2	no	—
169	C₂×C₈×C₃₁	—	2	8	no	—
170	C₂×C₈×C₃₁	—	4	4	no	—
171	C₂×C₈×C₃₁	—	4	2	no	—
172	C₂×C₈×C₃₁	—	4	1	no	—
173	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
174	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
176	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₃₁	—	2	4	no	—
178	C₄⋊C₄×C₃₁	—	2	8	no	—
179	C₄⋊C₄×C₃₁	—	2	4	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	4	no	—
186	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
187	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
189	C₂²⋊C₄×C₃₁	—	2	4	no	—
190	C₂²⋊C₄×C₃₁	—	2	8	no	—
191	C₂²⋊C₄×C₃₁	—	2	4	no	—
192	C₂²⋊C₄×C₃₁	—	2	8	no	—
193	C₂²⋊C₄×C₃₁	—	2	4	no	—
194	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
195	C₄²×C₃₁	—	2	4	no	—
196	C₄²×C₃₁	C₂×D₄	1	2	yes	—
197	C₄²×C₃₁	C₂⁴	1	2	yes	—
198	#62243	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂³×C₃₁	—	2	4	no	—
200	C₂³×C₃₁	—	4	2	no	—
201	C₂³×C₃₁	—	4	2	no	—
202	C₂³×C₃₁	—	4	8	no	—
203	C₂³×C₃₁	—	8	4	no	—
204	C₂³×C₃₁	—	8	4	no	—
205	C₂³×C₃₁	C₂×SD₁₆	1	8	yes	complemented
206	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
207	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
208	Q₈×C₃₁	—	2	4	no	—
209	Q₈×C₃₁	—	2	4	no	—
210	Q₈×C₃₁	—	4	4	no	—
211	Q₈×C₃₁	—	8	2	no	—
212	Q₈×C₃₁	—	8	4	no	—
213	D₄×C₃₁	—	2	4	no	—
214	D₄×C₃₁	—	2	4	no	—
215	D₄×C₃₁	—	4	16	no	—
216	D₄×C₃₁	—	4	8	no	—
217	D₄×C₃₁	—	4	4	no	—
218	D₄×C₃₁	—	8	2	no	—
219	D₄×C₃₁	—	8	4	no	—
220	D₄×C₃₁	C₂×SD₁₆	1	16	yes	complemented
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	4	4	no	—
229	C₂×C₄×C₃₁	—	4	2	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	4	8	no	—
232	C₂×C₄×C₃₁	—	4	8	no	—
233	C₂×C₄×C₃₁	—	8	4	no	—
234	C₂×C₄×C₃₁	—	8	4	no	—
235	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	complemented
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
241	C₈×C₃₁	—	2	1	no	—
242	C₈×C₃₁	—	2	4	no	—
243	C₈×C₃₁	—	2	2	no	—
244	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	complemented
245	#3501	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3496	C₂×C₃₁	1	1	yes	characteristic
247	#3335	C₂×C₃₁	1	4	yes	complemented
248	#3298	C₂×C₃₁	1	4	yes	complemented
249	#3295	C₂×C₃₁	1	4	yes	complemented
250	D₄×SD₁₆	C₂×C₃₁	1	4	yes	complemented
251	#3246	C₂×C₃₁	1	2	yes	complemented
252	#3187	C₂×C₃₁	1	4	yes	complemented
253	#3167	C₂×C₃₁	1	2	yes	complemented
254	C₂×#465	C₂×C₃₁	1	4	yes	complemented
255	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₂²×C₃₁	—	2	1	no	—
257	C₂²×C₃₁	—	2	4	no	—
258	C₂²×C₃₁	—	2	1	no	—
259	C₂²×C₃₁	—	2	4	no	—
260	C₂²×C₃₁	—	4	8	no	—
261	C₂²×C₃₁	—	4	4	no	—
262	C₂²×C₃₁	—	4	4	no	—
263	C₂²×C₃₁	—	4	4	no	—
264	C₂²×C₃₁	—	16	8	no	—
265	C₂²×C₃₁	C₂²×SD₁₆	1	8	yes	—
266	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
267	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
268	C₄×C₃₁	—	2	2	no	—
269	C₄×C₃₁	—	2	1	no	—
270	C₄×C₃₁	—	2	4	no	—
271	C₄×C₃₁	—	4	4	no	—
272	C₄×C₃₁	—	4	4	no	—
273	C₄×C₃₁	—	4	4	no	—
274	C₄×C₃₁	C₂²×SD₁₆	1	4	yes	—
275	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
276	C₄×C₃₁	#585	1	1	yes	characteristic
277	C₂×H(5,𝔽₂)	—	2	1	no	—
278	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
279	C₂³×C₈	C₂²×C₃₁	1	2	yes	complemented
280	#554	—	2	4	no	—
281	#554	C₂²×C₃₁	1	4	yes	complemented
282	D₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
283	#547	C₂²×C₃₁	1	4	yes	—
284	#546	—	2	4	no	—
285	D₄²	C₂²×C₃₁	1	4	yes	—
286	#544	—	2	1	no	—
287	#543	—	2	2	no	—
288	#537	—	2	1	no	—
289	#534	—	2	2	no	—
290	#533	C₂²×C₃₁	1	2	yes	complemented
291	#532	C₂²×C₃₁	1	2	yes	—
292	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
293	C₂×#121	C₂²×C₃₁	1	4	yes	—
294	#518	—	2	1	no	—
295	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
296	#498	C₂²×C₃₁	1	2	yes	complemented
297	#492	C₂²×C₃₁	1	2	yes	complemented
298	#487	C₂²×C₃₁	1	2	yes	complemented
299	#486	C₂²×C₃₁	1	2	yes	complemented
300	#483	C₂²×C₃₁	1	8	yes	complemented
301	#481	C₂²×C₃₁	1	8	yes	complemented
302	#476	C₂²×C₃₁	1	4	yes	complemented
303	#475	C₂²×C₃₁	1	4	yes	complemented
304	#465	C₂²×C₃₁	1	8	yes	complemented
305	#465	C₂²×C₃₁	1	4	yes	complemented
306	#465	C₂²×C₃₁	1	4	yes	complemented
307	#465	C₂²×C₃₁	1	8	yes	complemented
308	#465	C₂²×C₃₁	1	8	yes	complemented
309	#461	C₂²×C₃₁	1	4	yes	complemented
310	#460	C₂²×C₃₁	1	4	yes	complemented
311	#450	C₂²×C₃₁	1	8	yes	complemented
312	#448	C₂²×C₃₁	1	8	yes	complemented
313	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
314	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
315	C₈×D₄	C₂²×C₃₁	1	4	yes	complemented
316	C₈×D₄	C₂²×C₃₁	1	4	yes	complemented
317	C₂×#106	C₂²×C₃₁	1	4	yes	complemented
318	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
319	C₂×#103	C₂²×C₃₁	1	4	yes	complemented
320	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
321	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
322	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
323	C₂×C₃₁	—	2	2	no	—
324	C₂×C₃₁	—	2	8	no	—
325	C₂×C₃₁	—	8	4	no	—
326	C₂×C₃₁	#3496	1	1	yes	characteristic
327	C₂×C₃₁	C₂³×SD₁₆	1	1	yes	characteristic
328	C₂×C₃₁	#3584	1	1	yes	characteristic
329	H(5,𝔽₂)	—	4	8	no	—
330	C₂×#41	—	2	4	no	—
331	C₂×#41	—	2	2	no	—
332	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
333	C₂²×D₄	—	2	4	no	—
334	C₂²×D₄	—	2	4	no	—
335	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
336	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
337	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
338	C₂×SD₁₆	—	4	8	no	—
339	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
340	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
341	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
342	C₂²×C₈	—	4	4	no	—
343	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
345	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
346	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
347	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
348	C₄⋊Q₈	—	2	4	no	—
349	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
350	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
351	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
352	#127	C₂³×C₃₁	1	2	yes	—
353	#125	—	2	4	no	—
354	#125	—	2	2	no	—
355	#124	—	2	2	no	—
356	#123	—	2	8	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₂²⋊Q₈	—	2	2	no	—
360	C₂²⋊Q₈	—	2	4	no	—
361	C₂²⋊Q₈	—	2	8	no	—
362	C₂²⋊Q₈	—	2	8	no	—
363	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
364	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
365	#121	—	2	4	no	—
366	#121	—	2	4	no	—
367	#121	—	2	2	no	—
368	#121	—	2	4	no	—
369	#121	C₂³×C₃₁	1	4	yes	—
370	#121	C₂³×C₃₁	1	8	yes	—
371	#120	—	2	8	no	—
372	C₄×Q₈	—	2	2	no	—
373	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	—	2	4	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	D₄×C₃₁	1	4	yes	—
378	C₄×D₄	C₂³×C₃₁	1	4	yes	—
379	C₄×D₄	C₂³×C₃₁	1	4	yes	—
380	#117	—	2	2	no	—
381	C₂×C₄⋊C₄	—	2	4	no	—
382	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
384	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
385	C₂×C₂²⋊C₄	—	2	4	no	—
386	C₂×C₂²⋊C₄	—	2	4	no	—
387	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
388	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
389	#106	C₂³×C₃₁	1	4	yes	—
390	#106	C₂³×C₃₁	1	8	yes	—
391	#106	C₂³×C₃₁	1	4	yes	—
392	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
393	#103	C₂³×C₃₁	1	4	yes	—
394	#103	C₂³×C₃₁	1	8	yes	—
395	#103	C₂³×C₃₁	1	4	yes	—
396	#102	C₂³×C₃₁	1	4	yes	—
397	#102	C₂³×C₃₁	1	8	yes	—
398	#102	C₂³×C₃₁	1	4	yes	—
399	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
400	C₄×C₈	C₂³×C₃₁	1	2	yes	—
401	C₄×C₈	C₂³×C₃₁	1	2	yes	—
402	C₃₁	#62243	1	1	yes	characteristic, direct product factor, Sylow
403	C₂⁴	—	2	4	no	—
404	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
405	#41	—	4	16	no	—
406	#41	—	4	8	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	—	4	4	no	—
411	C₂×Q₈	—	4	4	no	—
412	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	4	4	no	—
419	C₂×D₄	—	4	4	no	—
420	C₂×D₄	—	4	16	no	—
421	C₂×D₄	—	4	16	no	—
422	C₂×D₄	SD₁₆×C₃₁	1	8	yes	complemented
423	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
424	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	4	2	no	—
430	C₂²×C₄	—	4	2	no	—
431	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
433	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
434	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
436	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
437	SD₁₆	—	2	8	no	—
438	SD₁₆	—	4	8	no	—
439	SD₁₆	—	4	4	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	8	no	—
442	C₂×C₈	—	4	4	no	—
443	C₂×C₈	—	4	2	no	—
444	C₂×C₈	—	4	1	no	—
445	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
446	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
447	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
448	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
459	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
460	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	8	no	—
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	—	2	8	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
467	C₄²	—	2	4	no	—
468	C₄²	C₂×D₄×C₃₁	1	2	yes	—
469	C₄²	C₂⁴×C₃₁	1	2	yes	—
470	C₂³	—	2	4	no	—
471	C₂³	—	4	2	no	—
472	C₂³	—	4	2	no	—
473	C₂³	—	4	8	no	—
474	C₂³	—	8	4	no	—
475	C₂³	—	8	4	no	—
476	C₂³	C₂×SD₁₆×C₃₁	1	8	yes	complemented
477	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
478	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
479	Q₈	—	2	4	no	—
480	Q₈	—	2	4	no	—
481	Q₈	—	4	4	no	—
482	Q₈	—	8	2	no	—
483	Q₈	—	8	4	no	—
484	D₄	—	2	4	no	—
485	D₄	—	2	4	no	—
486	D₄	—	4	16	no	—
487	D₄	—	4	8	no	—
488	D₄	—	4	4	no	—
489	D₄	—	8	2	no	—
490	D₄	—	8	4	no	—
491	D₄	C₂×SD₁₆×C₃₁	1	16	yes	complemented
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	8	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	—	8	4	no	—
505	C₂×C₄	—	8	4	no	—
506	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	complemented
507	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
509	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
510	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
511	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
512	C₈	—	2	1	no	—
513	C₈	—	2	4	no	—
514	C₈	—	2	2	no	—
515	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	complemented
516	C₂²	—	2	1	no	—
517	C₂²	—	2	4	no	—
518	C₂²	—	2	1	no	—
519	C₂²	—	2	4	no	—
520	C₂²	—	4	8	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	4	4	no	—
524	C₂²	—	16	8	no	—
525	C₂²	C₂²×SD₁₆×C₃₁	1	8	yes	—
526	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
527	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
528	C₄	—	2	1	no	—
529	C₄	—	2	4	no	—
530	C₄	—	2	2	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	4	no	—
534	C₄	C₂²×SD₁₆×C₃₁	1	4	yes	—
535	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
536	C₄	C₃₁×#585	1	1	yes	characteristic
537	C₂	—	2	2	no	—
538	C₂	—	2	8	no	—
539	C₂	—	8	4	no	—
540	C₂	C₃₁×#3496	1	1	yes	characteristic
541	C₂	C₂³×SD₁₆×C₃₁	1	1	yes	characteristic
542	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3496	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3501	C₂	semidirect
3. C₃₁×#3335	C₂	semidirect
4. C₃₁×#3298	C₂	semidirect
5. C₃₁×#3295	C₂	semidirect
6. D₄×SD₁₆×C₃₁	C₂	semidirect
7. C₃₁×#3246	C₂	semidirect
8. C₃₁×#3187	C₂	semidirect
9. C₃₁×#3167	C₂	semidirect
10. C₂×C₃₁×#465	C₂	semidirect
11. C₂×C₈×D₄×C₃₁	C₂	semidirect
13. C₂²×SD₁₆×C₃₁	C₂²	semidirect
14. C₂³×C₈×C₃₁	C₂²	semidirect
16. C₃₁×#554	C₂²	semidirect
17. D₄×Q₈×C₃₁	C₂²	semidirect
25. C₃₁×#533	C₂²	semidirect
27. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
31. C₃₁×#498	C₂²	semidirect
32. C₃₁×#492	C₂²	semidirect
33. C₃₁×#487	C₂²	semidirect
34. C₃₁×#486	C₂²	semidirect
35. C₃₁×#483	C₂²	semidirect
36. C₃₁×#481	C₂²	semidirect
37. C₃₁×#476	C₂²	semidirect
38. C₃₁×#475	C₂²	semidirect
39. C₃₁×#465	C₂²	semidirect
40. C₃₁×#465	C₂²	semidirect
41. C₃₁×#465	C₂²	semidirect
42. C₃₁×#465	C₂²	semidirect
43. C₃₁×#465	C₂²	semidirect
44. C₃₁×#461	C₂²	semidirect
45. C₃₁×#460	C₂²	semidirect
46. C₃₁×#450	C₂²	semidirect
47. C₃₁×#448	C₂²	semidirect
48. C₄×SD₁₆×C₃₁	C₂²	semidirect
49. C₄×SD₁₆×C₃₁	C₂²	semidirect
50. C₈×D₄×C₃₁	C₂²	semidirect
51. C₈×D₄×C₃₁	C₂²	semidirect
52. C₂×C₃₁×#106	C₂²	semidirect
53. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
54. C₂×C₃₁×#103	C₂²	semidirect
55. C₂×C₃₁×#102	C₂²	semidirect
56. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
57. C₂×C₄×C₈×C₃₁	C₂²	semidirect
150. C₂×D₄×C₃₁	SD₁₆	semidirect
198. #62243	C₃₁	direct (N × H)
205. C₂³×C₃₁	C₂×SD₁₆	semidirect
220. D₄×C₃₁	C₂×SD₁₆	semidirect
235. C₂×C₄×C₃₁	C₂×SD₁₆	semidirect
244. C₈×C₃₁	H(5,𝔽₂)	semidirect
245. #3501	C₂×C₃₁	semidirect
247. #3335	C₂×C₃₁	semidirect
248. #3298	C₂×C₃₁	semidirect
249. #3295	C₂×C₃₁	semidirect
250. D₄×SD₁₆	C₂×C₃₁	semidirect
251. #3246	C₂×C₃₁	semidirect
252. #3187	C₂×C₃₁	semidirect
253. #3167	C₂×C₃₁	semidirect
254. C₂×#465	C₂×C₃₁	semidirect
255. C₂×C₈×D₄	C₂×C₃₁	semidirect
278. C₂²×SD₁₆	C₂²×C₃₁	semidirect
279. C₂³×C₈	C₂²×C₃₁	semidirect
281. #554	C₂²×C₃₁	semidirect
282. D₄×Q₈	C₂²×C₃₁	semidirect
290. #533	C₂²×C₃₁	semidirect
292. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
296. #498	C₂²×C₃₁	semidirect
297. #492	C₂²×C₃₁	semidirect
298. #487	C₂²×C₃₁	semidirect
299. #486	C₂²×C₃₁	semidirect
300. #483	C₂²×C₃₁	semidirect
301. #481	C₂²×C₃₁	semidirect
302. #476	C₂²×C₃₁	semidirect
303. #475	C₂²×C₃₁	semidirect
304. #465	C₂²×C₃₁	semidirect
305. #465	C₂²×C₃₁	semidirect
306. #465	C₂²×C₃₁	semidirect
307. #465	C₂²×C₃₁	semidirect
308. #465	C₂²×C₃₁	semidirect
309. #461	C₂²×C₃₁	semidirect
310. #460	C₂²×C₃₁	semidirect
311. #450	C₂²×C₃₁	semidirect
312. #448	C₂²×C₃₁	semidirect
313. C₄×SD₁₆	C₂²×C₃₁	semidirect
314. C₄×SD₁₆	C₂²×C₃₁	semidirect
315. C₈×D₄	C₂²×C₃₁	semidirect
316. C₈×D₄	C₂²×C₃₁	semidirect
317. C₂×#106	C₂²×C₃₁	semidirect
318. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
319. C₂×#103	C₂²×C₃₁	semidirect
320. C₂×#102	C₂²×C₃₁	semidirect
321. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
322. C₂×C₄×C₈	C₂²×C₃₁	semidirect
402. C₃₁	#62243	direct (N × H)
422. C₂×D₄	SD₁₆×C₃₁	semidirect
476. C₂³	C₂×SD₁₆×C₃₁	semidirect
491. D₄	C₂×SD₁₆×C₃₁	semidirect
506. C₂×C₄	C₂×SD₁₆×C₃₁	semidirect
515. C₈	C₃₁×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3501	2
C₃₁×#3496	2
C₃₁×#3335	2
C₃₁×#3298	2
C₃₁×#3295	2
D₄×SD₁₆×C₃₁	2
C₃₁×#3246	2
C₃₁×#3187	2
C₃₁×#3167	2
C₂×C₃₁×#465	2
C₂×C₈×D₄×C₃₁	2
#62243	31

Subgroups — C31×#62243 #21519002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62243 #21519002