Subgroups — C₃₁×#21782 #21478541

Order 7936.

1382 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	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2888	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2883	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1911	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1910	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1887	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1884	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1827	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1795	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
12	C₃₁×#568	C₂²	1	2	yes	complemented
13	C₃₁×#550	—	2	2	no	—
14	D₄²×C₃₁	—	2	4	no	—
15	D₄²×C₃₁	—	2	2	no	—
16	C₃₁×#535	C₄	1	2	yes	complemented
17	C₃₁×#535	C₄	1	2	yes	complemented
18	C₃₁×#535	C₂²	1	1	yes	characteristic
19	C₃₁×#535	C₂²	1	1	yes	characteristic
20	C₃₁×#532	C₄	1	2	yes	complemented
21	C₃₁×#532	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#127	C₄	1	2	yes	complemented
23	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
24	C₃₁×#518	C₂²	1	2	yes	complemented
25	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#421	C₂²	1	4	yes	complemented
27	C₃₁×#418	C₂²	1	4	yes	complemented
28	C₃₁×#413	—	2	2	no	—
29	C₃₁×#413	C₂²	1	2	yes	—
30	C₂×C₃₁×#100	—	2	2	no	—
31	C₂×C₃₁×#100	C₂²	1	2	yes	complemented
32	C₃₁×#410	C₂²	1	2	yes	—
33	C₃₁×#410	C₂²	1	2	yes	complemented
34	C₂×C₃₁×#99	—	2	4	no	—
35	C₃₁×#339	C₂²	1	4	yes	complemented
36	C₃₁×H(3,ℤ/4)	C₂²	1	4	yes	complemented
37	C₃₁×H(5,𝔽₂)	—	4	4	no	—
38	C₃₁×H(5,𝔽₂)	—	4	4	no	—
39	C₂×#41×C₃₁	—	2	1	no	—
40	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×#41×C₃₁	C₂×C₄	1	2	yes	complemented
42	C₂×#41×C₃₁	D₄	1	2	yes	—
43	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	—	2	4	no	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₂²×D₄×C₃₁	—	2	2	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
52	C₂²×D₄×C₃₁	D₄	1	4	yes	—
53	C₂²×D₄×C₃₁	D₄	1	2	yes	—
54	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#131	—	4	4	no	—
57	C₂×M₁₆×C₃₁	—	2	4	no	—
58	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
59	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
60	C₃₁×#127	—	2	4	no	—
61	C₃₁×#127	—	2	2	no	—
62	C₃₁×#127	C₂×C₄	1	2	yes	—
63	C₃₁×#127	C₂×C₄	1	2	yes	—
64	C₃₁×#127	C₂×C₄	1	2	yes	—
65	C₃₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
66	C₃₁×#127	C₂×C₄	1	2	yes	complemented
67	C₃₁×#124	C₂×C₄	1	2	yes	—
68	C₃₁×#124	C₂×C₄	1	2	yes	complemented
69	C₃₁×#124	C₂×C₄	1	1	yes	characteristic, complemented
70	C₃₁×#124	C₂×C₄	1	1	yes	characteristic, complemented
71	C₃₁×#121	—	2	4	no	—
72	C₃₁×#121	—	2	4	no	—
73	C₃₁×#121	—	2	2	no	—
74	C₃₁×#121	—	2	2	no	—
75	C₃₁×#121	—	2	4	no	—
76	C₃₁×#121	—	2	2	no	—
77	C₃₁×#120	—	2	4	no	—
78	C₃₁×#120	—	2	2	no	—
79	C₃₁×#120	—	2	4	no	—
80	C₃₁×#120	—	2	2	no	—
81	C₄×Q₈×C₃₁	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	2	no	—
83	C₄×D₄×C₃₁	—	2	1	no	—
84	C₄×D₄×C₃₁	—	2	4	no	—
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	4	4	no	—
87	C₃₁×#117	C₂×C₄	1	2	yes	complemented
88	C₃₁×#117	C₂³	1	4	yes	—
89	C₃₁×#117	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
91	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	complemented
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#104	—	4	4	no	—
94	C₃₁×#102	—	4	4	no	—
95	C₃₁×#101	—	2	2	no	—
96	C₃₁×#100	—	2	2	no	—
97	C₃₁×#100	—	2	4	no	—
98	C₃₁×#99	—	2	4	no	—
99	C₃₁×#99	—	2	4	no	—
100	C₂⁴×C₃₁	—	2	2	no	—
101	C₂⁴×C₃₁	—	2	1	no	—
102	C₂⁴×C₃₁	C₂²⋊C₄	1	4	yes	—
103	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
104	#41×C₃₁	—	2	4	no	—
105	#41×C₃₁	—	2	4	no	—
106	#41×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	4	4	no	—
108	#41×C₃₁	—	4	4	no	—
109	#41×C₃₁	—	4	4	no	—
110	#41×C₃₁	—	4	4	no	—
111	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×D₄×C₃₁	—	2	4	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	4	no	—
119	C₂×D₄×C₃₁	—	2	4	no	—
120	C₂×D₄×C₃₁	—	2	4	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	4	4	no	—
126	C₂×D₄×C₃₁	—	4	4	no	—
127	C₂×D₄×C₃₁	—	4	4	no	—
128	C₂×D₄×C₃₁	—	4	8	no	—
129	C₂×D₄×C₃₁	—	4	8	no	—
130	C₂×D₄×C₃₁	—	4	4	no	—
131	C₂×D₄×C₃₁	—	8	4	no	—
132	C₂×D₄×C₃₁	—	8	8	no	—
133	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
134	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
135	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
138	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
140	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₃₁	—	2	2	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	1	no	—
147	C₂²×C₄×C₃₁	—	4	4	no	—
148	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
149	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
152	M₁₆×C₃₁	—	2	4	no	—
153	M₁₆×C₃₁	—	2	4	no	—
154	M₁₆×C₃₁	—	4	4	no	—
155	C₂×C₈×C₃₁	—	2	4	no	—
156	C₂×C₈×C₃₁	—	4	4	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₂²⋊C₄×C₃₁	—	2	2	no	—
162	C₂²⋊C₄×C₃₁	—	2	4	no	—
163	C₂²⋊C₄×C₃₁	—	2	2	no	—
164	C₂²⋊C₄×C₃₁	—	2	1	no	—
165	C₂²⋊C₄×C₃₁	—	2	4	no	—
166	C₂²⋊C₄×C₃₁	—	2	2	no	—
167	C₂²⋊C₄×C₃₁	—	2	4	no	—
168	C₂²⋊C₄×C₃₁	—	4	4	no	—
169	C₄²×C₃₁	—	2	2	no	—
170	C₄²×C₃₁	—	2	1	no	—
171	C₄²×C₃₁	—	2	4	no	—
172	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
174	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
175	#21782	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	4	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	1	no	—
179	C₂³×C₃₁	—	4	4	no	—
180	C₂³×C₃₁	—	4	4	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	8	2	no	—
183	C₂³×C₃₁	—	8	4	no	—
184	C₂³×C₃₁	—	8	2	no	—
185	C₂³×C₃₁	—	8	4	no	—
186	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
188	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
189	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	Q₈×C₃₁	—	2	2	no	—
191	Q₈×C₃₁	—	2	2	no	—
192	Q₈×C₃₁	—	2	2	no	—
193	D₄×C₃₁	—	2	4	no	—
194	D₄×C₃₁	—	2	4	no	—
195	D₄×C₃₁	—	2	4	no	—
196	D₄×C₃₁	—	2	2	no	—
197	D₄×C₃₁	—	2	4	no	—
198	D₄×C₃₁	—	2	2	no	—
199	D₄×C₃₁	—	2	4	no	—
200	D₄×C₃₁	—	2	2	no	—
201	D₄×C₃₁	—	4	4	no	—
202	D₄×C₃₁	—	4	4	no	—
203	D₄×C₃₁	—	4	4	no	—
204	D₄×C₃₁	—	4	8	no	—
205	D₄×C₃₁	—	4	4	no	—
206	D₄×C₃₁	—	4	4	no	—
207	D₄×C₃₁	—	4	2	no	—
208	D₄×C₃₁	—	4	2	no	—
209	D₄×C₃₁	—	8	4	no	—
210	D₄×C₃₁	—	8	4	no	—
211	D₄×C₃₁	—	8	4	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	4	4	no	—
219	C₂×C₄×C₃₁	—	4	4	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	4	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	4	no	—
224	C₂×C₄×C₃₁	—	16	4	no	—
225	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
226	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
230	C₈×C₃₁	—	4	4	no	—
231	#3498	C₂×C₃₁	1	1	yes	characteristic
232	#2888	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#2883	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1911	C₂×C₃₁	1	2	yes	complemented
235	#1910	C₂×C₃₁	1	2	yes	complemented
236	#1887	C₂×C₃₁	1	2	yes	complemented
237	#1884	C₂×C₃₁	1	2	yes	complemented
238	#1827	C₂×C₃₁	1	2	yes	complemented
239	#1795	C₂×C₃₁	1	2	yes	complemented
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	2	2	no	—
242	C₂²×C₃₁	—	2	4	no	—
243	C₂²×C₃₁	—	4	1	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	—	4	4	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	—	4	4	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	—	4	2	no	—
250	C₂²×C₃₁	—	4	1	no	—
251	C₂²×C₃₁	—	8	4	no	—
252	C₂²×C₃₁	—	16	2	no	—
253	C₂²×C₃₁	—	16	4	no	—
254	C₂²×C₃₁	—	16	4	no	—
255	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
256	C₂²×C₃₁	#534	1	2	yes	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	2	2	no	—
259	C₄×C₃₁	—	2	2	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	—	4	1	no	—
262	C₄×C₃₁	—	4	1	no	—
263	C₄×C₃₁	—	8	4	no	—
264	C₄×C₃₁	#518	1	2	yes	—
265	C₄×C₃₁	#534	1	2	yes	—
266	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
268	#568	C₂²×C₃₁	1	2	yes	complemented
269	#550	—	2	2	no	—
270	D₄²	—	2	4	no	—
271	D₄²	—	2	2	no	—
272	#535	C₄×C₃₁	1	2	yes	complemented
273	#535	C₄×C₃₁	1	2	yes	complemented
274	#535	C₂²×C₃₁	1	1	yes	characteristic
275	#535	C₂²×C₃₁	1	1	yes	characteristic
276	#532	C₄×C₃₁	1	2	yes	complemented
277	#532	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#127	C₄×C₃₁	1	2	yes	complemented
279	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
280	#518	C₂²×C₃₁	1	2	yes	complemented
281	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
282	#421	C₂²×C₃₁	1	4	yes	complemented
283	#418	C₂²×C₃₁	1	4	yes	complemented
284	#413	—	2	2	no	—
285	#413	C₂²×C₃₁	1	2	yes	—
286	C₂×#100	—	2	2	no	—
287	C₂×#100	C₂²×C₃₁	1	2	yes	complemented
288	#410	C₂²×C₃₁	1	2	yes	—
289	#410	C₂²×C₃₁	1	2	yes	complemented
290	C₂×#99	—	2	4	no	—
291	#339	C₂²×C₃₁	1	4	yes	complemented
292	H(3,ℤ/4)	C₂²×C₃₁	1	4	yes	complemented
293	C₂×C₃₁	—	2	1	no	—
294	C₂×C₃₁	—	2	2	no	—
295	C₂×C₃₁	—	4	4	no	—
296	C₂×C₃₁	—	4	2	no	—
297	C₂×C₃₁	—	4	2	no	—
298	C₂×C₃₁	—	8	2	no	—
299	C₂×C₃₁	—	8	1	no	—
300	C₂×C₃₁	#2308	1	1	yes	characteristic
301	H(5,𝔽₂)	—	4	4	no	—
302	H(5,𝔽₂)	—	4	4	no	—
303	C₂×#41	—	2	1	no	—
304	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
305	C₂×#41	C₂×C₄×C₃₁	1	2	yes	complemented
306	C₂×#41	D₄×C₃₁	1	2	yes	—
307	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
308	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
310	C₂²×D₄	—	2	4	no	—
311	C₂²×D₄	—	2	2	no	—
312	C₂²×D₄	—	2	2	no	—
313	C₂²×D₄	—	2	1	no	—
314	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
315	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
316	C₂²×D₄	D₄×C₃₁	1	4	yes	—
317	C₂²×D₄	D₄×C₃₁	1	2	yes	—
318	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
320	#131	—	4	4	no	—
321	C₂×M₁₆	—	2	4	no	—
322	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
323	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
324	#127	—	2	4	no	—
325	#127	—	2	2	no	—
326	#127	C₂×C₄×C₃₁	1	2	yes	—
327	#127	C₂×C₄×C₃₁	1	2	yes	—
328	#127	C₂×C₄×C₃₁	1	2	yes	—
329	#127	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
330	#127	C₂×C₄×C₃₁	1	2	yes	complemented
331	#124	C₂×C₄×C₃₁	1	2	yes	—
332	#124	C₂×C₄×C₃₁	1	2	yes	complemented
333	#124	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
334	#124	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
335	#121	—	2	4	no	—
336	#121	—	2	4	no	—
337	#121	—	2	2	no	—
338	#121	—	2	2	no	—
339	#121	—	2	4	no	—
340	#121	—	2	2	no	—
341	#120	—	2	4	no	—
342	#120	—	2	2	no	—
343	#120	—	2	4	no	—
344	#120	—	2	2	no	—
345	C₄×Q₈	—	2	1	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	—	2	1	no	—
348	C₄×D₄	—	2	4	no	—
349	C₄×D₄	—	2	2	no	—
350	C₄×D₄	—	4	4	no	—
351	#117	C₂×C₄×C₃₁	1	2	yes	complemented
352	#117	C₂³×C₃₁	1	4	yes	—
353	#117	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×C₂²⋊C₄	—	2	4	no	—
355	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	complemented
356	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
357	#104	—	4	4	no	—
358	#102	—	4	4	no	—
359	#101	—	2	2	no	—
360	#100	—	2	2	no	—
361	#100	—	2	4	no	—
362	#99	—	2	4	no	—
363	#99	—	2	4	no	—
364	C₃₁	#21782	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	—	2	2	no	—
366	C₂⁴	—	2	1	no	—
367	C₂⁴	C₂²⋊C₄×C₃₁	1	4	yes	—
368	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
369	#41	—	2	4	no	—
370	#41	—	2	4	no	—
371	#41	—	2	4	no	—
372	#41	—	4	4	no	—
373	#41	—	4	4	no	—
374	#41	—	4	4	no	—
375	#41	—	4	4	no	—
376	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
377	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
378	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂×D₄	—	2	4	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	4	4	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	4	no	—
393	C₂×D₄	—	4	8	no	—
394	C₂×D₄	—	4	8	no	—
395	C₂×D₄	—	4	4	no	—
396	C₂×D₄	—	8	4	no	—
397	C₂×D₄	—	8	8	no	—
398	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
399	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
400	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
401	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
402	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
403	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
404	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
405	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
406	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
407	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	4	4	no	—
413	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
414	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
417	M₁₆	—	2	4	no	—
418	M₁₆	—	2	4	no	—
419	M₁₆	—	4	4	no	—
420	C₂×C₈	—	2	4	no	—
421	C₂×C₈	—	4	4	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	4	no	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	2	1	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	2	4	no	—
433	C₂²⋊C₄	—	4	4	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	4	no	—
437	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
438	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
439	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
440	C₂³	—	2	4	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	4	4	no	—
444	C₂³	—	4	4	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	8	2	no	—
447	C₂³	—	8	4	no	—
448	C₂³	—	8	2	no	—
449	C₂³	—	8	4	no	—
450	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
451	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
452	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
453	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
454	Q₈	—	2	2	no	—
455	Q₈	—	2	2	no	—
456	Q₈	—	2	2	no	—
457	D₄	—	2	4	no	—
458	D₄	—	2	4	no	—
459	D₄	—	2	4	no	—
460	D₄	—	2	2	no	—
461	D₄	—	2	4	no	—
462	D₄	—	2	2	no	—
463	D₄	—	2	4	no	—
464	D₄	—	2	2	no	—
465	D₄	—	4	4	no	—
466	D₄	—	4	4	no	—
467	D₄	—	4	4	no	—
468	D₄	—	4	8	no	—
469	D₄	—	4	4	no	—
470	D₄	—	4	4	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	2	no	—
473	D₄	—	8	4	no	—
474	D₄	—	8	4	no	—
475	D₄	—	8	4	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	4	4	no	—
483	C₂×C₄	—	4	4	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	16	4	no	—
489	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
490	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
491	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
494	C₈	—	4	4	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	8	4	no	—
507	C₂²	—	16	2	no	—
508	C₂²	—	16	4	no	—
509	C₂²	—	16	4	no	—
510	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
511	C₂²	C₃₁×#534	1	2	yes	—
512	C₄	—	2	2	no	—
513	C₄	—	2	2	no	—
514	C₄	—	2	2	no	—
515	C₄	—	4	2	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	8	4	no	—
519	C₄	C₃₁×#518	1	2	yes	—
520	C₄	C₃₁×#534	1	2	yes	—
521	C₂	—	2	1	no	—
522	C₂	—	2	2	no	—
523	C₂	—	4	4	no	—
524	C₂	—	4	2	no	—
525	C₂	—	4	2	no	—
526	C₂	—	8	2	no	—
527	C₂	—	8	1	no	—
528	C₂	C₃₁×#2308	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2308
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3498	C₂
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
2. C₃₁×#2888	C₂	semidirect
3. C₃₁×#2883	C₂	semidirect
4. C₃₁×#1911	C₂	semidirect
5. C₃₁×#1910	C₂	semidirect
6. C₃₁×#1887	C₂	semidirect
7. C₃₁×#1884	C₂	semidirect
8. C₃₁×#1827	C₂	semidirect
9. C₃₁×#1795	C₂	semidirect
12. C₃₁×#568	C₂²	semidirect
16. C₃₁×#535	C₄	semidirect
17. C₃₁×#535	C₄	semidirect
20. C₃₁×#532	C₄	semidirect
22. C₂×C₃₁×#127	C₄	semidirect
24. C₃₁×#518	C₂²	semidirect
26. C₃₁×#421	C₂²	semidirect
27. C₃₁×#418	C₂²	semidirect
31. C₂×C₃₁×#100	C₂²	semidirect
33. C₃₁×#410	C₂²	semidirect
35. C₃₁×#339	C₂²	semidirect
36. C₃₁×H(3,ℤ/4)	C₂²	semidirect
41. C₂×#41×C₃₁	C₂×C₄	semidirect
51. C₂²×D₄×C₃₁	C₂×C₄	semidirect
59. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
65. C₃₁×#127	C₂×C₄	semidirect
66. C₃₁×#127	C₂×C₄	semidirect
68. C₃₁×#124	C₂×C₄	semidirect
69. C₃₁×#124	C₂×C₄	semidirect
70. C₃₁×#124	C₂×C₄	semidirect
87. C₃₁×#117	C₂×C₄	semidirect
91. C₂×C₄²×C₃₁	C₂×C₄	semidirect
175. #21782	C₃₁	direct (N × H)
232. #2888	C₂×C₃₁	semidirect
233. #2883	C₂×C₃₁	semidirect
234. #1911	C₂×C₃₁	semidirect
235. #1910	C₂×C₃₁	semidirect
236. #1887	C₂×C₃₁	semidirect
237. #1884	C₂×C₃₁	semidirect
238. #1827	C₂×C₃₁	semidirect
239. #1795	C₂×C₃₁	semidirect
268. #568	C₂²×C₃₁	semidirect
272. #535	C₄×C₃₁	semidirect
273. #535	C₄×C₃₁	semidirect
276. #532	C₄×C₃₁	semidirect
278. C₂×#127	C₄×C₃₁	semidirect
280. #518	C₂²×C₃₁	semidirect
282. #421	C₂²×C₃₁	semidirect
283. #418	C₂²×C₃₁	semidirect
287. C₂×#100	C₂²×C₃₁	semidirect
289. #410	C₂²×C₃₁	semidirect
291. #339	C₂²×C₃₁	semidirect
292. H(3,ℤ/4)	C₂²×C₃₁	semidirect
305. C₂×#41	C₂×C₄×C₃₁	semidirect
315. C₂²×D₄	C₂×C₄×C₃₁	semidirect
323. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
329. #127	C₂×C₄×C₃₁	semidirect
330. #127	C₂×C₄×C₃₁	semidirect
332. #124	C₂×C₄×C₃₁	semidirect
333. #124	C₂×C₄×C₃₁	semidirect
334. #124	C₂×C₄×C₃₁	semidirect
351. #117	C₂×C₄×C₃₁	semidirect
355. C₂×C₄²	C₂×C₄×C₃₁	semidirect
364. C₃₁	#21782	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×#3498	2
C₃₁×#2888	2
C₃₁×#2883	2
C₃₁×#1911	2
C₃₁×#1910	2
C₃₁×#1887	2
C₃₁×#1884	2
C₃₁×#1827	2
C₃₁×#1795	2
#21782	31

Subgroups — C31×#21782 #21478541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21782 #21478541