Subgroups — C₃₁×#22286 #21479045

Order 7936.

820 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3155	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1911	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1910	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1904	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1903	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1833	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#568	C₂²	1	2	yes	—
16	C₃₁×#535	C₂²	1	2	yes	—
17	C₃₁×#518	C₂²	1	2	yes	—
18	C₃₁×#496	C₄	1	4	yes	complemented
19	C₃₁×#495	C₄	1	2	yes	complemented
20	C₃₁×#493	C₄	1	2	yes	complemented
21	C₃₁×#444	—	2	2	no	—
22	C₃₁×#443	—	2	2	no	—
23	C₃₁×#442	—	2	2	no	—
24	C₄×D₈×C₃₁	—	2	2	no	—
25	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
26	C₃₁×#427	C₂²	1	1	yes	characteristic
27	C₃₁×#421	C₂²	1	2	yes	—
28	C₃₁×#421	C₂²	1	2	yes	—
29	C₃₁×#418	C₂²	1	2	yes	—
30	C₃₁×#418	C₂²	1	2	yes	—
31	C₂×C₃₁×#100	C₂²	1	2	yes	—
32	C₃₁×#410	C₂²	1	2	yes	—
33	C₃₁×#405	C₂²	1	1	yes	characteristic
34	C₃₁×#405	C₂²	1	1	yes	characteristic
35	C₃₁×#405	C₂²	1	1	yes	characteristic
36	C₃₁×#344	C₂²	1	2	yes	—
37	C₃₁×#338	C₂²	1	1	yes	characteristic
38	C₃₁×H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₃₁×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
40	C₃₁×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
41	C₃₁×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
42	C₂×#41×C₃₁	C₂³	1	2	yes	—
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
45	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
46	C₃₁×#135	—	2	1	no	—
47	C₃₁×#135	—	2	1	no	—
48	C₃₁×#135	—	2	1	no	—
49	C₃₁×#135	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	complemented
52	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₂×D₈×C₃₁	—	2	1	no	—
55	C₂×D₈×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	2	1	no	—
57	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
59	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
60	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
61	C₃₁×#131	—	2	4	no	—
62	C₃₁×#131	—	4	2	no	—
63	C₂×M₁₆×C₃₁	—	4	2	no	—
64	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
66	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×#127	C₂×C₄	1	4	yes	complemented
70	C₃₁×#124	C₂×C₄	1	4	yes	complemented
71	C₃₁×#121	—	4	2	no	—
72	C₃₁×#120	—	4	2	no	—
73	C₄×D₄×C₃₁	—	2	4	no	—
74	C₄×D₄×C₃₁	—	4	2	no	—
75	C₃₁×#117	C₂³	1	1	yes	characteristic
76	C₃₁×#117	C₂³	1	2	yes	—
77	C₃₁×#117	C₂³	1	1	yes	characteristic
78	C₃₁×#117	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
80	C₃₁×#108	—	2	1	no	—
81	C₃₁×#108	—	2	1	no	—
82	C₃₁×#107	—	2	1	no	—
83	C₃₁×#106	—	2	1	no	—
84	C₃₁×#104	—	2	4	no	—
85	C₃₁×#104	—	4	2	no	—
86	C₃₁×#102	—	2	4	no	—
87	C₃₁×#102	—	4	2	no	—
88	C₃₁×#100	—	4	2	no	—
89	C₃₁×#99	—	4	2	no	—
90	C₃₁×#97	—	2	1	no	—
91	C₃₁×#97	C₂×C₄	1	2	yes	complemented
92	C₃₁×#97	D₄	1	2	yes	complemented
93	C₄×C₈×C₃₁	—	2	1	no	—
94	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	complemented
95	C₄×C₈×C₃₁	D₄	1	2	yes	complemented
96	C₂⁴×C₃₁	—	2	2	no	—
97	#41×C₃₁	—	2	2	no	—
98	#41×C₃₁	—	2	2	no	—
99	#41×C₃₁	—	2	2	no	—
100	#41×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	4	4	no	—
109	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
110	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	4	4	no	—
113	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	Q₁₆×C₃₁	—	2	1	no	—
115	Q₁₆×C₃₁	—	2	1	no	—
116	SD₁₆×C₃₁	—	2	2	no	—
117	SD₁₆×C₃₁	—	2	2	no	—
118	SD₁₆×C₃₁	—	4	4	no	—
119	D₈×C₃₁	—	2	2	no	—
120	D₈×C₃₁	—	2	2	no	—
121	D₈×C₃₁	—	2	2	no	—
122	D₈×C₃₁	—	2	2	no	—
123	D₈×C₃₁	—	2	1	no	—
124	D₈×C₃₁	—	2	1	no	—
125	D₈×C₃₁	—	4	4	no	—
126	M₁₆×C₃₁	—	2	1	no	—
127	M₁₆×C₃₁	—	2	1	no	—
128	M₁₆×C₃₁	—	2	1	no	—
129	M₁₆×C₃₁	—	2	1	no	—
130	M₁₆×C₃₁	—	2	2	no	—
131	M₁₆×C₃₁	—	2	1	no	—
132	M₁₆×C₃₁	—	2	2	no	—
133	M₁₆×C₃₁	—	2	1	no	—
134	M₁₆×C₃₁	—	4	4	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	4	4	no	—
138	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
145	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₂²⋊C₄×C₃₁	—	2	1	no	—
151	C₂²⋊C₄×C₃₁	—	2	1	no	—
152	C₂²⋊C₄×C₃₁	—	2	1	no	—
153	C₂²⋊C₄×C₃₁	—	2	2	no	—
154	C₂²⋊C₄×C₃₁	—	4	4	no	—
155	C₂²⋊C₄×C₃₁	—	4	4	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
159	C₄²×C₃₁	C₂×D₄	1	2	yes	—
160	#22286	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	4	no	—
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	16	2	no	—
166	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
167	Q₈×C₃₁	—	2	2	no	—
168	Q₈×C₃₁	—	2	2	no	—
169	D₄×C₃₁	—	2	4	no	—
170	D₄×C₃₁	—	2	2	no	—
171	D₄×C₃₁	—	2	2	no	—
172	D₄×C₃₁	—	2	4	no	—
173	D₄×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	16	4	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	16	4	no	—
186	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
190	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
192	C₈×C₃₁	—	2	2	no	—
193	C₈×C₃₁	—	2	2	no	—
194	C₈×C₃₁	—	4	2	no	—
195	C₈×C₃₁	—	4	1	no	—
196	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
197	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
198	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
199	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
200	#3155	C₂×C₃₁	1	1	yes	characteristic
201	#2957	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1976	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1911	C₂×C₃₁	1	2	yes	complemented
206	#1910	C₂×C₃₁	1	2	yes	complemented
207	#1904	C₂×C₃₁	1	2	yes	complemented
208	#1903	C₂×C₃₁	1	2	yes	complemented
209	#1833	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1747	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂²×C₃₁	—	4	1	no	—
212	C₂²×C₃₁	—	4	2	no	—
213	C₂²×C₃₁	—	4	4	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	—	8	4	no	—
216	C₂²×C₃₁	—	8	4	no	—
217	C₂²×C₃₁	—	16	2	no	—
218	C₂²×C₃₁	#518	1	1	yes	characteristic
219	C₂²×C₃₁	#535	1	1	yes	characteristic
220	C₂²×C₃₁	#540	1	1	yes	characteristic
221	C₄×C₃₁	—	4	2	no	—
222	C₄×C₃₁	—	4	2	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	8	1	no	—
225	C₄×C₃₁	—	8	2	no	—
226	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₃₁	#518	1	1	yes	characteristic
228	C₄×C₃₁	#535	1	1	yes	characteristic
229	C₄×C₃₁	#540	1	1	yes	characteristic
230	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
231	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
232	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
233	#568	C₂²×C₃₁	1	2	yes	—
234	#535	C₂²×C₃₁	1	2	yes	—
235	#518	C₂²×C₃₁	1	2	yes	—
236	#496	C₄×C₃₁	1	4	yes	complemented
237	#495	C₄×C₃₁	1	2	yes	complemented
238	#493	C₄×C₃₁	1	2	yes	complemented
239	#444	—	2	2	no	—
240	#443	—	2	2	no	—
241	#442	—	2	2	no	—
242	C₄×D₈	—	2	2	no	—
243	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
244	#427	C₂²×C₃₁	1	1	yes	characteristic
245	#421	C₂²×C₃₁	1	2	yes	—
246	#421	C₂²×C₃₁	1	2	yes	—
247	#418	C₂²×C₃₁	1	2	yes	—
248	#418	C₂²×C₃₁	1	2	yes	—
249	C₂×#100	C₂²×C₃₁	1	2	yes	—
250	#410	C₂²×C₃₁	1	2	yes	—
251	#405	C₂²×C₃₁	1	1	yes	characteristic
252	#405	C₂²×C₃₁	1	1	yes	characteristic
253	#405	C₂²×C₃₁	1	1	yes	characteristic
254	#344	C₂²×C₃₁	1	2	yes	—
255	#338	C₂²×C₃₁	1	1	yes	characteristic
256	H(3,ℤ/4)	C₂²×C₃₁	1	2	yes	—
257	H(3,ℤ/4)	C₂²×C₃₁	1	1	yes	characteristic
258	H(3,ℤ/4)	C₂²×C₃₁	1	1	yes	characteristic
259	H(3,ℤ/4)	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₃₁	—	2	1	no	—
261	C₂×C₃₁	—	2	1	no	—
262	C₂×C₃₁	—	2	1	no	—
263	C₂×C₃₁	—	8	2	no	—
264	C₂×C₃₁	—	8	4	no	—
265	C₂×C₃₁	#2328	1	1	yes	characteristic
266	C₂×#41	C₂³×C₃₁	1	2	yes	—
267	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
268	AGL(1,ℤ/8)	—	4	2	no	—
269	AGL(1,ℤ/8)	—	4	2	no	—
270	#135	—	2	1	no	—
271	#135	—	2	1	no	—
272	#135	—	2	1	no	—
273	#135	—	2	1	no	—
274	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
275	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	complemented
276	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
277	C₂×D₈	—	2	1	no	—
278	C₂×D₈	—	2	1	no	—
279	C₂×D₈	—	2	1	no	—
280	C₂×D₈	—	2	1	no	—
281	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
282	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
283	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
284	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
285	#131	—	2	4	no	—
286	#131	—	4	2	no	—
287	C₂×M₁₆	—	4	2	no	—
288	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
289	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
291	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	#127	C₂×C₄×C₃₁	1	4	yes	complemented
294	#124	C₂×C₄×C₃₁	1	4	yes	complemented
295	#121	—	4	2	no	—
296	#120	—	4	2	no	—
297	C₄×D₄	—	2	4	no	—
298	C₄×D₄	—	4	2	no	—
299	#117	C₂³×C₃₁	1	2	yes	—
300	#117	C₂³×C₃₁	1	1	yes	characteristic
301	#117	C₂³×C₃₁	1	1	yes	characteristic
302	#117	C₂³×C₃₁	1	1	yes	characteristic
303	C₂×C₂²⋊C₄	—	4	2	no	—
304	#108	—	2	1	no	—
305	#108	—	2	1	no	—
306	#107	—	2	1	no	—
307	#106	—	2	1	no	—
308	#104	—	2	4	no	—
309	#104	—	4	2	no	—
310	#102	—	2	4	no	—
311	#102	—	4	2	no	—
312	#100	—	4	2	no	—
313	#99	—	4	2	no	—
314	#97	—	2	1	no	—
315	#97	C₂×C₄×C₃₁	1	2	yes	complemented
316	#97	D₄×C₃₁	1	2	yes	complemented
317	C₄×C₈	—	2	1	no	—
318	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	complemented
319	C₄×C₈	D₄×C₃₁	1	2	yes	complemented
320	C₃₁	#22286	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	—	2	2	no	—
322	#41	—	2	2	no	—
323	#41	—	2	2	no	—
324	#41	—	2	2	no	—
325	#41	—	2	2	no	—
326	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
335	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	4	no	—
338	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
339	Q₁₆	—	2	1	no	—
340	Q₁₆	—	2	1	no	—
341	SD₁₆	—	2	2	no	—
342	SD₁₆	—	2	2	no	—
343	SD₁₆	—	4	4	no	—
344	D₈	—	2	2	no	—
345	D₈	—	2	2	no	—
346	D₈	—	2	2	no	—
347	D₈	—	2	2	no	—
348	D₈	—	2	1	no	—
349	D₈	—	2	1	no	—
350	D₈	—	4	4	no	—
351	M₁₆	—	2	1	no	—
352	M₁₆	—	2	1	no	—
353	M₁₆	—	2	1	no	—
354	M₁₆	—	2	1	no	—
355	M₁₆	—	2	2	no	—
356	M₁₆	—	2	1	no	—
357	M₁₆	—	2	2	no	—
358	M₁₆	—	2	1	no	—
359	M₁₆	—	4	4	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	4	4	no	—
363	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
364	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
365	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
366	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
367	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
370	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₂²⋊C₄	—	2	1	no	—
376	C₂²⋊C₄	—	2	1	no	—
377	C₂²⋊C₄	—	2	1	no	—
378	C₂²⋊C₄	—	2	2	no	—
379	C₂²⋊C₄	—	4	4	no	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	2	2	no	—
383	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
384	C₄²	C₂×D₄×C₃₁	1	2	yes	—
385	C₂³	—	2	4	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	16	2	no	—
390	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
391	Q₈	—	2	2	no	—
392	Q₈	—	2	2	no	—
393	D₄	—	2	4	no	—
394	D₄	—	2	2	no	—
395	D₄	—	2	2	no	—
396	D₄	—	2	4	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	16	4	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	16	4	no	—
410	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
411	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
414	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
415	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	4	2	no	—
419	C₈	—	4	1	no	—
420	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
421	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
422	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
423	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
424	C₂²	—	4	1	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	4	no	—
427	C₂²	—	8	2	no	—
428	C₂²	—	8	4	no	—
429	C₂²	—	8	4	no	—
430	C₂²	—	16	2	no	—
431	C₂²	C₃₁×#518	1	1	yes	characteristic
432	C₂²	C₃₁×#535	1	1	yes	characteristic
433	C₂²	C₃₁×#540	1	1	yes	characteristic
434	C₄	—	4	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	2	no	—
437	C₄	—	8	2	no	—
438	C₄	—	8	1	no	—
439	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
440	C₄	C₃₁×#518	1	1	yes	characteristic
441	C₄	C₃₁×#535	1	1	yes	characteristic
442	C₄	C₃₁×#540	1	1	yes	characteristic
443	C₂	—	2	1	no	—
444	C₂	—	2	1	no	—
445	C₂	—	2	1	no	—
446	C₂	—	8	4	no	—
447	C₂	—	8	2	no	—
448	C₂	C₃₁×#2328	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2328
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3155	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×#2957	C₂	semidirect
3. C₃₁×#2948	C₂	semidirect
4. C₃₁×#1978	C₂	semidirect
5. C₃₁×#1976	C₂	semidirect
6. C₃₁×#1911	C₂	semidirect
7. C₃₁×#1910	C₂	semidirect
8. C₃₁×#1904	C₂	semidirect
9. C₃₁×#1903	C₂	semidirect
10. C₃₁×#1833	C₂	semidirect
11. C₃₁×#1747	C₂	semidirect
18. C₃₁×#496	C₄	semidirect
19. C₃₁×#495	C₄	semidirect
20. C₃₁×#493	C₄	semidirect
50. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
51. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
52. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
59. C₂×D₈×C₃₁	C₂×C₄	semidirect
60. C₂×D₈×C₃₁	C₂×C₄	semidirect
69. C₃₁×#127	C₂×C₄	semidirect
70. C₃₁×#124	C₂×C₄	semidirect
91. C₃₁×#97	C₂×C₄	semidirect
92. C₃₁×#97	D₄	semidirect
94. C₄×C₈×C₃₁	C₂×C₄	semidirect
95. C₄×C₈×C₃₁	D₄	semidirect
160. #22286	C₃₁	direct (N × H)
201. #2957	C₂×C₃₁	semidirect
202. #2948	C₂×C₃₁	semidirect
203. #1978	C₂×C₃₁	semidirect
204. #1976	C₂×C₃₁	semidirect
205. #1911	C₂×C₃₁	semidirect
206. #1910	C₂×C₃₁	semidirect
207. #1904	C₂×C₃₁	semidirect
208. #1903	C₂×C₃₁	semidirect
209. #1833	C₂×C₃₁	semidirect
210. #1747	C₂×C₃₁	semidirect
236. #496	C₄×C₃₁	semidirect
237. #495	C₄×C₃₁	semidirect
238. #493	C₄×C₃₁	semidirect
274. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
275. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
276. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
283. C₂×D₈	C₂×C₄×C₃₁	semidirect
284. C₂×D₈	C₂×C₄×C₃₁	semidirect
293. #127	C₂×C₄×C₃₁	semidirect
294. #124	C₂×C₄×C₃₁	semidirect
315. #97	C₂×C₄×C₃₁	semidirect
316. #97	D₄×C₃₁	semidirect
318. C₄×C₈	C₂×C₄×C₃₁	semidirect
319. C₄×C₈	D₄×C₃₁	semidirect
320. C₃₁	#22286	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×#3155	2
C₃₁×#2957	2
C₃₁×#2948	2
C₃₁×#1978	2
C₃₁×#1976	2
C₃₁×#1911	2
C₃₁×#1910	2
C₃₁×#1904	2
C₃₁×#1903	2
C₃₁×#1833	2
C₃₁×#1747	2
#22286	31

Subgroups — C31×#22286 #21479045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22286 #21479045