Subgroups — C₃₁×#21244 #21478003

Order 7936.

1460 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#575	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#421	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#410	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1976	C₂	1	8	yes	maximal, complemented
7	C₂²×#41×C₃₁	C₂²	1	2	yes	—
8	C₃₁×#575	—	2	2	no	—
9	C₃₁×#575	—	2	2	no	—
10	C₃₁×#575	—	2	4	no	—
11	C₂×C₃₁×#137	C₄	1	4	yes	complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₄	1	4	yes	complemented
13	C₂×C₃₁×#135	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₃₁×#428	—	2	4	no	—
19	C₃₁×#427	C₂²	1	4	yes	complemented
20	C₃₁×#421	—	2	4	no	—
21	C₃₁×#421	C₂²	1	8	yes	complemented
22	C₂×C₃₁×#104	—	2	4	no	—
23	C₃₁×#410	—	2	4	no	—
24	C₃₁×#410	C₂²	1	8	yes	complemented
25	C₂×C₃₁×#99	—	2	4	no	—
26	C₃₁×#405	—	2	2	no	—
27	C₃₁×#405	C₂²	1	4	yes	complemented
28	C₄×M₁₆×C₃₁	—	2	2	no	—
29	C₂×#41×C₃₁	—	2	2	no	—
30	C₂×#41×C₃₁	—	2	2	no	—
31	C₂×#41×C₃₁	—	2	2	no	—
32	C₂×#41×C₃₁	—	2	2	no	—
33	C₂×#41×C₃₁	—	2	2	no	—
34	C₂×#41×C₃₁	—	2	2	no	—
35	C₂×#41×C₃₁	—	2	2	no	—
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂×#41×C₃₁	—	2	2	no	—
38	C₂×#41×C₃₁	—	2	2	no	—
39	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
40	C₂×#41×C₃₁	C₂³	1	4	yes	—
41	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂³×C₄×C₃₁	—	2	2	no	—
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#137	—	2	8	no	—
47	C₃₁×#137	—	2	8	no	—
48	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
49	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
50	C₃₁×#135	—	2	8	no	—
51	C₃₁×#135	—	2	8	no	—
52	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
53	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
54	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
55	C₂×M₁₆×C₃₁	—	2	1	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
59	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
60	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
61	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
63	C₂²×C₈×C₃₁	—	2	1	no	—
64	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
65	C₃₁×#117	—	2	1	no	—
66	C₃₁×#117	—	2	1	no	—
67	C₃₁×#117	—	2	2	no	—
68	C₃₁×#117	—	2	2	no	—
69	C₃₁×#117	—	2	2	no	—
70	C₃₁×#117	D₄	1	2	yes	—
71	C₃₁×#117	C₂³	1	4	yes	—
72	C₃₁×#117	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
77	C₂×C₄²×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	—	2	2	no	—
79	C₃₁×#107	—	2	4	no	—
80	C₃₁×#106	—	2	4	no	—
81	C₃₁×#104	—	2	16	no	—
82	C₃₁×#99	—	2	16	no	—
83	C₃₁×#97	—	2	4	no	—
84	C₄×C₈×C₃₁	—	2	4	no	—
85	C₂⁴×C₃₁	—	2	2	no	—
86	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
87	#41×C₃₁	—	2	4	no	—
88	#41×C₃₁	—	2	4	no	—
89	#41×C₃₁	—	2	4	no	—
90	#41×C₃₁	—	2	4	no	—
91	#41×C₃₁	—	2	4	no	—
92	#41×C₃₁	—	2	4	no	—
93	#41×C₃₁	—	2	4	no	—
94	#41×C₃₁	—	2	4	no	—
95	#41×C₃₁	—	4	2	no	—
96	#41×C₃₁	—	4	2	no	—
97	#41×C₃₁	—	4	2	no	—
98	#41×C₃₁	—	4	2	no	—
99	#41×C₃₁	—	4	2	no	—
100	#41×C₃₁	—	4	2	no	—
101	#41×C₃₁	—	4	2	no	—
102	#41×C₃₁	—	4	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
109	C₂×D₄×C₃₁	—	2	4	no	—
110	C₂×D₄×C₃₁	—	2	4	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	4	2	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	—	4	2	no	—
119	C₂×D₄×C₃₁	—	4	2	no	—
120	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	4	no	—
125	C₂²×C₄×C₃₁	—	4	2	no	—
126	C₂²×C₄×C₃₁	—	4	2	no	—
127	C₂²×C₄×C₃₁	—	4	2	no	—
128	C₂²×C₄×C₃₁	—	4	2	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	#41	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₃₁	—	2	8	no	—
141	SD₁₆×C₃₁	—	2	16	no	—
142	D₈×C₃₁	—	2	8	no	—
143	M₁₆×C₃₁	—	2	4	no	—
144	M₁₆×C₃₁	—	2	2	no	—
145	M₁₆×C₃₁	C₂²⋊C₄	1	8	yes	—
146	M₁₆×C₃₁	C₂²⋊C₄	1	8	yes	—
147	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
148	C₂×C₈×C₃₁	—	2	4	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
151	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
152	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	4	2	no	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₂²⋊C₄×C₃₁	—	4	2	no	—
160	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
161	C₄²×C₃₁	—	2	2	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	4	no	—
164	C₄²×C₃₁	—	4	2	no	—
165	#21244	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₃₁	—	2	4	no	—
167	C₂³×C₃₁	—	4	2	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	—	8	1	no	—
170	C₂³×C₃₁	—	8	4	no	—
171	C₂³×C₃₁	C₄×D₄	1	2	yes	—
172	C₂³×C₃₁	#121	1	1	yes	characteristic
173	C₂³×C₃₁	#121	1	1	yes	characteristic
174	C₂³×C₃₁	#127	1	2	yes	—
175	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	Q₈×C₃₁	—	2	4	no	—
177	Q₈×C₃₁	—	2	4	no	—
178	Q₈×C₃₁	—	4	2	no	—
179	Q₈×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	2	4	no	—
181	D₄×C₃₁	—	2	4	no	—
182	D₄×C₃₁	—	4	4	no	—
183	D₄×C₃₁	—	4	4	no	—
184	D₄×C₃₁	—	4	4	no	—
185	D₄×C₃₁	—	4	4	no	—
186	D₄×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	4	no	—
193	C₂×C₄×C₃₁	—	4	4	no	—
194	C₂×C₄×C₃₁	—	4	4	no	—
195	C₂×C₄×C₃₁	—	4	4	no	—
196	C₂×C₄×C₃₁	—	4	4	no	—
197	C₂×C₄×C₃₁	—	8	2	no	—
198	C₂×C₄×C₃₁	—	8	4	no	—
199	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
200	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
201	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
202	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
203	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
204	C₂×C₄×C₃₁	#121	1	2	yes	—
205	C₂×C₄×C₃₁	#121	1	2	yes	—
206	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
207	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
208	C₂×C₄×C₃₁	#124	1	2	yes	—
209	C₂×C₄×C₃₁	#124	1	2	yes	—
210	C₂×C₄×C₃₁	#127	1	2	yes	—
211	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₈×C₃₁	—	2	4	no	—
218	C₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
219	C₂×#575	C₂×C₃₁	1	1	yes	characteristic
220	#2911	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂×#421	C₂×C₃₁	1	2	yes	complemented
222	C₂×#410	C₂×C₃₁	1	2	yes	complemented
223	#2873	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#1976	C₂×C₃₁	1	8	yes	complemented
225	C₂²×C₃₁	—	4	1	no	—
226	C₂²×C₃₁	—	4	2	no	—
227	C₂²×C₃₁	—	4	2	no	—
228	C₂²×C₃₁	—	4	1	no	—
229	C₂²×C₃₁	—	4	1	no	—
230	C₂²×C₃₁	—	4	4	no	—
231	C₂²×C₃₁	—	8	4	no	—
232	C₂²×C₃₁	—	8	4	no	—
233	C₂²×C₃₁	—	8	4	no	—
234	C₂²×C₃₁	#390	1	2	yes	—
235	C₂²×C₃₁	#390	1	2	yes	—
236	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
237	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
238	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
239	C₄×C₃₁	—	4	4	no	—
240	C₄×C₃₁	—	4	4	no	—
241	C₄×C₃₁	—	8	4	no	—
242	C₄×C₃₁	#390	1	2	yes	—
243	C₄×C₃₁	#390	1	2	yes	—
244	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
245	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
246	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
247	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
248	C₂²×#41	C₂²×C₃₁	1	2	yes	—
249	#575	—	2	2	no	—
250	#575	—	2	2	no	—
251	#575	—	2	4	no	—
252	C₂×#137	C₄×C₃₁	1	4	yes	complemented
253	C₂×AGL(1,ℤ/8)	C₄×C₃₁	1	4	yes	complemented
254	C₂×#135	C₂²×C₃₁	1	4	yes	—
255	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
256	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×#117	C₂²×C₃₁	1	2	yes	—
258	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
259	#428	—	2	4	no	—
260	#427	C₂²×C₃₁	1	4	yes	complemented
261	#421	—	2	4	no	—
262	#421	C₂²×C₃₁	1	8	yes	complemented
263	C₂×#104	—	2	4	no	—
264	#410	—	2	4	no	—
265	#410	C₂²×C₃₁	1	8	yes	complemented
266	C₂×#99	—	2	4	no	—
267	#405	—	2	2	no	—
268	#405	C₂²×C₃₁	1	4	yes	complemented
269	C₄×M₁₆	—	2	2	no	—
270	C₂×C₃₁	—	2	1	no	—
271	C₂×C₃₁	—	2	1	no	—
272	C₂×C₃₁	—	2	1	no	—
273	C₂×C₃₁	—	2	2	no	—
274	C₂×C₃₁	—	2	2	no	—
275	C₂×C₃₁	—	8	4	no	—
276	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
277	C₂×#41	—	2	2	no	—
278	C₂×#41	—	2	2	no	—
279	C₂×#41	—	2	2	no	—
280	C₂×#41	—	2	2	no	—
281	C₂×#41	—	2	2	no	—
282	C₂×#41	—	2	2	no	—
283	C₂×#41	—	2	2	no	—
284	C₂×#41	—	2	2	no	—
285	C₂×#41	—	2	2	no	—
286	C₂×#41	—	2	2	no	—
287	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
288	C₂×#41	C₂³×C₃₁	1	4	yes	—
289	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
290	C₂²×D₄	—	2	2	no	—
291	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
292	C₂³×C₄	—	2	2	no	—
293	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
294	#137	—	2	8	no	—
295	#137	—	2	8	no	—
296	AGL(1,ℤ/8)	—	2	8	no	—
297	AGL(1,ℤ/8)	—	2	8	no	—
298	#135	—	2	8	no	—
299	#135	—	2	8	no	—
300	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
301	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
302	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
303	C₂×M₁₆	—	2	1	no	—
304	C₂×M₁₆	—	2	2	no	—
305	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
306	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
307	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
308	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
309	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
310	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
311	C₂²×C₈	—	2	1	no	—
312	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
313	#117	—	2	1	no	—
314	#117	—	2	1	no	—
315	#117	—	2	2	no	—
316	#117	—	2	2	no	—
317	#117	—	2	2	no	—
318	#117	D₄×C₃₁	1	2	yes	—
319	#117	C₂³×C₃₁	1	4	yes	—
320	#117	C₂³×C₃₁	1	2	yes	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	—	2	2	no	—
324	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	—	2	2	no	—
327	#107	—	2	4	no	—
328	#106	—	2	4	no	—
329	#104	—	2	16	no	—
330	#99	—	2	16	no	—
331	#97	—	2	4	no	—
332	C₄×C₈	—	2	4	no	—
333	C₃₁	#21244	1	1	yes	characteristic, direct product factor, Sylow
334	C₂⁴	—	2	2	no	—
335	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
336	#41	—	2	4	no	—
337	#41	—	2	4	no	—
338	#41	—	2	4	no	—
339	#41	—	2	4	no	—
340	#41	—	2	4	no	—
341	#41	—	2	4	no	—
342	#41	—	2	4	no	—
343	#41	—	2	4	no	—
344	#41	—	4	2	no	—
345	#41	—	4	2	no	—
346	#41	—	4	2	no	—
347	#41	—	4	2	no	—
348	#41	—	4	2	no	—
349	#41	—	4	2	no	—
350	#41	—	4	2	no	—
351	#41	—	4	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
358	C₂×D₄	—	2	4	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	4	2	no	—
366	C₂×D₄	—	4	2	no	—
367	C₂×D₄	—	4	2	no	—
368	C₂×D₄	—	4	2	no	—
369	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	—	4	2	no	—
376	C₂²×C₄	—	4	2	no	—
377	C₂²×C₄	—	4	2	no	—
378	C₂²×C₄	—	4	2	no	—
379	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
380	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
385	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
387	C₂²×C₄	#41×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
389	Q₁₆	—	2	8	no	—
390	SD₁₆	—	2	16	no	—
391	D₈	—	2	8	no	—
392	M₁₆	—	2	4	no	—
393	M₁₆	—	2	2	no	—
394	M₁₆	C₂²⋊C₄×C₃₁	1	8	yes	—
395	M₁₆	C₂²⋊C₄×C₃₁	1	8	yes	—
396	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
397	C₂×C₈	—	2	4	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
400	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
401	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	4	no	—
408	C₂²⋊C₄	—	4	2	no	—
409	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
410	C₄²	—	2	2	no	—
411	C₄²	—	2	2	no	—
412	C₄²	—	2	4	no	—
413	C₄²	—	4	2	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	8	1	no	—
418	C₂³	—	8	4	no	—
419	C₂³	C₄×D₄×C₃₁	1	2	yes	—
420	C₂³	C₃₁×#121	1	1	yes	characteristic
421	C₂³	C₃₁×#121	1	1	yes	characteristic
422	C₂³	C₃₁×#127	1	2	yes	—
423	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
424	Q₈	—	2	4	no	—
425	Q₈	—	2	4	no	—
426	Q₈	—	4	2	no	—
427	Q₈	—	4	2	no	—
428	D₄	—	2	4	no	—
429	D₄	—	2	4	no	—
430	D₄	—	4	4	no	—
431	D₄	—	4	4	no	—
432	D₄	—	4	4	no	—
433	D₄	—	4	4	no	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	4	no	—
442	C₂×C₄	—	4	4	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	—	4	4	no	—
445	C₂×C₄	—	8	2	no	—
446	C₂×C₄	—	8	4	no	—
447	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
448	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
449	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
450	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
451	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
452	C₂×C₄	C₃₁×#121	1	2	yes	—
453	C₂×C₄	C₃₁×#121	1	2	yes	—
454	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
455	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
456	C₂×C₄	C₃₁×#124	1	2	yes	—
457	C₂×C₄	C₃₁×#124	1	2	yes	—
458	C₂×C₄	C₃₁×#127	1	2	yes	—
459	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
464	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
465	C₈	—	2	4	no	—
466	C₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	1	no	—
471	C₂²	—	4	1	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	8	4	no	—
474	C₂²	—	8	4	no	—
475	C₂²	—	8	4	no	—
476	C₂²	C₃₁×#390	1	2	yes	—
477	C₂²	C₃₁×#390	1	2	yes	—
478	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
479	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
480	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
481	C₄	—	4	4	no	—
482	C₄	—	4	4	no	—
483	C₄	—	8	4	no	—
484	C₄	C₃₁×#390	1	2	yes	—
485	C₄	C₃₁×#390	1	2	yes	—
486	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
487	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
488	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
489	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
490	C₂	—	2	1	no	—
491	C₂	—	2	1	no	—
492	C₂	—	2	1	no	—
493	C₂	—	2	2	no	—
494	C₂	—	2	2	no	—
495	C₂	—	8	4	no	—
496	C₂	C₂×C₃₁×#390	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×#390
prime radical	C₂×C₃₁×#575	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2911	C₂	semidirect
3. C₂×C₃₁×#421	C₂	semidirect
4. C₂×C₃₁×#410	C₂	semidirect
5. C₃₁×#2873	C₂	semidirect
6. C₃₁×#1976	C₂	semidirect
11. C₂×C₃₁×#137	C₄	semidirect
12. C₂×C₃₁×AGL(1,ℤ/8)	C₄	semidirect
19. C₃₁×#427	C₂²	semidirect
21. C₃₁×#421	C₂²	semidirect
24. C₃₁×#410	C₂²	semidirect
27. C₃₁×#405	C₂²	semidirect
52. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
53. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
54. C₂×D₈×C₃₁	C₂×C₄	semidirect
165. #21244	C₃₁	direct (N × H)
220. #2911	C₂×C₃₁	semidirect
221. C₂×#421	C₂×C₃₁	semidirect
222. C₂×#410	C₂×C₃₁	semidirect
223. #2873	C₂×C₃₁	semidirect
224. #1976	C₂×C₃₁	semidirect
252. C₂×#137	C₄×C₃₁	semidirect
253. C₂×AGL(1,ℤ/8)	C₄×C₃₁	semidirect
260. #427	C₂²×C₃₁	semidirect
262. #421	C₂²×C₃₁	semidirect
265. #410	C₂²×C₃₁	semidirect
268. #405	C₂²×C₃₁	semidirect
300. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
301. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
302. C₂×D₈	C₂×C₄×C₃₁	semidirect
333. C₃₁	#21244	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#575	2
C₃₁×#2911	2
C₂×C₃₁×#421	2
C₂×C₃₁×#410	2
C₃₁×#2873	2
C₃₁×#1976	2
#21244	31

Subgroups — C31×#21244 #21478003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21244 #21478003