Subgroups — C₃₁×#17685 #21474444

Order 7936.

1152 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3500	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2970	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1533	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1530	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1527	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1525	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#329	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×2₋¹⁺⁴	C₄	1	1	yes	characteristic
10	C₂×C₃₁×H(5,𝔽₂)	C₄	1	1	yes	characteristic
11	C₃₁×#548	—	2	2	no	—
12	C₃₁×#546	—	2	2	no	—
13	C₃₁×#546	—	2	4	no	—
14	C₃₁×#536	C₄	1	1	yes	characteristic
15	C₃₁×#536	C₄	1	2	yes	—
16	C₃₁×#535	C₄	1	1	yes	characteristic
17	C₃₁×#535	C₄	1	2	yes	—
18	C₃₁×#532	C₂²	1	1	yes	characteristic
19	C₃₁×#532	C₂²	1	2	yes	—
20	C₂×C₃₁×#124	C₂²	1	2	yes	—
21	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
22	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#446	—	2	2	no	—
24	C₃₁×#435	—	2	4	no	—
25	C₈×D₄×C₃₁	—	2	2	no	—
26	C₃₁×#432	C₂²	1	2	yes	complemented
27	C₃₁×#432	C₂²	1	4	yes	complemented
28	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
29	C₂×C₃₁×#97	C₂²	1	4	yes	complemented
30	C₃₁×#329	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#329	C₂²	1	4	yes	complemented
32	C₃₁×#329	C₂²	1	2	yes	complemented
33	C₃₁×#329	C₂²	1	2	yes	complemented
34	C₃₁×#329	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#329	C₂²	1	4	yes	complemented
36	C₃₁×#329	C₂²	1	2	yes	complemented
37	C₃₁×2₋¹⁺⁴	—	4	4	no	—
38	C₃₁×H(5,𝔽₂)	—	4	4	no	—
39	C₂×#41×C₃₁	—	2	1	no	—
40	C₂×#41×C₃₁	—	2	2	no	—
41	C₂×#41×C₃₁	—	2	1	no	—
42	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
50	C₂²×D₄×C₃₁	—	2	2	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	2	4	no	—
56	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
58	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
59	C₃₁×#127	C₂×C₄	1	2	yes	—
60	C₃₁×#124	—	2	2	no	—
61	C₃₁×#124	—	2	4	no	—
62	C₃₁×#124	C₂×C₄	1	2	yes	—
63	C₃₁×#124	C₂×C₄	1	2	yes	—
64	C₃₁×#124	C₂×C₄	1	2	yes	—
65	C₃₁×#124	C₂³	1	1	yes	characteristic
66	C₃₁×#124	C₂³	1	2	yes	—
67	C₃₁×#124	C₂³	1	1	yes	characteristic
68	C₃₁×#123	—	2	2	no	—
69	C₃₁×#123	—	2	4	no	—
70	C₃₁×C₂²⋊Q₈	—	2	4	no	—
71	C₃₁×C₂²⋊Q₈	—	2	2	no	—
72	C₃₁×#121	—	2	4	no	—
73	C₃₁×#121	—	2	2	no	—
74	C₃₁×#121	—	2	4	no	—
75	C₃₁×#121	—	2	2	no	—
76	C₃₁×#120	—	2	2	no	—
77	C₃₁×#120	—	2	4	no	—
78	C₄×Q₈×C₃₁	D₄	1	2	yes	—
79	C₄×D₄×C₃₁	—	2	2	no	—
80	C₄×D₄×C₃₁	—	2	4	no	—
81	C₄×D₄×C₃₁	D₄	1	2	yes	—
82	C₄×D₄×C₃₁	D₄	1	4	yes	—
83	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
84	C₃₁×#117	C₂×C₄	1	2	yes	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×C₄⋊C₈	—	2	4	no	—
91	C₃₁×C₂²⋊C₈	—	2	4	no	—
92	C₃₁×C₂²⋊C₈	—	2	2	no	—
93	C₃₁×#97	—	2	4	no	—
94	C₃₁×#97	C₂³	1	2	yes	—
95	C₃₁×#97	C₂³	1	4	yes	—
96	C₃₁×#97	C₂³	1	2	yes	—
97	C₄×C₈×C₃₁	—	2	2	no	—
98	C₂⁴×C₃₁	—	2	1	no	—
99	C₂⁴×C₃₁	—	2	2	no	—
100	#41×C₃₁	—	2	4	no	—
101	#41×C₃₁	—	4	4	no	—
102	#41×C₃₁	—	4	4	no	—
103	#41×C₃₁	—	4	2	no	—
104	#41×C₃₁	—	4	2	no	—
105	#41×C₃₁	—	4	8	no	—
106	#41×C₃₁	—	4	4	no	—
107	#41×C₃₁	—	4	4	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	4	2	no	—
111	C₂×Q₈×C₃₁	—	4	4	no	—
112	C₂×Q₈×C₃₁	—	4	4	no	—
113	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
115	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	4	4	no	—
126	C₂×D₄×C₃₁	—	4	2	no	—
127	C₂×D₄×C₃₁	—	4	8	no	—
128	C₂×D₄×C₃₁	—	4	4	no	—
129	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
130	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
131	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
133	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
144	M₁₆×C₃₁	—	4	4	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	4	no	—
148	C₂×C₈×C₃₁	—	4	8	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
152	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	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	2	no	—
157	C₂²⋊C₄×C₃₁	—	2	4	no	—
158	C₂²⋊C₄×C₃₁	—	2	2	no	—
159	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
160	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
161	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
162	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₃₁	C₂×D₄	1	2	yes	—
164	C₄²×C₃₁	C₂×D₄	1	4	yes	—
165	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
166	#17685	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	2	no	—
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	—	4	4	no	—
172	C₂³×C₃₁	—	8	4	no	—
173	C₂³×C₃₁	—	8	2	no	—
174	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
176	Q₈×C₃₁	—	2	2	no	—
177	Q₈×C₃₁	—	4	2	no	—
178	Q₈×C₃₁	—	4	1	no	—
179	Q₈×C₃₁	—	4	1	no	—
180	Q₈×C₃₁	—	4	4	no	—
181	Q₈×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	2	4	no	—
183	D₄×C₃₁	—	4	1	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	4	1	no	—
186	D₄×C₃₁	—	4	4	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	2	no	—
189	D₄×C₃₁	—	4	4	no	—
190	D₄×C₃₁	—	4	4	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	8	4	no	—
193	D₄×C₃₁	—	8	2	no	—
194	D₄×C₃₁	#100	1	4	yes	—
195	C₂×C₄×C₃₁	—	2	4	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	4	4	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	4	no	—
203	C₂×C₄×C₃₁	—	4	4	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	4	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
209	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
210	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
211	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
214	C₈×C₃₁	—	2	4	no	—
215	C₈×C₃₁	—	4	4	no	—
216	C₈×C₃₁	—	4	2	no	—
217	#3500	C₂×C₃₁	1	1	yes	characteristic
218	#2970	C₂×C₃₁	1	2	yes	complemented
219	#1536	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1533	C₂×C₃₁	1	2	yes	complemented
221	#1530	C₂×C₃₁	1	4	yes	complemented
222	#1527	C₂×C₃₁	1	2	yes	complemented
223	#1525	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂×#329	C₂×C₃₁	1	2	yes	complemented
225	C₂²×C₃₁	—	2	4	no	—
226	C₂²×C₃₁	—	2	1	no	—
227	C₂²×C₃₁	—	2	2	no	—
228	C₂²×C₃₁	—	4	1	no	—
229	C₂²×C₃₁	—	4	2	no	—
230	C₂²×C₃₁	—	4	1	no	—
231	C₂²×C₃₁	—	4	2	no	—
232	C₂²×C₃₁	—	4	2	no	—
233	C₂²×C₃₁	—	4	4	no	—
234	C₂²×C₃₁	—	16	4	no	—
235	C₂²×C₃₁	—	16	2	no	—
236	C₂²×C₃₁	C₂×#100	1	4	yes	—
237	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
238	C₄×C₃₁	—	2	1	no	—
239	C₄×C₃₁	—	2	2	no	—
240	C₄×C₃₁	—	2	2	no	—
241	C₄×C₃₁	—	2	4	no	—
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	4	1	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	1	no	—
246	C₄×C₃₁	C₂×#100	1	2	yes	—
247	C₂×2₋¹⁺⁴	C₄×C₃₁	1	1	yes	characteristic
248	C₂×H(5,𝔽₂)	C₄×C₃₁	1	1	yes	characteristic
249	#548	—	2	2	no	—
250	#546	—	2	2	no	—
251	#546	—	2	4	no	—
252	#536	C₄×C₃₁	1	1	yes	characteristic
253	#536	C₄×C₃₁	1	2	yes	—
254	#535	C₄×C₃₁	1	1	yes	characteristic
255	#535	C₄×C₃₁	1	2	yes	—
256	#532	C₂²×C₃₁	1	1	yes	characteristic
257	#532	C₂²×C₃₁	1	2	yes	—
258	C₂×#124	C₂²×C₃₁	1	2	yes	—
259	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
260	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
261	#446	—	2	2	no	—
262	#435	—	2	4	no	—
263	C₈×D₄	—	2	2	no	—
264	#432	C₂²×C₃₁	1	2	yes	complemented
265	#432	C₂²×C₃₁	1	4	yes	complemented
266	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
267	C₂×#97	C₂²×C₃₁	1	4	yes	complemented
268	#329	C₂²×C₃₁	1	2	yes	complemented
269	#329	C₂²×C₃₁	1	2	yes	complemented
270	#329	C₂²×C₃₁	1	1	yes	characteristic, complemented
271	#329	C₂²×C₃₁	1	4	yes	complemented
272	#329	C₂²×C₃₁	1	2	yes	complemented
273	#329	C₂²×C₃₁	1	1	yes	characteristic, complemented
274	#329	C₂²×C₃₁	1	4	yes	complemented
275	C₂×C₃₁	—	2	4	no	—
276	C₂×C₃₁	—	4	1	no	—
277	C₂×C₃₁	—	8	1	no	—
278	C₂×C₃₁	—	8	1	no	—
279	C₂×C₃₁	—	8	2	no	—
280	C₂×C₃₁	C₂²×#100	1	1	yes	characteristic
281	C₂×C₃₁	#2901	1	2	yes	—
282	2₋¹⁺⁴	—	4	4	no	—
283	H(5,𝔽₂)	—	4	4	no	—
284	C₂×#41	—	2	1	no	—
285	C₂×#41	—	2	2	no	—
286	C₂×#41	—	2	1	no	—
287	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
288	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
289	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
290	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
291	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
294	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
295	C₂²×D₄	—	2	2	no	—
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
298	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
299	C₂×M₁₆	—	2	2	no	—
300	C₂²×C₈	—	2	4	no	—
301	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
303	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
304	#127	C₂×C₄×C₃₁	1	2	yes	—
305	#124	—	2	2	no	—
306	#124	—	2	4	no	—
307	#124	C₂×C₄×C₃₁	1	2	yes	—
308	#124	C₂×C₄×C₃₁	1	2	yes	—
309	#124	C₂×C₄×C₃₁	1	2	yes	—
310	#124	C₂³×C₃₁	1	1	yes	characteristic
311	#124	C₂³×C₃₁	1	2	yes	—
312	#124	C₂³×C₃₁	1	1	yes	characteristic
313	#123	—	2	2	no	—
314	#123	—	2	4	no	—
315	C₂²⋊Q₈	—	2	4	no	—
316	C₂²⋊Q₈	—	2	2	no	—
317	#121	—	2	4	no	—
318	#121	—	2	2	no	—
319	#121	—	2	4	no	—
320	#121	—	2	2	no	—
321	#120	—	2	2	no	—
322	#120	—	2	4	no	—
323	C₄×Q₈	D₄×C₃₁	1	2	yes	—
324	C₄×D₄	—	2	2	no	—
325	C₄×D₄	—	2	4	no	—
326	C₄×D₄	D₄×C₃₁	1	2	yes	—
327	C₄×D₄	D₄×C₃₁	1	4	yes	—
328	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
329	#117	C₂×C₄×C₃₁	1	2	yes	—
330	C₂×C₂²⋊C₄	—	2	4	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
333	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
334	C₄⋊C₈	—	2	2	no	—
335	C₄⋊C₈	—	2	4	no	—
336	C₂²⋊C₈	—	2	4	no	—
337	C₂²⋊C₈	—	2	2	no	—
338	#97	—	2	4	no	—
339	#97	C₂³×C₃₁	1	2	yes	—
340	#97	C₂³×C₃₁	1	4	yes	—
341	#97	C₂³×C₃₁	1	2	yes	—
342	C₄×C₈	—	2	2	no	—
343	C₃₁	#17685	1	1	yes	characteristic, direct product factor, Sylow
344	C₂⁴	—	2	1	no	—
345	C₂⁴	—	2	2	no	—
346	#41	—	2	4	no	—
347	#41	—	4	4	no	—
348	#41	—	4	4	no	—
349	#41	—	4	2	no	—
350	#41	—	4	2	no	—
351	#41	—	4	8	no	—
352	#41	—	4	4	no	—
353	#41	—	4	4	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	4	2	no	—
357	C₂×Q₈	—	4	4	no	—
358	C₂×Q₈	—	4	4	no	—
359	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
360	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
361	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
362	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	—	4	2	no	—
373	C₂×D₄	—	4	8	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
376	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
377	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
378	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
379	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
390	M₁₆	—	4	4	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	4	no	—
394	C₂×C₈	—	4	8	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
398	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
399	C₂²⋊C₄	—	2	4	no	—
400	C₂²⋊C₄	—	2	4	no	—
401	C₂²⋊C₄	—	2	4	no	—
402	C₂²⋊C₄	—	2	2	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	2	2	no	—
405	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
406	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
407	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
408	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
409	C₄²	C₂×D₄×C₃₁	1	2	yes	—
410	C₄²	C₂×D₄×C₃₁	1	4	yes	—
411	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
412	C₂³	—	2	2	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	4	no	—
417	C₂³	—	8	4	no	—
418	C₂³	—	8	2	no	—
419	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
420	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
421	Q₈	—	2	2	no	—
422	Q₈	—	4	2	no	—
423	Q₈	—	4	1	no	—
424	Q₈	—	4	1	no	—
425	Q₈	—	4	4	no	—
426	Q₈	—	4	2	no	—
427	D₄	—	2	4	no	—
428	D₄	—	4	1	no	—
429	D₄	—	4	2	no	—
430	D₄	—	4	1	no	—
431	D₄	—	4	4	no	—
432	D₄	—	4	2	no	—
433	D₄	—	4	2	no	—
434	D₄	—	4	4	no	—
435	D₄	—	4	4	no	—
436	D₄	—	4	2	no	—
437	D₄	—	8	4	no	—
438	D₄	—	8	2	no	—
439	D₄	C₃₁×#100	1	4	yes	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	4	no	—
448	C₂×C₄	—	4	4	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
454	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
455	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
456	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
459	C₈	—	2	4	no	—
460	C₈	—	4	4	no	—
461	C₈	—	4	2	no	—
462	C₂²	—	2	4	no	—
463	C₂²	—	2	1	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	4	1	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	16	4	no	—
472	C₂²	—	16	2	no	—
473	C₂²	C₂×C₃₁×#100	1	4	yes	—
474	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	2	4	no	—
479	C₄	—	2	2	no	—
480	C₄	—	4	1	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	1	no	—
483	C₄	C₂×C₃₁×#100	1	2	yes	—
484	C₂	—	2	4	no	—
485	C₂	—	4	1	no	—
486	C₂	—	8	1	no	—
487	C₂	—	8	2	no	—
488	C₂	—	8	1	no	—
489	C₂	C₂²×C₃₁×#100	1	1	yes	characteristic
490	C₂	C₃₁×#2901	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2970	C₂	semidirect
3. C₃₁×#1536	C₂	semidirect
4. C₃₁×#1533	C₂	semidirect
5. C₃₁×#1530	C₂	semidirect
6. C₃₁×#1527	C₂	semidirect
7. C₃₁×#1525	C₂	semidirect
8. C₂×C₃₁×#329	C₂	semidirect
26. C₃₁×#432	C₂²	semidirect
27. C₃₁×#432	C₂²	semidirect
28. C₄×M₁₆×C₃₁	C₂²	semidirect
29. C₂×C₃₁×#97	C₂²	semidirect
30. C₃₁×#329	C₂²	semidirect
31. C₃₁×#329	C₂²	semidirect
32. C₃₁×#329	C₂²	semidirect
33. C₃₁×#329	C₂²	semidirect
34. C₃₁×#329	C₂²	semidirect
35. C₃₁×#329	C₂²	semidirect
36. C₃₁×#329	C₂²	semidirect
166. #17685	C₃₁	direct (N × H)
218. #2970	C₂×C₃₁	semidirect
219. #1536	C₂×C₃₁	semidirect
220. #1533	C₂×C₃₁	semidirect
221. #1530	C₂×C₃₁	semidirect
222. #1527	C₂×C₃₁	semidirect
223. #1525	C₂×C₃₁	semidirect
224. C₂×#329	C₂×C₃₁	semidirect
264. #432	C₂²×C₃₁	semidirect
265. #432	C₂²×C₃₁	semidirect
266. C₄×M₁₆	C₂²×C₃₁	semidirect
267. C₂×#97	C₂²×C₃₁	semidirect
268. #329	C₂²×C₃₁	semidirect
269. #329	C₂²×C₃₁	semidirect
270. #329	C₂²×C₃₁	semidirect
271. #329	C₂²×C₃₁	semidirect
272. #329	C₂²×C₃₁	semidirect
273. #329	C₂²×C₃₁	semidirect
274. #329	C₂²×C₃₁	semidirect
343. C₃₁	#17685	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×#3500	2
C₃₁×#2970	2
C₃₁×#1536	2
C₃₁×#1533	2
C₃₁×#1530	2
C₃₁×#1527	2
C₃₁×#1525	2
C₂×C₃₁×#329	2
#17685	31

Subgroups — C31×#17685 #21474444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17685 #21474444