Subgroups — C₃₁×#23343 #21480102

Order 7936.

700 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1984	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1940	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1924	C₂	1	2	yes	maximal
5	C₃₁×#1923	C₂	1	2	yes	maximal
6	C₃₁×#1917	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1916	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1817	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#391	C₄	1	1	yes	characteristic
22	C₃₁×#389	C₄	1	1	yes	characteristic
23	C₃₁×#387	C₄	1	2	yes	—
24	C₃₁×#385	C₄	1	2	yes	—
25	C₃₁×#380	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#336	C₂²	1	2	yes	—
30	C₃₁×#336	C₂²	1	1	yes	characteristic
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	1	yes	characteristic
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₃₁×C₂²⋊Q₈	—	2	4	no	—
46	C₄×Q₈×C₃₁	—	4	2	no	—
47	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₃₁	—	4	2	no	—
56	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊C₈	—	4	2	no	—
63	C₃₁×C₄⋊C₈	—	4	2	no	—
64	C₃₁×C₄⋊C₈	—	4	2	no	—
65	C₃₁×C₂²⋊C₈	—	4	2	no	—
66	C₃₁×C₂²⋊C₈	—	4	1	no	—
67	C₃₁×C₂²⋊C₈	—	4	1	no	—
68	C₃₁×#97	—	4	2	no	—
69	C₄×C₈×C₃₁	—	4	2	no	—
70	C₃₁×#95	C₂×C₄	1	2	yes	—
71	C₃₁×#95	C₂×C₄	1	2	yes	—
72	C₃₁×#95	C₂×C₄	1	2	yes	—
73	C₃₁×#95	C₂×C₄	1	2	yes	—
74	C₃₁×#95	C₂×C₄	1	2	yes	—
75	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×Q₈×C₃₁	—	4	1	no	—
77	C₂×Q₈×C₃₁	—	4	1	no	—
78	C₂×Q₈×C₃₁	M₁₆	1	4	yes	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	1	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	1	no	—
86	C₂²×C₄×C₃₁	—	4	1	no	—
87	C₂²×C₄×C₃₁	—	4	1	no	—
88	C₂²×C₄×C₃₁	—	4	1	no	—
89	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₄⋊C₄×C₃₁	—	2	4	no	—
114	C₄⋊C₄×C₃₁	—	4	1	no	—
115	C₄⋊C₄×C₃₁	—	4	1	no	—
116	C₄⋊C₄×C₃₁	—	4	2	no	—
117	C₂²⋊C₄×C₃₁	—	2	4	no	—
118	C₂²⋊C₄×C₃₁	—	2	4	no	—
119	C₄²×C₃₁	—	2	2	no	—
120	C₄²×C₃₁	—	2	2	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	4	1	no	—
126	C₄²×C₃₁	—	4	1	no	—
127	#23343	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	—	2	1	no	—
129	C₂³×C₃₁	—	2	1	no	—
130	C₂³×C₃₁	—	2	1	no	—
131	C₂³×C₃₁	—	2	1	no	—
132	C₂³×C₃₁	—	2	1	no	—
133	C₂³×C₃₁	—	2	1	no	—
134	C₂³×C₃₁	#131	1	2	yes	—
135	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
136	Q₈×C₃₁	—	4	4	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	4	4	no	—
162	C₂×C₄×C₃₁	—	4	4	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
166	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
167	C₂×C₄×C₃₁	#131	1	2	yes	—
168	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
170	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
171	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
172	C₈×C₃₁	—	4	2	no	—
173	C₈×C₃₁	—	4	2	no	—
174	C₈×C₃₁	—	4	2	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
177	#1984	C₂×C₃₁	1	2	yes	complemented
178	#1940	C₂×C₃₁	1	2	yes	complemented
179	#1924	C₂×C₃₁	1	2	yes	—
180	#1923	C₂×C₃₁	1	2	yes	—
181	#1917	C₂×C₃₁	1	2	yes	complemented
182	#1916	C₂×C₃₁	1	2	yes	complemented
183	#1817	C₂×C₃₁	1	1	yes	characteristic
184	#1797	C₂×C₃₁	1	1	yes	characteristic
185	C₂²×C₃₁	—	2	2	no	—
186	C₂²×C₃₁	—	2	2	no	—
187	C₂²×C₃₁	—	2	2	no	—
188	C₂²×C₃₁	—	4	1	no	—
189	C₂²×C₃₁	—	4	1	no	—
190	C₂²×C₃₁	—	4	1	no	—
191	C₂²×C₃₁	—	4	1	no	—
192	C₂²×C₃₁	#518	1	1	yes	characteristic
193	C₂²×C₃₁	#519	1	1	yes	characteristic
194	C₂²×C₃₁	#520	1	1	yes	characteristic
195	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
196	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
197	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
198	C₂²×C₃₁	#568	1	1	yes	characteristic
199	C₄×C₃₁	—	2	2	no	—
200	C₄×C₃₁	—	2	4	no	—
201	C₄×C₃₁	—	2	2	no	—
202	C₄×C₃₁	—	4	1	no	—
203	C₄×C₃₁	—	4	1	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	—	4	2	no	—
208	C₄×C₃₁	#562	1	2	yes	—
209	C₄×C₃₁	#563	1	2	yes	—
210	C₂×C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
212	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
214	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
215	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
216	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
219	C₂×#97	C₂²×C₃₁	1	2	yes	—
220	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
221	#391	C₄×C₃₁	1	1	yes	characteristic
222	#389	C₄×C₃₁	1	1	yes	characteristic
223	#387	C₄×C₃₁	1	2	yes	—
224	#385	C₄×C₃₁	1	2	yes	—
225	#380	C₄×C₃₁	1	1	yes	characteristic
226	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
227	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
228	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
229	#336	C₂²×C₃₁	1	2	yes	—
230	#336	C₂²×C₃₁	1	1	yes	characteristic
231	#336	C₂²×C₃₁	1	2	yes	—
232	#336	C₂²×C₃₁	1	1	yes	characteristic
233	#336	C₂²×C₃₁	1	2	yes	—
234	#336	C₂²×C₃₁	1	2	yes	—
235	#336	C₂²×C₃₁	1	2	yes	—
236	#336	C₂²×C₃₁	1	2	yes	—
237	C₂×C₃₁	—	4	2	no	—
238	C₂×C₃₁	#2378	1	1	yes	characteristic
239	C₂×C₃₁	#2973	1	1	yes	characteristic
240	C₂×C₃₁	#2977	1	1	yes	characteristic
241	C₂×C₃₁	#2984	1	1	yes	characteristic
242	C₂×C₃₁	#2985	1	1	yes	characteristic
243	C₂×C₃₁	#2991	1	1	yes	characteristic
244	C₂×C₃₁	#2992	1	1	yes	characteristic
245	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
246	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
247	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
248	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
250	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
252	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
253	C₂²⋊Q₈	—	2	4	no	—
254	C₄×Q₈	—	4	2	no	—
255	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
256	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
258	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
260	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
262	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
263	C₂×C₄²	—	4	2	no	—
264	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
268	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
269	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
270	C₄⋊C₈	—	4	2	no	—
271	C₄⋊C₈	—	4	2	no	—
272	C₄⋊C₈	—	4	2	no	—
273	C₂²⋊C₈	—	4	2	no	—
274	C₂²⋊C₈	—	4	1	no	—
275	C₂²⋊C₈	—	4	1	no	—
276	#97	—	4	2	no	—
277	C₄×C₈	—	4	2	no	—
278	#95	C₂×C₄×C₃₁	1	2	yes	—
279	#95	C₂×C₄×C₃₁	1	2	yes	—
280	#95	C₂×C₄×C₃₁	1	2	yes	—
281	#95	C₂×C₄×C₃₁	1	2	yes	—
282	#95	C₂×C₄×C₃₁	1	2	yes	—
283	C₃₁	#23343	1	1	yes	characteristic, direct product factor, Sylow
284	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
285	C₂×Q₈	—	4	1	no	—
286	C₂×Q₈	—	4	1	no	—
287	C₂×Q₈	M₁₆×C₃₁	1	4	yes	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
300	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
302	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
306	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	2	4	no	—
327	C₂²⋊C₄	—	2	4	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	4	1	no	—
335	C₄²	—	4	1	no	—
336	C₂³	—	2	1	no	—
337	C₂³	—	2	1	no	—
338	C₂³	—	2	1	no	—
339	C₂³	—	2	1	no	—
340	C₂³	—	2	1	no	—
341	C₂³	—	2	1	no	—
342	C₂³	C₃₁×#131	1	2	yes	—
343	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
344	Q₈	—	4	4	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
374	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
375	C₂×C₄	C₃₁×#131	1	2	yes	—
376	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
377	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
378	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
379	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
380	C₈	—	4	2	no	—
381	C₈	—	4	2	no	—
382	C₈	—	4	2	no	—
383	C₈	—	4	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	C₃₁×#518	1	1	yes	characteristic
392	C₂²	C₃₁×#519	1	1	yes	characteristic
393	C₂²	C₃₁×#520	1	1	yes	characteristic
394	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
395	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
396	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
397	C₂²	C₃₁×#568	1	1	yes	characteristic
398	C₄	—	2	2	no	—
399	C₄	—	2	4	no	—
400	C₄	—	2	2	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	4	2	no	—
407	C₄	C₃₁×#562	1	2	yes	—
408	C₄	C₃₁×#563	1	2	yes	—
409	C₂	—	4	2	no	—
410	C₂	C₃₁×#2378	1	1	yes	characteristic
411	C₂	C₃₁×#2973	1	1	yes	characteristic
412	C₂	C₃₁×#2977	1	1	yes	characteristic
413	C₂	C₃₁×#2984	1	1	yes	characteristic
414	C₂	C₃₁×#2985	1	1	yes	characteristic
415	C₂	C₃₁×#2991	1	1	yes	characteristic
416	C₂	C₃₁×#2992	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1984	C₂	semidirect
3. C₃₁×#1940	C₂	semidirect
6. C₃₁×#1917	C₂	semidirect
7. C₃₁×#1916	C₂	semidirect
127. #23343	C₃₁	direct (N × H)
177. #1984	C₂×C₃₁	semidirect
178. #1940	C₂×C₃₁	semidirect
181. #1917	C₂×C₃₁	semidirect
182. #1916	C₂×C₃₁	semidirect
283. C₃₁	#23343	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#1984	2
C₃₁×#1940	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1817	2
C₃₁×#1797	2
#23343	31

Subgroups — C31×#23343 #21480102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23343 #21480102