Subgroups — C₃₁×#21997 #21478756

Order 7936.

800 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2290	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1923	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1916	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1869	C₂	1	2	yes	maximal
5	C₃₁×#1850	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1840	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1818	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1798	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#388	C₄	1	2	yes	—
23	C₃₁×#387	C₄	1	2	yes	—
24	C₃₁×#386	C₄	1	2	yes	—
25	C₃₁×#385	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#376	C₂²	1	1	yes	characteristic
30	C₃₁×#336	C₂²	1	1	yes	characteristic
31	C₃₁×#336	C₂²	1	1	yes	characteristic
32	C₃₁×#336	C₂²	1	1	yes	characteristic
33	C₃₁×#336	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	—	2	4	no	—
44	C₂²×C₈×C₃₁	—	4	2	no	—
45	C₂²×C₈×C₃₁	—	4	2	no	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₄×D₄×C₃₁	—	2	4	no	—
51	C₃₁×#117	—	4	2	no	—
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₃₁	—	4	2	no	—
61	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊C₈	—	4	2	no	—
68	C₃₁×C₄⋊C₈	—	4	2	no	—
69	C₃₁×C₂²⋊C₈	—	2	4	no	—
70	C₃₁×C₂²⋊C₈	—	4	2	no	—
71	C₃₁×C₂²⋊C₈	—	4	2	no	—
72	C₄×C₈×C₃₁	—	4	2	no	—
73	C₃₁×#95	C₂×C₄	1	2	yes	—
74	C₃₁×#95	C₂×C₄	1	2	yes	—
75	C₃₁×#95	C₂×C₄	1	2	yes	—
76	C₃₁×#95	C₂×C₄	1	2	yes	—
77	C₃₁×#95	C₂×C₄	1	2	yes	—
78	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	2	4	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	1	no	—
84	C₂²×C₄×C₃₁	—	2	1	no	—
85	C₂²×C₄×C₃₁	—	2	4	no	—
86	C₂²×C₄×C₃₁	—	2	4	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	1	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	1	no	—
91	C₂²×C₄×C₃₁	—	4	2	no	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₃₁	#41	1	4	yes	—
101	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
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	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	4	8	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	—	4	2	no	—
118	C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂²⋊C₄×C₃₁	—	2	4	no	—
120	C₂²⋊C₄×C₃₁	—	2	2	no	—
121	C₂²⋊C₄×C₃₁	—	2	4	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	1	no	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	2	1	no	—
129	C₄²×C₃₁	—	4	2	no	—
130	#21997	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂³×C₃₁	—	2	1	no	—
132	C₂³×C₃₁	—	2	1	no	—
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	—	2	1	no	—
135	C₂³×C₃₁	—	2	1	no	—
136	C₂³×C₃₁	—	2	1	no	—
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	#117	1	2	yes	—
140	C₂³×C₃₁	#131	1	4	yes	—
141	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
142	D₄×C₃₁	—	4	4	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	4	no	—
148	C₂×C₄×C₃₁	—	2	4	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	4	8	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	#117	1	2	yes	—
169	C₂×C₄×C₃₁	#131	1	4	yes	—
170	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
171	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
173	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
174	C₈×C₃₁	—	4	2	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₈×C₃₁	—	4	2	no	—
177	C₈×C₃₁	—	4	2	no	—
178	#2290	C₂×C₃₁	1	1	yes	characteristic
179	#1923	C₂×C₃₁	1	2	yes	complemented
180	#1916	C₂×C₃₁	1	2	yes	complemented
181	#1869	C₂×C₃₁	1	2	yes	—
182	#1850	C₂×C₃₁	1	2	yes	complemented
183	#1840	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1832	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1818	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1798	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
188	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₂²×C₃₁	—	2	2	no	—
190	C₂²×C₃₁	—	2	2	no	—
191	C₂²×C₃₁	—	2	4	no	—
192	C₂²×C₃₁	—	2	4	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	—	4	1	no	—
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	#433	1	4	yes	—
199	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
200	C₂²×C₃₁	#518	1	1	yes	characteristic
201	C₂²×C₃₁	#520	1	1	yes	characteristic
202	C₂²×C₃₁	C₂×#131	1	2	yes	—
203	C₂²×C₃₁	C₂×#131	1	2	yes	—
204	C₄×C₃₁	—	2	2	no	—
205	C₄×C₃₁	—	2	4	no	—
206	C₄×C₃₁	—	4	1	no	—
207	C₄×C₃₁	—	4	1	no	—
208	C₄×C₃₁	—	4	1	no	—
209	C₄×C₃₁	—	4	2	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	4	2	no	—
212	C₄×C₃₁	#538	1	2	yes	—
213	C₄×C₃₁	#539	1	2	yes	—
214	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
215	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
217	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
219	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
220	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
221	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
222	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
223	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
224	#388	C₄×C₃₁	1	2	yes	—
225	#387	C₄×C₃₁	1	2	yes	—
226	#386	C₄×C₃₁	1	2	yes	—
227	#385	C₄×C₃₁	1	2	yes	—
228	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
229	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
230	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
231	#376	C₂²×C₃₁	1	1	yes	characteristic
232	#336	C₂²×C₃₁	1	1	yes	characteristic
233	#336	C₂²×C₃₁	1	1	yes	characteristic
234	#336	C₂²×C₃₁	1	1	yes	characteristic
235	#336	C₂²×C₃₁	1	1	yes	characteristic
236	#336	C₂²×C₃₁	1	2	yes	—
237	#336	C₂²×C₃₁	1	2	yes	—
238	#336	C₂²×C₃₁	1	2	yes	—
239	#336	C₂²×C₃₁	1	2	yes	—
240	#336	C₂²×C₃₁	1	2	yes	—
241	C₂×C₃₁	—	2	4	no	—
242	C₂×C₃₁	—	8	1	no	—
243	C₂×C₃₁	#2318	1	1	yes	characteristic
244	C₂×C₃₁	C₂×#433	1	1	yes	characteristic
245	C₂×C₃₁	#2922	1	1	yes	characteristic
246	C₂×C₃₁	#2977	1	2	yes	—
247	C₂×C₃₁	#2994	1	2	yes	—
248	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
249	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
250	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
251	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
252	C₂²×C₈	—	2	4	no	—
253	C₂²×C₈	—	4	2	no	—
254	C₂²×C₈	—	4	2	no	—
255	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
256	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
257	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
258	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
259	C₄×D₄	—	2	4	no	—
260	#117	—	4	2	no	—
261	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
263	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
264	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
267	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
268	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
269	C₂×C₄²	—	4	2	no	—
270	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
271	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
272	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
276	C₄⋊C₈	—	4	2	no	—
277	C₄⋊C₈	—	4	2	no	—
278	C₂²⋊C₈	—	2	4	no	—
279	C₂²⋊C₈	—	4	2	no	—
280	C₂²⋊C₈	—	4	2	no	—
281	C₄×C₈	—	4	2	no	—
282	#95	C₂×C₄×C₃₁	1	2	yes	—
283	#95	C₂×C₄×C₃₁	1	2	yes	—
284	#95	C₂×C₄×C₃₁	1	2	yes	—
285	#95	C₂×C₄×C₃₁	1	2	yes	—
286	#95	C₂×C₄×C₃₁	1	2	yes	—
287	C₃₁	#21997	1	1	yes	characteristic, direct product factor, Sylow
288	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
289	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	4	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
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	2	yes	—
307	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
310	C₂²×C₄	#41×C₃₁	1	4	yes	—
311	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
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	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	4	8	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	4	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	2	4	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	4	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	1	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	4	2	no	—
340	C₂³	—	2	1	no	—
341	C₂³	—	2	1	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	1	no	—
344	C₂³	—	2	1	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	C₃₁×#117	1	2	yes	—
349	C₂³	C₃₁×#131	1	4	yes	—
350	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
351	D₄	—	4	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	4	8	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	C₃₁×#117	1	2	yes	—
378	C₂×C₄	C₃₁×#131	1	4	yes	—
379	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
380	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
381	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
382	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
383	C₈	—	4	2	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	4	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	4	2	no	—
396	C₂²	C₃₁×#433	1	4	yes	—
397	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
398	C₂²	C₃₁×#518	1	1	yes	characteristic
399	C₂²	C₃₁×#520	1	1	yes	characteristic
400	C₂²	C₂×C₃₁×#131	1	2	yes	—
401	C₂²	C₂×C₃₁×#131	1	2	yes	—
402	C₄	—	2	2	no	—
403	C₄	—	2	4	no	—
404	C₄	—	4	1	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	1	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	2	no	—
410	C₄	C₃₁×#538	1	2	yes	—
411	C₄	C₃₁×#539	1	2	yes	—
412	C₂	—	2	4	no	—
413	C₂	—	8	1	no	—
414	C₂	C₃₁×#2318	1	1	yes	characteristic
415	C₂	C₂×C₃₁×#433	1	1	yes	characteristic
416	C₂	C₃₁×#2922	1	1	yes	characteristic
417	C₂	C₃₁×#2977	1	2	yes	—
418	C₂	C₃₁×#2994	1	2	yes	—

	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	C₂²×D₄×C₃₁	C₂×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×#1923	C₂	semidirect
3. C₃₁×#1916	C₂	semidirect
5. C₃₁×#1850	C₂	semidirect
6. C₃₁×#1840	C₂	semidirect
7. C₃₁×#1832	C₂	semidirect
8. C₃₁×#1818	C₂	semidirect
9. C₃₁×#1798	C₂	semidirect
10. C₄×C₃₁×C₂²⋊C₈	C₂	semidirect
11. C₂×C₃₁×#336	C₂	semidirect
130. #21997	C₃₁	direct (N × H)
179. #1923	C₂×C₃₁	semidirect
180. #1916	C₂×C₃₁	semidirect
182. #1850	C₂×C₃₁	semidirect
183. #1840	C₂×C₃₁	semidirect
184. #1832	C₂×C₃₁	semidirect
185. #1818	C₂×C₃₁	semidirect
186. #1798	C₂×C₃₁	semidirect
187. C₄×C₂²⋊C₈	C₂×C₃₁	semidirect
188. C₂×#336	C₂×C₃₁	semidirect
287. C₃₁	#21997	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×#2290	2
C₃₁×#1923	2
C₃₁×#1916	2
C₃₁×#1869	2
C₃₁×#1850	2
C₃₁×#1840	2
C₃₁×#1832	2
C₃₁×#1818	2
C₃₁×#1798	2
C₄×C₃₁×C₂²⋊C₈	2
C₂×C₃₁×#336	2
#21997	31

Subgroups — C31×#21997 #21478756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21997 #21478756