Subgroups — C₃₁×#21051 #21477810

Order 7936.

1142 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2297	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1924	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1923	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#423	—	2	4	no	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
19	C₃₁×#408	—	2	4	no	—
20	C₃₁×#408	—	2	4	no	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₃₁×#405	—	2	4	no	—
25	C₂×C₃₁×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#391	C₄	1	1	yes	characteristic
28	C₃₁×#391	C₄	1	1	yes	characteristic
29	C₃₁×#390	C₄	1	1	yes	characteristic
30	C₃₁×#390	C₄	1	1	yes	characteristic
31	C₃₁×#386	C₄	1	4	yes	—
32	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	4	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
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	—	2	2	no	—
40	C₂×#41×C₃₁	—	2	2	no	—
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	D₄	1	4	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#117	—	2	2	no	—
66	C₃₁×#117	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
71	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
77	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
78	C₃₁×C₄⋊C₈	—	2	8	no	—
79	C₃₁×C₂²⋊C₈	—	2	8	no	—
80	C₃₁×C₂²⋊C₈	—	2	4	no	—
81	C₃₁×C₂²⋊C₈	—	2	4	no	—
82	C₃₁×#97	—	2	4	no	—
83	C₄×C₈×C₃₁	—	2	4	no	—
84	C₃₁×#95	C₂×C₄	1	4	yes	—
85	C₃₁×#95	C₂×C₄	1	4	yes	—
86	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
87	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	#41×C₃₁	—	4	4	no	—
89	#41×C₃₁	—	4	4	no	—
90	#41×C₃₁	—	4	4	no	—
91	#41×C₃₁	—	4	4	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	2	4	no	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	4	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₃₁	#41	1	2	yes	—
119	C₂²×C₄×C₃₁	#41	1	2	yes	—
120	C₂²×C₄×C₃₁	#41	1	2	yes	—
121	C₂²×C₄×C₃₁	#41	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₃₁	—	2	8	no	—
124	M₁₆×C₃₁	—	4	4	no	—
125	C₂×C₈×C₃₁	—	2	8	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	4	4	no	—
139	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₂²⋊C₄×C₃₁	—	2	4	no	—
145	C₂²⋊C₄×C₃₁	—	2	8	no	—
146	C₂²⋊C₄×C₃₁	—	4	4	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	2	2	no	—
152	C₄²×C₃₁	—	2	2	no	—
153	#21051	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	2	2	no	—
156	C₂³×C₃₁	—	2	2	no	—
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	—	2	2	no	—
159	C₂³×C₃₁	—	2	4	no	—
160	C₂³×C₃₁	—	2	4	no	—
161	C₂³×C₃₁	#117	1	2	yes	—
162	C₂³×C₃₁	C₄×D₄	1	2	yes	—
163	C₂³×C₃₁	C₄×D₄	1	2	yes	—
164	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
165	Q₈×C₃₁	—	4	4	no	—
166	D₄×C₃₁	—	4	8	no	—
167	D₄×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	4	no	—
180	C₂×C₄×C₃₁	—	4	8	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	2	no	—
185	C₂×C₄×C₃₁	#117	1	2	yes	—
186	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
187	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
188	C₂×C₄×C₃₁	#121	1	4	yes	—
189	C₂×C₄×C₃₁	#123	1	4	yes	—
190	C₂×C₄×C₃₁	#124	1	4	yes	—
191	C₂×C₄×C₃₁	#126	1	4	yes	—
192	C₂×C₄×C₃₁	#131	1	4	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₈×C₃₁	—	2	4	no	—
200	C₈×C₃₁	—	2	2	no	—
201	C₈×C₃₁	—	2	2	no	—
202	C₈×C₃₁	—	4	2	no	—
203	C₈×C₃₁	—	4	2	no	—
204	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₂×#408	C₂×C₃₁	1	1	yes	characteristic
206	C₂×#408	C₂×C₃₁	1	1	yes	characteristic
207	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#2297	C₂×C₃₁	1	1	yes	characteristic
209	#1924	C₂×C₃₁	1	4	yes	complemented
210	#1923	C₂×C₃₁	1	4	yes	complemented
211	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	2	4	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	2	4	no	—
217	C₂²×C₃₁	—	4	4	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	—	4	4	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	#388	1	4	yes	—
223	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
224	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
226	C₂²×C₃₁	C₂×#131	1	2	yes	—
227	C₂²×C₃₁	#568	1	2	yes	—
228	C₄×C₃₁	—	2	4	no	—
229	C₄×C₃₁	—	4	2	no	—
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	#388	1	4	yes	—
234	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
235	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
236	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
237	C₄×C₃₁	C₂×#126	1	1	yes	characteristic
238	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
239	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
240	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
241	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
244	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
245	#423	—	2	4	no	—
246	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
247	#408	—	2	4	no	—
248	#408	—	2	4	no	—
249	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
250	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
251	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
252	#405	—	2	4	no	—
253	C₂×#97	C₂²×C₃₁	1	2	yes	—
254	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
255	#391	C₄×C₃₁	1	1	yes	characteristic
256	#391	C₄×C₃₁	1	1	yes	characteristic
257	#390	C₄×C₃₁	1	1	yes	characteristic
258	#390	C₄×C₃₁	1	1	yes	characteristic
259	#386	C₄×C₃₁	1	4	yes	—
260	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
261	#336	C₂²×C₃₁	1	2	yes	—
262	#336	C₂²×C₃₁	1	4	yes	—
263	#336	C₂²×C₃₁	1	2	yes	—
264	C₂×C₃₁	—	2	4	no	—
265	C₂×C₃₁	—	8	2	no	—
266	C₂×C₃₁	C₂×#388	1	1	yes	characteristic
267	C₂×C₃₁	#2925	1	2	yes	—
268	C₂×C₃₁	#2931	1	1	yes	characteristic
269	C₂×C₃₁	#2932	1	1	yes	characteristic
270	C₂×C₃₁	#2933	1	2	yes	—
271	C₂×#41	—	2	2	no	—
272	C₂×#41	—	2	2	no	—
273	C₂×#41	—	2	2	no	—
274	C₂×#41	—	2	2	no	—
275	C₂×#41	—	2	2	no	—
276	C₂×#41	—	2	2	no	—
277	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
278	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
279	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
280	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
281	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
282	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×M₁₆	—	2	2	no	—
284	C₂×M₁₆	—	2	2	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	—	2	2	no	—
287	C₂×M₁₆	—	2	2	no	—
288	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
289	C₂²×C₈	—	2	2	no	—
290	C₂²×C₈	—	2	2	no	—
291	C₂²×C₈	—	2	2	no	—
292	C₂²×C₈	—	2	2	no	—
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	D₄×C₃₁	1	4	yes	—
295	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
296	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
300	#117	—	2	2	no	—
301	#117	—	2	2	no	—
302	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
304	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
305	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
306	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
307	C₂×C₄²	—	2	2	no	—
308	C₂×C₄²	—	2	2	no	—
309	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
312	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
313	C₄⋊C₈	—	2	8	no	—
314	C₂²⋊C₈	—	2	8	no	—
315	C₂²⋊C₈	—	2	4	no	—
316	C₂²⋊C₈	—	2	4	no	—
317	#97	—	2	4	no	—
318	C₄×C₈	—	2	4	no	—
319	#95	C₂×C₄×C₃₁	1	4	yes	—
320	#95	C₂×C₄×C₃₁	1	4	yes	—
321	C₃₁	#21051	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
323	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
324	#41	—	4	4	no	—
325	#41	—	4	4	no	—
326	#41	—	4	4	no	—
327	#41	—	4	4	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×D₄	—	2	4	no	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	4	2	no	—
346	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
347	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
351	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
354	C₂²×C₄	#41×C₃₁	1	2	yes	—
355	C₂²×C₄	#41×C₃₁	1	2	yes	—
356	C₂²×C₄	#41×C₃₁	1	2	yes	—
357	C₂²×C₄	#41×C₃₁	1	2	yes	—
358	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
359	M₁₆	—	2	8	no	—
360	M₁₆	—	4	4	no	—
361	C₂×C₈	—	2	8	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	4	4	no	—
375	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
376	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
377	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₂²⋊C₄	—	2	4	no	—
381	C₂²⋊C₄	—	2	8	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	2	no	—
394	C₂³	—	2	4	no	—
395	C₂³	—	2	4	no	—
396	C₂³	C₃₁×#117	1	2	yes	—
397	C₂³	C₄×D₄×C₃₁	1	2	yes	—
398	C₂³	C₄×D₄×C₃₁	1	2	yes	—
399	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
400	Q₈	—	4	4	no	—
401	D₄	—	4	8	no	—
402	D₄	—	4	4	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	4	8	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	C₃₁×#117	1	2	yes	—
421	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
422	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
423	C₂×C₄	C₃₁×#121	1	4	yes	—
424	C₂×C₄	C₃₁×#123	1	4	yes	—
425	C₂×C₄	C₃₁×#124	1	4	yes	—
426	C₂×C₄	C₃₁×#126	1	4	yes	—
427	C₂×C₄	C₃₁×#131	1	4	yes	—
428	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
434	C₈	—	2	4	no	—
435	C₈	—	2	2	no	—
436	C₈	—	2	2	no	—
437	C₈	—	4	2	no	—
438	C₈	—	4	2	no	—
439	C₂²	—	2	4	no	—
440	C₂²	—	2	2	no	—
441	C₂²	—	2	2	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	2	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	2	no	—
448	C₂²	C₃₁×#388	1	4	yes	—
449	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
450	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
451	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
452	C₂²	C₂×C₃₁×#131	1	2	yes	—
453	C₂²	C₃₁×#568	1	2	yes	—
454	C₄	—	2	4	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	4	2	no	—
459	C₄	C₃₁×#388	1	4	yes	—
460	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
461	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
462	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
463	C₄	C₂×C₃₁×#126	1	1	yes	characteristic
464	C₂	—	2	4	no	—
465	C₂	—	8	2	no	—
466	C₂	C₂×C₃₁×#388	1	1	yes	characteristic
467	C₂	C₃₁×#2925	1	2	yes	—
468	C₂	C₃₁×#2931	1	1	yes	characteristic
469	C₂	C₃₁×#2932	1	1	yes	characteristic
470	C₂	C₃₁×#2933	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×#41×C₃₁	C₂²
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
1. C₂×C₃₁×#423	C₂	semidirect
4. C₂×C₃₁×#405	C₂	semidirect
6. C₃₁×#1924	C₂	semidirect
7. C₃₁×#1923	C₂	semidirect
8. C₃₁×#1729	C₂	semidirect
9. C₂×C₃₁×#336	C₂	semidirect
153. #21051	C₃₁	direct (N × H)
204. C₂×#423	C₂×C₃₁	semidirect
207. C₂×#405	C₂×C₃₁	semidirect
209. #1924	C₂×C₃₁	semidirect
210. #1923	C₂×C₃₁	semidirect
211. #1729	C₂×C₃₁	semidirect
212. C₂×#336	C₂×C₃₁	semidirect
321. C₃₁	#21051	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×#423	2
C₂×C₃₁×#408	2
C₂×C₃₁×#405	2
C₃₁×#2297	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1729	2
C₂×C₃₁×#336	2
#21051	31

Subgroups — C31×#21051 #21477810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21051 #21477810