Subgroups — C₃₁×#22206 #21478965

Order 7936.

774 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3050	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2930	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1929	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1922	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1897	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1876	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1866	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#131	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#525	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₃₁×#467	C₄	1	1	yes	characteristic, complemented
17	C₃₁×#467	C₄	1	1	yes	characteristic, complemented
18	C₃₁×#466	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#466	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#465	C₄	1	2	yes	complemented
21	C₃₁×#465	C₄	1	2	yes	complemented
22	C₃₁×#436	—	2	2	no	—
23	C₃₁×#436	—	2	2	no	—
24	C₃₁×#435	—	2	2	no	—
25	C₃₁×#435	—	2	2	no	—
26	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
28	C₃₁×#427	C₂²	1	1	yes	characteristic
29	C₃₁×#424	C₂²	1	2	yes	—
30	C₂×C₃₁×#104	C₂²	1	2	yes	—
31	C₂×C₃₁×#104	C₂²	1	2	yes	—
32	C₃₁×#416	C₂²	1	2	yes	—
33	C₃₁×#408	C₂²	1	2	yes	—
34	C₃₁×#407	C₂²	1	2	yes	—
35	C₃₁×#405	C₂²	1	2	yes	—
36	C₃₁×#341	C₂²	1	2	yes	—
37	C₃₁×#339	C₂²	1	2	yes	—
38	C₃₁×#339	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₂×#41×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#135	—	4	1	no	—
44	C₃₁×#135	—	4	1	no	—
45	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₃₁×#131	—	4	2	no	—
49	C₃₁×#131	—	4	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	4	1	no	—
52	C₂×M₁₆×C₃₁	—	4	1	no	—
53	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
55	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	4	1	no	—
60	C₂²×C₈×C₃₁	—	4	1	no	—
61	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
68	C₃₁×#123	—	2	2	no	—
69	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
70	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
71	C₃₁×#121	C₂×C₄	1	2	yes	—
72	C₃₁×#121	C₂×C₄	1	2	yes	—
73	C₃₁×#120	—	2	2	no	—
74	C₄×D₄×C₃₁	—	2	2	no	—
75	C₄×D₄×C₃₁	—	2	2	no	—
76	C₃₁×#117	—	4	1	no	—
77	C₃₁×#117	—	4	1	no	—
78	C₃₁×#117	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
82	C₃₁×#108	—	4	1	no	—
83	C₃₁×#108	—	4	1	no	—
84	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
85	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
86	C₃₁×#106	C₂×C₄	1	2	yes	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	2	no	—
89	C₃₁×#104	—	4	2	no	—
90	C₃₁×#104	—	4	2	no	—
91	C₃₁×#103	C₂×C₄	1	2	yes	—
92	C₃₁×#103	C₂×C₄	1	2	yes	—
93	C₃₁×#102	C₂×C₄	1	2	yes	—
94	C₃₁×#102	C₂×C₄	1	2	yes	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	—	2	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	2	no	—
98	C₃₁×C₂²⋊C₈	—	2	2	no	—
99	C₃₁×#97	—	2	2	no	—
100	C₄×C₈×C₃₁	—	2	2	no	—
101	C₂⁴×C₃₁	#41	1	1	yes	characteristic
102	#41×C₃₁	—	2	2	no	—
103	#41×C₃₁	—	2	2	no	—
104	#41×C₃₁	—	2	2	no	—
105	#41×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	Q₁₆×C₃₁	—	4	1	no	—
124	SD₁₆×C₃₁	—	4	2	no	—
125	D₈×C₃₁	—	4	1	no	—
126	M₁₆×C₃₁	—	2	2	no	—
127	M₁₆×C₃₁	—	2	2	no	—
128	M₁₆×C₃₁	—	2	2	no	—
129	M₁₆×C₃₁	—	2	2	no	—
130	M₁₆×C₃₁	—	4	1	no	—
131	M₁₆×C₃₁	—	4	2	no	—
132	M₁₆×C₃₁	—	4	1	no	—
133	M₁₆×C₃₁	—	4	4	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	4	1	no	—
143	C₂×C₈×C₃₁	—	4	1	no	—
144	C₂×C₈×C₃₁	—	4	1	no	—
145	C₂×C₈×C₃₁	—	4	1	no	—
146	C₂×C₈×C₃₁	—	4	4	no	—
147	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
152	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	4	2	no	—
156	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
157	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
158	C₂²⋊C₄×C₃₁	—	2	2	no	—
159	C₂²⋊C₄×C₃₁	—	2	2	no	—
160	C₂²⋊C₄×C₃₁	—	2	2	no	—
161	C₂²⋊C₄×C₃₁	—	2	2	no	—
162	C₂²⋊C₄×C₃₁	—	2	2	no	—
163	C₂²⋊C₄×C₃₁	—	4	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	#22206	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	2	2	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	—	4	1	no	—
173	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
174	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
175	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
176	Q₈×C₃₁	—	2	2	no	—
177	Q₈×C₃₁	—	2	2	no	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	2	2	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	8	2	no	—
183	D₄×C₃₁	—	8	2	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
209	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
210	C₈×C₃₁	—	2	1	no	—
211	C₈×C₃₁	—	2	1	no	—
212	C₈×C₃₁	—	2	1	no	—
213	C₈×C₃₁	—	2	1	no	—
214	C₈×C₃₁	—	2	2	no	—
215	C₈×C₃₁	—	2	2	no	—
216	C₈×C₃₁	—	4	2	no	—
217	C₈×C₃₁	—	4	2	no	—
218	#3050	C₂×C₃₁	1	1	yes	characteristic
219	#2930	C₂×C₃₁	1	2	yes	complemented
220	#1929	C₂×C₃₁	1	2	yes	complemented
221	#1922	C₂×C₃₁	1	2	yes	complemented
222	#1897	C₂×C₃₁	1	2	yes	complemented
223	#1876	C₂×C₃₁	1	2	yes	complemented
224	#1866	C₂×C₃₁	1	2	yes	complemented
225	#1810	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂²×C₃₁	—	2	1	no	—
228	C₂²×C₃₁	—	2	1	no	—
229	C₂²×C₃₁	—	2	1	no	—
230	C₂²×C₃₁	—	2	1	no	—
231	C₂²×C₃₁	—	2	1	no	—
232	C₂²×C₃₁	—	2	1	no	—
233	C₂²×C₃₁	—	2	1	no	—
234	C₂²×C₃₁	—	2	1	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
241	C₂²×C₃₁	#538	1	1	yes	characteristic
242	C₂²×C₃₁	#539	1	1	yes	characteristic
243	C₄×C₃₁	—	2	1	no	—
244	C₄×C₃₁	—	2	1	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	—	4	2	no	—
248	C₄×C₃₁	—	8	2	no	—
249	C₄×C₃₁	#518	1	1	yes	characteristic
250	C₄×C₃₁	#520	1	1	yes	characteristic
251	C₄×C₃₁	#534	1	1	yes	characteristic
252	C₄×C₃₁	#537	1	1	yes	characteristic
253	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×#131	C₂²×C₃₁	1	2	yes	—
255	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
256	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
257	#525	C₂²×C₃₁	1	2	yes	—
258	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
259	#467	C₄×C₃₁	1	1	yes	characteristic, complemented
260	#467	C₄×C₃₁	1	1	yes	characteristic, complemented
261	#466	C₄×C₃₁	1	1	yes	characteristic, complemented
262	#466	C₄×C₃₁	1	1	yes	characteristic, complemented
263	#465	C₄×C₃₁	1	2	yes	complemented
264	#465	C₄×C₃₁	1	2	yes	complemented
265	#436	—	2	2	no	—
266	#436	—	2	2	no	—
267	#435	—	2	2	no	—
268	#435	—	2	2	no	—
269	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
271	#427	C₂²×C₃₁	1	1	yes	characteristic
272	#424	C₂²×C₃₁	1	2	yes	—
273	C₂×#104	C₂²×C₃₁	1	2	yes	—
274	C₂×#104	C₂²×C₃₁	1	2	yes	—
275	#416	C₂²×C₃₁	1	2	yes	—
276	#408	C₂²×C₃₁	1	2	yes	—
277	#407	C₂²×C₃₁	1	2	yes	—
278	#405	C₂²×C₃₁	1	2	yes	—
279	#341	C₂²×C₃₁	1	2	yes	—
280	#339	C₂²×C₃₁	1	2	yes	—
281	#339	C₂²×C₃₁	1	2	yes	—
282	#336	C₂²×C₃₁	1	2	yes	—
283	#336	C₂²×C₃₁	1	2	yes	—
284	C₂×C₃₁	—	2	1	no	—
285	C₂×C₃₁	—	2	1	no	—
286	C₂×C₃₁	—	4	1	no	—
287	C₂×C₃₁	—	4	1	no	—
288	C₂×C₃₁	—	8	2	no	—
289	C₂×C₃₁	#2325	1	1	yes	characteristic
290	C₂×C₃₁	#2957	1	1	yes	characteristic
291	C₂×C₃₁	#2957	1	1	yes	characteristic
292	C₂×#41	C₂³×C₃₁	1	2	yes	—
293	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
294	#135	—	4	1	no	—
295	#135	—	4	1	no	—
296	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
297	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
298	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
299	#131	—	4	2	no	—
300	#131	—	4	2	no	—
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	—	4	1	no	—
303	C₂×M₁₆	—	4	1	no	—
304	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
305	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
306	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
307	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
308	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
309	C₂²×C₈	—	2	2	no	—
310	C₂²×C₈	—	4	1	no	—
311	C₂²×C₈	—	4	1	no	—
312	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
313	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
319	#123	—	2	2	no	—
320	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
321	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
322	#121	C₂×C₄×C₃₁	1	2	yes	—
323	#121	C₂×C₄×C₃₁	1	2	yes	—
324	#120	—	2	2	no	—
325	C₄×D₄	—	2	2	no	—
326	C₄×D₄	—	2	2	no	—
327	#117	—	4	1	no	—
328	#117	—	4	1	no	—
329	#117	C₂³×C₃₁	1	2	yes	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
333	#108	—	4	1	no	—
334	#108	—	4	1	no	—
335	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
336	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
337	#106	C₂×C₄×C₃₁	1	2	yes	—
338	C₄⋊C₈	—	2	2	no	—
339	C₄⋊C₈	—	2	2	no	—
340	#104	—	4	2	no	—
341	#104	—	4	2	no	—
342	#103	C₂×C₄×C₃₁	1	2	yes	—
343	#103	C₂×C₄×C₃₁	1	2	yes	—
344	#102	C₂×C₄×C₃₁	1	2	yes	—
345	#102	C₂×C₄×C₃₁	1	2	yes	—
346	C₂²⋊C₈	—	2	2	no	—
347	C₂²⋊C₈	—	2	2	no	—
348	C₂²⋊C₈	—	2	2	no	—
349	C₂²⋊C₈	—	2	2	no	—
350	#97	—	2	2	no	—
351	C₄×C₈	—	2	2	no	—
352	C₃₁	#22206	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁴	#41×C₃₁	1	1	yes	characteristic
354	#41	—	2	2	no	—
355	#41	—	2	2	no	—
356	#41	—	2	2	no	—
357	#41	—	2	2	no	—
358	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
375	Q₁₆	—	4	1	no	—
376	SD₁₆	—	4	2	no	—
377	D₈	—	4	1	no	—
378	M₁₆	—	2	2	no	—
379	M₁₆	—	2	2	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	2	2	no	—
382	M₁₆	—	4	1	no	—
383	M₁₆	—	4	2	no	—
384	M₁₆	—	4	1	no	—
385	M₁₆	—	4	4	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	4	1	no	—
395	C₂×C₈	—	4	1	no	—
396	C₂×C₈	—	4	1	no	—
397	C₂×C₈	—	4	1	no	—
398	C₂×C₈	—	4	4	no	—
399	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
400	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
401	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
402	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
403	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
404	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	4	2	no	—
408	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
409	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	2	2	no	—
415	C₂²⋊C₄	—	4	2	no	—
416	C₄²	—	2	2	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	2	2	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	1	no	—
424	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
425	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
426	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
427	Q₈	—	2	2	no	—
428	Q₈	—	2	2	no	—
429	D₄	—	2	2	no	—
430	D₄	—	2	2	no	—
431	D₄	—	4	2	no	—
432	D₄	—	4	2	no	—
433	D₄	—	8	2	no	—
434	D₄	—	8	2	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
454	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
456	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
458	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
459	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
460	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	2	1	no	—
464	C₈	—	2	1	no	—
465	C₈	—	2	2	no	—
466	C₈	—	2	2	no	—
467	C₈	—	4	2	no	—
468	C₈	—	4	2	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	1	no	—
473	C₂²	—	2	1	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
483	C₂²	C₃₁×#538	1	1	yes	characteristic
484	C₂²	C₃₁×#539	1	1	yes	characteristic
485	C₄	—	2	1	no	—
486	C₄	—	2	1	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	8	2	no	—
491	C₄	C₃₁×#518	1	1	yes	characteristic
492	C₄	C₃₁×#520	1	1	yes	characteristic
493	C₄	C₃₁×#534	1	1	yes	characteristic
494	C₄	C₃₁×#537	1	1	yes	characteristic
495	C₂	—	2	1	no	—
496	C₂	—	2	1	no	—
497	C₂	—	4	1	no	—
498	C₂	—	4	1	no	—
499	C₂	—	8	2	no	—
500	C₂	C₃₁×#2325	1	1	yes	characteristic
501	C₂	C₃₁×#2957	1	1	yes	characteristic
502	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3050	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2930	C₂	semidirect
3. C₃₁×#1929	C₂	semidirect
4. C₃₁×#1922	C₂	semidirect
5. C₃₁×#1897	C₂	semidirect
6. C₃₁×#1876	C₂	semidirect
7. C₃₁×#1866	C₂	semidirect
8. C₃₁×#1810	C₂	semidirect
9. C₃₁×#1729	C₂	semidirect
16. C₃₁×#467	C₄	semidirect
17. C₃₁×#467	C₄	semidirect
18. C₃₁×#466	C₄	semidirect
19. C₃₁×#466	C₄	semidirect
20. C₃₁×#465	C₄	semidirect
21. C₃₁×#465	C₄	semidirect
167. #22206	C₃₁	direct (N × H)
219. #2930	C₂×C₃₁	semidirect
220. #1929	C₂×C₃₁	semidirect
221. #1922	C₂×C₃₁	semidirect
222. #1897	C₂×C₃₁	semidirect
223. #1876	C₂×C₃₁	semidirect
224. #1866	C₂×C₃₁	semidirect
225. #1810	C₂×C₃₁	semidirect
226. #1729	C₂×C₃₁	semidirect
259. #467	C₄×C₃₁	semidirect
260. #467	C₄×C₃₁	semidirect
261. #466	C₄×C₃₁	semidirect
262. #466	C₄×C₃₁	semidirect
263. #465	C₄×C₃₁	semidirect
264. #465	C₄×C₃₁	semidirect
352. C₃₁	#22206	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×#3050	2
C₃₁×#2930	2
C₃₁×#1929	2
C₃₁×#1922	2
C₃₁×#1897	2
C₃₁×#1876	2
C₃₁×#1866	2
C₃₁×#1810	2
C₃₁×#1729	2
#22206	31

Subgroups — C31×#22206 #21478965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22206 #21478965