Subgroups — C₃₁×#22207 #21478966

Order 7936.

806 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3054	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×#1896	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×#1811	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1728	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	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#525	C₂²	1	2	yes	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#470	C₄	1	1	yes	characteristic, complemented
17	C₃₁×#470	C₄	1	1	yes	characteristic, complemented
18	C₃₁×#469	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#469	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#468	C₄	1	2	yes	complemented
21	C₃₁×#468	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	—	2	1	no	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×#135	—	4	1	no	—
45	C₃₁×#135	—	4	1	no	—
46	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	complemented
48	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
49	C₃₁×#131	—	4	2	no	—
50	C₃₁×#131	—	4	2	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	4	1	no	—
53	C₂×M₁₆×C₃₁	—	4	1	no	—
54	C₂×M₁₆×C₃₁	—	4	1	no	—
55	C₂×M₁₆×C₃₁	—	4	1	no	—
56	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
62	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
63	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
65	C₂²×C₈×C₃₁	—	2	2	no	—
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
69	C₃₁×#123	—	2	2	no	—
70	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
71	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
72	C₃₁×#121	C₂×C₄	1	2	yes	—
73	C₃₁×#121	C₂×C₄	1	2	yes	—
74	C₃₁×#120	—	2	2	no	—
75	C₄×D₄×C₃₁	—	2	2	no	—
76	C₄×D₄×C₃₁	—	2	2	no	—
77	C₃₁×#117	C₂³	1	2	yes	—
78	C₂×C₄²×C₃₁	—	2	1	no	—
79	C₂×C₄²×C₃₁	—	4	1	no	—
80	C₂×C₄²×C₃₁	—	4	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	2	yes	complemented
85	C₃₁×#106	C₂×C₄	1	1	yes	characteristic, complemented
86	C₃₁×#106	C₂×C₄	1	1	yes	characteristic, complemented
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	complemented
92	C₃₁×#103	C₂×C₄	1	2	yes	complemented
93	C₃₁×#102	C₂×C₄	1	2	yes	complemented
94	C₃₁×#102	C₂×C₄	1	2	yes	complemented
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	—	4	4	no	—
118	C₂²×C₄×C₃₁	—	4	2	no	—
119	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	Q₁₆×C₃₁	—	4	1	no	—
126	SD₁₆×C₃₁	—	4	2	no	—
127	D₈×C₃₁	—	4	1	no	—
128	M₁₆×C₃₁	—	2	2	no	—
129	M₁₆×C₃₁	—	2	2	no	—
130	M₁₆×C₃₁	—	2	2	no	—
131	M₁₆×C₃₁	—	2	2	no	—
132	M₁₆×C₃₁	—	4	1	no	—
133	M₁₆×C₃₁	—	4	1	no	—
134	M₁₆×C₃₁	—	4	1	no	—
135	M₁₆×C₃₁	—	4	1	no	—
136	M₁₆×C₃₁	—	4	1	no	—
137	M₁₆×C₃₁	—	4	2	no	—
138	M₁₆×C₃₁	—	4	1	no	—
139	M₁₆×C₃₁	—	4	4	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	4	4	no	—
149	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
158	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	—	2	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	4	2	no	—
168	#22207	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	4	2	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	4	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	8	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂²×D₄	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	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
211	C₈×C₃₁	—	2	1	no	—
212	C₈×C₃₁	—	2	1	no	—
213	C₈×C₃₁	—	2	1	no	—
214	C₈×C₃₁	—	2	1	no	—
215	C₈×C₃₁	—	2	2	no	—
216	C₈×C₃₁	—	2	2	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₈×C₃₁	—	4	2	no	—
219	#3054	C₂×C₃₁	1	1	yes	characteristic
220	#2930	C₂×C₃₁	1	2	yes	complemented
221	#1929	C₂×C₃₁	1	2	yes	complemented
222	#1922	C₂×C₃₁	1	2	yes	complemented
223	#1896	C₂×C₃₁	1	2	yes	complemented
224	#1876	C₂×C₃₁	1	2	yes	complemented
225	#1866	C₂×C₃₁	1	2	yes	complemented
226	#1811	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1728	C₂×C₃₁	1	1	yes	characteristic, complemented
228	C₂²×C₃₁	—	2	2	no	—
229	C₂²×C₃₁	—	2	1	no	—
230	C₂²×C₃₁	—	2	1	no	—
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
239	C₂²×C₃₁	#538	1	1	yes	characteristic
240	C₂²×C₃₁	#539	1	1	yes	characteristic
241	C₄×C₃₁	—	2	1	no	—
242	C₄×C₃₁	—	2	1	no	—
243	C₄×C₃₁	—	4	4	no	—
244	C₄×C₃₁	—	4	2	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	4	2	no	—
247	C₄×C₃₁	#518	1	1	yes	characteristic
248	C₄×C₃₁	#520	1	1	yes	characteristic
249	C₄×C₃₁	#534	1	1	yes	characteristic
250	C₄×C₃₁	#537	1	1	yes	characteristic
251	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×#131	C₂²×C₃₁	1	2	yes	—
253	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
254	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
255	#525	C₂²×C₃₁	1	2	yes	—
256	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
257	#470	C₄×C₃₁	1	1	yes	characteristic, complemented
258	#470	C₄×C₃₁	1	1	yes	characteristic, complemented
259	#469	C₄×C₃₁	1	1	yes	characteristic, complemented
260	#469	C₄×C₃₁	1	1	yes	characteristic, complemented
261	#468	C₄×C₃₁	1	2	yes	complemented
262	#468	C₄×C₃₁	1	2	yes	complemented
263	#436	—	2	2	no	—
264	#436	—	2	2	no	—
265	#435	—	2	2	no	—
266	#435	—	2	2	no	—
267	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
269	#427	C₂²×C₃₁	1	1	yes	characteristic
270	#424	C₂²×C₃₁	1	2	yes	—
271	C₂×#104	C₂²×C₃₁	1	2	yes	—
272	C₂×#104	C₂²×C₃₁	1	2	yes	—
273	#416	C₂²×C₃₁	1	2	yes	—
274	#408	C₂²×C₃₁	1	2	yes	—
275	#407	C₂²×C₃₁	1	2	yes	—
276	#405	C₂²×C₃₁	1	2	yes	—
277	#341	C₂²×C₃₁	1	2	yes	—
278	#339	C₂²×C₃₁	1	2	yes	—
279	#339	C₂²×C₃₁	1	2	yes	—
280	#336	C₂²×C₃₁	1	2	yes	—
281	#336	C₂²×C₃₁	1	2	yes	—
282	C₂×C₃₁	—	2	1	no	—
283	C₂×C₃₁	—	2	1	no	—
284	C₂×C₃₁	—	4	1	no	—
285	C₂×C₃₁	—	4	1	no	—
286	C₂×C₃₁	—	8	2	no	—
287	C₂×C₃₁	#2325	1	1	yes	characteristic
288	C₂×C₃₁	#2956	1	2	yes	—
289	C₂×#41	C₂³×C₃₁	1	2	yes	—
290	C₂³×C₄	—	2	1	no	—
291	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
292	#135	—	4	1	no	—
293	#135	—	4	1	no	—
294	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
295	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	complemented
296	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
297	#131	—	4	2	no	—
298	#131	—	4	2	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂×M₁₆	—	4	1	no	—
301	C₂×M₁₆	—	4	1	no	—
302	C₂×M₁₆	—	4	1	no	—
303	C₂×M₁₆	—	4	1	no	—
304	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
305	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
306	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
307	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
308	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
309	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
310	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
311	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
312	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
313	C₂²×C₈	—	2	2	no	—
314	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
317	#123	—	2	2	no	—
318	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
319	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
320	#121	C₂×C₄×C₃₁	1	2	yes	—
321	#121	C₂×C₄×C₃₁	1	2	yes	—
322	#120	—	2	2	no	—
323	C₄×D₄	—	2	2	no	—
324	C₄×D₄	—	2	2	no	—
325	#117	C₂³×C₃₁	1	2	yes	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	—	4	1	no	—
328	C₂×C₄²	—	4	1	no	—
329	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
330	#108	—	4	1	no	—
331	#108	—	4	1	no	—
332	#107	C₂×C₄×C₃₁	1	2	yes	complemented
333	#106	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
334	#106	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
335	C₄⋊C₈	—	2	2	no	—
336	C₄⋊C₈	—	2	2	no	—
337	#104	—	4	2	no	—
338	#104	—	4	2	no	—
339	#103	C₂×C₄×C₃₁	1	2	yes	complemented
340	#103	C₂×C₄×C₃₁	1	2	yes	complemented
341	#102	C₂×C₄×C₃₁	1	2	yes	complemented
342	#102	C₂×C₄×C₃₁	1	2	yes	complemented
343	C₂²⋊C₈	—	2	2	no	—
344	C₂²⋊C₈	—	2	2	no	—
345	C₂²⋊C₈	—	2	2	no	—
346	C₂²⋊C₈	—	2	2	no	—
347	#97	—	2	2	no	—
348	C₄×C₈	—	2	2	no	—
349	C₃₁	#22207	1	1	yes	characteristic, direct product factor, Sylow
350	C₂⁴	#41×C₃₁	1	1	yes	characteristic
351	#41	—	2	2	no	—
352	#41	—	2	2	no	—
353	#41	—	2	2	no	—
354	#41	—	2	2	no	—
355	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
374	Q₁₆	—	4	1	no	—
375	SD₁₆	—	4	2	no	—
376	D₈	—	4	1	no	—
377	M₁₆	—	2	2	no	—
378	M₁₆	—	2	2	no	—
379	M₁₆	—	2	2	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	4	1	no	—
382	M₁₆	—	4	1	no	—
383	M₁₆	—	4	1	no	—
384	M₁₆	—	4	1	no	—
385	M₁₆	—	4	1	no	—
386	M₁₆	—	4	2	no	—
387	M₁₆	—	4	1	no	—
388	M₁₆	—	4	4	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	4	4	no	—
398	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
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	C₂×D₄×C₃₁	1	2	yes	—
403	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
407	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	—	2	2	no	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	2	2	no	—
415	C₄²	—	2	2	no	—
416	C₄²	—	4	2	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
422	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
423	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
424	Q₈	—	2	2	no	—
425	Q₈	—	2	2	no	—
426	D₄	—	2	2	no	—
427	D₄	—	2	2	no	—
428	D₄	—	4	2	no	—
429	D₄	—	4	2	no	—
430	D₄	—	8	2	no	—
431	D₄	—	8	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	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	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	8	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
452	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
454	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
456	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
457	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
458	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
459	C₈	—	2	1	no	—
460	C₈	—	2	1	no	—
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	2	2	no	—
464	C₈	—	2	2	no	—
465	C₈	—	4	2	no	—
466	C₈	—	4	2	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
478	C₂²	C₃₁×#538	1	1	yes	characteristic
479	C₂²	C₃₁×#539	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	4	4	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	2	no	—
486	C₄	C₃₁×#518	1	1	yes	characteristic
487	C₄	C₃₁×#520	1	1	yes	characteristic
488	C₄	C₃₁×#534	1	1	yes	characteristic
489	C₄	C₃₁×#537	1	1	yes	characteristic
490	C₂	—	2	1	no	—
491	C₂	—	2	1	no	—
492	C₂	—	4	1	no	—
493	C₂	—	4	1	no	—
494	C₂	—	8	2	no	—
495	C₂	C₃₁×#2325	1	1	yes	characteristic
496	C₂	C₃₁×#2956	1	2	yes	—

	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×#3054	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×#1896	C₂	semidirect
6. C₃₁×#1876	C₂	semidirect
7. C₃₁×#1866	C₂	semidirect
8. C₃₁×#1811	C₂	semidirect
9. C₃₁×#1728	C₂	semidirect
16. C₃₁×#470	C₄	semidirect
17. C₃₁×#470	C₄	semidirect
18. C₃₁×#469	C₄	semidirect
19. C₃₁×#469	C₄	semidirect
20. C₃₁×#468	C₄	semidirect
21. C₃₁×#468	C₄	semidirect
46. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
47. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
48. C₂×D₈×C₃₁	C₂×C₄	semidirect
84. C₃₁×#107	C₂×C₄	semidirect
85. C₃₁×#106	C₂×C₄	semidirect
86. C₃₁×#106	C₂×C₄	semidirect
91. C₃₁×#103	C₂×C₄	semidirect
92. C₃₁×#103	C₂×C₄	semidirect
93. C₃₁×#102	C₂×C₄	semidirect
94. C₃₁×#102	C₂×C₄	semidirect
168. #22207	C₃₁	direct (N × H)
220. #2930	C₂×C₃₁	semidirect
221. #1929	C₂×C₃₁	semidirect
222. #1922	C₂×C₃₁	semidirect
223. #1896	C₂×C₃₁	semidirect
224. #1876	C₂×C₃₁	semidirect
225. #1866	C₂×C₃₁	semidirect
226. #1811	C₂×C₃₁	semidirect
227. #1728	C₂×C₃₁	semidirect
257. #470	C₄×C₃₁	semidirect
258. #470	C₄×C₃₁	semidirect
259. #469	C₄×C₃₁	semidirect
260. #469	C₄×C₃₁	semidirect
261. #468	C₄×C₃₁	semidirect
262. #468	C₄×C₃₁	semidirect
294. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
295. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
296. C₂×D₈	C₂×C₄×C₃₁	semidirect
332. #107	C₂×C₄×C₃₁	semidirect
333. #106	C₂×C₄×C₃₁	semidirect
334. #106	C₂×C₄×C₃₁	semidirect
339. #103	C₂×C₄×C₃₁	semidirect
340. #103	C₂×C₄×C₃₁	semidirect
341. #102	C₂×C₄×C₃₁	semidirect
342. #102	C₂×C₄×C₃₁	semidirect
349. C₃₁	#22207	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×#3054	2
C₃₁×#2930	2
C₃₁×#1929	2
C₃₁×#1922	2
C₃₁×#1896	2
C₃₁×#1876	2
C₃₁×#1866	2
C₃₁×#1811	2
C₃₁×#1728	2
#22207	31

Subgroups — C31×#22207 #21478966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22207 #21478966