Subgroups — C₃₁×#22204 #21478963

Order 7936.

838 conjugacy classes of subgroups.

The subgroup lattice has 510 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×#2929	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×#1810	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#336	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	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#525	C₂²	1	2	yes	—
15	C₂²×C₄²×C₃₁	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×#435	—	2	2	no	—
23	C₃₁×#435	—	2	2	no	—
24	C₈×D₄×C₃₁	—	2	2	no	—
25	C₈×D₄×C₃₁	—	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	C₂³	1	2	yes	—
53	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	4	1	no	—
56	C₂²×C₈×C₃₁	—	4	1	no	—
57	C₂²×C₈×C₃₁	—	4	1	no	—
58	C₂²×C₈×C₃₁	—	4	1	no	—
59	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	D₄	1	2	yes	—
65	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
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	1	yes	characteristic, complemented
85	C₃₁×#107	C₂×C₄	1	1	yes	characteristic, complemented
86	C₃₁×#106	C₂×C₄	1	2	yes	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	4	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	4	2	no	—
119	C₂²×C₄×C₃₁	—	4	2	no	—
120	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	Q₁₆×C₃₁	—	4	1	no	—
127	SD₁₆×C₃₁	—	4	2	no	—
128	D₈×C₃₁	—	4	1	no	—
129	M₁₆×C₃₁	—	2	2	no	—
130	M₁₆×C₃₁	—	2	2	no	—
131	M₁₆×C₃₁	—	2	2	no	—
132	M₁₆×C₃₁	—	2	2	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	2	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	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	1	no	—
147	C₂×C₈×C₃₁	—	4	2	no	—
148	C₂×C₈×C₃₁	—	4	1	no	—
149	C₂×C₈×C₃₁	—	4	4	no	—
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	C₂²×C₄	1	1	yes	characteristic
154	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
159	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	C₂²⋊C₄×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	4	2	no	—
169	#22204	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
176	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
177	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
178	Q₈×C₃₁	—	2	2	no	—
179	Q₈×C₃₁	—	2	2	no	—
180	D₄×C₃₁	—	2	2	no	—
181	D₄×C₃₁	—	2	2	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	—	4	4	no	—
185	D₄×C₃₁	—	4	4	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	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	4	8	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
213	C₈×C₃₁	—	2	1	no	—
214	C₈×C₃₁	—	2	1	no	—
215	C₈×C₃₁	—	2	1	no	—
216	C₈×C₃₁	—	2	1	no	—
217	C₈×C₃₁	—	2	2	no	—
218	C₈×C₃₁	—	2	2	no	—
219	C₈×C₃₁	—	4	2	no	—
220	C₈×C₃₁	—	4	2	no	—
221	#3050	C₂×C₃₁	1	1	yes	characteristic
222	#2929	C₂×C₃₁	1	2	yes	complemented
223	#1929	C₂×C₃₁	1	2	yes	complemented
224	#1922	C₂×C₃₁	1	2	yes	complemented
225	#1896	C₂×C₃₁	1	2	yes	complemented
226	#1876	C₂×C₃₁	1	2	yes	complemented
227	#1866	C₂×C₃₁	1	2	yes	complemented
228	#1810	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
230	C₂²×C₃₁	—	2	4	no	—
231	C₂²×C₃₁	—	2	4	no	—
232	C₂²×C₃₁	—	2	1	no	—
233	C₂²×C₃₁	—	2	1	no	—
234	C₂²×C₃₁	—	2	1	no	—
235	C₂²×C₃₁	—	2	1	no	—
236	C₂²×C₃₁	—	2	1	no	—
237	C₂²×C₃₁	—	2	1	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	#443	1	2	yes	—
242	C₂²×C₃₁	#444	1	2	yes	—
243	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂²×C₃₁	#538	1	1	yes	characteristic
245	C₂²×C₃₁	#539	1	1	yes	characteristic
246	C₄×C₃₁	—	2	1	no	—
247	C₄×C₃₁	—	2	1	no	—
248	C₄×C₃₁	—	4	4	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	#518	1	1	yes	characteristic
253	C₄×C₃₁	#520	1	1	yes	characteristic
254	C₄×C₃₁	#534	1	1	yes	characteristic
255	C₄×C₃₁	#537	1	1	yes	characteristic
256	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×#131	C₂²×C₃₁	1	2	yes	—
258	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
259	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	#525	C₂²×C₃₁	1	2	yes	—
261	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
262	#467	C₄×C₃₁	1	1	yes	characteristic, complemented
263	#467	C₄×C₃₁	1	1	yes	characteristic, complemented
264	#466	C₄×C₃₁	1	1	yes	characteristic, complemented
265	#466	C₄×C₃₁	1	1	yes	characteristic, complemented
266	#465	C₄×C₃₁	1	2	yes	complemented
267	#465	C₄×C₃₁	1	2	yes	complemented
268	#435	—	2	2	no	—
269	#435	—	2	2	no	—
270	C₈×D₄	—	2	2	no	—
271	C₈×D₄	—	2	2	no	—
272	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
274	#427	C₂²×C₃₁	1	1	yes	characteristic
275	#424	C₂²×C₃₁	1	2	yes	—
276	C₂×#104	C₂²×C₃₁	1	2	yes	—
277	C₂×#104	C₂²×C₃₁	1	2	yes	—
278	#416	C₂²×C₃₁	1	2	yes	—
279	#408	C₂²×C₃₁	1	2	yes	—
280	#407	C₂²×C₃₁	1	2	yes	—
281	#405	C₂²×C₃₁	1	2	yes	—
282	#341	C₂²×C₃₁	1	2	yes	—
283	#339	C₂²×C₃₁	1	2	yes	—
284	#339	C₂²×C₃₁	1	2	yes	—
285	#336	C₂²×C₃₁	1	2	yes	—
286	#336	C₂²×C₃₁	1	2	yes	—
287	C₂×C₃₁	—	2	1	no	—
288	C₂×C₃₁	—	2	1	no	—
289	C₂×C₃₁	—	2	2	no	—
290	C₂×C₃₁	—	2	2	no	—
291	C₂×C₃₁	—	8	2	no	—
292	C₂×C₃₁	#2325	1	1	yes	characteristic
293	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
294	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
295	C₂×#41	C₂³×C₃₁	1	2	yes	—
296	C₂³×C₄	—	2	1	no	—
297	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
298	#135	—	4	1	no	—
299	#135	—	4	1	no	—
300	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
301	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	complemented
302	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
303	#131	—	4	2	no	—
304	#131	—	4	2	no	—
305	C₂×M₁₆	—	2	2	no	—
306	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
307	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
308	C₂²×C₈	—	2	2	no	—
309	C₂²×C₈	—	4	1	no	—
310	C₂²×C₈	—	4	1	no	—
311	C₂²×C₈	—	4	1	no	—
312	C₂²×C₈	—	4	1	no	—
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	C₂×C₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
318	C₂²×C₈	D₄×C₃₁	1	2	yes	—
319	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
323	#123	—	2	2	no	—
324	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
325	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
326	#121	C₂×C₄×C₃₁	1	2	yes	—
327	#121	C₂×C₄×C₃₁	1	2	yes	—
328	#120	—	2	2	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	—	2	2	no	—
331	#117	C₂³×C₃₁	1	2	yes	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	—	4	1	no	—
334	C₂×C₄²	—	4	1	no	—
335	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
336	#108	—	4	1	no	—
337	#108	—	4	1	no	—
338	#107	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
339	#107	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
340	#106	C₂×C₄×C₃₁	1	2	yes	complemented
341	C₄⋊C₈	—	2	2	no	—
342	C₄⋊C₈	—	2	2	no	—
343	#104	—	4	2	no	—
344	#104	—	4	2	no	—
345	#103	C₂×C₄×C₃₁	1	2	yes	complemented
346	#103	C₂×C₄×C₃₁	1	2	yes	complemented
347	#102	C₂×C₄×C₃₁	1	2	yes	complemented
348	#102	C₂×C₄×C₃₁	1	2	yes	complemented
349	C₂²⋊C₈	—	2	2	no	—
350	C₂²⋊C₈	—	2	2	no	—
351	C₂²⋊C₈	—	2	2	no	—
352	C₂²⋊C₈	—	2	2	no	—
353	#97	—	2	2	no	—
354	C₄×C₈	—	2	2	no	—
355	C₃₁	#22204	1	1	yes	characteristic, direct product factor, Sylow
356	C₂⁴	#41×C₃₁	1	1	yes	characteristic
357	#41	—	2	2	no	—
358	#41	—	2	2	no	—
359	#41	—	2	2	no	—
360	#41	—	2	2	no	—
361	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	4	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
381	Q₁₆	—	4	1	no	—
382	SD₁₆	—	4	2	no	—
383	D₈	—	4	1	no	—
384	M₁₆	—	2	2	no	—
385	M₁₆	—	2	2	no	—
386	M₁₆	—	2	2	no	—
387	M₁₆	—	2	2	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	1	no	—
398	C₂×C₈	—	4	1	no	—
399	C₂×C₈	—	4	1	no	—
400	C₂×C₈	—	4	1	no	—
401	C₂×C₈	—	4	1	no	—
402	C₂×C₈	—	4	2	no	—
403	C₂×C₈	—	4	1	no	—
404	C₂×C₈	—	4	4	no	—
405	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
406	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
407	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
408	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
409	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
410	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
414	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	4	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	1	no	—
427	C₂³	—	2	1	no	—
428	C₂³	—	2	2	no	—
429	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
430	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
431	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
432	Q₈	—	2	2	no	—
433	Q₈	—	2	2	no	—
434	D₄	—	2	2	no	—
435	D₄	—	2	2	no	—
436	D₄	—	4	4	no	—
437	D₄	—	4	4	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	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	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	4	8	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
464	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
465	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
466	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
467	C₈	—	2	1	no	—
468	C₈	—	2	1	no	—
469	C₈	—	2	1	no	—
470	C₈	—	2	1	no	—
471	C₈	—	2	2	no	—
472	C₈	—	2	2	no	—
473	C₈	—	4	2	no	—
474	C₈	—	4	2	no	—
475	C₂²	—	2	4	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	C₃₁×#443	1	2	yes	—
487	C₂²	C₃₁×#444	1	2	yes	—
488	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
489	C₂²	C₃₁×#538	1	1	yes	characteristic
490	C₂²	C₃₁×#539	1	1	yes	characteristic
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	4	4	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	C₃₁×#518	1	1	yes	characteristic
498	C₄	C₃₁×#520	1	1	yes	characteristic
499	C₄	C₃₁×#534	1	1	yes	characteristic
500	C₄	C₃₁×#537	1	1	yes	characteristic
501	C₂	—	2	1	no	—
502	C₂	—	2	2	no	—
503	C₂	—	2	2	no	—
504	C₂	—	2	1	no	—
505	C₂	—	8	2	no	—
506	C₂	C₃₁×#2325	1	1	yes	characteristic
507	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
508	C₂	C₂×C₃₁×#444	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 ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2929	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×#1810	C₂	semidirect
9. C₂×C₃₁×#336	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
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×#107	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
169. #22204	C₃₁	direct (N × H)
222. #2929	C₂×C₃₁	semidirect
223. #1929	C₂×C₃₁	semidirect
224. #1922	C₂×C₃₁	semidirect
225. #1896	C₂×C₃₁	semidirect
226. #1876	C₂×C₃₁	semidirect
227. #1866	C₂×C₃₁	semidirect
228. #1810	C₂×C₃₁	semidirect
229. C₂×#336	C₂×C₃₁	semidirect
262. #467	C₄×C₃₁	semidirect
263. #467	C₄×C₃₁	semidirect
264. #466	C₄×C₃₁	semidirect
265. #466	C₄×C₃₁	semidirect
266. #465	C₄×C₃₁	semidirect
267. #465	C₄×C₃₁	semidirect
300. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
301. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
302. C₂×D₈	C₂×C₄×C₃₁	semidirect
338. #107	C₂×C₄×C₃₁	semidirect
339. #107	C₂×C₄×C₃₁	semidirect
340. #106	C₂×C₄×C₃₁	semidirect
345. #103	C₂×C₄×C₃₁	semidirect
346. #103	C₂×C₄×C₃₁	semidirect
347. #102	C₂×C₄×C₃₁	semidirect
348. #102	C₂×C₄×C₃₁	semidirect
355. C₃₁	#22204	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×#2929	2
C₃₁×#1929	2
C₃₁×#1922	2
C₃₁×#1896	2
C₃₁×#1876	2
C₃₁×#1866	2
C₃₁×#1810	2
C₂×C₃₁×#336	2
#22204	31

Subgroups — C31×#22204 #21478963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22204 #21478963