Subgroups — C₃₁×#22430 #21479189

Order 7936.

1426 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#419	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1894	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1873	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1787	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	D₄×Q₈×C₃₁	—	2	4	no	—
15	D₄×Q₈×C₃₁	C₄	1	8	yes	complemented
16	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	—	2	1	no	—
22	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#117	C₂²	1	2	yes	—
24	C₃₁×#436	—	2	4	no	—
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₃₁×#419	—	2	8	no	—
27	C₂×C₃₁×#103	C₂²	1	2	yes	—
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#103	C₂²	1	2	yes	—
33	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#390	—	2	1	no	—
37	C₃₁×#388	—	2	2	no	—
38	C₃₁×#384	C₂²	1	1	yes	characteristic
39	C₃₁×#380	C₂²	1	2	yes	—
40	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	1	yes	characteristic
44	C₃₁×#340	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₃₁	—	2	2	no	—
46	C₂²×Q₈×C₃₁	—	2	4	no	—
47	C₂²×Q₈×C₃₁	—	2	4	no	—
48	C₂²×Q₈×C₃₁	—	2	4	no	—
49	C₂²×Q₈×C₃₁	—	4	2	no	—
50	C₂²×Q₈×C₃₁	—	4	2	no	—
51	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	—
52	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	—	2	2	no	—
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂×M₁₆×C₃₁	—	2	4	no	—
58	C₂×M₁₆×C₃₁	—	2	4	no	—
59	C₂×M₁₆×C₃₁	—	4	2	no	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	—	2	8	no	—
64	C₃₁×C₄⋊Q₈	—	2	2	no	—
65	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
66	C₃₁×C₂²⋊Q₈	—	2	4	no	—
67	C₃₁×C₂²⋊Q₈	—	2	16	no	—
68	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
69	C₄×Q₈×C₃₁	—	2	2	no	—
70	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
71	C₄×D₄×C₃₁	—	2	8	no	—
72	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
73	C₄×D₄×C₃₁	D₄	1	4	yes	—
74	C₃₁×#117	—	2	4	no	—
75	C₃₁×#117	—	4	2	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
91	C₂×C₄²×C₃₁	—	2	1	no	—
92	C₂×C₄²×C₃₁	—	2	2	no	—
93	C₂×C₄²×C₃₁	—	2	1	no	—
94	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
95	C₃₁×C₄⋊C₈	—	2	2	no	—
96	C₃₁×#103	—	2	4	no	—
97	C₃₁×#103	—	2	2	no	—
98	C₃₁×#103	—	2	4	no	—
99	C₃₁×#103	—	2	4	no	—
100	C₃₁×#103	—	4	2	no	—
101	C₃₁×#103	—	4	2	no	—
102	C₃₁×#103	D₄	1	4	yes	complemented
103	C₃₁×C₂²⋊C₈	—	2	4	no	—
104	C₄×C₈×C₃₁	—	2	2	no	—
105	C₃₁×#95	—	2	2	no	—
106	C₃₁×#95	—	2	1	no	—
107	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×Q₈×C₃₁	—	2	8	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	16	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	—	2	2	no	—
117	C₂×Q₈×C₃₁	—	2	2	no	—
118	C₂×Q₈×C₃₁	—	4	4	no	—
119	C₂×Q₈×C₃₁	—	4	4	no	—
120	C₂×Q₈×C₃₁	—	4	8	no	—
121	C₂×Q₈×C₃₁	—	4	8	no	—
122	C₂×Q₈×C₃₁	—	4	8	no	—
123	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
124	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
125	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×Q₈×C₃₁	#41	1	2	yes	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	2	no	—
135	C₂²×C₄×C₃₁	—	2	8	no	—
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	4	4	no	—
142	C₂²×C₄×C₃₁	—	4	4	no	—
143	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
147	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
149	M₁₆×C₃₁	—	4	8	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	4	no	—
162	C₄⋊C₄×C₃₁	—	2	8	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	4	2	no	—
170	C₄⋊C₄×C₃₁	—	8	1	no	—
171	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
172	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²⋊C₄×C₃₁	—	2	8	no	—
175	C₂²⋊C₄×C₃₁	—	4	4	no	—
176	C₂²⋊C₄×C₃₁	—	8	2	no	—
177	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
178	C₄²×C₃₁	—	2	4	no	—
179	C₄²×C₃₁	—	4	4	no	—
180	C₄²×C₃₁	—	4	2	no	—
181	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
182	C₄²×C₃₁	C₂×D₄	1	2	yes	—
183	#22430	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₃₁	—	2	2	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	—	2	2	no	—
188	C₂³×C₃₁	—	2	2	no	—
189	C₂³×C₃₁	—	2	2	no	—
190	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
191	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	Q₈×C₃₁	—	2	2	no	—
193	Q₈×C₃₁	—	2	2	no	—
194	Q₈×C₃₁	—	2	2	no	—
195	Q₈×C₃₁	—	2	4	no	—
196	Q₈×C₃₁	—	2	4	no	—
197	Q₈×C₃₁	—	4	16	no	—
198	Q₈×C₃₁	—	4	2	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	C₄×D₄	1	4	yes	—
201	D₄×C₃₁	—	2	8	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	4	16	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
230	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
231	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
232	C₂×C₄×C₃₁	#137	1	2	yes	—
233	C₂×C₄×C₃₁	#137	1	2	yes	—
234	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
239	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
240	C₈×C₃₁	—	2	2	no	—
241	C₈×C₃₁	—	4	2	no	—
242	C₈×C₃₁	—	4	1	no	—
243	C₂×D₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
244	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
245	C₂×#419	C₂×C₃₁	1	2	yes	complemented
246	#2354	C₂×C₃₁	1	1	yes	characteristic
247	#1981	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#1956	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1894	C₂×C₃₁	1	2	yes	complemented
250	#1873	C₂×C₃₁	1	2	yes	complemented
251	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1787	C₂×C₃₁	1	2	yes	complemented
253	C₄×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂²×C₃₁	—	2	4	no	—
255	C₂²×C₃₁	—	2	4	no	—
256	C₂²×C₃₁	—	2	4	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	4	2	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
262	C₂²×C₃₁	D₄²	1	1	yes	characteristic
263	C₂²×C₃₁	#546	1	1	yes	characteristic
264	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
265	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
266	C₂²×C₃₁	#577	1	1	yes	characteristic
267	C₂²×C₃₁	#578	1	1	yes	characteristic
268	C₄×C₃₁	—	2	4	no	—
269	C₄×C₃₁	—	2	4	no	—
270	C₄×C₃₁	—	2	2	no	—
271	C₄×C₃₁	—	4	4	no	—
272	C₄×C₃₁	—	4	2	no	—
273	C₄×C₃₁	—	4	2	no	—
274	C₄×C₃₁	—	8	1	no	—
275	C₄×C₃₁	—	8	2	no	—
276	C₄×C₃₁	#419	1	4	yes	complemented
277	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
278	C₄×C₃₁	#518	1	1	yes	characteristic
279	C₄×C₃₁	D₄²	1	1	yes	characteristic
280	C₄×C₃₁	#546	1	1	yes	characteristic
281	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
282	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
283	D₄×Q₈	—	2	4	no	—
284	D₄×Q₈	C₄×C₃₁	1	8	yes	complemented
285	C₂×C₄⋊Q₈	—	2	1	no	—
286	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₂²⋊Q₈	—	2	2	no	—
288	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
289	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄×D₄	—	2	1	no	—
291	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#117	C₂²×C₃₁	1	2	yes	—
293	#436	—	2	4	no	—
294	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
295	#419	—	2	8	no	—
296	C₂×#103	C₂²×C₃₁	1	2	yes	—
297	C₂×#103	C₂²×C₃₁	1	2	yes	—
298	C₂×#103	C₂²×C₃₁	1	2	yes	—
299	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×#103	C₂²×C₃₁	1	2	yes	—
301	C₂×#103	C₂²×C₃₁	1	2	yes	—
302	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
304	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
305	#390	—	2	1	no	—
306	#388	—	2	2	no	—
307	#384	C₂²×C₃₁	1	1	yes	characteristic
308	#380	C₂²×C₃₁	1	2	yes	—
309	C₄×C₄⋊C₄	—	2	1	no	—
310	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
311	#340	C₂²×C₃₁	1	2	yes	—
312	#340	C₂²×C₃₁	1	1	yes	characteristic
313	#340	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×C₃₁	—	4	4	no	—
315	C₂×C₃₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
316	C₂×C₃₁	C₂×#419	1	1	yes	characteristic
317	C₂×C₃₁	#2898	1	1	yes	characteristic
318	C₂×C₃₁	#3276	1	1	yes	characteristic
319	C₂×C₃₁	#3278	1	1	yes	characteristic
320	C₂×C₃₁	#3328	1	1	yes	characteristic
321	C₂×C₃₁	#3329	1	1	yes	characteristic
322	C₂²×Q₈	—	2	2	no	—
323	C₂²×Q₈	—	2	4	no	—
324	C₂²×Q₈	—	2	4	no	—
325	C₂²×Q₈	—	2	4	no	—
326	C₂²×Q₈	—	4	2	no	—
327	C₂²×Q₈	—	4	2	no	—
328	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	—
329	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
330	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
332	C₂³×C₄	—	2	2	no	—
333	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
334	C₂×M₁₆	—	2	4	no	—
335	C₂×M₁₆	—	2	4	no	—
336	C₂×M₁₆	—	4	2	no	—
337	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
338	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
339	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
340	C₄⋊Q₈	—	2	8	no	—
341	C₄⋊Q₈	—	2	2	no	—
342	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	—	2	16	no	—
345	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
348	C₄×D₄	—	2	8	no	—
349	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
350	C₄×D₄	D₄×C₃₁	1	4	yes	—
351	#117	—	2	4	no	—
352	#117	—	4	2	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	—	2	2	no	—
357	C₂×C₄⋊C₄	—	2	1	no	—
358	C₂×C₄⋊C₄	—	2	1	no	—
359	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
360	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
361	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
364	C₂×C₂²⋊C₄	—	2	2	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₂²⋊C₄	—	2	2	no	—
367	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
368	C₂×C₄²	—	2	1	no	—
369	C₂×C₄²	—	2	2	no	—
370	C₂×C₄²	—	2	1	no	—
371	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
372	C₄⋊C₈	—	2	2	no	—
373	#103	—	2	4	no	—
374	#103	—	2	2	no	—
375	#103	—	2	4	no	—
376	#103	—	2	4	no	—
377	#103	—	4	2	no	—
378	#103	—	4	2	no	—
379	#103	D₄×C₃₁	1	4	yes	complemented
380	C₂²⋊C₈	—	2	4	no	—
381	C₄×C₈	—	2	2	no	—
382	#95	—	2	2	no	—
383	#95	—	2	1	no	—
384	C₃₁	#22430	1	1	yes	characteristic, direct product factor, Sylow
385	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
386	C₂×Q₈	—	2	8	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	1	no	—
391	C₂×Q₈	—	2	16	no	—
392	C₂×Q₈	—	2	2	no	—
393	C₂×Q₈	—	2	2	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	—	4	4	no	—
397	C₂×Q₈	—	4	4	no	—
398	C₂×Q₈	—	4	8	no	—
399	C₂×Q₈	—	4	8	no	—
400	C₂×Q₈	—	4	8	no	—
401	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
402	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
403	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
404	C₂×Q₈	#41×C₃₁	1	2	yes	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
408	C₂²×C₄	—	2	1	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	C₂²×C₄	—	2	8	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	4	4	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
422	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
423	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
425	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
427	M₁₆	—	4	8	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	4	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	4	2	no	—
448	C₄⋊C₄	—	8	1	no	—
449	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
450	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
451	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	4	4	no	—
454	C₂²⋊C₄	—	8	2	no	—
455	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
456	C₄²	—	2	4	no	—
457	C₄²	—	4	4	no	—
458	C₄²	—	4	2	no	—
459	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
460	C₄²	C₂×D₄×C₃₁	1	2	yes	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
468	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	Q₈	—	2	2	no	—
470	Q₈	—	2	2	no	—
471	Q₈	—	2	2	no	—
472	Q₈	—	2	4	no	—
473	Q₈	—	2	4	no	—
474	Q₈	—	4	16	no	—
475	Q₈	—	4	2	no	—
476	Q₈	—	4	2	no	—
477	Q₈	C₄×D₄×C₃₁	1	4	yes	—
478	D₄	—	2	8	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	4	16	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
507	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
508	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
509	C₂×C₄	C₃₁×#137	1	2	yes	—
510	C₂×C₄	C₃₁×#137	1	2	yes	—
511	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
513	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
514	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
515	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
516	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
517	C₈	—	2	2	no	—
518	C₈	—	4	2	no	—
519	C₈	—	4	1	no	—
520	C₂²	—	2	4	no	—
521	C₂²	—	2	4	no	—
522	C₂²	—	2	4	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	4	2	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
528	C₂²	D₄²×C₃₁	1	1	yes	characteristic
529	C₂²	C₃₁×#546	1	1	yes	characteristic
530	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
531	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
532	C₂²	C₃₁×#577	1	1	yes	characteristic
533	C₂²	C₃₁×#578	1	1	yes	characteristic
534	C₄	—	2	4	no	—
535	C₄	—	2	4	no	—
536	C₄	—	2	2	no	—
537	C₄	—	4	4	no	—
538	C₄	—	4	2	no	—
539	C₄	—	4	2	no	—
540	C₄	—	8	2	no	—
541	C₄	—	8	1	no	—
542	C₄	C₃₁×#419	1	4	yes	complemented
543	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
544	C₄	C₃₁×#518	1	1	yes	characteristic
545	C₄	D₄²×C₃₁	1	1	yes	characteristic
546	C₄	C₃₁×#546	1	1	yes	characteristic
547	C₂	—	4	4	no	—
548	C₂	D₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
549	C₂	C₂×C₃₁×#419	1	1	yes	characteristic
550	C₂	C₃₁×#2898	1	1	yes	characteristic
551	C₂	C₃₁×#3276	1	1	yes	characteristic
552	C₂	C₃₁×#3278	1	1	yes	characteristic
553	C₂	C₃₁×#3328	1	1	yes	characteristic
554	C₂	C₃₁×#3329	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#419	C₂	semidirect
5. C₃₁×#1981	C₂	semidirect
6. C₃₁×#1956	C₂	semidirect
7. C₃₁×#1894	C₂	semidirect
8. C₃₁×#1873	C₂	semidirect
9. C₃₁×#1803	C₂	semidirect
10. C₃₁×#1787	C₂	semidirect
11. C₄×C₃₁×#103	C₂	semidirect
15. D₄×Q₈×C₃₁	C₄	semidirect
102. C₃₁×#103	D₄	semidirect
183. #22430	C₃₁	direct (N × H)
245. C₂×#419	C₂×C₃₁	semidirect
247. #1981	C₂×C₃₁	semidirect
248. #1956	C₂×C₃₁	semidirect
249. #1894	C₂×C₃₁	semidirect
250. #1873	C₂×C₃₁	semidirect
251. #1803	C₂×C₃₁	semidirect
252. #1787	C₂×C₃₁	semidirect
253. C₄×#103	C₂×C₃₁	semidirect
276. C₄×C₃₁	#419	semidirect
284. D₄×Q₈	C₄×C₃₁	semidirect
379. #103	D₄×C₃₁	semidirect
384. C₃₁	#22430	direct (N × H)
542. C₄	C₃₁×#419	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄×Q₈×C₃₁	2
C₂×C₃₁×#436	2
C₂×C₃₁×#419	2
C₃₁×#2354	2
C₃₁×#1981	2
C₃₁×#1956	2
C₃₁×#1894	2
C₃₁×#1873	2
C₃₁×#1803	2
C₃₁×#1787	2
C₄×C₃₁×#103	2
#22430	31

Subgroups — C31×#22430 #21479189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22430 #21479189