Subgroups — C₃₁×#20873 #21477632

Order 7936.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1864	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1862	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₃₁	—	2	1	no	—
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	—	2	4	no	—
15	C₂×C₄×D₄×C₃₁	—	2	4	no	—
16	C₂×C₄×D₄×C₃₁	—	2	2	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
18	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
22	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#428	—	2	4	no	—
24	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
25	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
26	C₃₁×#418	—	2	4	no	—
27	C₃₁×#418	C₄	1	8	yes	complemented
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	4	yes	—
31	C₃₁×#389	C₂²	1	2	yes	complemented
32	C₃₁×#389	C₂²	1	2	yes	complemented
33	C₃₁×#385	—	2	1	no	—
34	C₃₁×#385	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₃₁×#340	C₂²	1	4	yes	complemented
37	C₃₁×#336	C₂²	1	4	yes	complemented
38	C₂⁵×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
44	C₂²×D₄×C₃₁	D₄	1	4	yes	—
45	C₂²×D₄×C₃₁	Q₈	1	4	yes	—
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	—	2	8	no	—
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂³×C₄×C₃₁	—	2	4	no	—
50	C₂³×C₄×C₃₁	—	4	8	no	—
51	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
60	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
62	C₄×D₄×C₃₁	—	2	16	no	—
63	C₄×D₄×C₃₁	—	2	8	no	—
64	C₄×D₄×C₃₁	—	2	8	no	—
65	C₃₁×#117	—	2	2	no	—
66	C₃₁×#117	—	2	2	no	—
67	C₃₁×#117	—	2	2	no	—
68	C₃₁×#117	C₂×C₄	1	4	yes	complemented
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	D₄	1	4	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	—	2	2	no	—
84	C₂×C₄²×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	D₄	1	2	yes	—
87	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
88	C₃₁×#107	—	2	4	no	—
89	C₃₁×#106	—	2	4	no	—
90	C₃₁×#102	—	2	4	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	C₂×C₄	1	16	yes	complemented
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₂⁴×C₃₁	—	2	4	no	—
96	C₂⁴×C₃₁	—	2	4	no	—
97	C₂⁴×C₃₁	—	2	4	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	C₂⁴×C₃₁	—	4	4	no	—
100	C₂⁴×C₃₁	—	4	4	no	—
101	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	—	2	8	no	—
103	C₂×D₄×C₃₁	—	2	4	no	—
104	C₂×D₄×C₃₁	—	2	4	no	—
105	C₂×D₄×C₃₁	—	2	8	no	—
106	C₂×D₄×C₃₁	—	2	4	no	—
107	C₂×D₄×C₃₁	—	2	8	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	C₄⋊C₄	1	16	yes	complemented
112	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
113	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
114	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	16	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	4	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	4	2	no	—
125	C₂²×C₄×C₃₁	—	4	32	no	—
126	C₂²×C₄×C₃₁	—	4	8	no	—
127	C₂²×C₄×C₃₁	—	4	4	no	—
128	C₂²×C₄×C₃₁	—	4	8	no	—
129	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
136	C₂²×C₄×C₃₁	#41	1	2	yes	—
137	C₂²×C₄×C₃₁	#41	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
139	M₁₆×C₃₁	—	2	8	no	—
140	M₁₆×C₃₁	—	4	4	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	complemented
147	C₄⋊C₄×C₃₁	—	2	4	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	—	4	4	no	—
155	C₄⋊C₄×C₃₁	—	4	4	no	—
156	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	complemented
157	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
158	C₂²⋊C₄×C₃₁	—	2	16	no	—
159	C₂²⋊C₄×C₃₁	—	2	4	no	—
160	C₂²⋊C₄×C₃₁	—	4	2	no	—
161	C₄²×C₃₁	—	2	4	no	—
162	C₄²×C₃₁	—	2	4	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	4	2	no	—
166	#20873	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	2	no	—
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	2	4	no	—
170	C₂³×C₃₁	—	2	4	no	—
171	C₂³×C₃₁	—	2	4	no	—
172	C₂³×C₃₁	—	2	16	no	—
173	C₂³×C₃₁	—	4	8	no	—
174	C₂³×C₃₁	—	4	16	no	—
175	C₂³×C₃₁	—	4	16	no	—
176	C₂³×C₃₁	—	4	8	no	—
177	C₂³×C₃₁	—	4	4	no	—
178	C₂³×C₃₁	—	4	4	no	—
179	C₂³×C₃₁	C₄×D₄	1	2	yes	—
180	C₂³×C₃₁	#120	1	2	yes	—
181	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
182	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	D₄×C₃₁	—	2	4	no	—
184	D₄×C₃₁	—	2	4	no	—
185	D₄×C₃₁	—	2	4	no	—
186	D₄×C₃₁	—	2	16	no	—
187	D₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
188	C₂×C₄×C₃₁	—	2	8	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	4	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	4	32	no	—
197	C₂×C₄×C₃₁	—	4	4	no	—
198	C₂×C₄×C₃₁	—	4	8	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	4	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
207	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
208	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₃₁	#120	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
211	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
212	C₂×C₄×C₃₁	#123	1	4	yes	—
213	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
214	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
220	C₈×C₃₁	—	2	4	no	—
221	C₈×C₃₁	—	4	2	no	—
222	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
223	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
224	C₂×#418	C₂×C₃₁	1	1	yes	characteristic
225	C₂×#418	C₂×C₃₁	1	1	yes	characteristic
226	#2296	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1864	C₂×C₃₁	1	4	yes	complemented
228	#1862	C₂×C₃₁	1	4	yes	complemented
229	#1736	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1728	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	2	4	no	—
232	C₂²×C₃₁	—	2	4	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	2	8	no	—
236	C₂²×C₃₁	—	4	32	no	—
237	C₂²×C₃₁	—	4	8	no	—
238	C₂²×C₃₁	—	4	8	no	—
239	C₂²×C₃₁	—	4	4	no	—
240	C₂²×C₃₁	—	4	4	no	—
241	C₂²×C₃₁	#385	1	4	yes	—
242	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
243	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
244	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
246	C₂²×C₃₁	#575	1	2	yes	—
247	C₄×C₃₁	—	2	4	no	—
248	C₄×C₃₁	—	4	4	no	—
249	C₄×C₃₁	—	4	4	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	8	1	no	—
252	C₄×C₃₁	—	8	1	no	—
253	C₄×C₃₁	#385	1	4	yes	—
254	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
255	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
256	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
257	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
258	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
259	C₂⁴×C₄	—	2	1	no	—
260	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
261	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄×D₄	—	2	4	no	—
263	C₂×C₄×D₄	—	2	4	no	—
264	C₂×C₄×D₄	—	2	2	no	—
265	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
266	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
269	C₂²×C₂²⋊C₄	—	2	1	no	—
270	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
271	#428	—	2	4	no	—
272	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
273	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
274	#418	—	2	4	no	—
275	#418	C₄×C₃₁	1	8	yes	complemented
276	C₂×#102	C₂²×C₃₁	1	2	yes	—
277	C₂×#102	C₂²×C₃₁	1	2	yes	—
278	C₂×#102	C₂²×C₃₁	1	4	yes	—
279	#389	C₂²×C₃₁	1	2	yes	complemented
280	#389	C₂²×C₃₁	1	2	yes	complemented
281	#385	—	2	1	no	—
282	#385	—	2	1	no	—
283	C₄×C₄⋊C₄	—	2	2	no	—
284	#340	C₂²×C₃₁	1	4	yes	complemented
285	#336	C₂²×C₃₁	1	4	yes	complemented
286	C₂×C₃₁	—	2	4	no	—
287	C₂×C₃₁	—	4	8	no	—
288	C₂×C₃₁	C₂×#385	1	1	yes	characteristic
289	C₂×C₃₁	C₄×AGL(1,ℤ/8)	1	2	yes	—
290	C₂×C₃₁	#3003	1	1	yes	characteristic
291	C₂×C₃₁	#3005	1	1	yes	characteristic
292	C₂×C₃₁	#3077	1	2	yes	—
293	C₂⁵	—	2	1	no	—
294	C₂²×D₄	—	2	1	no	—
295	C₂²×D₄	—	2	2	no	—
296	C₂²×D₄	—	2	2	no	—
297	C₂²×D₄	—	2	2	no	—
298	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
299	C₂²×D₄	D₄×C₃₁	1	4	yes	—
300	C₂²×D₄	Q₈×C₃₁	1	4	yes	—
301	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
302	C₂³×C₄	—	2	8	no	—
303	C₂³×C₄	—	2	1	no	—
304	C₂³×C₄	—	2	4	no	—
305	C₂³×C₄	—	4	8	no	—
306	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
307	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
308	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×M₁₆	—	2	2	no	—
310	C₂×M₁₆	—	2	2	no	—
311	C₂×M₁₆	—	2	2	no	—
312	C₂×M₁₆	—	2	2	no	—
313	C₂×M₁₆	—	2	2	no	—
314	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
315	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
316	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
317	C₄×D₄	—	2	16	no	—
318	C₄×D₄	—	2	8	no	—
319	C₄×D₄	—	2	8	no	—
320	#117	—	2	2	no	—
321	#117	—	2	2	no	—
322	#117	—	2	2	no	—
323	#117	C₂×C₄×C₃₁	1	4	yes	complemented
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	2	no	—
328	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
333	C₂×C₂²⋊C₄	—	2	8	no	—
334	C₂×C₂²⋊C₄	—	2	1	no	—
335	C₂×C₂²⋊C₄	—	2	4	no	—
336	C₂×C₂²⋊C₄	—	2	1	no	—
337	C₂×C₄²	—	2	1	no	—
338	C₂×C₄²	—	2	2	no	—
339	C₂×C₄²	—	2	2	no	—
340	C₂×C₄²	—	2	1	no	—
341	C₂×C₄²	D₄×C₃₁	1	2	yes	—
342	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
343	#107	—	2	4	no	—
344	#106	—	2	4	no	—
345	#102	—	2	4	no	—
346	#102	—	2	4	no	—
347	#102	C₂×C₄×C₃₁	1	16	yes	complemented
348	#95	—	2	2	no	—
349	#95	—	2	2	no	—
350	C₃₁	#20873	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	—	2	4	no	—
352	C₂⁴	—	2	4	no	—
353	C₂⁴	—	2	4	no	—
354	C₂⁴	—	2	2	no	—
355	C₂⁴	—	4	4	no	—
356	C₂⁴	—	4	4	no	—
357	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₂×D₄	—	2	8	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	4	no	—
361	C₂×D₄	—	2	8	no	—
362	C₂×D₄	—	2	4	no	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	C₄⋊C₄×C₃₁	1	16	yes	complemented
368	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
369	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
370	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	16	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	4	2	no	—
381	C₂²×C₄	—	4	32	no	—
382	C₂²×C₄	—	4	8	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	8	no	—
385	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
391	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
392	C₂²×C₄	#41×C₃₁	1	2	yes	—
393	C₂²×C₄	#41×C₃₁	1	2	yes	—
394	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
395	M₁₆	—	2	8	no	—
396	M₁₆	—	4	4	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	complemented
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	4	4	no	—
411	C₄⋊C₄	—	4	4	no	—
412	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	complemented
413	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
414	C₂²⋊C₄	—	2	16	no	—
415	C₂²⋊C₄	—	2	4	no	—
416	C₂²⋊C₄	—	4	2	no	—
417	C₄²	—	2	4	no	—
418	C₄²	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	4	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	4	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	2	4	no	—
427	C₂³	—	2	16	no	—
428	C₂³	—	4	8	no	—
429	C₂³	—	4	16	no	—
430	C₂³	—	4	16	no	—
431	C₂³	—	4	8	no	—
432	C₂³	—	4	4	no	—
433	C₂³	—	4	4	no	—
434	C₂³	C₄×D₄×C₃₁	1	2	yes	—
435	C₂³	C₃₁×#120	1	2	yes	—
436	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
437	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
438	D₄	—	2	4	no	—
439	D₄	—	2	4	no	—
440	D₄	—	2	4	no	—
441	D₄	—	2	16	no	—
442	D₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	4	32	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	8	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
462	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
463	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
464	C₂×C₄	C₃₁×#120	1	2	yes	—
465	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
466	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
467	C₂×C₄	C₃₁×#123	1	4	yes	—
468	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
469	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
473	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
474	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
475	C₈	—	2	4	no	—
476	C₈	—	4	2	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	8	no	—
482	C₂²	—	4	32	no	—
483	C₂²	—	4	8	no	—
484	C₂²	—	4	8	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	C₃₁×#385	1	4	yes	—
488	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
489	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
490	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
491	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
492	C₂²	C₃₁×#575	1	2	yes	—
493	C₄	—	2	4	no	—
494	C₄	—	4	4	no	—
495	C₄	—	4	4	no	—
496	C₄	—	4	2	no	—
497	C₄	—	8	1	no	—
498	C₄	—	8	1	no	—
499	C₄	C₃₁×#385	1	4	yes	—
500	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
501	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
502	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
503	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
504	C₂	—	2	4	no	—
505	C₂	—	4	8	no	—
506	C₂	C₂×C₃₁×#385	1	1	yes	characteristic
507	C₂	C₄×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
508	C₂	C₃₁×#3003	1	1	yes	characteristic
509	C₂	C₃₁×#3005	1	1	yes	characteristic
510	C₂	C₃₁×#3077	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#428	C₂	semidirect
5. C₃₁×#2296	C₂	semidirect
6. C₃₁×#1864	C₂	semidirect
7. C₃₁×#1862	C₂	semidirect
8. C₃₁×#1736	C₂	semidirect
9. C₃₁×#1728	C₂	semidirect
24. C₂×C₃₁×#107	C₂²	semidirect
25. C₂×C₃₁×#106	C₂²	semidirect
27. C₃₁×#418	C₄	semidirect
31. C₃₁×#389	C₂²	semidirect
32. C₃₁×#389	C₂²	semidirect
36. C₃₁×#340	C₂²	semidirect
37. C₃₁×#336	C₂²	semidirect
59. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
68. C₃₁×#117	C₂×C₄	semidirect
92. C₃₁×#102	C₂×C₄	semidirect
111. C₂×D₄×C₃₁	C₄⋊C₄	semidirect
146. C₂×C₈×C₃₁	C₂²×C₄	semidirect
156. C₄⋊C₄×C₃₁	C₂²×C₄	semidirect
166. #20873	C₃₁	direct (N × H)
223. C₂×#428	C₂×C₃₁	semidirect
226. #2296	C₂×C₃₁	semidirect
227. #1864	C₂×C₃₁	semidirect
228. #1862	C₂×C₃₁	semidirect
229. #1736	C₂×C₃₁	semidirect
230. #1728	C₂×C₃₁	semidirect
272. C₂×#107	C₂²×C₃₁	semidirect
273. C₂×#106	C₂²×C₃₁	semidirect
275. #418	C₄×C₃₁	semidirect
279. #389	C₂²×C₃₁	semidirect
280. #389	C₂²×C₃₁	semidirect
284. #340	C₂²×C₃₁	semidirect
285. #336	C₂²×C₃₁	semidirect
314. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
323. #117	C₂×C₄×C₃₁	semidirect
347. #102	C₂×C₄×C₃₁	semidirect
350. C₃₁	#20873	direct (N × H)
367. C₂×D₄	C₄⋊C₄×C₃₁	semidirect
402. C₂×C₈	C₂²×C₄×C₃₁	semidirect
412. C₄⋊C₄	C₂²×C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₃₁	2
C₂×C₃₁×#428	2
C₂×C₃₁×#418	2
C₃₁×#2296	2
C₃₁×#1864	2
C₃₁×#1862	2
C₃₁×#1736	2
C₃₁×#1728	2
#20873	31

Subgroups — C31×#20873 #21477632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20873 #21477632