Subgroups — C₃₁×#60161 #21516920

Order 7936.

2706 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#575	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3428	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2947	C₂	1	8	yes	maximal, complemented
4	C₃₁×#2946	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2943	C₂	1	4	yes	maximal, complemented
6	C₄×C₃₁×#135	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#416	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#575	C₄	1	16	yes	complemented
12	C₂×C₃₁×#137	C₂²	1	4	yes	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
14	C₂×C₃₁×#135	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#520	—	2	4	no	—
18	C₃₁×#520	C₂²	1	8	yes	—
19	C₃₁×#519	C₂²	1	4	yes	complemented
20	C₃₁×#518	—	2	4	no	—
21	C₃₁×#518	C₂²	1	4	yes	complemented
22	C₄×#41×C₃₁	—	2	4	no	—
23	C₄×#41×C₃₁	C₂²	1	8	yes	complemented
24	C₂×C₃₁×#117	—	2	4	no	—
25	C₂×C₃₁×#117	C₂²	1	2	yes	—
26	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
27	C₃₁×#442	C₂²	1	8	yes	complemented
28	C₃₁×#441	C₂²	1	8	yes	complemented
29	C₃₁×#440	C₂²	1	16	yes	complemented
30	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
31	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
32	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
33	C₃₁×#428	C₂²	1	8	yes	complemented
34	C₃₁×#427	C₂²	1	4	yes	complemented
35	C₃₁×#416	C₂²	1	8	yes	—
36	C₃₁×#416	C₂²	1	4	yes	complemented
37	C₃₁×#416	C₂²	1	4	yes	complemented
38	C₂×C₃₁×#103	C₂²	1	4	yes	complemented
39	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
40	C₃₁×#405	C₂²	1	8	yes	complemented
41	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
42	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₂×#41×C₃₁	—	2	2	no	—
45	C₂×#41×C₃₁	—	2	8	no	—
46	C₂×#41×C₃₁	C₂×C₄	1	16	yes	complemented
47	C₂×#41×C₃₁	C₂³	1	4	yes	—
48	C₂×#41×C₃₁	C₂³	1	4	yes	—
49	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₃₁	—	2	2	no	—
51	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₃₁	—	2	2	no	—
53	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#137	C₂×C₄	1	32	yes	complemented
55	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	32	yes	complemented
56	C₃₁×#135	C₂×C₄	1	32	yes	complemented
57	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
58	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
59	C₂×D₈×C₃₁	C₂³	1	4	yes	—
60	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
61	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
62	C₂²×C₈×C₃₁	—	2	4	no	—
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
66	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
67	C₄×Q₈×C₃₁	—	2	8	no	—
68	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
69	C₄×D₄×C₃₁	—	2	16	no	—
70	C₄×D₄×C₃₁	—	2	8	no	—
71	C₄×D₄×C₃₁	—	2	16	no	—
72	C₄×D₄×C₃₁	C₂³	1	16	yes	—
73	C₃₁×#117	—	2	2	no	—
74	C₃₁×#117	—	2	2	no	—
75	C₃₁×#117	—	2	8	no	—
76	C₃₁×#117	—	2	8	no	—
77	C₃₁×#117	—	2	8	no	—
78	C₃₁×#117	D₄	1	4	yes	—
79	C₃₁×#117	C₂³	1	4	yes	—
80	C₃₁×#117	C₂³	1	4	yes	—
81	C₃₁×#117	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
83	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
87	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
88	C₂×C₄²×C₃₁	—	2	2	no	—
89	C₂×C₄²×C₃₁	—	2	8	no	—
90	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
91	C₃₁×#107	C₂³	1	8	yes	—
92	C₃₁×#106	C₂³	1	8	yes	—
93	C₃₁×#103	C₂³	1	8	yes	—
94	C₃₁×#103	C₂³	1	8	yes	—
95	C₃₁×#102	C₂³	1	8	yes	—
96	C₃₁×#102	C₂³	1	8	yes	—
97	C₃₁×#97	C₂³	1	8	yes	—
98	C₄×C₈×C₃₁	C₂³	1	8	yes	—
99	C₂⁴×C₃₁	—	2	2	no	—
100	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
101	#41×C₃₁	—	2	16	no	—
102	#41×C₃₁	—	2	16	no	—
103	#41×C₃₁	—	2	8	no	—
104	#41×C₃₁	—	2	8	no	—
105	#41×C₃₁	C₂²×C₄	1	16	yes	—
106	#41×C₃₁	C₂²×C₄	1	16	yes	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	4	no	—
109	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
110	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
111	C₂×D₄×C₃₁	—	2	16	no	—
112	C₂×D₄×C₃₁	—	2	4	no	—
113	C₂×D₄×C₃₁	—	2	4	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	4	no	—
116	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
117	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	16	no	—
120	C₂²×C₄×C₃₁	—	2	4	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	16	no	—
124	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₃₁	#41	1	4	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
131	Q₁₆×C₃₁	C₂²×C₄	1	16	yes	—
132	SD₁₆×C₃₁	C₂²×C₄	1	32	yes	—
133	D₈×C₃₁	C₂²×C₄	1	16	yes	—
134	M₁₆×C₃₁	C₂²×C₄	1	16	yes	—
135	C₂×C₈×C₃₁	—	2	8	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
139	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
140	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
141	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
142	C₄⋊C₄×C₃₁	—	2	4	no	—
143	C₄⋊C₄×C₃₁	—	2	16	no	—
144	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
145	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
146	C₂²⋊C₄×C₃₁	—	2	4	no	—
147	C₂²⋊C₄×C₃₁	—	2	4	no	—
148	C₂²⋊C₄×C₃₁	—	2	16	no	—
149	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
150	C₄²×C₃₁	—	2	16	no	—
151	C₄²×C₃₁	—	2	4	no	—
152	C₄²×C₃₁	—	2	4	no	—
153	C₄²×C₃₁	C₂×D₄	1	4	yes	—
154	C₄²×C₃₁	C₂⁴	1	4	yes	—
155	#60161	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	4	no	—
157	C₂³×C₃₁	—	2	8	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	8	4	no	—
160	C₂³×C₃₁	C₄×D₄	1	4	yes	—
161	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
163	Q₈×C₃₁	—	2	8	no	—
164	Q₈×C₃₁	—	2	4	no	—
165	Q₈×C₃₁	C₂³×C₄	1	8	yes	—
166	D₄×C₃₁	—	2	16	no	—
167	D₄×C₃₁	—	2	16	no	—
168	D₄×C₃₁	—	2	8	no	—
169	D₄×C₃₁	—	2	4	no	—
170	D₄×C₃₁	C₂³×C₄	1	8	yes	—
171	C₂×C₄×C₃₁	—	2	8	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	16	no	—
174	C₂×C₄×C₃₁	—	2	16	no	—
175	C₂×C₄×C₃₁	—	2	8	no	—
176	C₂×C₄×C₃₁	—	4	4	no	—
177	C₂×C₄×C₃₁	—	8	16	no	—
178	C₂×C₄×C₃₁	—	8	4	no	—
179	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
180	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
181	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
182	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
183	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
184	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
187	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
188	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	2	2	no	—
191	C₈×C₃₁	C₂³×C₄	1	8	yes	—
192	C₂×#575	C₂×C₃₁	1	1	yes	characteristic
193	#3428	C₂×C₃₁	1	2	yes	complemented
194	#2947	C₂×C₃₁	1	8	yes	complemented
195	#2946	C₂×C₃₁	1	8	yes	complemented
196	#2943	C₂×C₃₁	1	4	yes	complemented
197	C₄×#135	C₂×C₃₁	1	4	yes	complemented
198	#2911	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂×#416	C₂×C₃₁	1	2	yes	complemented
200	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂²×C₃₁	—	2	2	no	—
202	C₂²×C₃₁	—	2	4	no	—
203	C₂²×C₃₁	—	2	8	no	—
204	C₂²×C₃₁	—	4	4	no	—
205	C₂²×C₃₁	—	4	8	no	—
206	C₂²×C₃₁	—	8	8	no	—
207	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
208	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
209	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
210	C₄×C₃₁	—	2	8	no	—
211	C₄×C₃₁	—	2	8	no	—
212	C₄×C₃₁	—	4	8	no	—
213	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
214	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
215	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
216	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
217	C₂²×#41	C₂²×C₃₁	1	2	yes	—
218	#575	C₄×C₃₁	1	16	yes	complemented
219	C₂×#137	C₂²×C₃₁	1	4	yes	—
220	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
221	C₂×#135	C₂²×C₃₁	1	4	yes	—
222	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
223	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
224	#520	—	2	4	no	—
225	#520	C₂²×C₃₁	1	8	yes	—
226	#519	C₂²×C₃₁	1	4	yes	complemented
227	#518	—	2	4	no	—
228	#518	C₂²×C₃₁	1	4	yes	complemented
229	C₄×#41	—	2	4	no	—
230	C₄×#41	C₂²×C₃₁	1	8	yes	complemented
231	C₂×#117	—	2	4	no	—
232	C₂×#117	C₂²×C₃₁	1	2	yes	—
233	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
234	#442	C₂²×C₃₁	1	8	yes	complemented
235	#441	C₂²×C₃₁	1	8	yes	complemented
236	#440	C₂²×C₃₁	1	16	yes	complemented
237	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
238	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
239	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
240	#428	C₂²×C₃₁	1	8	yes	complemented
241	#427	C₂²×C₃₁	1	4	yes	complemented
242	#416	C₂²×C₃₁	1	8	yes	—
243	#416	C₂²×C₃₁	1	4	yes	complemented
244	#416	C₂²×C₃₁	1	4	yes	complemented
245	C₂×#103	C₂²×C₃₁	1	4	yes	complemented
246	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
247	#405	C₂²×C₃₁	1	8	yes	complemented
248	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
249	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
250	C₂×C₃₁	—	2	4	no	—
251	C₂×C₃₁	—	2	2	no	—
252	C₂×C₃₁	—	4	8	no	—
253	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
254	C₂×C₃₁	#3584	1	2	yes	—
255	C₂×#41	—	2	2	no	—
256	C₂×#41	—	2	2	no	—
257	C₂×#41	—	2	8	no	—
258	C₂×#41	C₂×C₄×C₃₁	1	16	yes	complemented
259	C₂×#41	C₂³×C₃₁	1	4	yes	—
260	C₂×#41	C₂³×C₃₁	1	4	yes	—
261	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
262	C₂²×D₄	—	2	2	no	—
263	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
264	C₂³×C₄	—	2	2	no	—
265	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
266	#137	C₂×C₄×C₃₁	1	32	yes	complemented
267	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	32	yes	complemented
268	#135	C₂×C₄×C₃₁	1	32	yes	complemented
269	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
270	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
271	C₂×D₈	C₂³×C₃₁	1	4	yes	—
272	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
273	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
274	C₂²×C₈	—	2	4	no	—
275	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
278	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
279	C₄×Q₈	—	2	8	no	—
280	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
281	C₄×D₄	—	2	16	no	—
282	C₄×D₄	—	2	8	no	—
283	C₄×D₄	—	2	16	no	—
284	C₄×D₄	C₂³×C₃₁	1	16	yes	—
285	#117	—	2	2	no	—
286	#117	—	2	2	no	—
287	#117	—	2	8	no	—
288	#117	—	2	8	no	—
289	#117	—	2	8	no	—
290	#117	D₄×C₃₁	1	4	yes	—
291	#117	C₂³×C₃₁	1	4	yes	—
292	#117	C₂³×C₃₁	1	4	yes	—
293	#117	C₂³×C₃₁	1	4	yes	—
294	C₂×C₄⋊C₄	—	2	8	no	—
295	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
297	C₂×C₂²⋊C₄	—	2	2	no	—
298	C₂×C₂²⋊C₄	—	2	8	no	—
299	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
300	C₂×C₄²	—	2	2	no	—
301	C₂×C₄²	—	2	8	no	—
302	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
303	#107	C₂³×C₃₁	1	8	yes	—
304	#106	C₂³×C₃₁	1	8	yes	—
305	#103	C₂³×C₃₁	1	8	yes	—
306	#103	C₂³×C₃₁	1	8	yes	—
307	#102	C₂³×C₃₁	1	8	yes	—
308	#102	C₂³×C₃₁	1	8	yes	—
309	#97	C₂³×C₃₁	1	8	yes	—
310	C₄×C₈	C₂³×C₃₁	1	8	yes	—
311	C₃₁	#60161	1	1	yes	characteristic, direct product factor, Sylow
312	C₂⁴	—	2	2	no	—
313	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
314	#41	—	2	16	no	—
315	#41	—	2	16	no	—
316	#41	—	2	8	no	—
317	#41	—	2	8	no	—
318	#41	C₂²×C₄×C₃₁	1	16	yes	—
319	#41	C₂²×C₄×C₃₁	1	16	yes	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	4	no	—
322	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
323	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
324	C₂×D₄	—	2	16	no	—
325	C₂×D₄	—	2	4	no	—
326	C₂×D₄	—	2	4	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
330	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	16	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	16	no	—
337	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
338	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
339	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	#41×C₃₁	1	4	yes	—
342	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
344	Q₁₆	C₂²×C₄×C₃₁	1	16	yes	—
345	SD₁₆	C₂²×C₄×C₃₁	1	32	yes	—
346	D₈	C₂²×C₄×C₃₁	1	16	yes	—
347	M₁₆	C₂²×C₄×C₃₁	1	16	yes	—
348	C₂×C₈	—	2	8	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
352	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
353	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
354	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
355	C₄⋊C₄	—	2	4	no	—
356	C₄⋊C₄	—	2	16	no	—
357	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
358	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
359	C₂²⋊C₄	—	2	4	no	—
360	C₂²⋊C₄	—	2	4	no	—
361	C₂²⋊C₄	—	2	16	no	—
362	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
363	C₄²	—	2	4	no	—
364	C₄²	—	2	16	no	—
365	C₄²	—	2	4	no	—
366	C₄²	C₂×D₄×C₃₁	1	4	yes	—
367	C₄²	C₂⁴×C₃₁	1	4	yes	—
368	C₂³	—	2	4	no	—
369	C₂³	—	2	8	no	—
370	C₂³	—	4	2	no	—
371	C₂³	—	8	4	no	—
372	C₂³	C₄×D₄×C₃₁	1	4	yes	—
373	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
374	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
375	Q₈	—	2	8	no	—
376	Q₈	—	2	4	no	—
377	Q₈	C₂³×C₄×C₃₁	1	8	yes	—
378	D₄	—	2	16	no	—
379	D₄	—	2	16	no	—
380	D₄	—	2	8	no	—
381	D₄	—	2	4	no	—
382	D₄	C₂³×C₄×C₃₁	1	8	yes	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	16	no	—
386	C₂×C₄	—	2	16	no	—
387	C₂×C₄	—	2	8	no	—
388	C₂×C₄	—	4	4	no	—
389	C₂×C₄	—	8	16	no	—
390	C₂×C₄	—	8	4	no	—
391	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
392	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
393	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
394	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
395	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
396	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
397	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
398	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
399	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
400	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
401	C₈	—	2	2	no	—
402	C₈	—	2	2	no	—
403	C₈	C₂³×C₄×C₃₁	1	8	yes	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	4	no	—
406	C₂²	—	2	8	no	—
407	C₂²	—	4	4	no	—
408	C₂²	—	4	8	no	—
409	C₂²	—	8	8	no	—
410	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
411	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
412	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
413	C₄	—	2	8	no	—
414	C₄	—	2	8	no	—
415	C₄	—	4	8	no	—
416	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
417	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
418	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
419	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
420	C₂	—	2	4	no	—
421	C₂	—	2	2	no	—
422	C₂	—	4	8	no	—
423	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
424	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₂×C₃₁×#575	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3428	C₂	semidirect
3. C₃₁×#2947	C₂	semidirect
4. C₃₁×#2946	C₂	semidirect
5. C₃₁×#2943	C₂	semidirect
6. C₄×C₃₁×#135	C₂	semidirect
7. C₃₁×#2911	C₂	semidirect
8. C₂×C₃₁×#416	C₂	semidirect
9. C₂×C₃₁×#405	C₂	semidirect
11. C₃₁×#575	C₄	semidirect
19. C₃₁×#519	C₂²	semidirect
21. C₃₁×#518	C₂²	semidirect
23. C₄×#41×C₃₁	C₂²	semidirect
27. C₃₁×#442	C₂²	semidirect
28. C₃₁×#441	C₂²	semidirect
29. C₃₁×#440	C₂²	semidirect
30. C₄×Q₁₆×C₃₁	C₂²	semidirect
31. C₄×SD₁₆×C₃₁	C₂²	semidirect
32. C₄×D₈×C₃₁	C₂²	semidirect
33. C₃₁×#428	C₂²	semidirect
34. C₃₁×#427	C₂²	semidirect
36. C₃₁×#416	C₂²	semidirect
37. C₃₁×#416	C₂²	semidirect
38. C₂×C₃₁×#103	C₂²	semidirect
39. C₂×C₃₁×#102	C₂²	semidirect
40. C₃₁×#405	C₂²	semidirect
41. C₂×C₃₁×#97	C₂²	semidirect
42. C₂×C₄×C₈×C₃₁	C₂²	semidirect
46. C₂×#41×C₃₁	C₂×C₄	semidirect
54. C₃₁×#137	C₂×C₄	semidirect
55. C₃₁×AGL(1,ℤ/8)	C₂×C₄	semidirect
56. C₃₁×#135	C₂×C₄	semidirect
60. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
155. #60161	C₃₁	direct (N × H)
193. #3428	C₂×C₃₁	semidirect
194. #2947	C₂×C₃₁	semidirect
195. #2946	C₂×C₃₁	semidirect
196. #2943	C₂×C₃₁	semidirect
197. C₄×#135	C₂×C₃₁	semidirect
198. #2911	C₂×C₃₁	semidirect
199. C₂×#416	C₂×C₃₁	semidirect
200. C₂×#405	C₂×C₃₁	semidirect
218. #575	C₄×C₃₁	semidirect
226. #519	C₂²×C₃₁	semidirect
228. #518	C₂²×C₃₁	semidirect
230. C₄×#41	C₂²×C₃₁	semidirect
234. #442	C₂²×C₃₁	semidirect
235. #441	C₂²×C₃₁	semidirect
236. #440	C₂²×C₃₁	semidirect
237. C₄×Q₁₆	C₂²×C₃₁	semidirect
238. C₄×SD₁₆	C₂²×C₃₁	semidirect
239. C₄×D₈	C₂²×C₃₁	semidirect
240. #428	C₂²×C₃₁	semidirect
241. #427	C₂²×C₃₁	semidirect
243. #416	C₂²×C₃₁	semidirect
244. #416	C₂²×C₃₁	semidirect
245. C₂×#103	C₂²×C₃₁	semidirect
246. C₂×#102	C₂²×C₃₁	semidirect
247. #405	C₂²×C₃₁	semidirect
248. C₂×#97	C₂²×C₃₁	semidirect
249. C₂×C₄×C₈	C₂²×C₃₁	semidirect
258. C₂×#41	C₂×C₄×C₃₁	semidirect
266. #137	C₂×C₄×C₃₁	semidirect
267. AGL(1,ℤ/8)	C₂×C₄×C₃₁	semidirect
268. #135	C₂×C₄×C₃₁	semidirect
272. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
311. C₃₁	#60161	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#575	2
C₃₁×#3428	2
C₃₁×#2947	2
C₃₁×#2946	2
C₃₁×#2943	2
C₄×C₃₁×#135	2
C₃₁×#2911	2
C₂×C₃₁×#416	2
C₂×C₃₁×#405	2
#60161	31

Subgroups — C31×#60161 #21516920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60161 #21516920