Subgroups — C₃₁×#17997 #21474756

Order 7936.

670 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2935	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1590	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1583	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1582	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#481	C₄	1	2	yes	—
16	C₃₁×#475	C₄	1	1	yes	characteristic
17	C₃₁×#475	C₄	1	1	yes	characteristic
18	C₃₁×#460	C₄	1	1	yes	characteristic
19	C₃₁×#460	C₄	1	1	yes	characteristic
20	C₃₁×#448	C₄	1	2	yes	—
21	C₃₁×#446	—	2	2	no	—
22	C₃₁×#446	C₂²	1	2	yes	—
23	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
25	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#436	C₂²	1	2	yes	—
27	C₃₁×#435	—	2	2	no	—
28	C₈×D₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#431	C₂²	1	1	yes	characteristic
30	C₃₁×#423	—	2	1	no	—
31	C₃₁×#423	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃₁×#408	—	2	1	no	—
34	C₄×M₁₆×C₃₁	—	2	1	no	—
35	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#334	C₂²	1	2	yes	—
38	C₃₁×#334	C₂²	1	2	yes	—
39	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
40	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
44	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
45	C₃₁×#322	C₂²	1	2	yes	—
46	C₈²×C₃₁	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
51	C₃₁×#131	—	4	4	no	—
52	C₂×M₁₆×C₃₁	—	2	1	no	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
60	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
61	C₃₁×#124	—	2	1	no	—
62	C₃₁×#121	—	2	1	no	—
63	C₃₁×#121	C₂×C₄	1	2	yes	—
64	C₄×Q₈×C₃₁	—	2	1	no	—
65	C₄×Q₈×C₃₁	—	2	1	no	—
66	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#106	C₂×C₄	1	2	yes	—
77	C₃₁×#106	C₂×C₄	1	2	yes	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
83	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
84	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
87	C₃₁×#103	C₂×C₄	1	2	yes	—
88	C₃₁×#103	C₂×C₄	1	2	yes	—
89	C₃₁×#102	C₂×C₄	1	2	yes	—
90	C₃₁×#102	C₂×C₄	1	2	yes	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	2	no	—
93	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
94	C₃₁×#97	—	2	2	no	—
95	C₃₁×#97	D₄	1	2	yes	—
96	C₄×C₈×C₃₁	—	2	2	no	—
97	C₄×C₈×C₃₁	—	2	1	no	—
98	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	2	yes	—
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	#41×C₃₁	—	4	2	no	—
103	#41×C₃₁	—	4	2	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	4	1	no	—
108	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
120	SD₁₆×C₃₁	—	2	4	no	—
121	SD₁₆×C₃₁	—	4	2	no	—
122	M₁₆×C₃₁	—	2	2	no	—
123	M₁₆×C₃₁	—	4	2	no	—
124	M₁₆×C₃₁	—	4	4	no	—
125	M₁₆×C₃₁	SD₁₆	1	4	yes	complemented
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	4	2	no	—
132	C₂×C₈×C₃₁	—	4	2	no	—
133	C₂×C₈×C₃₁	—	4	4	no	—
134	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
135	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
145	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
146	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
147	C₂²⋊C₄×C₃₁	—	2	2	no	—
148	C₂²⋊C₄×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	—	2	1	no	—
153	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₃₁	#41	1	1	yes	characteristic
156	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
157	#17997	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂³×C₃₁	—	2	2	no	—
159	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
160	Q₈×C₃₁	—	2	1	no	—
161	Q₈×C₃₁	—	2	1	no	—
162	Q₈×C₃₁	—	2	1	no	—
163	Q₈×C₃₁	—	4	2	no	—
164	Q₈×C₃₁	—	4	2	no	—
165	Q₈×C₃₁	#131	1	2	yes	complemented
166	D₄×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	2	2	no	—
168	D₄×C₃₁	—	4	4	no	—
169	D₄×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	8	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	8	1	no	—
183	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
187	C₂×C₄×C₃₁	#135	1	2	yes	—
188	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
191	C₈×C₃₁	—	2	2	no	—
192	C₈×C₃₁	—	2	2	no	—
193	C₈×C₃₁	—	2	2	no	—
194	C₈×C₃₁	—	2	2	no	—
195	C₈×C₃₁	—	4	1	no	—
196	C₈×C₃₁	—	4	1	no	—
197	C₈×C₃₁	—	4	1	no	—
198	C₈×C₃₁	—	4	2	no	—
199	C₈×C₃₁	C₂×SD₁₆	1	4	yes	—
200	#3103	C₂×C₃₁	1	1	yes	characteristic
201	Q₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#2935	C₂×C₃₁	1	1	yes	characteristic
203	#1590	C₂×C₃₁	1	2	yes	complemented
204	#1583	C₂×C₃₁	1	2	yes	complemented
205	#1582	C₂×C₃₁	1	2	yes	complemented
206	C₈×SD₁₆	C₂×C₃₁	1	2	yes	complemented
207	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1492	C₂×C₃₁	1	1	yes	characteristic
209	C₂×Q₈⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
210	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂²×C₃₁	—	2	1	no	—
212	C₂²×C₃₁	—	2	1	no	—
213	C₂²×C₃₁	—	4	2	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	C₄×SD₁₆	1	2	yes	—
217	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	2	1	no	—
221	C₄×C₃₁	—	2	1	no	—
222	C₄×C₃₁	—	4	1	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	1	no	—
225	C₄×C₃₁	—	4	1	no	—
226	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
227	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
228	C₄×C₃₁	#568	1	1	yes	characteristic
229	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
230	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
231	C₂×#131	—	2	1	no	—
232	#532	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
234	#481	C₄×C₃₁	1	2	yes	—
235	#475	C₄×C₃₁	1	1	yes	characteristic
236	#475	C₄×C₃₁	1	1	yes	characteristic
237	#460	C₄×C₃₁	1	1	yes	characteristic
238	#460	C₄×C₃₁	1	1	yes	characteristic
239	#448	C₄×C₃₁	1	2	yes	—
240	#446	—	2	2	no	—
241	#446	C₂²×C₃₁	1	2	yes	—
242	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
243	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
244	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
245	#436	C₂²×C₃₁	1	2	yes	—
246	#435	—	2	2	no	—
247	C₈×D₄	C₂²×C₃₁	1	2	yes	—
248	#431	C₂²×C₃₁	1	1	yes	characteristic
249	#423	—	2	1	no	—
250	#423	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
252	#408	—	2	1	no	—
253	C₄×M₁₆	—	2	1	no	—
254	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
256	#334	C₂²×C₃₁	1	2	yes	—
257	#334	C₂²×C₃₁	1	2	yes	—
258	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
259	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
260	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
261	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
262	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
263	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
264	#322	C₂²×C₃₁	1	2	yes	—
265	C₈²	C₂²×C₃₁	1	2	yes	—
266	C₂×C₃₁	—	2	2	no	—
267	C₂×C₃₁	—	8	2	no	—
268	C₂×C₃₁	#2933	1	1	yes	characteristic
269	C₂×C₃₁	C₂×C₄×SD₁₆	1	1	yes	characteristic
270	C₂×C₃₁	#2956	1	1	yes	characteristic
271	C₂×#41	—	2	1	no	—
272	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
273	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
274	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
275	#131	—	4	4	no	—
276	C₂×M₁₆	—	2	1	no	—
277	C₂×M₁₆	—	2	2	no	—
278	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
279	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
280	C₂²×C₈	—	2	1	no	—
281	C₂²×C₈	—	2	1	no	—
282	C₂²×C₈	—	2	2	no	—
283	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
284	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
285	#124	—	2	1	no	—
286	#121	—	2	1	no	—
287	#121	C₂×C₄×C₃₁	1	2	yes	—
288	C₄×Q₈	—	2	1	no	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
291	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
292	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
294	C₄×D₄	—	2	1	no	—
295	C₄×D₄	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
298	C₂×C₄²	—	2	1	no	—
299	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
300	#106	C₂×C₄×C₃₁	1	2	yes	—
301	#106	C₂×C₄×C₃₁	1	2	yes	—
302	C₄⋊C₈	—	2	2	no	—
303	C₄⋊C₈	—	2	2	no	—
304	C₄⋊C₈	—	2	1	no	—
305	C₄⋊C₈	—	2	1	no	—
306	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
307	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
308	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
309	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
310	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
311	#103	C₂×C₄×C₃₁	1	2	yes	—
312	#103	C₂×C₄×C₃₁	1	2	yes	—
313	#102	C₂×C₄×C₃₁	1	2	yes	—
314	#102	C₂×C₄×C₃₁	1	2	yes	—
315	C₂²⋊C₈	—	2	2	no	—
316	C₂²⋊C₈	—	2	2	no	—
317	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
318	#97	—	2	2	no	—
319	#97	D₄×C₃₁	1	2	yes	—
320	C₄×C₈	—	2	2	no	—
321	C₄×C₈	—	2	1	no	—
322	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
323	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
324	C₄×C₈	C₂³×C₃₁	1	2	yes	—
325	C₄×C₈	C₂³×C₃₁	1	2	yes	—
326	C₃₁	#17997	1	1	yes	characteristic, direct product factor, Sylow
327	#41	—	4	2	no	—
328	#41	—	4	2	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	4	1	no	—
333	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
334	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
345	SD₁₆	—	2	4	no	—
346	SD₁₆	—	4	2	no	—
347	M₁₆	—	2	2	no	—
348	M₁₆	—	4	2	no	—
349	M₁₆	—	4	4	no	—
350	M₁₆	SD₁₆×C₃₁	1	4	yes	complemented
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	4	2	no	—
357	C₂×C₈	—	4	2	no	—
358	C₂×C₈	—	4	4	no	—
359	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
360	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
361	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
362	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
370	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
371	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
372	C₂²⋊C₄	—	2	2	no	—
373	C₂²⋊C₄	—	2	2	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	1	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	2	1	no	—
378	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
379	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₄²	#41×C₃₁	1	1	yes	characteristic
381	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
382	C₂³	—	2	2	no	—
383	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
384	Q₈	—	2	1	no	—
385	Q₈	—	2	1	no	—
386	Q₈	—	2	1	no	—
387	Q₈	—	4	2	no	—
388	Q₈	—	4	2	no	—
389	Q₈	C₃₁×#131	1	2	yes	complemented
390	D₄	—	2	2	no	—
391	D₄	—	2	2	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	2	no	—
394	D₄	—	8	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	8	1	no	—
407	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
409	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
410	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
411	C₂×C₄	C₃₁×#135	1	2	yes	—
412	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
415	C₈	—	2	2	no	—
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	2	2	no	—
419	C₈	—	4	1	no	—
420	C₈	—	4	1	no	—
421	C₈	—	4	1	no	—
422	C₈	—	4	2	no	—
423	C₈	C₂×SD₁₆×C₃₁	1	4	yes	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	C₄×SD₁₆×C₃₁	1	2	yes	—
430	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
431	C₄	—	2	1	no	—
432	C₄	—	2	2	no	—
433	C₄	—	2	1	no	—
434	C₄	—	2	1	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	1	no	—
439	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
440	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
441	C₄	C₃₁×#568	1	1	yes	characteristic
442	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
443	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
444	C₂	—	2	2	no	—
445	C₂	—	8	2	no	—
446	C₂	C₃₁×#2933	1	1	yes	characteristic
447	C₂	C₂×C₄×SD₁₆×C₃₁	1	1	yes	characteristic
448	C₂	C₃₁×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1590	C₂	semidirect
5. C₃₁×#1583	C₂	semidirect
6. C₃₁×#1582	C₂	semidirect
7. C₈×SD₁₆×C₃₁	C₂	semidirect
8. C₃₁×#1570	C₂	semidirect
10. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
125. M₁₆×C₃₁	SD₁₆	semidirect
157. #17997	C₃₁	direct (N × H)
165. Q₈×C₃₁	#131	semidirect
201. Q₈×M₁₆	C₂×C₃₁	semidirect
203. #1590	C₂×C₃₁	semidirect
204. #1583	C₂×C₃₁	semidirect
205. #1582	C₂×C₃₁	semidirect
206. C₈×SD₁₆	C₂×C₃₁	semidirect
207. #1570	C₂×C₃₁	semidirect
209. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
210. C₈×M₁₆	C₂×C₃₁	semidirect
326. C₃₁	#17997	direct (N × H)
350. M₁₆	SD₁₆×C₃₁	semidirect
389. Q₈	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3103	2
Q₈×M₁₆×C₃₁	2
C₃₁×#2935	2
C₃₁×#1590	2
C₃₁×#1583	2
C₃₁×#1582	2
C₈×SD₁₆×C₃₁	2
C₃₁×#1570	2
C₃₁×#1492	2
C₂×C₃₁×Q₈⋊C₈	2
C₈×M₁₆×C₃₁	2
#17997	31

Subgroups — C31×#17997 #21474756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17997 #21474756