Subgroups — C₃₁×#17327 #21474086

Order 7936.

836 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1453	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1451	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1451	C₂	1	2	yes	maximal, complemented
7	C₈×M₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆×C₃₁	C₄	1	2	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#432	C₄	1	2	yes	—
12	C₃₁×#432	C₄	1	2	yes	—
13	C₃₁×#431	C₄	1	2	yes	—
14	C₃₁×#431	C₄	1	2	yes	—
15	C₃₁×#431	C₂²	1	1	yes	characteristic
16	C₃₁×#431	C₂²	1	1	yes	characteristic
17	C₃₁×#431	C₂²	1	2	yes	—
18	C₃₁×#431	C₂²	1	2	yes	—
19	C₃₁×#431	C₂²	1	2	yes	—
20	C₃₁×#423	C₄	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
22	C₃₁×#407	C₄	1	2	yes	—
23	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
24	C₄×M₁₆×C₃₁	C₄	1	2	yes	—
25	C₄×M₁₆×C₃₁	C₄	1	2	yes	—
26	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
29	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
30	C₂×C₃₁×#97	C₄	1	2	yes	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
33	C₃₁×#322	C₂²	1	4	yes	complemented
34	C₃₁×#322	C₂²	1	4	yes	complemented
35	C₃₁×#322	C₂²	1	4	yes	complemented
36	C₈²×C₃₁	C₂²	1	4	yes	complemented
37	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₃₁	—	2	8	no	—
41	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
44	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
62	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
64	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
67	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
69	C₃₁×#97	C₂×C₄	1	2	yes	—
70	C₃₁×#97	C₂×C₄	1	4	yes	—
71	C₃₁×#97	C₂×C₄	1	2	yes	—
72	C₃₁×#97	C₂×C₄	1	4	yes	—
73	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
74	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
75	C₄×C₈×C₃₁	C₂³	1	2	yes	—
76	C₄×C₈×C₃₁	C₂³	1	2	yes	—
77	C₄×C₈×C₃₁	C₂³	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	2	yes	—
79	C₄×C₈×C₃₁	C₂³	1	4	yes	—
80	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	—	2	4	no	—
82	C₂²×C₄×C₃₁	—	2	4	no	—
83	C₂²×C₄×C₃₁	—	4	2	no	—
84	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	M₁₆×C₃₁	—	2	4	no	—
103	M₁₆×C₃₁	—	2	8	no	—
104	M₁₆×C₃₁	—	2	8	no	—
105	M₁₆×C₃₁	M₁₆	1	8	yes	complemented
106	C₂×C₈×C₃₁	—	2	8	no	—
107	C₂×C₈×C₃₁	—	4	4	no	—
108	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
110	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
113	C₄²×C₃₁	C₄²	1	1	yes	characteristic
114	C₄²×C₃₁	C₄²	1	1	yes	characteristic
115	C₄²×C₃₁	C₄²	1	1	yes	characteristic
116	C₄²×C₃₁	C₄²	1	1	yes	characteristic
117	C₄²×C₃₁	C₄²	1	1	yes	characteristic
118	C₄²×C₃₁	C₄²	1	1	yes	characteristic
119	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
129	#17327	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	2	2	no	—
131	C₂³×C₃₁	—	2	2	no	—
132	C₂³×C₃₁	—	4	1	no	—
133	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
134	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
135	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	4	1	no	—
150	C₂×C₄×C₃₁	—	4	1	no	—
151	C₂×C₄×C₃₁	—	4	1	no	—
152	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
162	C₂×C₄×C₃₁	#131	1	4	yes	—
163	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
166	C₈×C₃₁	—	2	4	no	—
167	C₈×C₃₁	—	2	8	no	—
168	C₈×C₃₁	—	2	4	no	—
169	C₈×C₃₁	—	2	4	no	—
170	C₈×C₃₁	C₂×M₁₆	1	8	yes	—
171	#2920	C₂×C₃₁	1	1	yes	characteristic
172	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
173	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
174	#1453	C₂×C₃₁	1	4	yes	complemented
175	#1451	C₂×C₃₁	1	2	yes	complemented
176	#1451	C₂×C₃₁	1	2	yes	complemented
177	C₈×M₁₆	C₂×C₃₁	1	2	yes	complemented
178	C₈×M₁₆	C₂×C₃₁	1	2	yes	complemented
179	C₂²×C₃₁	—	2	1	no	—
180	C₂²×C₃₁	—	2	1	no	—
181	C₂²×C₃₁	—	2	1	no	—
182	C₂²×C₃₁	—	2	1	no	—
183	C₂²×C₃₁	—	2	1	no	—
184	C₂²×C₃₁	—	2	1	no	—
185	C₂²×C₃₁	—	2	1	no	—
186	C₂²×C₃₁	—	4	4	no	—
187	C₂²×C₃₁	C₄×M₁₆	1	2	yes	—
188	C₂²×C₃₁	#405	1	2	yes	—
189	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
190	C₄×C₃₁	—	2	1	no	—
191	C₄×C₃₁	—	2	1	no	—
192	C₄×C₃₁	—	2	1	no	—
193	C₄×C₃₁	—	2	1	no	—
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	2	1	no	—
196	C₄×C₃₁	—	2	1	no	—
197	C₄×C₃₁	C₄×M₁₆	1	2	yes	—
198	C₄×C₃₁	#405	1	2	yes	—
199	C₄×C₃₁	C₂²×M₁₆	1	2	yes	—
200	C₄×C₃₁	C₂×#131	1	2	yes	—
201	C₄×C₃₁	#568	1	2	yes	—
202	C₂²×M₁₆	C₄×C₃₁	1	2	yes	—
203	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
204	#432	C₄×C₃₁	1	2	yes	—
205	#432	C₄×C₃₁	1	2	yes	—
206	#431	C₄×C₃₁	1	2	yes	—
207	#431	C₄×C₃₁	1	2	yes	—
208	#431	C₂²×C₃₁	1	1	yes	characteristic
209	#431	C₂²×C₃₁	1	1	yes	characteristic
210	#431	C₂²×C₃₁	1	2	yes	—
211	#431	C₂²×C₃₁	1	2	yes	—
212	#431	C₂²×C₃₁	1	2	yes	—
213	#423	C₄×C₃₁	1	2	yes	—
214	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
215	#407	C₄×C₃₁	1	2	yes	—
216	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
217	C₄×M₁₆	C₄×C₃₁	1	2	yes	—
218	C₄×M₁₆	C₄×C₃₁	1	2	yes	—
219	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
220	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
221	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
222	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
223	C₂×#97	C₄×C₃₁	1	2	yes	—
224	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
225	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
226	#322	C₂²×C₃₁	1	4	yes	complemented
227	#322	C₂²×C₃₁	1	4	yes	complemented
228	#322	C₂²×C₃₁	1	4	yes	complemented
229	C₈²	C₂²×C₃₁	1	4	yes	complemented
230	C₂×C₃₁	—	2	2	no	—
231	C₂×C₃₁	—	2	2	no	—
232	C₂×C₃₁	—	4	1	no	—
233	C₂×C₃₁	C₂×C₄×M₁₆	1	1	yes	characteristic
234	C₂×C₃₁	C₂×#405	1	1	yes	characteristic
235	C₂×C₃₁	#2873	1	1	yes	characteristic
236	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
237	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
238	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
239	C₂×M₁₆	—	2	8	no	—
240	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
241	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
242	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
243	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
244	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
245	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
246	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
254	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
256	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
257	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
258	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
259	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
260	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
261	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
262	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
263	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
264	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
265	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
266	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
267	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
268	#97	C₂×C₄×C₃₁	1	2	yes	—
269	#97	C₂×C₄×C₃₁	1	4	yes	—
270	#97	C₂×C₄×C₃₁	1	2	yes	—
271	#97	C₂×C₄×C₃₁	1	4	yes	—
272	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
273	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
274	C₄×C₈	C₂³×C₃₁	1	2	yes	—
275	C₄×C₈	C₂³×C₃₁	1	2	yes	—
276	C₄×C₈	C₂³×C₃₁	1	4	yes	—
277	C₄×C₈	C₂³×C₃₁	1	2	yes	—
278	C₄×C₈	C₂³×C₃₁	1	2	yes	—
279	C₃₁	#17327	1	1	yes	characteristic, direct product factor, Sylow
280	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
285	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
286	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
288	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
289	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
290	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
291	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
292	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
293	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
294	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
295	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
297	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
298	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
302	M₁₆	—	2	4	no	—
303	M₁₆	—	2	8	no	—
304	M₁₆	—	2	8	no	—
305	M₁₆	M₁₆×C₃₁	1	8	yes	complemented
306	C₂×C₈	—	2	8	no	—
307	C₂×C₈	—	4	4	no	—
308	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
309	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
310	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
311	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
312	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
313	C₄²	C₄²×C₃₁	1	1	yes	characteristic
314	C₄²	C₄²×C₃₁	1	1	yes	characteristic
315	C₄²	C₄²×C₃₁	1	1	yes	characteristic
316	C₄²	C₄²×C₃₁	1	1	yes	characteristic
317	C₄²	C₄²×C₃₁	1	1	yes	characteristic
318	C₄²	C₄²×C₃₁	1	1	yes	characteristic
319	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
320	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
321	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
322	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
323	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
324	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
325	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
326	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
328	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	4	1	no	—
332	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
333	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
334	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
352	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
353	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
354	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
356	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
357	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
358	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
359	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
360	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
361	C₂×C₄	C₃₁×#131	1	4	yes	—
362	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
364	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
365	C₈	—	2	4	no	—
366	C₈	—	2	8	no	—
367	C₈	—	2	4	no	—
368	C₈	—	2	4	no	—
369	C₈	C₂×M₁₆×C₃₁	1	8	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	4	4	no	—
378	C₂²	C₄×M₁₆×C₃₁	1	2	yes	—
379	C₂²	C₃₁×#405	1	2	yes	—
380	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	C₄×M₁₆×C₃₁	1	2	yes	—
389	C₄	C₃₁×#405	1	2	yes	—
390	C₄	C₂²×M₁₆×C₃₁	1	2	yes	—
391	C₄	C₂×C₃₁×#131	1	2	yes	—
392	C₄	C₃₁×#568	1	2	yes	—
393	C₂	—	2	2	no	—
394	C₂	—	2	2	no	—
395	C₂	—	4	1	no	—
396	C₂	C₂×C₄×M₁₆×C₃₁	1	1	yes	characteristic
397	C₂	C₂×C₃₁×#405	1	1	yes	characteristic
398	C₂	C₃₁×#2873	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1453	C₂	semidirect
5. C₃₁×#1451	C₂	semidirect
6. C₃₁×#1451	C₂	semidirect
7. C₈×M₁₆×C₃₁	C₂	semidirect
8. C₈×M₁₆×C₃₁	C₂	semidirect
33. C₃₁×#322	C₂²	semidirect
34. C₃₁×#322	C₂²	semidirect
35. C₃₁×#322	C₂²	semidirect
36. C₈²×C₃₁	C₂²	semidirect
105. M₁₆×C₃₁	M₁₆	semidirect
129. #17327	C₃₁	direct (N × H)
174. #1453	C₂×C₃₁	semidirect
175. #1451	C₂×C₃₁	semidirect
176. #1451	C₂×C₃₁	semidirect
177. C₈×M₁₆	C₂×C₃₁	semidirect
178. C₈×M₁₆	C₂×C₃₁	semidirect
226. #322	C₂²×C₃₁	semidirect
227. #322	C₂²×C₃₁	semidirect
228. #322	C₂²×C₃₁	semidirect
229. C₈²	C₂²×C₃₁	semidirect
279. C₃₁	#17327	direct (N × H)
305. M₁₆	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2920	2
C₂×C₃₁×#431	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1453	2
C₃₁×#1451	2
C₈×M₁₆×C₃₁	2
#17327	31

Subgroups — C31×#17327 #21474086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17327 #21474086