Subgroups — C₃₁×#17340 #21474099

Order 7936.

704 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1455	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1454	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1454	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#446	C₄	1	2	yes	—
18	C₃₁×#446	C₄	1	2	yes	—
19	C₃₁×#436	C₄	1	2	yes	—
20	C₃₁×#436	C₄	1	2	yes	—
21	C₃₁×#435	C₄	1	2	yes	—
22	C₃₁×#435	C₄	1	2	yes	—
23	C₃₁×#435	C₄	1	2	yes	—
24	C₃₁×#435	C₄	1	2	yes	—
25	C₃₁×#433	C₄	1	2	yes	—
26	C₃₁×#433	C₄	1	2	yes	—
27	C₃₁×#431	C₂²	1	1	yes	characteristic
28	C₃₁×#431	C₂²	1	1	yes	characteristic
29	C₃₁×#431	C₂²	1	1	yes	characteristic
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#431	C₂²	1	1	yes	characteristic
33	C₃₁×#431	C₂²	1	1	yes	characteristic
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#424	C₄	1	2	yes	—
38	C₃₁×#408	C₄	1	2	yes	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
42	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
47	C₃₁×#322	C₂²	1	1	yes	characteristic
48	C₃₁×#322	C₂²	1	1	yes	characteristic
49	C₃₁×#322	C₂²	1	1	yes	characteristic
50	C₃₁×#322	C₂²	1	1	yes	characteristic
51	C₃₁×#322	C₂²	1	1	yes	characteristic
52	C₃₁×#322	C₂²	1	1	yes	characteristic
53	C₃₁×#322	C₂²	1	1	yes	characteristic
54	C₃₁×#322	C₂²	1	1	yes	characteristic
55	C₃₁×#322	C₂²	1	1	yes	characteristic
56	C₃₁×#322	C₂²	1	1	yes	characteristic
57	C₃₁×#322	C₂²	1	1	yes	characteristic
58	C₃₁×#322	C₂²	1	1	yes	characteristic
59	C₈²×C₃₁	C₂²	1	1	yes	characteristic
60	C₈²×C₃₁	C₂²	1	1	yes	characteristic
61	C₈²×C₃₁	C₂²	1	1	yes	characteristic
62	C₈²×C₃₁	C₂²	1	1	yes	characteristic
63	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
66	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
67	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
68	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
69	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
70	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
71	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
72	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
78	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
81	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
82	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
87	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
88	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
89	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
90	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
91	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
92	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
93	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
94	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
95	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
96	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
97	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
98	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
99	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
100	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
101	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
102	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
103	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
104	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
105	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
106	C₃₁×#97	C₂×C₄	1	2	yes	—
107	C₃₁×#97	C₂×C₄	1	2	yes	—
108	C₃₁×#97	C₂×C₄	1	2	yes	—
109	C₃₁×#97	C₂×C₄	1	2	yes	—
110	C₃₁×#97	C₂×C₄	1	2	yes	—
111	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
112	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
113	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
114	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
115	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
116	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
117	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
118	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
119	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
120	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
121	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
122	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
123	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	#41×C₃₁	—	4	2	no	—
127	C₂×Q₈×C₃₁	C₄²	1	1	yes	characteristic
128	C₂×D₄×C₃₁	C₄²	1	1	yes	characteristic
129	C₂×D₄×C₃₁	C₄²	1	1	yes	characteristic
130	C₂×D₄×C₃₁	C₄²	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	M₁₆×C₃₁	—	2	4	no	—
141	M₁₆×C₃₁	—	2	4	no	—
142	M₁₆×C₃₁	—	4	2	no	—
143	M₁₆×C₃₁	—	4	2	no	—
144	M₁₆×C₃₁	—	4	2	no	—
145	C₂×C₈×C₃₁	—	2	4	no	—
146	C₂×C₈×C₃₁	—	4	2	no	—
147	C₂×C₈×C₃₁	—	4	2	no	—
148	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
149	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
150	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
151	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
152	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
153	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
154	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
155	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
156	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
157	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
158	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
159	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
160	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
161	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
162	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
163	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
164	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
165	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
166	C₂²⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
167	C₂²⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
168	C₂²⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
169	C₂²⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
170	C₂²⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
171	C₂²⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
172	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
174	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
175	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
176	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
182	#17340	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
184	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
185	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
186	Q₈×C₃₁	—	4	1	no	—
187	D₄×C₃₁	—	4	1	no	—
188	D₄×C₃₁	—	4	1	no	—
189	D₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
211	C₈×C₃₁	—	2	2	no	—
212	C₈×C₃₁	—	2	2	no	—
213	C₈×C₃₁	—	2	2	no	—
214	C₈×C₃₁	—	2	2	no	—
215	C₈×C₃₁	—	2	2	no	—
216	C₈×C₃₁	—	2	2	no	—
217	C₈×C₃₁	—	2	2	no	—
218	C₈×C₃₁	—	2	2	no	—
219	C₈×C₃₁	—	2	2	no	—
220	C₈×C₃₁	—	2	2	no	—
221	C₈×C₃₁	—	2	2	no	—
222	C₈×C₃₁	#131	1	4	yes	—
223	#2965	C₂×C₃₁	1	1	yes	characteristic
224	#2965	C₂×C₃₁	1	1	yes	characteristic
225	#2922	C₂×C₃₁	1	1	yes	characteristic
226	#1455	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1454	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#1454	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×C₃₁	—	2	1	no	—
239	C₂²×C₃₁	—	2	1	no	—
240	C₂²×C₃₁	—	2	1	no	—
241	C₂²×C₃₁	—	2	1	no	—
242	C₂²×C₃₁	—	2	1	no	—
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
248	C₄×C₃₁	—	2	1	no	—
249	C₄×C₃₁	—	2	1	no	—
250	C₄×C₃₁	—	2	1	no	—
251	C₄×C₃₁	—	2	1	no	—
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	1	no	—
254	C₄×C₃₁	—	2	1	no	—
255	C₄×C₃₁	—	2	1	no	—
256	C₄×C₃₁	—	2	1	no	—
257	C₄×C₃₁	C₂×#131	1	2	yes	—
258	C₄×C₃₁	#568	1	2	yes	—
259	C₄×C₃₁	#568	1	2	yes	—
260	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
261	#446	C₄×C₃₁	1	2	yes	—
262	#446	C₄×C₃₁	1	2	yes	—
263	#436	C₄×C₃₁	1	2	yes	—
264	#436	C₄×C₃₁	1	2	yes	—
265	#435	C₄×C₃₁	1	2	yes	—
266	#435	C₄×C₃₁	1	2	yes	—
267	#435	C₄×C₃₁	1	2	yes	—
268	#435	C₄×C₃₁	1	2	yes	—
269	#433	C₄×C₃₁	1	2	yes	—
270	#433	C₄×C₃₁	1	2	yes	—
271	#431	C₂²×C₃₁	1	1	yes	characteristic
272	#431	C₂²×C₃₁	1	1	yes	characteristic
273	#431	C₂²×C₃₁	1	1	yes	characteristic
274	#431	C₂²×C₃₁	1	1	yes	characteristic
275	#431	C₂²×C₃₁	1	1	yes	characteristic
276	#431	C₂²×C₃₁	1	1	yes	characteristic
277	#431	C₂²×C₃₁	1	1	yes	characteristic
278	#431	C₂²×C₃₁	1	1	yes	characteristic
279	#431	C₂²×C₃₁	1	1	yes	characteristic
280	#431	C₂²×C₃₁	1	1	yes	characteristic
281	#424	C₄×C₃₁	1	2	yes	—
282	#408	C₄×C₃₁	1	2	yes	—
283	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
284	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
285	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
286	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
287	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
291	#322	C₂²×C₃₁	1	1	yes	characteristic
292	#322	C₂²×C₃₁	1	1	yes	characteristic
293	#322	C₂²×C₃₁	1	1	yes	characteristic
294	#322	C₂²×C₃₁	1	1	yes	characteristic
295	#322	C₂²×C₃₁	1	1	yes	characteristic
296	#322	C₂²×C₃₁	1	1	yes	characteristic
297	#322	C₂²×C₃₁	1	1	yes	characteristic
298	#322	C₂²×C₃₁	1	1	yes	characteristic
299	#322	C₂²×C₃₁	1	1	yes	characteristic
300	#322	C₂²×C₃₁	1	1	yes	characteristic
301	#322	C₂²×C₃₁	1	1	yes	characteristic
302	#322	C₂²×C₃₁	1	1	yes	characteristic
303	C₈²	C₂²×C₃₁	1	1	yes	characteristic
304	C₈²	C₂²×C₃₁	1	1	yes	characteristic
305	C₈²	C₂²×C₃₁	1	1	yes	characteristic
306	C₈²	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×C₃₁	—	4	1	no	—
308	C₂×C₃₁	—	4	1	no	—
309	C₂×C₃₁	—	4	1	no	—
310	C₂×C₃₁	#2873	1	1	yes	characteristic
311	C₂×C₃₁	C₄×#131	1	1	yes	characteristic
312	C₂×C₃₁	#2875	1	1	yes	characteristic
313	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
314	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
315	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
316	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
317	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
318	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
319	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
320	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
321	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
322	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
323	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
324	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
325	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
326	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
327	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
328	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
329	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
330	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
331	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
332	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
336	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
337	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
338	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
339	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
340	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
341	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
342	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
343	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
344	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
345	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
346	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
347	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
348	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
349	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
350	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
351	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
352	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
353	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
354	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
355	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
356	#97	C₂×C₄×C₃₁	1	2	yes	—
357	#97	C₂×C₄×C₃₁	1	2	yes	—
358	#97	C₂×C₄×C₃₁	1	2	yes	—
359	#97	C₂×C₄×C₃₁	1	2	yes	—
360	#97	C₂×C₄×C₃₁	1	2	yes	—
361	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
362	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
363	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
364	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
365	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
366	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
367	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
368	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
370	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
372	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
373	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
374	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
375	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₃₁	#17340	1	1	yes	characteristic, direct product factor, Sylow
377	#41	—	4	2	no	—
378	C₂×Q₈	C₄²×C₃₁	1	1	yes	characteristic
379	C₂×D₄	C₄²×C₃₁	1	1	yes	characteristic
380	C₂×D₄	C₄²×C₃₁	1	1	yes	characteristic
381	C₂×D₄	C₄²×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
387	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
391	M₁₆	—	2	4	no	—
392	M₁₆	—	2	4	no	—
393	M₁₆	—	4	2	no	—
394	M₁₆	—	4	2	no	—
395	M₁₆	—	4	2	no	—
396	C₂×C₈	—	2	4	no	—
397	C₂×C₈	—	4	2	no	—
398	C₂×C₈	—	4	2	no	—
399	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
400	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
401	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
402	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
403	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
404	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
405	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
406	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
407	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
408	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
409	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
410	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
411	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
412	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
413	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
414	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
415	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
416	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
417	C₂²⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
418	C₂²⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
419	C₂²⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
420	C₂²⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
421	C₂²⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
422	C₂²⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
423	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
424	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
425	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
426	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
427	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
428	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
429	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
430	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
431	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
432	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
433	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
434	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
435	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
436	Q₈	—	4	1	no	—
437	D₄	—	4	1	no	—
438	D₄	—	4	1	no	—
439	D₄	—	4	1	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
450	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
451	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
452	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
454	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
455	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
456	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
458	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
461	C₈	—	2	2	no	—
462	C₈	—	2	2	no	—
463	C₈	—	2	2	no	—
464	C₈	—	2	2	no	—
465	C₈	—	2	2	no	—
466	C₈	—	2	2	no	—
467	C₈	—	2	2	no	—
468	C₈	—	2	2	no	—
469	C₈	—	2	2	no	—
470	C₈	—	2	2	no	—
471	C₈	—	2	2	no	—
472	C₈	C₃₁×#131	1	4	yes	—
473	C₂²	—	2	1	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
483	C₄	—	2	1	no	—
484	C₄	—	2	1	no	—
485	C₄	—	2	1	no	—
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	2	1	no	—
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	C₂×C₃₁×#131	1	2	yes	—
493	C₄	C₃₁×#568	1	2	yes	—
494	C₄	C₃₁×#568	1	2	yes	—
495	C₂	—	4	1	no	—
496	C₂	—	4	1	no	—
497	C₂	—	4	1	no	—
498	C₂	C₃₁×#2873	1	1	yes	characteristic
499	C₂	C₄×C₃₁×#131	1	1	yes	characteristic
500	C₂	C₃₁×#2875	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	C₂×#41×C₃₁	C₂×C₄
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×#1455	C₂	semidirect
5. C₃₁×#1454	C₂	semidirect
6. C₃₁×#1454	C₂	semidirect
7. C₃₁×#1453	C₂	semidirect
8. C₃₁×#1453	C₂	semidirect
9. C₃₁×#1453	C₂	semidirect
10. C₃₁×#1452	C₂	semidirect
11. C₃₁×#1451	C₂	semidirect
12. C₃₁×#1451	C₂	semidirect
13. C₃₁×#1450	C₂	semidirect
14. C₈×M₁₆×C₃₁	C₂	semidirect
15. C₈×M₁₆×C₃₁	C₂	semidirect
182. #17340	C₃₁	direct (N × H)
226. #1455	C₂×C₃₁	semidirect
227. #1454	C₂×C₃₁	semidirect
228. #1454	C₂×C₃₁	semidirect
229. #1453	C₂×C₃₁	semidirect
230. #1453	C₂×C₃₁	semidirect
231. #1453	C₂×C₃₁	semidirect
232. #1452	C₂×C₃₁	semidirect
233. #1451	C₂×C₃₁	semidirect
234. #1451	C₂×C₃₁	semidirect
235. #1450	C₂×C₃₁	semidirect
236. C₈×M₁₆	C₂×C₃₁	semidirect
237. C₈×M₁₆	C₂×C₃₁	semidirect
376. C₃₁	#17340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2965	2
C₃₁×#2922	2
C₃₁×#1455	2
C₃₁×#1454	2
C₃₁×#1453	2
C₃₁×#1452	2
C₃₁×#1451	2
C₃₁×#1450	2
C₈×M₁₆×C₃₁	2
#17340	31

Subgroups — C31×#17340 #21474099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17340 #21474099