Subgroups — C₃₁×#22698 #21479457

Order 7936.

1032 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#435	C₂	1	2	yes	maximal
2	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1940	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1850	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#435	—	2	8	no	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×#97	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄³×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	1	yes	characteristic
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₃₁	—	2	4	no	—
52	C₂×M₁₆×C₃₁	—	2	4	no	—
53	C₂×M₁₆×C₃₁	—	4	2	no	—
54	C₂²×C₈×C₃₁	—	2	4	no	—
55	C₂²×C₈×C₃₁	—	2	4	no	—
56	C₂²×C₈×C₃₁	—	4	2	no	—
57	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
61	C₄×D₄×C₃₁	—	2	4	no	—
62	C₄×D₄×C₃₁	—	2	4	no	—
63	C₄×D₄×C₃₁	D₄	1	4	yes	—
64	C₄×D₄×C₃₁	Q₈	1	4	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	—	4	2	no	—
75	C₂×C₄²×C₃₁	—	4	2	no	—
76	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₃₁×C₄⋊C₈	—	2	4	no	—
90	C₃₁×C₄⋊C₈	—	2	4	no	—
91	C₃₁×C₄⋊C₈	—	4	2	no	—
92	C₃₁×C₄⋊C₈	—	4	2	no	—
93	C₃₁×C₂²⋊C₈	—	2	4	no	—
94	C₃₁×C₂²⋊C₈	—	2	4	no	—
95	C₃₁×#97	—	2	4	no	—
96	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
97	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
98	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
99	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
100	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
101	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
102	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	4	1	no	—
117	C₂²×C₄×C₃₁	—	4	1	no	—
118	C₂²×C₄×C₃₁	—	4	1	no	—
119	C₂²×C₄×C₃₁	—	4	1	no	—
120	C₂²×C₄×C₃₁	—	4	1	no	—
121	C₂²×C₄×C₃₁	—	4	1	no	—
122	C₂²×C₄×C₃₁	M₁₆	1	4	yes	complemented
123	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂²×C₄	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	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	M₁₆×C₃₁	—	4	8	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	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	4	8	no	—
153	C₂×C₈×C₃₁	#41	1	4	yes	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
157	C₂²⋊C₄×C₃₁	—	2	2	no	—
158	C₂²⋊C₄×C₃₁	—	2	2	no	—
159	C₂²⋊C₄×C₃₁	—	2	2	no	—
160	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
161	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
162	C₂²⋊C₄×C₃₁	M₁₆	1	4	yes	complemented
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₄²×C₃₁	—	2	2	no	—
170	C₄²×C₃₁	—	4	1	no	—
171	C₄²×C₃₁	—	4	1	no	—
172	C₄²×C₃₁	—	4	1	no	—
173	C₄²×C₃₁	—	4	1	no	—
174	C₄²×C₃₁	—	4	1	no	—
175	C₄²×C₃₁	C₂×D₄	1	2	yes	—
176	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
177	#22698	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂³×C₃₁	—	2	1	no	—
179	C₂³×C₃₁	—	2	1	no	—
180	C₂³×C₃₁	—	2	1	no	—
181	C₂³×C₃₁	—	2	1	no	—
182	C₂³×C₃₁	—	2	1	no	—
183	C₂³×C₃₁	—	2	1	no	—
184	C₂³×C₃₁	—	2	1	no	—
185	C₂³×C₃₁	—	2	1	no	—
186	C₂³×C₃₁	—	2	1	no	—
187	C₂³×C₃₁	—	2	1	no	—
188	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
189	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
190	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
191	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
192	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
193	D₄×C₃₁	—	2	8	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	4	4	no	—
226	C₂×C₄×C₃₁	—	4	4	no	—
227	C₂×C₄×C₃₁	—	4	4	no	—
228	C₂×C₄×C₃₁	—	4	4	no	—
229	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
230	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
231	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
232	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
233	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
234	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
235	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
236	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
237	C₂×C₄×C₃₁	#131	1	2	yes	—
238	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
241	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
242	C₈×C₃₁	—	2	4	no	—
243	C₈×C₃₁	—	4	2	no	—
244	C₈×C₃₁	—	4	2	no	—
245	C₈×C₃₁	—	4	2	no	—
246	C₂×#435	C₂×C₃₁	1	2	yes	—
247	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
248	#1940	C₂×C₃₁	1	2	yes	complemented
249	#1851	C₂×C₃₁	1	2	yes	complemented
250	#1850	C₂×C₃₁	1	2	yes	complemented
251	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1728	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂²×C₃₁	—	2	2	no	—
258	C₂²×C₃₁	—	2	2	no	—
259	C₂²×C₃₁	—	2	2	no	—
260	C₂²×C₃₁	—	2	2	no	—
261	C₂²×C₃₁	—	2	2	no	—
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	2	2	no	—
265	C₂²×C₃₁	—	2	2	no	—
266	C₂²×C₃₁	—	4	1	no	—
267	C₂²×C₃₁	—	4	1	no	—
268	C₂²×C₃₁	—	4	1	no	—
269	C₂²×C₃₁	—	4	1	no	—
270	C₂²×C₃₁	#423	1	4	yes	—
271	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
272	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
273	C₂²×C₃₁	#518	1	1	yes	characteristic
274	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
275	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
276	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
277	C₂²×C₃₁	#568	1	1	yes	characteristic
278	C₄×C₃₁	—	2	2	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	2	no	—
281	C₄×C₃₁	—	2	2	no	—
282	C₄×C₃₁	—	2	2	no	—
283	C₄×C₃₁	—	2	2	no	—
284	C₄×C₃₁	—	2	2	no	—
285	C₄×C₃₁	—	2	2	no	—
286	C₄×C₃₁	—	2	2	no	—
287	C₄×C₃₁	—	4	1	no	—
288	C₄×C₃₁	—	4	1	no	—
289	C₄×C₃₁	—	4	1	no	—
290	C₄×C₃₁	#546	1	2	yes	—
291	C₄×C₃₁	#554	1	2	yes	—
292	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
293	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
294	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
297	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
298	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
299	#435	—	2	8	no	—
300	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
301	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
302	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
303	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
304	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
305	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
306	C₂×#97	C₂²×C₃₁	1	2	yes	—
307	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
308	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
309	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
310	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
311	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
312	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
313	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
314	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
315	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
316	C₄³	C₂²×C₃₁	1	1	yes	characteristic
317	#336	C₂²×C₃₁	1	1	yes	characteristic
318	#336	C₂²×C₃₁	1	1	yes	characteristic
319	#336	C₂²×C₃₁	1	1	yes	characteristic
320	#336	C₂²×C₃₁	1	2	yes	—
321	#336	C₂²×C₃₁	1	2	yes	—
322	#336	C₂²×C₃₁	1	2	yes	—
323	#336	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×C₃₁	—	2	4	no	—
325	C₂×C₃₁	—	4	2	no	—
326	C₂×C₃₁	#2349	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#423	1	1	yes	characteristic
328	C₂×C₃₁	#2906	1	1	yes	characteristic
329	C₂×C₃₁	#2965	1	1	yes	characteristic
330	C₂×C₃₁	#2970	1	1	yes	characteristic
331	C₂×C₃₁	#2973	1	1	yes	characteristic
332	C₂×C₃₁	#2974	1	1	yes	characteristic
333	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
334	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
335	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
336	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
337	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₂×M₁₆	—	2	4	no	—
341	C₂×M₁₆	—	2	4	no	—
342	C₂×M₁₆	—	4	2	no	—
343	C₂²×C₈	—	2	4	no	—
344	C₂²×C₈	—	2	4	no	—
345	C₂²×C₈	—	4	2	no	—
346	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
347	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
348	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
349	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
350	C₄×D₄	—	2	4	no	—
351	C₄×D₄	—	2	4	no	—
352	C₄×D₄	D₄×C₃₁	1	4	yes	—
353	C₄×D₄	Q₈×C₃₁	1	4	yes	—
354	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
357	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
358	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
359	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
360	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
361	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
362	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄²	—	4	2	no	—
364	C₂×C₄²	—	4	2	no	—
365	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
367	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
368	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
369	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
370	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
371	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
374	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
378	C₄⋊C₈	—	2	4	no	—
379	C₄⋊C₈	—	2	4	no	—
380	C₄⋊C₈	—	4	2	no	—
381	C₄⋊C₈	—	4	2	no	—
382	C₂²⋊C₈	—	2	4	no	—
383	C₂²⋊C₈	—	2	4	no	—
384	#97	—	2	4	no	—
385	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
386	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
387	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
388	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
389	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
390	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
391	C₃₁	#22698	1	1	yes	characteristic, direct product factor, Sylow
392	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
393	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	4	1	no	—
407	C₂²×C₄	—	4	1	no	—
408	C₂²×C₄	—	4	1	no	—
409	C₂²×C₄	—	4	1	no	—
410	C₂²×C₄	—	4	1	no	—
411	C₂²×C₄	—	4	1	no	—
412	C₂²×C₄	M₁₆×C₃₁	1	4	yes	complemented
413	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
414	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
415	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
416	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
418	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
419	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
428	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
429	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
430	M₁₆	—	4	8	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	4	8	no	—
443	C₂×C₈	#41×C₃₁	1	4	yes	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
451	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
452	C₂²⋊C₄	M₁₆×C₃₁	1	4	yes	complemented
453	C₄²	—	2	2	no	—
454	C₄²	—	2	2	no	—
455	C₄²	—	2	2	no	—
456	C₄²	—	2	2	no	—
457	C₄²	—	2	2	no	—
458	C₄²	—	2	2	no	—
459	C₄²	—	2	2	no	—
460	C₄²	—	4	1	no	—
461	C₄²	—	4	1	no	—
462	C₄²	—	4	1	no	—
463	C₄²	—	4	1	no	—
464	C₄²	—	4	1	no	—
465	C₄²	C₂×D₄×C₃₁	1	2	yes	—
466	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
467	C₂³	—	2	1	no	—
468	C₂³	—	2	1	no	—
469	C₂³	—	2	1	no	—
470	C₂³	—	2	1	no	—
471	C₂³	—	2	1	no	—
472	C₂³	—	2	1	no	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	1	no	—
475	C₂³	—	2	1	no	—
476	C₂³	—	2	1	no	—
477	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
478	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
479	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
480	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
481	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
482	D₄	—	2	8	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	4	no	—
518	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
519	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
520	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
521	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
522	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
523	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
524	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
525	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
526	C₂×C₄	C₃₁×#131	1	2	yes	—
527	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
530	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
531	C₈	—	2	4	no	—
532	C₈	—	4	2	no	—
533	C₈	—	4	2	no	—
534	C₈	—	4	2	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	4	1	no	—
548	C₂²	C₃₁×#423	1	4	yes	—
549	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
550	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
551	C₂²	C₃₁×#518	1	1	yes	characteristic
552	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
553	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
554	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
555	C₂²	C₃₁×#568	1	1	yes	characteristic
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	2	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	2	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	C₃₁×#546	1	2	yes	—
569	C₄	C₃₁×#554	1	2	yes	—
570	C₂	—	2	4	no	—
571	C₂	—	4	2	no	—
572	C₂	C₃₁×#2349	1	1	yes	characteristic
573	C₂	C₂×C₃₁×#423	1	1	yes	characteristic
574	C₂	C₃₁×#2906	1	1	yes	characteristic
575	C₂	C₃₁×#2965	1	1	yes	characteristic
576	C₂	C₃₁×#2970	1	1	yes	characteristic
577	C₂	C₃₁×#2973	1	1	yes	characteristic
578	C₂	C₃₁×#2974	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1940	C₂	semidirect
4. C₃₁×#1851	C₂	semidirect
5. C₃₁×#1850	C₂	semidirect
6. C₃₁×#1831	C₂	semidirect
7. C₃₁×#1797	C₂	semidirect
8. C₃₁×#1797	C₂	semidirect
9. C₃₁×#1767	C₂	semidirect
10. C₃₁×#1728	C₂	semidirect
11. C₂×C₃₁×#336	C₂	semidirect
122. C₂²×C₄×C₃₁	M₁₆	semidirect
162. C₂²⋊C₄×C₃₁	M₁₆	semidirect
177. #22698	C₃₁	direct (N × H)
248. #1940	C₂×C₃₁	semidirect
249. #1851	C₂×C₃₁	semidirect
250. #1850	C₂×C₃₁	semidirect
251. #1831	C₂×C₃₁	semidirect
252. #1797	C₂×C₃₁	semidirect
253. #1797	C₂×C₃₁	semidirect
254. #1767	C₂×C₃₁	semidirect
255. #1728	C₂×C₃₁	semidirect
256. C₂×#336	C₂×C₃₁	semidirect
391. C₃₁	#22698	direct (N × H)
412. C₂²×C₄	M₁₆×C₃₁	semidirect
452. C₂²⋊C₄	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#435	2
C₄²×D₄×C₃₁	2
C₃₁×#1940	2
C₃₁×#1851	2
C₃₁×#1850	2
C₃₁×#1831	2
C₃₁×#1797	2
C₃₁×#1767	2
C₃₁×#1728	2
C₂×C₃₁×#336	2
#22698	31

Subgroups — C31×#22698 #21479457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22698 #21479457