Subgroups — C₃₁×#20556 #21477315

Order 7936.

1098 conjugacy classes of subgroups.

The subgroup lattice has 444 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×#1771	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1753	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1752	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1749	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1745	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	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#435	C₄	1	16	yes	complemented
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	2	yes	—
23	C₂×C₃₁×#97	C₂²	1	2	yes	—
24	C₂×C₃₁×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
29	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
30	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₄³×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#336	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	1	yes	characteristic
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	1	yes	characteristic
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	1	yes	characteristic
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₃₁	—	4	2	no	—
46	C₂×M₁₆×C₃₁	—	4	2	no	—
47	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
48	C₂²×C₈×C₃₁	—	4	2	no	—
49	C₂²×C₈×C₃₁	—	4	2	no	—
50	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₄×D₄×C₃₁	—	2	8	no	—
56	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	—	4	2	no	—
64	C₂×C₄²×C₃₁	—	4	2	no	—
65	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
75	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
76	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
77	C₃₁×#97	—	4	2	no	—
78	C₃₁×#97	—	4	2	no	—
79	C₃₁×#97	C₂×C₄	1	8	yes	complemented
80	C₄×C₈×C₃₁	—	4	2	no	—
81	C₃₁×#95	C₂×C₄	1	2	yes	—
82	C₃₁×#95	C₂×C₄	1	2	yes	—
83	C₃₁×#95	C₂×C₄	1	2	yes	—
84	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×D₄×C₃₁	—	2	4	no	—
87	C₂×D₄×C₃₁	C₄²	1	4	yes	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	1	no	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	4	no	—
95	C₂²×C₄×C₃₁	—	4	2	no	—
96	C₂²×C₄×C₃₁	—	4	2	no	—
97	C₂²×C₄×C₃₁	—	4	2	no	—
98	C₂²×C₄×C₃₁	—	4	2	no	—
99	C₂²×C₄×C₃₁	M₁₆	1	8	yes	complemented
100	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
107	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
108	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
111	M₁₆×C₃₁	—	4	8	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	4	8	no	—
121	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
122	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
125	C₄⋊C₄×C₃₁	—	2	4	no	—
126	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
127	C₂²⋊C₄×C₃₁	—	2	4	no	—
128	C₂²⋊C₄×C₃₁	—	2	4	no	—
129	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
130	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
131	C₄²×C₃₁	—	2	4	no	—
132	C₄²×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	—	2	1	no	—
134	C₄²×C₃₁	—	2	2	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	2	1	no	—
137	C₄²×C₃₁	—	4	2	no	—
138	C₄²×C₃₁	—	4	2	no	—
139	C₄²×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
141	#20556	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	—	2	2	no	—
149	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
150	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
151	C₂³×C₃₁	C₂×M₁₆	1	4	yes	—
152	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
153	D₄×C₃₁	—	2	8	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	4	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	4	8	no	—
175	C₂×C₄×C₃₁	—	4	8	no	—
176	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
177	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
178	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
179	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
180	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
181	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
182	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
183	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
185	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
186	C₈×C₃₁	—	2	4	no	—
187	C₈×C₃₁	—	2	4	no	—
188	C₈×C₃₁	—	2	4	no	—
189	C₈×C₃₁	—	2	4	no	—
190	C₂×#435	C₂×C₃₁	1	2	yes	—
191	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
192	#1771	C₂×C₃₁	1	2	yes	complemented
193	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1753	C₂×C₃₁	1	2	yes	complemented
195	#1752	C₂×C₃₁	1	2	yes	complemented
196	#1749	C₂×C₃₁	1	1	yes	characteristic, complemented
197	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1745	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1728	C₂×C₃₁	1	1	yes	characteristic, complemented
200	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂²×C₃₁	—	2	2	no	—
202	C₂²×C₃₁	—	2	4	no	—
203	C₂²×C₃₁	—	2	4	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	2	2	no	—
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	4	2	no	—
209	C₂²×C₃₁	—	4	2	no	—
210	C₂²×C₃₁	C₄×M₁₆	1	4	yes	—
211	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
212	C₂²×C₃₁	#518	1	1	yes	characteristic
213	C₂²×C₃₁	#520	1	1	yes	characteristic
214	C₂²×C₃₁	C₂²×M₁₆	1	2	yes	—
215	C₂²×C₃₁	#568	1	2	yes	—
216	C₄×C₃₁	—	2	2	no	—
217	C₄×C₃₁	—	2	2	no	—
218	C₄×C₃₁	—	2	4	no	—
219	C₄×C₃₁	—	2	4	no	—
220	C₄×C₃₁	—	2	2	no	—
221	C₄×C₃₁	—	2	2	no	—
222	C₄×C₃₁	—	4	2	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	#518	1	2	yes	—
225	C₄×C₃₁	#520	1	2	yes	—
226	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
227	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
228	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
229	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
230	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
231	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
232	#435	C₄×C₃₁	1	16	yes	complemented
233	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
234	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
235	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
236	C₂×#97	C₂²×C₃₁	1	2	yes	—
237	C₂×#97	C₂²×C₃₁	1	2	yes	—
238	C₂×#97	C₂²×C₃₁	1	2	yes	—
239	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
240	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
241	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
242	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
243	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
244	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
245	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
246	C₄³	C₂²×C₃₁	1	1	yes	characteristic
247	#336	C₂²×C₃₁	1	1	yes	characteristic
248	#336	C₂²×C₃₁	1	1	yes	characteristic
249	#336	C₂²×C₃₁	1	2	yes	—
250	#336	C₂²×C₃₁	1	1	yes	characteristic
251	#336	C₂²×C₃₁	1	2	yes	—
252	#336	C₂²×C₃₁	1	1	yes	characteristic
253	#336	C₂²×C₃₁	1	2	yes	—
254	C₂×C₃₁	—	2	4	no	—
255	C₂×C₃₁	—	4	2	no	—
256	C₂×C₃₁	#2275	1	1	yes	characteristic
257	C₂×C₃₁	C₂×C₄×M₁₆	1	1	yes	characteristic
258	C₂×C₃₁	#2875	1	1	yes	characteristic
259	C₂×C₃₁	#2973	1	2	yes	—
260	C₂×C₃₁	#2989	1	2	yes	—
261	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
262	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
263	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
264	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
265	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
266	C₂×M₁₆	—	4	2	no	—
267	C₂×M₁₆	—	4	2	no	—
268	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
269	C₂²×C₈	—	4	2	no	—
270	C₂²×C₈	—	4	2	no	—
271	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
272	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
273	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
274	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
275	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
276	C₄×D₄	—	2	8	no	—
277	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
278	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
279	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
281	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
282	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
283	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄²	—	4	2	no	—
285	C₂×C₄²	—	4	2	no	—
286	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
287	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
288	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
289	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
290	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
291	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
293	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
295	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
296	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
297	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
298	#97	—	4	2	no	—
299	#97	—	4	2	no	—
300	#97	C₂×C₄×C₃₁	1	8	yes	complemented
301	C₄×C₈	—	4	2	no	—
302	#95	C₂×C₄×C₃₁	1	2	yes	—
303	#95	C₂×C₄×C₃₁	1	2	yes	—
304	#95	C₂×C₄×C₃₁	1	2	yes	—
305	C₃₁	#20556	1	1	yes	characteristic, direct product factor, Sylow
306	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
307	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
308	C₂×D₄	—	2	4	no	—
309	C₂×D₄	C₄²×C₃₁	1	4	yes	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	—	4	2	no	—
320	C₂²×C₄	—	4	2	no	—
321	C₂²×C₄	M₁₆×C₃₁	1	8	yes	complemented
322	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
323	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
324	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
329	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
330	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
332	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
333	M₁₆	—	4	8	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	4	8	no	—
343	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
344	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
345	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
346	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
349	C₂²⋊C₄	—	2	4	no	—
350	C₂²⋊C₄	—	2	4	no	—
351	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
352	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
353	C₄²	—	2	4	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	2	2	no	—
358	C₄²	—	2	1	no	—
359	C₄²	—	4	2	no	—
360	C₄²	—	4	2	no	—
361	C₄²	—	4	2	no	—
362	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
363	C₂³	—	2	1	no	—
364	C₂³	—	2	1	no	—
365	C₂³	—	2	2	no	—
366	C₂³	—	2	1	no	—
367	C₂³	—	2	1	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	2	no	—
370	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
371	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
372	C₂³	C₂×M₁₆×C₃₁	1	4	yes	—
373	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
374	D₄	—	2	8	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	4	8	no	—
396	C₂×C₄	—	4	8	no	—
397	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
398	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
399	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
400	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
401	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
402	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
403	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
404	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
405	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
406	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
407	C₈	—	2	4	no	—
408	C₈	—	2	4	no	—
409	C₈	—	2	4	no	—
410	C₈	—	2	4	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	4	no	—
413	C₂²	—	2	4	no	—
414	C₂²	—	2	2	no	—
415	C₂²	—	2	2	no	—
416	C₂²	—	2	2	no	—
417	C₂²	—	2	2	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	4	2	no	—
420	C₂²	C₄×M₁₆×C₃₁	1	4	yes	—
421	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
422	C₂²	C₃₁×#518	1	1	yes	characteristic
423	C₂²	C₃₁×#520	1	1	yes	characteristic
424	C₂²	C₂²×M₁₆×C₃₁	1	2	yes	—
425	C₂²	C₃₁×#568	1	2	yes	—
426	C₄	—	2	2	no	—
427	C₄	—	2	2	no	—
428	C₄	—	2	4	no	—
429	C₄	—	2	4	no	—
430	C₄	—	2	2	no	—
431	C₄	—	2	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	2	no	—
434	C₄	C₃₁×#518	1	2	yes	—
435	C₄	C₃₁×#520	1	2	yes	—
436	C₂	—	2	4	no	—
437	C₂	—	4	2	no	—
438	C₂	C₃₁×#2275	1	1	yes	characteristic
439	C₂	C₂×C₄×M₁₆×C₃₁	1	1	yes	characteristic
440	C₂	C₃₁×#2875	1	1	yes	characteristic
441	C₂	C₃₁×#2973	1	2	yes	—
442	C₂	C₃₁×#2989	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂² ▷ 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×#1771	C₂	semidirect
4. C₃₁×#1767	C₂	semidirect
5. C₃₁×#1753	C₂	semidirect
6. C₃₁×#1752	C₂	semidirect
7. C₃₁×#1749	C₂	semidirect
8. C₄×C₃₁×C₂²⋊C₈	C₂	semidirect
9. C₃₁×#1745	C₂	semidirect
10. C₃₁×#1728	C₂	semidirect
11. C₂×C₃₁×#336	C₂	semidirect
18. C₃₁×#435	C₄	semidirect
47. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
50. C₂²×C₈×C₃₁	C₂×C₄	semidirect
74. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
75. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
76. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
79. C₃₁×#97	C₂×C₄	semidirect
99. C₂²×C₄×C₃₁	M₁₆	semidirect
141. #20556	C₃₁	direct (N × H)
192. #1771	C₂×C₃₁	semidirect
193. #1767	C₂×C₃₁	semidirect
194. #1753	C₂×C₃₁	semidirect
195. #1752	C₂×C₃₁	semidirect
196. #1749	C₂×C₃₁	semidirect
197. C₄×C₂²⋊C₈	C₂×C₃₁	semidirect
198. #1745	C₂×C₃₁	semidirect
199. #1728	C₂×C₃₁	semidirect
200. C₂×#336	C₂×C₃₁	semidirect
232. #435	C₄×C₃₁	semidirect
268. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
271. C₂²×C₈	C₂×C₄×C₃₁	semidirect
295. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
296. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
297. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
300. #97	C₂×C₄×C₃₁	semidirect
305. C₃₁	#20556	direct (N × H)
321. 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×#1771	2
C₃₁×#1767	2
C₃₁×#1753	2
C₃₁×#1752	2
C₃₁×#1749	2
C₄×C₃₁×C₂²⋊C₈	2
C₃₁×#1745	2
C₃₁×#1728	2
C₂×C₃₁×#336	2
#20556	31

Subgroups — C31×#20556 #21477315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20556 #21477315