Subgroups — C₃₁×#22168 #21478927

Order 7936.

1046 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
2	C₄×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1923	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1916	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1869	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1869	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1850	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#121	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₈×D₄×C₃₁	—	2	8	no	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#390	C₄	1	1	yes	characteristic
27	C₃₁×#389	C₄	1	1	yes	characteristic
28	C₃₁×#388	C₄	1	2	yes	—
29	C₃₁×#386	C₄	1	2	yes	—
30	C₃₁×#380	C₄	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	—	2	4	no	—
46	C₂²×C₈×C₃₁	—	4	1	no	—
47	C₂²×C₈×C₃₁	—	4	1	no	—
48	C₂²×C₈×C₃₁	—	4	2	no	—
49	C₂²×C₈×C₃₁	—	4	2	no	—
50	C₂²×C₈×C₃₁	—	4	1	no	—
51	C₂²×C₈×C₃₁	—	4	1	no	—
52	C₂²×C₈×C₃₁	D₄	1	4	yes	complemented
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₃₁×#121	C₈	1	8	yes	complemented
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	—	2	4	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₄×D₄×C₃₁	—	4	1	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
71	C₂×C₄²×C₃₁	—	4	1	no	—
72	C₂×C₄²×C₃₁	—	4	1	no	—
73	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	—	2	4	no	—
79	C₃₁×C₄⋊C₈	—	4	1	no	—
80	C₃₁×C₄⋊C₈	—	4	1	no	—
81	C₃₁×C₂²⋊C₈	—	2	4	no	—
82	C₃₁×C₂²⋊C₈	—	2	4	no	—
83	C₃₁×C₂²⋊C₈	—	4	2	no	—
84	C₃₁×C₂²⋊C₈	—	4	2	no	—
85	C₄×C₈×C₃₁	—	2	4	no	—
86	C₃₁×#95	C₂×C₄	1	2	yes	—
87	C₃₁×#95	C₂×C₄	1	2	yes	—
88	C₃₁×#95	C₂×C₄	1	2	yes	—
89	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×D₄×C₃₁	—	2	4	no	—
92	C₂×D₄×C₃₁	—	2	4	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	C₂×C₈	1	8	yes	complemented
96	C₂×D₄×C₃₁	C₂×C₈	1	4	yes	complemented
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	4	1	no	—
109	C₂²×C₄×C₃₁	—	4	1	no	—
110	C₂²×C₄×C₃₁	—	4	1	no	—
111	C₂²×C₄×C₃₁	—	4	1	no	—
112	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	complemented
113	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	#41	1	2	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	4	4	no	—
136	C₂×C₈×C₃₁	—	4	8	no	—
137	C₂×C₈×C₃₁	—	4	4	no	—
138	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
139	C₄⋊C₄×C₃₁	—	2	4	no	—
140	C₄⋊C₄×C₃₁	—	4	1	no	—
141	C₄⋊C₄×C₃₁	—	4	1	no	—
142	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	complemented
143	C₂²⋊C₄×C₃₁	—	2	4	no	—
144	C₂²⋊C₄×C₃₁	—	2	4	no	—
145	C₂²⋊C₄×C₃₁	—	4	2	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₂²⋊C₄×C₃₁	C₂×C₈	1	8	yes	complemented
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	—	2	1	no	—
153	C₄²×C₃₁	—	2	1	no	—
154	C₄²×C₃₁	—	4	1	no	—
155	C₄²×C₃₁	—	4	1	no	—
156	#22168	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	2	1	no	—
158	C₂³×C₃₁	—	2	1	no	—
159	C₂³×C₃₁	—	2	1	no	—
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	2	2	no	—
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	2	2	no	—
167	C₂³×C₃₁	C₄×D₄	1	2	yes	—
168	C₂³×C₃₁	C₂²×C₈	1	4	yes	—
169	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
170	C₂³×C₃₁	#131	1	2	yes	—
171	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
172	D₄×C₃₁	—	2	16	no	—
173	D₄×C₃₁	—	4	4	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	4	4	no	—
199	C₂×C₄×C₃₁	—	4	4	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
205	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
206	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
207	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
208	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
214	C₈×C₃₁	—	2	4	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₈×C₃₁	—	4	2	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₂×C₈×D₄	C₂×C₃₁	1	2	yes	complemented
219	C₄×#121	C₂×C₃₁	1	1	yes	characteristic
220	#1923	C₂×C₃₁	1	2	yes	complemented
221	#1916	C₂×C₃₁	1	2	yes	complemented
222	#1869	C₂×C₃₁	1	2	yes	complemented
223	#1869	C₂×C₃₁	1	2	yes	complemented
224	#1850	C₂×C₃₁	1	2	yes	complemented
225	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
226	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂²×C₃₁	—	2	2	no	—
228	C₂²×C₃₁	—	2	2	no	—
229	C₂²×C₃₁	—	2	2	no	—
230	C₂²×C₃₁	—	2	2	no	—
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	4	no	—
234	C₂²×C₃₁	—	2	4	no	—
235	C₂²×C₃₁	—	2	4	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	C₈×D₄	1	4	yes	—
241	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
242	C₂²×C₃₁	#518	1	1	yes	characteristic
243	C₂²×C₃₁	#520	1	1	yes	characteristic
244	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
245	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
246	C₂²×C₃₁	#568	1	1	yes	characteristic
247	C₂²×C₃₁	#568	1	1	yes	characteristic
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	2	4	no	—
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	2	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	1	no	—
255	C₄×C₃₁	—	4	1	no	—
256	C₄×C₃₁	#534	1	1	yes	characteristic
257	C₄×C₃₁	#537	1	1	yes	characteristic
258	C₄×C₃₁	#538	1	1	yes	characteristic
259	C₄×C₃₁	#539	1	1	yes	characteristic
260	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
261	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#121	C₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
265	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
266	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
267	C₈×D₄	—	2	8	no	—
268	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
269	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
272	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
273	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
274	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
275	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
276	#390	C₄×C₃₁	1	1	yes	characteristic
277	#389	C₄×C₃₁	1	1	yes	characteristic
278	#388	C₄×C₃₁	1	2	yes	—
279	#386	C₄×C₃₁	1	2	yes	—
280	#380	C₄×C₃₁	1	1	yes	characteristic
281	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
282	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
283	#336	C₂²×C₃₁	1	2	yes	—
284	#336	C₂²×C₃₁	1	2	yes	—
285	#336	C₂²×C₃₁	1	2	yes	—
286	#336	C₂²×C₃₁	1	2	yes	—
287	#336	C₂²×C₃₁	1	2	yes	—
288	C₂×C₃₁	—	2	4	no	—
289	C₂×C₃₁	—	4	4	no	—
290	C₂×C₃₁	#2325	1	1	yes	characteristic
291	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
292	C₂×C₃₁	#2932	1	1	yes	characteristic
293	C₂×C₃₁	#2972	1	1	yes	characteristic
294	C₂×C₃₁	#2976	1	1	yes	characteristic
295	C₂×C₃₁	#2988	1	1	yes	characteristic
296	C₂×C₃₁	#2992	1	1	yes	characteristic
297	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
298	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
299	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
300	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
301	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
302	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
303	C₂²×C₈	—	2	2	no	—
304	C₂²×C₈	—	2	4	no	—
305	C₂²×C₈	—	4	1	no	—
306	C₂²×C₈	—	4	1	no	—
307	C₂²×C₈	—	4	2	no	—
308	C₂²×C₈	—	4	2	no	—
309	C₂²×C₈	—	4	1	no	—
310	C₂²×C₈	—	4	1	no	—
311	C₂²×C₈	D₄×C₃₁	1	4	yes	complemented
312	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
313	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
314	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
315	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
316	#121	C₈×C₃₁	1	8	yes	complemented
317	C₄×D₄	—	2	4	no	—
318	C₄×D₄	—	2	4	no	—
319	C₄×D₄	—	4	1	no	—
320	C₄×D₄	—	4	1	no	—
321	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
322	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
326	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
327	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
328	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
329	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
330	C₂×C₄²	—	4	1	no	—
331	C₂×C₄²	—	4	1	no	—
332	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
336	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
337	C₄⋊C₈	—	2	4	no	—
338	C₄⋊C₈	—	4	1	no	—
339	C₄⋊C₈	—	4	1	no	—
340	C₂²⋊C₈	—	2	4	no	—
341	C₂²⋊C₈	—	2	4	no	—
342	C₂²⋊C₈	—	4	2	no	—
343	C₂²⋊C₈	—	4	2	no	—
344	C₄×C₈	—	2	4	no	—
345	#95	C₂×C₄×C₃₁	1	2	yes	—
346	#95	C₂×C₄×C₃₁	1	2	yes	—
347	#95	C₂×C₄×C₃₁	1	2	yes	—
348	C₃₁	#22168	1	1	yes	characteristic, direct product factor, Sylow
349	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
351	C₂×D₄	—	2	4	no	—
352	C₂×D₄	—	2	4	no	—
353	C₂×D₄	—	4	1	no	—
354	C₂×D₄	—	4	1	no	—
355	C₂×D₄	C₂×C₈×C₃₁	1	8	yes	complemented
356	C₂×D₄	C₂×C₈×C₃₁	1	4	yes	complemented
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	4	1	no	—
369	C₂²×C₄	—	4	1	no	—
370	C₂²×C₄	—	4	1	no	—
371	C₂²×C₄	—	4	1	no	—
372	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	complemented
373	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
375	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
376	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
377	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
380	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
382	C₂²×C₄	#41×C₃₁	1	2	yes	—
383	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	4	4	no	—
396	C₂×C₈	—	4	8	no	—
397	C₂×C₈	—	4	4	no	—
398	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	4	1	no	—
401	C₄⋊C₄	—	4	1	no	—
402	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	complemented
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	—	4	2	no	—
406	C₂²⋊C₄	—	4	2	no	—
407	C₂²⋊C₄	C₂×C₈×C₃₁	1	8	yes	complemented
408	C₄²	—	2	2	no	—
409	C₄²	—	2	2	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	1	no	—
413	C₄²	—	2	1	no	—
414	C₄²	—	4	1	no	—
415	C₄²	—	4	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	C₄×D₄×C₃₁	1	2	yes	—
427	C₂³	C₂²×C₈×C₃₁	1	4	yes	—
428	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
429	C₂³	C₃₁×#131	1	2	yes	—
430	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
431	D₄	—	2	16	no	—
432	D₄	—	4	4	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	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	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	—	4	4	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
464	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
465	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
466	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
470	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
471	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
472	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
473	C₈	—	2	4	no	—
474	C₈	—	4	2	no	—
475	C₈	—	4	2	no	—
476	C₈	—	4	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	2	2	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	4	no	—
484	C₂²	—	2	4	no	—
485	C₂²	—	2	4	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	C₈×D₄×C₃₁	1	4	yes	—
491	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
492	C₂²	C₃₁×#518	1	1	yes	characteristic
493	C₂²	C₃₁×#520	1	1	yes	characteristic
494	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
495	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
496	C₂²	C₃₁×#568	1	1	yes	characteristic
497	C₂²	C₃₁×#568	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	1	no	—
505	C₄	—	4	1	no	—
506	C₄	C₃₁×#534	1	1	yes	characteristic
507	C₄	C₃₁×#537	1	1	yes	characteristic
508	C₄	C₃₁×#538	1	1	yes	characteristic
509	C₄	C₃₁×#539	1	1	yes	characteristic
510	C₂	—	2	4	no	—
511	C₂	—	4	4	no	—
512	C₂	C₃₁×#2325	1	1	yes	characteristic
513	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic
514	C₂	C₃₁×#2932	1	1	yes	characteristic
515	C₂	C₃₁×#2972	1	1	yes	characteristic
516	C₂	C₃₁×#2976	1	1	yes	characteristic
517	C₂	C₃₁×#2988	1	1	yes	characteristic
518	C₂	C₃₁×#2992	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂×C₃₁×#121	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
1. C₂×C₈×D₄×C₃₁	C₂	semidirect
3. C₃₁×#1923	C₂	semidirect
4. C₃₁×#1916	C₂	semidirect
5. C₃₁×#1869	C₂	semidirect
6. C₃₁×#1869	C₂	semidirect
7. C₃₁×#1850	C₂	semidirect
8. C₃₁×#1797	C₂	semidirect
9. C₂×C₃₁×#336	C₂	semidirect
52. C₂²×C₈×C₃₁	D₄	semidirect
57. C₃₁×#121	C₈	semidirect
95. C₂×D₄×C₃₁	C₂×C₈	semidirect
96. C₂×D₄×C₃₁	C₂×C₈	semidirect
112. C₂²×C₄×C₃₁	C₂×C₈	semidirect
142. C₄⋊C₄×C₃₁	C₂×C₈	semidirect
147. C₂²⋊C₄×C₃₁	C₂×C₈	semidirect
156. #22168	C₃₁	direct (N × H)
218. C₂×C₈×D₄	C₂×C₃₁	semidirect
220. #1923	C₂×C₃₁	semidirect
221. #1916	C₂×C₃₁	semidirect
222. #1869	C₂×C₃₁	semidirect
223. #1869	C₂×C₃₁	semidirect
224. #1850	C₂×C₃₁	semidirect
225. #1797	C₂×C₃₁	semidirect
226. C₂×#336	C₂×C₃₁	semidirect
311. C₂²×C₈	D₄×C₃₁	semidirect
316. #121	C₈×C₃₁	semidirect
348. C₃₁	#22168	direct (N × H)
355. C₂×D₄	C₂×C₈×C₃₁	semidirect
356. C₂×D₄	C₂×C₈×C₃₁	semidirect
372. C₂²×C₄	C₂×C₈×C₃₁	semidirect
402. C₄⋊C₄	C₂×C₈×C₃₁	semidirect
407. C₂²⋊C₄	C₂×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₃₁	2
C₄×C₃₁×#121	2
C₃₁×#1923	2
C₃₁×#1916	2
C₃₁×#1869	2
C₃₁×#1850	2
C₃₁×#1797	2
C₂×C₃₁×#336	2
#22168	31

Subgroups — C31×#22168 #21478927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22168 #21478927