Subgroups — C₃₁×#24780 #21481539

Order 7936.

1150 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2416	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2033	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1874	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#121	—	2	2	no	—
16	C₂×C₃₁×#121	C₂²	1	2	yes	—
17	C₂×C₃₁×#121	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#466	—	2	4	no	—
21	C₃₁×#466	—	2	4	no	—
22	C₈×D₄×C₃₁	—	2	4	no	—
23	C₂×C₃₁×#107	—	2	4	no	—
24	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#107	C₂²	1	2	yes	—
27	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#396	—	2	2	no	—
36	C₃₁×#394	—	2	2	no	—
37	C₃₁×#390	C₂²	1	2	yes	—
38	C₃₁×#389	C₂²	1	2	yes	—
39	C₃₁×#386	—	2	2	no	—
40	C₃₁×#380	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₂²×D₄×C₃₁	—	2	2	no	—
44	C₂²×D₄×C₃₁	—	2	2	no	—
45	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	—	2	2	no	—
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	D₄	1	4	yes	complemented
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#121	—	2	4	no	—
64	C₃₁×#121	—	2	4	no	—
65	C₃₁×#121	—	2	4	no	—
66	C₃₁×#121	—	4	2	no	—
67	C₃₁×#121	—	4	8	no	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
74	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#107	—	2	2	no	—
87	C₃₁×#107	—	2	2	no	—
88	C₃₁×#107	—	2	4	no	—
89	C₃₁×#107	D₄	1	4	yes	complemented
90	C₃₁×C₄⋊C₈	—	2	2	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	4	2	no	—
94	C₃₁×C₂²⋊C₈	—	2	2	no	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₄×C₈×C₃₁	—	2	2	no	—
97	C₃₁×#95	—	2	2	no	—
98	C₃₁×#95	—	2	2	no	—
99	C₂⁴×C₃₁	—	2	2	no	—
100	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	4	4	no	—
113	C₂×D₄×C₃₁	—	4	4	no	—
114	C₂×D₄×C₃₁	—	4	4	no	—
115	C₂×D₄×C₃₁	—	8	2	no	—
116	C₂×D₄×C₃₁	D₈	1	4	yes	complemented
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	4	4	no	—
131	C₂²×C₄×C₃₁	—	4	4	no	—
132	C₂²×C₄×C₃₁	—	8	2	no	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	#41	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	D₈×C₃₁	—	4	4	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	8	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	4	4	no	—
151	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
152	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
153	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₃₁	#41	1	2	yes	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	4	4	no	—
162	C₄⋊C₄×C₃₁	—	4	4	no	—
163	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
164	C₂²⋊C₄×C₃₁	—	2	2	no	—
165	C₂²⋊C₄×C₃₁	—	2	2	no	—
166	C₂²⋊C₄×C₃₁	—	4	4	no	—
167	C₂²⋊C₄×C₃₁	—	4	4	no	—
168	C₂²⋊C₄×C₃₁	—	4	4	no	—
169	C₂²⋊C₄×C₃₁	—	4	4	no	—
170	C₂²⋊C₄×C₃₁	—	8	2	no	—
171	C₄²×C₃₁	—	2	1	no	—
172	C₄²×C₃₁	—	2	1	no	—
173	#24780	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	1	no	—
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	—	4	2	no	—
185	C₂³×C₃₁	—	4	2	no	—
186	C₂³×C₃₁	—	4	2	no	—
187	C₂³×C₃₁	—	4	2	no	—
188	C₂³×C₃₁	—	4	2	no	—
189	C₂³×C₃₁	—	4	2	no	—
190	C₂³×C₃₁	—	4	2	no	—
191	C₂³×C₃₁	#121	1	2	yes	—
192	C₂³×C₃₁	C₂×D₈	1	2	yes	—
193	C₂³×C₃₁	C₂×D₈	1	2	yes	—
194	C₂³×C₃₁	#135	1	2	yes	—
195	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	D₄×C₃₁	—	2	8	no	—
197	D₄×C₃₁	—	4	2	no	—
198	D₄×C₃₁	—	4	2	no	—
199	D₄×C₃₁	—	4	2	no	—
200	D₄×C₃₁	—	4	2	no	—
201	D₄×C₃₁	—	8	8	no	—
202	D₄×C₃₁	—	8	8	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	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	2	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	2	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	—	4	2	no	—
230	C₂×C₄×C₃₁	—	8	8	no	—
231	C₂×C₄×C₃₁	#121	1	2	yes	—
232	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
238	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
239	C₈×C₃₁	—	2	2	no	—
240	C₈×C₃₁	—	4	1	no	—
241	C₈×C₃₁	—	4	1	no	—
242	C₈×C₃₁	#121	1	4	yes	complemented
243	C₂×#466	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂×#466	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
246	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#2416	C₂×C₃₁	1	2	yes	complemented
248	#2067	C₂×C₃₁	1	2	yes	complemented
249	#2033	C₂×C₃₁	1	2	yes	complemented
250	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1942	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1874	C₂×C₃₁	1	2	yes	complemented
253	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	2	2	no	—
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	—	4	1	no	—
264	C₂²×C₃₁	—	4	1	no	—
265	C₂²×C₃₁	—	4	1	no	—
266	C₂²×C₃₁	—	4	1	no	—
267	C₂²×C₃₁	—	8	2	no	—
268	C₂²×C₃₁	—	8	2	no	—
269	C₂²×C₃₁	—	8	2	no	—
270	C₂²×C₃₁	—	8	2	no	—
271	C₂²×C₃₁	—	8	2	no	—
272	C₂²×C₃₁	—	8	2	no	—
273	C₂²×C₃₁	—	8	2	no	—
274	C₂²×C₃₁	#466	1	4	yes	complemented
275	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
276	C₂²×C₃₁	D₄²	1	1	yes	characteristic
277	C₂²×C₃₁	#547	1	1	yes	characteristic
278	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
279	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
280	C₂²×C₃₁	#576	1	1	yes	characteristic
281	C₂²×C₃₁	#578	1	1	yes	characteristic
282	C₄×C₃₁	—	2	2	no	—
283	C₄×C₃₁	—	2	2	no	—
284	C₄×C₃₁	—	4	1	no	—
285	C₄×C₃₁	—	4	4	no	—
286	C₄×C₃₁	—	4	1	no	—
287	C₄×C₃₁	—	8	2	no	—
288	C₄×C₃₁	—	8	2	no	—
289	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
290	C₄×C₃₁	#537	1	1	yes	characteristic
291	C₄×C₃₁	#548	1	1	yes	characteristic
292	C₄×C₃₁	#553	1	1	yes	characteristic
293	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
294	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#121	—	2	2	no	—
297	C₂×#121	C₂²×C₃₁	1	2	yes	—
298	C₂×#121	C₂²×C₃₁	1	2	yes	—
299	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
300	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
301	#466	—	2	4	no	—
302	#466	—	2	4	no	—
303	C₈×D₄	—	2	4	no	—
304	C₂×#107	—	2	4	no	—
305	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×#107	C₂²×C₃₁	1	2	yes	—
308	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×#102	C₂²×C₃₁	1	2	yes	—
311	C₂×#102	C₂²×C₃₁	1	2	yes	—
312	C₂×#102	C₂²×C₃₁	1	2	yes	—
313	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
315	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
316	#396	—	2	2	no	—
317	#394	—	2	2	no	—
318	#390	C₂²×C₃₁	1	2	yes	—
319	#389	C₂²×C₃₁	1	2	yes	—
320	#386	—	2	2	no	—
321	#380	C₂²×C₃₁	1	2	yes	—
322	#340	C₂²×C₃₁	1	2	yes	—
323	#340	C₂²×C₃₁	1	2	yes	—
324	C₂×C₃₁	—	2	4	no	—
325	C₂×C₃₁	—	4	2	no	—
326	C₂×C₃₁	—	16	2	no	—
327	C₂×C₃₁	#2416	1	1	yes	characteristic
328	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
329	C₂×C₃₁	#3061	1	1	yes	characteristic
330	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
331	C₂×C₃₁	#3285	1	1	yes	characteristic
332	C₂×C₃₁	#3334	1	1	yes	characteristic
333	C₂×C₃₁	#3339	1	1	yes	characteristic
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	—	2	2	no	—
336	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
337	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
338	C₂³×C₄	—	2	2	no	—
339	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×D₈	—	2	2	no	—
342	C₂×D₈	—	2	2	no	—
343	C₂×D₈	—	2	2	no	—
344	C₂²×C₈	—	2	2	no	—
345	C₂²×C₈	—	2	2	no	—
346	C₂²×C₈	—	2	2	no	—
347	C₂²×C₈	—	2	2	no	—
348	C₂²×C₈	—	2	2	no	—
349	C₂²×C₈	D₄×C₃₁	1	4	yes	complemented
350	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
354	#121	—	2	4	no	—
355	#121	—	2	4	no	—
356	#121	—	2	4	no	—
357	#121	—	4	2	no	—
358	#121	—	4	8	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₂×C₄⋊C₄	—	2	4	no	—
362	C₂×C₄⋊C₄	—	2	2	no	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	4	2	no	—
365	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
377	#107	—	2	2	no	—
378	#107	—	2	2	no	—
379	#107	—	2	4	no	—
380	#107	D₄×C₃₁	1	4	yes	complemented
381	C₄⋊C₈	—	2	2	no	—
382	#102	—	2	4	no	—
383	#102	—	2	4	no	—
384	#102	—	4	2	no	—
385	C₂²⋊C₈	—	2	2	no	—
386	C₂²⋊C₈	—	2	2	no	—
387	C₄×C₈	—	2	2	no	—
388	#95	—	2	2	no	—
389	#95	—	2	2	no	—
390	C₃₁	#24780	1	1	yes	characteristic, direct product factor, Sylow
391	C₂⁴	—	2	2	no	—
392	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	4	4	no	—
403	C₂×D₄	—	4	4	no	—
404	C₂×D₄	—	4	4	no	—
405	C₂×D₄	—	4	4	no	—
406	C₂×D₄	—	4	4	no	—
407	C₂×D₄	—	8	2	no	—
408	C₂×D₄	D₈×C₃₁	1	4	yes	complemented
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	—	4	4	no	—
424	C₂²×C₄	—	8	2	no	—
425	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
427	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
428	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	#41×C₃₁	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
432	D₈	—	4	4	no	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	8	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	4	4	no	—
443	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
444	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
445	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
446	C₂×C₈	#41×C₃₁	1	2	yes	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	4	4	no	—
454	C₄⋊C₄	—	4	4	no	—
455	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
456	C₂²⋊C₄	—	2	2	no	—
457	C₂²⋊C₄	—	2	2	no	—
458	C₂²⋊C₄	—	4	4	no	—
459	C₂²⋊C₄	—	4	4	no	—
460	C₂²⋊C₄	—	4	4	no	—
461	C₂²⋊C₄	—	4	4	no	—
462	C₂²⋊C₄	—	8	2	no	—
463	C₄²	—	2	1	no	—
464	C₄²	—	2	1	no	—
465	C₂³	—	2	1	no	—
466	C₂³	—	2	1	no	—
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	—	4	2	no	—
476	C₂³	—	4	2	no	—
477	C₂³	—	4	2	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	4	2	no	—
481	C₂³	—	4	2	no	—
482	C₂³	C₃₁×#121	1	2	yes	—
483	C₂³	C₂×D₈×C₃₁	1	2	yes	—
484	C₂³	C₂×D₈×C₃₁	1	2	yes	—
485	C₂³	C₃₁×#135	1	2	yes	—
486	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
487	D₄	—	2	8	no	—
488	D₄	—	4	2	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	2	no	—
492	D₄	—	8	8	no	—
493	D₄	—	8	8	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	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	8	8	no	—
522	C₂×C₄	C₃₁×#121	1	2	yes	—
523	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
524	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
525	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
526	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
527	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
529	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
530	C₈	—	2	2	no	—
531	C₈	—	4	1	no	—
532	C₈	—	4	1	no	—
533	C₈	C₃₁×#121	1	4	yes	complemented
534	C₂²	—	2	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	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	8	2	no	—
548	C₂²	—	8	2	no	—
549	C₂²	—	8	2	no	—
550	C₂²	—	8	2	no	—
551	C₂²	—	8	2	no	—
552	C₂²	—	8	2	no	—
553	C₂²	—	8	2	no	—
554	C₂²	C₃₁×#466	1	4	yes	complemented
555	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
556	C₂²	D₄²×C₃₁	1	1	yes	characteristic
557	C₂²	C₃₁×#547	1	1	yes	characteristic
558	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
559	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
560	C₂²	C₃₁×#576	1	1	yes	characteristic
561	C₂²	C₃₁×#578	1	1	yes	characteristic
562	C₄	—	2	2	no	—
563	C₄	—	2	2	no	—
564	C₄	—	4	1	no	—
565	C₄	—	4	4	no	—
566	C₄	—	4	1	no	—
567	C₄	—	8	2	no	—
568	C₄	—	8	2	no	—
569	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
570	C₄	C₃₁×#537	1	1	yes	characteristic
571	C₄	C₃₁×#548	1	1	yes	characteristic
572	C₄	C₃₁×#553	1	1	yes	characteristic
573	C₂	—	2	4	no	—
574	C₂	—	4	2	no	—
575	C₂	—	16	2	no	—
576	C₂	C₃₁×#2416	1	1	yes	characteristic
577	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
578	C₂	C₃₁×#3061	1	1	yes	characteristic
579	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
580	C₂	C₃₁×#3285	1	1	yes	characteristic
581	C₂	C₃₁×#3334	1	1	yes	characteristic
582	C₂	C₃₁×#3339	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#466	C₂	semidirect
2. C₂×C₃₁×#466	C₂	semidirect
3. C₂×C₈×D₄×C₃₁	C₂	semidirect
4. C₂²×C₃₁×#107	C₂	semidirect
5. C₃₁×#2416	C₂	semidirect
6. C₃₁×#2067	C₂	semidirect
7. C₃₁×#2033	C₂	semidirect
8. C₃₁×#1970	C₂	semidirect
9. C₃₁×#1942	C₂	semidirect
10. C₃₁×#1874	C₂	semidirect
11. C₃₁×#1808	C₂	semidirect
58. C₂²×C₈×C₃₁	D₄	semidirect
89. C₃₁×#107	D₄	semidirect
116. C₂×D₄×C₃₁	D₈	semidirect
173. #24780	C₃₁	direct (N × H)
242. C₈×C₃₁	#121	semidirect
243. C₂×#466	C₂×C₃₁	semidirect
244. C₂×#466	C₂×C₃₁	semidirect
245. C₂×C₈×D₄	C₂×C₃₁	semidirect
246. C₂²×#107	C₂×C₃₁	semidirect
247. #2416	C₂×C₃₁	semidirect
248. #2067	C₂×C₃₁	semidirect
249. #2033	C₂×C₃₁	semidirect
250. #1970	C₂×C₃₁	semidirect
251. #1942	C₂×C₃₁	semidirect
252. #1874	C₂×C₃₁	semidirect
253. #1808	C₂×C₃₁	semidirect
274. C₂²×C₃₁	#466	semidirect
349. C₂²×C₈	D₄×C₃₁	semidirect
380. #107	D₄×C₃₁	semidirect
390. C₃₁	#24780	direct (N × H)
408. C₂×D₄	D₈×C₃₁	semidirect
533. C₈	C₃₁×#121	semidirect
554. C₂²	C₃₁×#466	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#466	2
C₂×C₈×D₄×C₃₁	2
C₂²×C₃₁×#107	2
C₃₁×#2416	2
C₃₁×#2067	2
C₃₁×#2033	2
C₃₁×#1970	2
C₃₁×#1942	2
C₃₁×#1874	2
C₃₁×#1808	2
#24780	31

Subgroups — C31×#24780 #21481539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24780 #21481539