Subgroups — C₂×C₃₁×#2599 #21491623

Order 7936.

2766 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2599	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#397	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#395	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#393	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#388	C₂	1	2	yes	maximal
9	C₂×C₃₁×#387	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#387	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#385	C₂	1	2	yes	maximal
12	C₂×C₃₁×#383	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₃₁	—	2	8	no	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	8	yes	complemented
17	C₂×C₃₁×#117	—	2	8	no	—
18	C₂×C₃₁×#117	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
31	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#398	C₂²	1	8	yes	complemented
33	C₃₁×#397	C₂²	1	8	yes	—
34	C₃₁×#395	C₂²	1	8	yes	complemented
35	C₃₁×#393	C₂²	1	8	yes	—
36	C₃₁×#388	C₂²	1	16	yes	—
37	C₃₁×#387	C₂²	1	16	yes	complemented
38	C₃₁×#387	C₂²	1	16	yes	complemented
39	C₃₁×#385	C₂²	1	16	yes	—
40	C₃₁×#383	C₂²	1	8	yes	complemented
41	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#95	C₂²	1	2	yes	—
43	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#95	C₂²	1	2	yes	—
45	C₂×C₃₁×#95	C₂²	1	2	yes	—
46	C₂×C₃₁×#95	C₂²	1	2	yes	—
47	C₂×C₃₁×#95	C₂²	1	2	yes	—
48	C₂×C₃₁×#95	C₂²	1	2	yes	—
49	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	—	2	4	no	—
51	C₂²×Q₈×C₃₁	—	2	8	no	—
52	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
53	C₂³×C₄×C₃₁	—	2	4	no	—
54	C₂³×C₄×C₃₁	—	2	4	no	—
55	C₂³×C₄×C₃₁	—	2	4	no	—
56	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₃₁	—	2	32	no	—
68	C₃₁×#117	—	2	32	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
82	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
83	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
86	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
89	C₂×C₄²×C₃₁	—	2	4	no	—
90	C₂×C₄²×C₃₁	—	2	8	no	—
91	C₂×C₄²×C₃₁	D₄	1	8	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
94	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
95	C₃₁×#95	C₂³	1	4	yes	—
96	C₃₁×#95	C₂³	1	8	yes	—
97	C₃₁×#95	C₂³	1	8	yes	—
98	C₃₁×#95	C₂³	1	4	yes	—
99	C₃₁×#95	C₂³	1	8	yes	—
100	C₃₁×#95	C₂³	1	8	yes	—
101	C₃₁×#95	C₂³	1	8	yes	—
102	C₃₁×#95	C₂³	1	8	yes	—
103	C₂⁴×C₃₁	—	2	2	no	—
104	C₂⁴×C₃₁	—	2	4	no	—
105	C₂⁴×C₃₁	—	2	2	no	—
106	C₂⁴×C₃₁	—	2	4	no	—
107	C₂⁴×C₃₁	#41	1	4	yes	—
108	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
109	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
110	C₂×Q₈×C₃₁	—	2	32	no	—
111	C₂×Q₈×C₃₁	—	2	8	no	—
112	C₂×Q₈×C₃₁	—	2	16	no	—
113	C₂²×C₄×C₃₁	—	2	32	no	—
114	C₂²×C₄×C₃₁	—	2	4	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	8	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	4	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	8	no	—
124	C₂²×C₄×C₃₁	—	2	8	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	4	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₃₁	#41	1	8	yes	—
142	C₂²×C₄×C₃₁	#41	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
153	C₄⋊C₄×C₃₁	—	2	8	no	—
154	C₄⋊C₄×C₃₁	—	2	8	no	—
155	C₄⋊C₄×C₃₁	—	2	16	no	—
156	C₄⋊C₄×C₃₁	—	2	8	no	—
157	C₄⋊C₄×C₃₁	—	2	16	no	—
158	C₄⋊C₄×C₃₁	—	2	16	no	—
159	C₂²⋊C₄×C₃₁	—	2	16	no	—
160	C₂²⋊C₄×C₃₁	—	2	8	no	—
161	C₂²⋊C₄×C₃₁	—	2	8	no	—
162	C₂²⋊C₄×C₃₁	—	2	16	no	—
163	C₄²×C₃₁	—	2	8	no	—
164	C₄²×C₃₁	—	2	16	no	—
165	C₄²×C₃₁	C₂×D₄	1	16	yes	—
166	C₂×#2599	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	8	no	—
168	C₂³×C₃₁	—	2	16	no	—
169	C₂³×C₃₁	—	2	2	no	—
170	C₂³×C₃₁	—	2	4	no	—
171	C₂³×C₃₁	—	2	2	no	—
172	C₂³×C₃₁	—	2	4	no	—
173	C₂³×C₃₁	—	4	4	no	—
174	C₂³×C₃₁	—	4	4	no	—
175	C₂³×C₃₁	—	4	8	no	—
176	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
177	C₂³×C₃₁	C₂×#41	1	4	yes	—
178	C₂³×C₃₁	C₂×#41	1	2	yes	—
179	C₂³×C₃₁	C₂×#41	1	4	yes	—
180	C₂³×C₃₁	C₂×#41	1	2	yes	—
181	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
182	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
183	Q₈×C₃₁	—	2	32	no	—
184	C₂×C₄×C₃₁	—	2	32	no	—
185	C₂×C₄×C₃₁	—	2	8	no	—
186	C₂×C₄×C₃₁	—	2	8	no	—
187	C₂×C₄×C₃₁	—	2	8	no	—
188	C₂×C₄×C₃₁	—	2	16	no	—
189	C₂×C₄×C₃₁	—	2	8	no	—
190	C₂×C₄×C₃₁	—	2	4	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	4	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	4	4	no	—
212	C₂×C₄×C₃₁	—	4	8	no	—
213	C₂×C₄×C₃₁	—	4	4	no	—
214	C₂×C₄×C₃₁	—	4	8	no	—
215	C₂×C₄×C₃₁	—	4	8	no	—
216	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
220	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
221	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic
223	#2599	C₂×C₃₁	1	16	yes	direct product factor
224	C₂×#398	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂×#397	C₂×C₃₁	1	1	yes	characteristic
226	C₂×#395	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂×#393	C₂×C₃₁	1	1	yes	characteristic
228	C₂×#388	C₂×C₃₁	1	2	yes	—
229	C₂×#387	C₂×C₃₁	1	2	yes	complemented
230	C₂×#387	C₂×C₃₁	1	2	yes	complemented
231	C₂×#385	C₂×C₃₁	1	2	yes	—
232	C₂×#383	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂²×C₃₁	—	2	4	no	—
234	C₂²×C₃₁	—	2	8	no	—
235	C₂²×C₃₁	—	4	16	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	4	no	—
239	C₂²×C₃₁	#525	1	4	yes	—
240	C₂²×C₃₁	#525	1	4	yes	—
241	C₂²×C₃₁	#536	1	4	yes	—
242	C₂²×C₃₁	#552	1	8	yes	—
243	C₂²×C₃₁	#556	1	8	yes	—
244	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
245	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
246	C₂²×C₃₁	C₂²×#41	1	2	yes	—
247	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
248	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
249	C₄×C₃₁	—	2	4	no	—
250	C₄×C₃₁	—	2	4	no	—
251	C₄×C₃₁	—	2	4	no	—
252	C₄×C₃₁	—	2	8	no	—
253	C₄×C₃₁	—	2	4	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	4	4	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	4	no	—
258	C₄×C₃₁	—	4	4	no	—
259	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄×Q₈	—	2	8	no	—
262	C₂×C₄×Q₈	C₂²×C₃₁	1	8	yes	complemented
263	C₂×#117	—	2	8	no	—
264	C₂×#117	C₂²×C₃₁	1	8	yes	—
265	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
266	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
267	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
268	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
269	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
270	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
271	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
272	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
273	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
274	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
275	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
276	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
277	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
278	#398	C₂²×C₃₁	1	8	yes	complemented
279	#397	C₂²×C₃₁	1	8	yes	—
280	#395	C₂²×C₃₁	1	8	yes	complemented
281	#393	C₂²×C₃₁	1	8	yes	—
282	#388	C₂²×C₃₁	1	16	yes	—
283	#387	C₂²×C₃₁	1	16	yes	complemented
284	#387	C₂²×C₃₁	1	16	yes	complemented
285	#385	C₂²×C₃₁	1	16	yes	—
286	#383	C₂²×C₃₁	1	8	yes	complemented
287	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#95	C₂²×C₃₁	1	2	yes	—
289	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#95	C₂²×C₃₁	1	2	yes	—
291	C₂×#95	C₂²×C₃₁	1	2	yes	—
292	C₂×#95	C₂²×C₃₁	1	2	yes	—
293	C₂×#95	C₂²×C₃₁	1	2	yes	—
294	C₂×#95	C₂²×C₃₁	1	2	yes	—
295	C₂×C₃₁	—	4	4	no	—
296	C₂×C₃₁	#2599	1	8	yes	direct product factor
297	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
298	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
299	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
300	C₂×C₃₁	C₂×#552	1	2	yes	—
301	C₂×C₃₁	C₂×#556	1	2	yes	—
302	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
303	C₂²×Q₈	—	2	4	no	—
304	C₂²×Q₈	—	2	8	no	—
305	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
306	C₂³×C₄	—	2	4	no	—
307	C₂³×C₄	—	2	4	no	—
308	C₂³×C₄	—	2	4	no	—
309	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
311	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
312	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
313	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
315	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
316	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
317	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
318	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
319	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
320	C₄×Q₈	—	2	32	no	—
321	#117	—	2	32	no	—
322	C₂×C₄⋊C₄	—	2	4	no	—
323	C₂×C₄⋊C₄	—	2	4	no	—
324	C₂×C₄⋊C₄	—	2	8	no	—
325	C₂×C₄⋊C₄	—	2	4	no	—
326	C₂×C₄⋊C₄	—	2	8	no	—
327	C₂×C₄⋊C₄	—	2	8	no	—
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
333	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
334	C₂×C₂²⋊C₄	—	2	8	no	—
335	C₂×C₂²⋊C₄	—	2	4	no	—
336	C₂×C₂²⋊C₄	—	2	4	no	—
337	C₂×C₂²⋊C₄	—	2	8	no	—
338	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
341	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
342	C₂×C₄²	—	2	4	no	—
343	C₂×C₄²	—	2	8	no	—
344	C₂×C₄²	D₄×C₃₁	1	8	yes	—
345	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
346	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
347	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
348	#95	C₂³×C₃₁	1	8	yes	—
349	#95	C₂³×C₃₁	1	8	yes	—
350	#95	C₂³×C₃₁	1	4	yes	—
351	#95	C₂³×C₃₁	1	4	yes	—
352	#95	C₂³×C₃₁	1	8	yes	—
353	#95	C₂³×C₃₁	1	8	yes	—
354	#95	C₂³×C₃₁	1	8	yes	—
355	#95	C₂³×C₃₁	1	8	yes	—
356	C₃₁	C₂×#2599	1	1	yes	characteristic, direct product factor, Sylow
357	C₂⁴	—	2	2	no	—
358	C₂⁴	—	2	4	no	—
359	C₂⁴	—	2	2	no	—
360	C₂⁴	—	2	4	no	—
361	C₂⁴	#41×C₃₁	1	4	yes	—
362	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
363	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
364	C₂×Q₈	—	2	32	no	—
365	C₂×Q₈	—	2	8	no	—
366	C₂×Q₈	—	2	16	no	—
367	C₂²×C₄	—	2	32	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	8	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	8	no	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
393	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
394	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
395	C₂²×C₄	#41×C₃₁	1	8	yes	—
396	C₂²×C₄	#41×C₃₁	1	4	yes	—
397	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
398	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
399	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
400	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
403	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
404	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
407	C₄⋊C₄	—	2	8	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	16	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	16	no	—
412	C₄⋊C₄	—	2	16	no	—
413	C₂²⋊C₄	—	2	16	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	2	16	no	—
417	C₄²	—	2	8	no	—
418	C₄²	—	2	16	no	—
419	C₄²	C₂×D₄×C₃₁	1	16	yes	—
420	C₂³	—	2	8	no	—
421	C₂³	—	2	16	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	—	4	4	no	—
428	C₂³	—	4	8	no	—
429	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
430	C₂³	C₂×#41×C₃₁	1	4	yes	—
431	C₂³	C₂×#41×C₃₁	1	2	yes	—
432	C₂³	C₂×#41×C₃₁	1	4	yes	—
433	C₂³	C₂×#41×C₃₁	1	2	yes	—
434	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
435	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
436	Q₈	—	2	32	no	—
437	C₂×C₄	—	2	32	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	8	no	—
441	C₂×C₄	—	2	16	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	4	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	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	8	no	—
466	C₂×C₄	—	4	4	no	—
467	C₂×C₄	—	4	8	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
470	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
471	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
472	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
473	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
474	C₂²	—	2	4	no	—
475	C₂²	—	2	8	no	—
476	C₂²	—	4	16	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	4	no	—
480	C₂²	C₃₁×#525	1	4	yes	—
481	C₂²	C₃₁×#525	1	4	yes	—
482	C₂²	C₃₁×#536	1	4	yes	—
483	C₂²	C₃₁×#552	1	8	yes	—
484	C₂²	C₃₁×#556	1	8	yes	—
485	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
486	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
487	C₂²	C₂²×#41×C₃₁	1	2	yes	—
488	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
489	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
490	C₄	—	2	4	no	—
491	C₄	—	2	4	no	—
492	C₄	—	2	4	no	—
493	C₄	—	2	8	no	—
494	C₄	—	2	4	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	4	no	—
499	C₄	—	4	4	no	—
500	C₂	—	4	4	no	—
501	C₂	C₃₁×#2599	1	8	yes	direct product factor
502	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
503	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
504	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
505	C₂	C₂×C₃₁×#552	1	2	yes	—
506	C₂	C₂×C₃₁×#556	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#2599	C₂	direct (N × H)
4. C₂×C₃₁×#398	C₂	semidirect
6. C₂×C₃₁×#395	C₂	semidirect
9. C₂×C₃₁×#387	C₂	semidirect
10. C₂×C₃₁×#387	C₂	semidirect
12. C₂×C₃₁×#383	C₂	semidirect
16. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
32. C₃₁×#398	C₂²	semidirect
34. C₃₁×#395	C₂²	semidirect
37. C₃₁×#387	C₂²	semidirect
38. C₃₁×#387	C₂²	semidirect
40. C₃₁×#383	C₂²	semidirect
166. C₂×#2599	C₃₁	direct (N × H)
221. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
223. #2599	C₂×C₃₁	direct (N × H)
224. C₂×#398	C₂×C₃₁	semidirect
226. C₂×#395	C₂×C₃₁	semidirect
229. C₂×#387	C₂×C₃₁	semidirect
230. C₂×#387	C₂×C₃₁	semidirect
232. C₂×#383	C₂×C₃₁	semidirect
262. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
278. #398	C₂²×C₃₁	semidirect
280. #395	C₂²×C₃₁	semidirect
283. #387	C₂²×C₃₁	semidirect
284. #387	C₂²×C₃₁	semidirect
286. #383	C₂²×C₃₁	semidirect
296. C₂×C₃₁	#2599	direct (N × H)
356. C₃₁	C₂×#2599	direct (N × H)
501. C₂	C₃₁×#2599	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₃₁	2
C₂²×C₃₁×#117	2
C₃₁×#2599	2
C₂×C₃₁×#398	2
C₂×C₃₁×#397	2
C₂×C₃₁×#395	2
C₂×C₃₁×#393	2
C₂×C₃₁×#388	2
C₂×C₃₁×#387	2
C₂×C₃₁×#385	2
C₂×C₃₁×#383	2
C₂×#2599	31

Subgroups — C2×C31×#2599 #21491623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2599 #21491623