Subgroups — C₂×C₃₁×#1866 #21477642

Order 7936.

1398 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₁×#104	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1866	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₃₁×#339	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#131	—	2	4	no	—
11	C₂×C₃₁×#131	—	2	4	no	—
12	C₂×C₃₁×#131	—	2	2	no	—
13	C₂×C₃₁×#131	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₃₁	—	2	1	no	—
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₃₁	—	2	1	no	—
18	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#108	—	2	4	no	—
23	C₂×C₃₁×#108	C₂²	1	4	yes	complemented
24	C₃₁×#424	—	2	2	no	—
25	C₃₁×#424	C₂²	1	4	yes	complemented
26	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
27	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
28	C₂×C₃₁×#104	—	2	4	no	—
29	C₂×C₃₁×#104	C₂²	1	4	yes	—
30	C₃₁×#416	C₄	1	4	yes	complemented
31	C₃₁×#416	C₂²	1	4	yes	—
32	C₂×C₃₁×#103	C₄	1	1	yes	characteristic, complemented
33	C₂×C₃₁×#103	C₄	1	1	yes	characteristic, complemented
34	C₂×C₃₁×#102	C₄	1	1	yes	characteristic, complemented
35	C₂×C₃₁×#102	C₄	1	1	yes	characteristic, complemented
36	C₃₁×#339	C₂²	1	4	yes	complemented
37	C₃₁×#336	C₂²	1	4	yes	complemented
38	C₂×#41×C₃₁	—	2	2	no	—
39	C₂×#41×C₃₁	—	2	1	no	—
40	C₂×#41×C₃₁	—	2	1	no	—
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×#41×C₃₁	D₄	1	4	yes	—
44	C₂×#41×C₃₁	Q₈	1	4	yes	—
45	C₂×#41×C₃₁	C₂³	1	2	yes	—
46	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#131	—	2	16	no	—
53	C₃₁×#131	—	2	8	no	—
54	C₃₁×#131	—	2	8	no	—
55	C₂×M₁₆×C₃₁	—	2	8	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂×M₁₆×C₃₁	—	2	4	no	—
60	C₂×M₁₆×C₃₁	—	2	2	no	—
61	C₂×M₁₆×C₃₁	—	2	2	no	—
62	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
63	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
65	C₂²×C₈×C₃₁	—	2	8	no	—
66	C₂²×C₈×C₃₁	—	2	2	no	—
67	C₂²×C₈×C₃₁	—	2	2	no	—
68	C₂²×C₈×C₃₁	—	2	2	no	—
69	C₂²×C₈×C₃₁	—	2	1	no	—
70	C₂²×C₈×C₃₁	—	2	2	no	—
71	C₂²×C₈×C₃₁	—	2	4	no	—
72	C₂²×C₈×C₃₁	—	2	1	no	—
73	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
76	C₂²×C₈×C₃₁	D₄	1	2	yes	—
77	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
80	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
81	C₃₁×#117	—	2	1	no	—
82	C₃₁×#117	—	2	1	no	—
83	C₃₁×#117	C₂×C₄	1	2	yes	—
84	C₃₁×#117	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₄²×C₃₁	—	2	1	no	—
89	C₂×C₄²×C₃₁	—	2	2	no	—
90	C₂×C₄²×C₃₁	—	2	2	no	—
91	C₂×C₄²×C₃₁	—	2	1	no	—
92	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
93	C₃₁×#108	—	2	8	no	—
94	C₃₁×C₄⋊C₈	—	2	4	no	—
95	C₃₁×C₄⋊C₈	—	2	4	no	—
96	C₃₁×#104	—	2	8	no	—
97	C₃₁×#103	C₂×C₄	1	4	yes	complemented
98	C₃₁×#103	C₂×C₄	1	4	yes	complemented
99	C₃₁×#102	C₂×C₄	1	4	yes	complemented
100	C₃₁×#102	C₂×C₄	1	4	yes	complemented
101	C₂⁴×C₃₁	—	2	1	no	—
102	C₂⁴×C₃₁	#41	1	1	yes	characteristic
103	#41×C₃₁	—	2	4	no	—
104	#41×C₃₁	—	2	4	no	—
105	#41×C₃₁	—	2	4	no	—
106	#41×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	2	4	no	—
108	#41×C₃₁	—	2	2	no	—
109	#41×C₃₁	—	2	2	no	—
110	#41×C₃₁	C₄⋊C₄	1	8	yes	—
111	#41×C₃₁	C₂×D₄	1	4	yes	—
112	#41×C₃₁	C₂×Q₈	1	4	yes	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
116	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	—	2	4	no	—
119	C₂×D₄×C₃₁	—	2	4	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
124	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₃₁	—	2	1	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	2	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	4	2	no	—
135	C₂²×C₄×C₃₁	—	4	2	no	—
136	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
143	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	#41	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
146	M₁₆×C₃₁	—	2	16	no	—
147	M₁₆×C₃₁	—	2	4	no	—
148	M₁₆×C₃₁	—	2	2	no	—
149	M₁₆×C₃₁	—	2	2	no	—
150	M₁₆×C₃₁	—	2	2	no	—
151	M₁₆×C₃₁	—	4	2	no	—
152	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
153	C₂×C₈×C₃₁	—	2	16	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	2	4	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	4	no	—
159	C₂×C₈×C₃₁	—	2	2	no	—
160	C₂×C₈×C₃₁	—	2	4	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	2	2	no	—
163	C₂×C₈×C₃₁	—	2	2	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	—	4	2	no	—
168	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
169	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
170	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
175	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
176	C₂²⋊C₄×C₃₁	—	2	2	no	—
177	C₂²⋊C₄×C₃₁	—	2	2	no	—
178	C₄²×C₃₁	—	2	2	no	—
179	C₄²×C₃₁	—	2	2	no	—
180	C₄²×C₃₁	—	2	2	no	—
181	C₄²×C₃₁	—	2	2	no	—
182	C₄²×C₃₁	—	4	2	no	—
183	C₂×#1866	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₃₁	—	2	2	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	—	2	4	no	—
188	C₂³×C₃₁	—	4	2	no	—
189	C₂³×C₃₁	—	4	2	no	—
190	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
191	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
192	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
193	C₂³×C₃₁	#123	1	2	yes	—
194	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
195	Q₈×C₃₁	—	2	2	no	—
196	Q₈×C₃₁	—	2	4	no	—
197	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
198	D₄×C₃₁	—	2	4	no	—
199	D₄×C₃₁	—	2	4	no	—
200	D₄×C₃₁	—	2	8	no	—
201	D₄×C₃₁	—	2	2	no	—
202	D₄×C₃₁	—	2	4	no	—
203	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
204	C₂×C₄×C₃₁	—	2	2	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	8	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	4	8	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
218	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
219	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
220	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	#120	1	2	yes	—
224	C₂×C₄×C₃₁	#120	1	2	yes	—
225	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
226	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
227	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
228	C₂×C₄×C₃₁	#123	1	2	yes	—
229	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
235	C₈×C₃₁	—	2	2	no	—
236	C₈×C₃₁	—	2	2	no	—
237	C₈×C₃₁	—	2	2	no	—
238	C₈×C₃₁	—	2	4	no	—
239	C₈×C₃₁	—	2	2	no	—
240	C₈×C₃₁	—	4	2	no	—
241	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic
242	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂×#424	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂²×#104	C₂×C₃₁	1	1	yes	characteristic
245	C₂×#416	C₂×C₃₁	1	1	yes	characteristic
246	#1866	C₂×C₃₁	1	8	yes	direct product factor
247	C₂×#339	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂²×C₃₁	—	2	4	no	—
250	C₂²×C₃₁	—	2	4	no	—
251	C₂²×C₃₁	—	2	1	no	—
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	2	1	no	—
254	C₂²×C₃₁	—	2	1	no	—
255	C₂²×C₃₁	—	2	4	no	—
256	C₂²×C₃₁	—	4	8	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	#385	1	2	yes	—
260	C₂²×C₃₁	#385	1	2	yes	—
261	C₂²×C₃₁	#443	1	2	yes	—
262	C₂²×C₃₁	#444	1	2	yes	—
263	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
264	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
265	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
266	C₄×C₃₁	—	2	4	no	—
267	C₄×C₃₁	—	4	4	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	#385	1	2	yes	—
271	C₄×C₃₁	#385	1	2	yes	—
272	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
273	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
274	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
275	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
276	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×#131	—	2	4	no	—
278	C₂×#131	—	2	4	no	—
279	C₂×#131	—	2	2	no	—
280	C₂×#131	C₂²×C₃₁	1	4	yes	—
281	C₂²×M₁₆	—	2	1	no	—
282	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
283	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
284	C₂³×C₈	—	2	1	no	—
285	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
286	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
288	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×#108	—	2	4	no	—
290	C₂×#108	C₂²×C₃₁	1	4	yes	complemented
291	#424	—	2	2	no	—
292	#424	C₂²×C₃₁	1	4	yes	complemented
293	C₂×C₄⋊C₈	—	2	1	no	—
294	C₂×C₄⋊C₈	—	2	1	no	—
295	C₂×#104	—	2	4	no	—
296	C₂×#104	C₂²×C₃₁	1	4	yes	—
297	#416	C₄×C₃₁	1	4	yes	complemented
298	#416	C₂²×C₃₁	1	4	yes	—
299	C₂×#103	C₄×C₃₁	1	1	yes	characteristic, complemented
300	C₂×#103	C₄×C₃₁	1	1	yes	characteristic, complemented
301	C₂×#102	C₄×C₃₁	1	1	yes	characteristic, complemented
302	C₂×#102	C₄×C₃₁	1	1	yes	characteristic, complemented
303	#339	C₂²×C₃₁	1	4	yes	complemented
304	#336	C₂²×C₃₁	1	4	yes	complemented
305	C₂×C₃₁	—	2	1	no	—
306	C₂×C₃₁	—	2	1	no	—
307	C₂×C₃₁	—	2	2	no	—
308	C₂×C₃₁	—	4	4	no	—
309	C₂×C₃₁	#1866	1	4	yes	direct product factor
310	C₂×C₃₁	C₂×#385	1	1	yes	characteristic
311	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
312	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
313	C₂×#41	—	2	2	no	—
314	C₂×#41	—	2	1	no	—
315	C₂×#41	—	2	1	no	—
316	C₂×#41	—	2	2	no	—
317	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
318	C₂×#41	D₄×C₃₁	1	4	yes	—
319	C₂×#41	Q₈×C₃₁	1	4	yes	—
320	C₂×#41	C₂³×C₃₁	1	2	yes	—
321	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
322	C₂²×D₄	—	2	1	no	—
323	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
324	C₂³×C₄	—	2	1	no	—
325	C₂³×C₄	—	2	1	no	—
326	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
327	#131	—	2	16	no	—
328	#131	—	2	8	no	—
329	#131	—	2	8	no	—
330	C₂×M₁₆	—	2	8	no	—
331	C₂×M₁₆	—	2	2	no	—
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	—	2	1	no	—
334	C₂×M₁₆	—	2	4	no	—
335	C₂×M₁₆	—	2	2	no	—
336	C₂×M₁₆	—	2	2	no	—
337	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
338	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
339	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
340	C₂²×C₈	—	2	8	no	—
341	C₂²×C₈	—	2	1	no	—
342	C₂²×C₈	—	2	2	no	—
343	C₂²×C₈	—	2	2	no	—
344	C₂²×C₈	—	2	2	no	—
345	C₂²×C₈	—	2	2	no	—
346	C₂²×C₈	—	2	4	no	—
347	C₂²×C₈	—	2	1	no	—
348	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
351	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₈	D₄×C₃₁	1	2	yes	—
354	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
355	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
356	#117	—	2	1	no	—
357	#117	—	2	1	no	—
358	#117	C₂×C₄×C₃₁	1	2	yes	—
359	#117	C₂³×C₃₁	1	2	yes	—
360	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
361	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
362	C₂×C₂²⋊C₄	—	2	1	no	—
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	—	2	2	no	—
365	C₂×C₄²	—	2	2	no	—
366	C₂×C₄²	—	2	1	no	—
367	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
368	#108	—	2	8	no	—
369	C₄⋊C₈	—	2	4	no	—
370	C₄⋊C₈	—	2	4	no	—
371	#104	—	2	8	no	—
372	#103	C₂×C₄×C₃₁	1	4	yes	complemented
373	#103	C₂×C₄×C₃₁	1	4	yes	complemented
374	#102	C₂×C₄×C₃₁	1	4	yes	complemented
375	#102	C₂×C₄×C₃₁	1	4	yes	complemented
376	C₃₁	C₂×#1866	1	1	yes	characteristic, direct product factor, Sylow
377	C₂⁴	—	2	1	no	—
378	C₂⁴	#41×C₃₁	1	1	yes	characteristic
379	#41	—	2	4	no	—
380	#41	—	2	4	no	—
381	#41	—	2	4	no	—
382	#41	—	2	4	no	—
383	#41	—	2	4	no	—
384	#41	—	2	2	no	—
385	#41	—	2	2	no	—
386	#41	C₄⋊C₄×C₃₁	1	8	yes	—
387	#41	C₂×D₄×C₃₁	1	4	yes	—
388	#41	C₂×Q₈×C₃₁	1	4	yes	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
392	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	2	4	no	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	1	no	—
399	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
400	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	4	2	no	—
411	C₂²×C₄	—	4	2	no	—
412	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
413	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
414	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
416	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
418	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
419	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	#41×C₃₁	1	2	yes	—
421	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
422	M₁₆	—	2	16	no	—
423	M₁₆	—	2	4	no	—
424	M₁₆	—	2	2	no	—
425	M₁₆	—	2	2	no	—
426	M₁₆	—	2	2	no	—
427	M₁₆	—	4	2	no	—
428	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
429	C₂×C₈	—	2	16	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	4	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	4	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	4	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	—	2	2	no	—
443	C₂×C₈	—	4	2	no	—
444	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
445	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
446	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
447	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
451	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
452	C₂²⋊C₄	—	2	2	no	—
453	C₂²⋊C₄	—	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	—	4	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	4	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	2	no	—
465	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
466	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
467	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
468	C₂³	C₃₁×#123	1	2	yes	—
469	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
470	Q₈	—	2	2	no	—
471	Q₈	—	2	4	no	—
472	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
473	D₄	—	2	4	no	—
474	D₄	—	2	4	no	—
475	D₄	—	2	8	no	—
476	D₄	—	2	2	no	—
477	D₄	—	2	4	no	—
478	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	8	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	4	8	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
493	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
494	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
495	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
496	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
497	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
498	C₂×C₄	C₃₁×#120	1	2	yes	—
499	C₂×C₄	C₃₁×#120	1	2	yes	—
500	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
501	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
502	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
503	C₂×C₄	C₃₁×#123	1	2	yes	—
504	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
509	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
510	C₈	—	2	2	no	—
511	C₈	—	2	2	no	—
512	C₈	—	2	2	no	—
513	C₈	—	2	4	no	—
514	C₈	—	2	2	no	—
515	C₈	—	4	2	no	—
516	C₂²	—	2	4	no	—
517	C₂²	—	2	4	no	—
518	C₂²	—	2	1	no	—
519	C₂²	—	2	1	no	—
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	4	no	—
523	C₂²	—	4	8	no	—
524	C₂²	—	4	2	no	—
525	C₂²	—	4	2	no	—
526	C₂²	C₃₁×#385	1	2	yes	—
527	C₂²	C₃₁×#385	1	2	yes	—
528	C₂²	C₃₁×#443	1	2	yes	—
529	C₂²	C₃₁×#444	1	2	yes	—
530	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
531	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
532	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
533	C₄	—	2	4	no	—
534	C₄	—	4	4	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	2	no	—
537	C₄	C₃₁×#385	1	2	yes	—
538	C₄	C₃₁×#385	1	2	yes	—
539	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
540	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
541	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
542	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
543	C₂	—	2	1	no	—
544	C₂	—	2	1	no	—
545	C₂	—	2	2	no	—
546	C₂	—	4	4	no	—
547	C₂	C₃₁×#1866	1	4	yes	direct product factor
548	C₂	C₂×C₃₁×#385	1	1	yes	characteristic
549	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
550	C₂	C₂×C₃₁×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₄×D₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#108	C₂	semidirect
3. C₂×C₃₁×#424	C₂	semidirect
6. C₃₁×#1866	C₂	direct (N × H)
7. C₂×C₃₁×#339	C₂	semidirect
8. C₂×C₃₁×#336	C₂	semidirect
23. C₂×C₃₁×#108	C₂²	semidirect
25. C₃₁×#424	C₂²	semidirect
30. C₃₁×#416	C₄	semidirect
32. C₂×C₃₁×#103	C₄	semidirect
33. C₂×C₃₁×#103	C₄	semidirect
34. C₂×C₃₁×#102	C₄	semidirect
35. C₂×C₃₁×#102	C₄	semidirect
36. C₃₁×#339	C₂²	semidirect
37. C₃₁×#336	C₂²	semidirect
97. C₃₁×#103	C₂×C₄	semidirect
98. C₃₁×#103	C₂×C₄	semidirect
99. C₃₁×#102	C₂×C₄	semidirect
100. C₃₁×#102	C₂×C₄	semidirect
183. C₂×#1866	C₃₁	direct (N × H)
242. C₂²×#108	C₂×C₃₁	semidirect
243. C₂×#424	C₂×C₃₁	semidirect
246. #1866	C₂×C₃₁	direct (N × H)
247. C₂×#339	C₂×C₃₁	semidirect
248. C₂×#336	C₂×C₃₁	semidirect
290. C₂×#108	C₂²×C₃₁	semidirect
292. #424	C₂²×C₃₁	semidirect
297. #416	C₄×C₃₁	semidirect
299. C₂×#103	C₄×C₃₁	semidirect
300. C₂×#103	C₄×C₃₁	semidirect
301. C₂×#102	C₄×C₃₁	semidirect
302. C₂×#102	C₄×C₃₁	semidirect
303. #339	C₂²×C₃₁	semidirect
304. #336	C₂²×C₃₁	semidirect
309. C₂×C₃₁	#1866	direct (N × H)
372. #103	C₂×C₄×C₃₁	semidirect
373. #103	C₂×C₄×C₃₁	semidirect
374. #102	C₂×C₄×C₃₁	semidirect
375. #102	C₂×C₄×C₃₁	semidirect
376. C₃₁	C₂×#1866	direct (N × H)
547. C₂	C₃₁×#1866	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×C₃₁×#131	2
C₂²×C₃₁×#108	2
C₂×C₃₁×#424	2
C₂²×C₃₁×#104	2
C₂×C₃₁×#416	2
C₃₁×#1866	2
C₂×C₃₁×#339	2
C₂×C₃₁×#336	2
C₂×#1866	31

Subgroups — C2×C31×#1866 #21477642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1866 #21477642