Subgroups — C₂×C₃₁×#3067 #21517225

Order 7936.

2650 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#575	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3067	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃₁×#473	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#472	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#471	C₂	1	4	yes	maximal, complemented
8	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#413	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×#41×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#575	—	2	8	no	—
13	C₃₁×#575	C₂²	1	16	yes	complemented
14	C₂×C₃₁×#137	—	2	2	no	—
15	C₂×C₃₁×#137	C₂²	1	4	yes	—
16	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
18	C₂×C₃₁×#135	—	2	4	no	—
19	C₂×C₃₁×#135	—	2	1	no	—
20	C₂×C₃₁×#135	—	2	1	no	—
21	C₂×C₃₁×#135	—	2	1	no	—
22	C₂×C₃₁×#135	—	2	1	no	—
23	C₂×C₃₁×#135	—	2	4	no	—
24	C₂×C₃₁×#135	C₂²	1	8	yes	complemented
25	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
26	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
27	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₃₁×#131	—	2	1	no	—
29	C₂×C₃₁×#131	—	2	1	no	—
30	C₂×C₃₁×#131	—	2	4	no	—
31	C₂×C₃₁×#131	C₂²	1	2	yes	—
32	C₂×C₃₁×#131	C₂²	1	2	yes	—
33	C₂×C₃₁×#131	C₂²	1	8	yes	complemented
34	C₂²×M₁₆×C₃₁	—	2	1	no	—
35	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
37	C₂³×C₈×C₃₁	—	2	1	no	—
38	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#473	C₂²	1	16	yes	complemented
40	C₃₁×#472	C₂²	1	16	yes	complemented
41	C₃₁×#471	C₂²	1	32	yes	complemented
42	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
43	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
44	C₂×C₃₁×#108	C₂²	1	8	yes	complemented
45	C₃₁×#413	C₂²	1	16	yes	complemented
46	C₂×C₃₁×#101	C₂²	1	4	yes	complemented
47	C₂×C₃₁×#100	C₂²	1	4	yes	complemented
48	C₂×#41×C₃₁	—	2	1	no	—
49	C₂×#41×C₃₁	—	2	2	no	—
50	C₂×#41×C₃₁	—	2	2	no	—
51	C₂×#41×C₃₁	—	2	1	no	—
52	C₂×#41×C₃₁	—	2	4	no	—
53	C₂×#41×C₃₁	—	2	2	no	—
54	C₂×#41×C₃₁	—	2	4	no	—
55	C₂×#41×C₃₁	—	2	2	no	—
56	C₂×#41×C₃₁	—	2	2	no	—
57	C₂×#41×C₃₁	—	2	2	no	—
58	C₂×#41×C₃₁	—	2	4	no	—
59	C₂×#41×C₃₁	D₄	1	2	yes	—
60	C₂×#41×C₃₁	D₄	1	4	yes	—
61	C₂×#41×C₃₁	C₂³	1	2	yes	—
62	C₂×#41×C₃₁	C₂³	1	4	yes	—
63	C₂×#41×C₃₁	C₂³	1	8	yes	—
64	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
65	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₃₁	—	2	1	no	—
67	C₂²×D₄×C₃₁	—	2	2	no	—
68	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	—	2	1	no	—
71	C₂³×C₄×C₃₁	—	2	2	no	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#137	—	2	8	no	—
74	C₃₁×#137	—	2	16	no	—
75	C₃₁×#137	C₂³	1	16	yes	—
76	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
77	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
78	C₃₁×AGL(1,ℤ/8)	C₂³	1	16	yes	—
79	C₃₁×#135	—	2	32	no	—
80	C₃₁×#135	—	2	8	no	—
81	C₃₁×#135	—	2	8	no	—
82	C₃₁×#135	—	2	8	no	—
83	C₃₁×#135	—	2	8	no	—
84	C₂×Q₁₆×C₃₁	—	2	4	no	—
85	C₂×Q₁₆×C₃₁	—	2	1	no	—
86	C₂×Q₁₆×C₃₁	—	2	1	no	—
87	C₂×Q₁₆×C₃₁	—	2	2	no	—
88	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	complemented
89	C₂×SD₁₆×C₃₁	—	2	4	no	—
90	C₂×SD₁₆×C₃₁	—	2	4	no	—
91	C₂×SD₁₆×C₃₁	—	2	2	no	—
92	C₂×SD₁₆×C₃₁	—	2	2	no	—
93	C₂×SD₁₆×C₃₁	—	2	4	no	—
94	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	complemented
95	C₂×D₈×C₃₁	—	2	4	no	—
96	C₂×D₈×C₃₁	—	2	1	no	—
97	C₂×D₈×C₃₁	—	2	1	no	—
98	C₂×D₈×C₃₁	—	2	2	no	—
99	C₂×D₈×C₃₁	C₂³	1	4	yes	complemented
100	C₃₁×#131	—	2	4	no	—
101	C₃₁×#131	—	2	4	no	—
102	C₃₁×#131	—	2	8	no	—
103	C₃₁×#131	—	2	8	no	—
104	C₃₁×#131	C₂³	1	8	yes	—
105	C₃₁×#131	C₂³	1	8	yes	—
106	C₂×M₁₆×C₃₁	—	2	1	no	—
107	C₂×M₁₆×C₃₁	—	2	2	no	—
108	C₂×M₁₆×C₃₁	—	2	2	no	—
109	C₂×M₁₆×C₃₁	—	2	4	no	—
110	C₂×M₁₆×C₃₁	—	2	2	no	—
111	C₂×M₁₆×C₃₁	—	2	2	no	—
112	C₂×M₁₆×C₃₁	—	2	8	no	—
113	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
114	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
115	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
116	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
117	C₂²×C₈×C₃₁	—	2	1	no	—
118	C₂²×C₈×C₃₁	—	2	2	no	—
119	C₂²×C₈×C₃₁	—	2	2	no	—
120	C₂²×C₈×C₃₁	—	2	2	no	—
121	C₂²×C₈×C₃₁	—	2	8	no	—
122	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
123	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
124	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
125	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
126	C₂²×C₈×C₃₁	D₄	1	4	yes	—
127	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
128	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
129	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
130	C₃₁×#108	C₂³	1	8	yes	complemented
131	C₃₁×#108	C₂³	1	8	yes	complemented
132	C₃₁×#101	C₂³	1	16	yes	complemented
133	C₃₁×#100	C₂³	1	16	yes	complemented
134	C₂⁴×C₃₁	—	2	1	no	—
135	C₂⁴×C₃₁	—	2	2	no	—
136	C₂⁴×C₃₁	#41	1	1	yes	characteristic
137	#41×C₃₁	—	2	4	no	—
138	#41×C₃₁	—	2	4	no	—
139	#41×C₃₁	—	2	4	no	—
140	#41×C₃₁	—	2	2	no	—
141	#41×C₃₁	—	2	4	no	—
142	#41×C₃₁	—	2	4	no	—
143	#41×C₃₁	—	2	2	no	—
144	#41×C₃₁	—	2	8	no	—
145	#41×C₃₁	—	2	8	no	—
146	#41×C₃₁	—	2	8	no	—
147	#41×C₃₁	—	2	8	no	—
148	#41×C₃₁	—	4	4	no	—
149	#41×C₃₁	—	4	4	no	—
150	#41×C₃₁	—	4	4	no	—
151	#41×C₃₁	—	4	4	no	—
152	#41×C₃₁	C₂×D₄	1	8	yes	—
153	#41×C₃₁	C₂×D₄	1	4	yes	—
154	#41×C₃₁	C₂⁴	1	4	yes	—
155	C₂×Q₈×C₃₁	—	2	1	no	—
156	C₂×Q₈×C₃₁	—	2	2	no	—
157	C₂×Q₈×C₃₁	—	2	2	no	—
158	C₂×Q₈×C₃₁	—	2	2	no	—
159	C₂×Q₈×C₃₁	—	2	2	no	—
160	C₂×Q₈×C₃₁	—	2	2	no	—
161	C₂×Q₈×C₃₁	—	2	2	no	—
162	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
163	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
164	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
165	C₂×D₄×C₃₁	—	2	2	no	—
166	C₂×D₄×C₃₁	—	2	4	no	—
167	C₂×D₄×C₃₁	—	2	2	no	—
168	C₂×D₄×C₃₁	—	2	4	no	—
169	C₂×D₄×C₃₁	—	2	8	no	—
170	C₂×D₄×C₃₁	—	2	1	no	—
171	C₂×D₄×C₃₁	—	2	2	no	—
172	C₂×D₄×C₃₁	—	2	2	no	—
173	C₂×D₄×C₃₁	—	2	2	no	—
174	C₂×D₄×C₃₁	—	2	2	no	—
175	C₂×D₄×C₃₁	—	2	2	no	—
176	C₂×D₄×C₃₁	—	2	2	no	—
177	C₂×D₄×C₃₁	—	4	2	no	—
178	C₂×D₄×C₃₁	—	4	2	no	—
179	C₂×D₄×C₃₁	—	4	4	no	—
180	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₃₁	—	2	1	no	—
184	C₂²×C₄×C₃₁	—	2	4	no	—
185	C₂²×C₄×C₃₁	—	2	2	no	—
186	C₂²×C₄×C₃₁	—	2	2	no	—
187	C₂²×C₄×C₃₁	—	2	4	no	—
188	C₂²×C₄×C₃₁	—	2	2	no	—
189	C₂²×C₄×C₃₁	—	2	8	no	—
190	C₂²×C₄×C₃₁	—	4	2	no	—
191	C₂²×C₄×C₃₁	—	4	4	no	—
192	C₂²×C₄×C₃₁	—	4	2	no	—
193	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₃₁	#41	1	2	yes	—
200	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
201	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
202	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
203	Q₁₆×C₃₁	—	2	16	no	—
204	Q₁₆×C₃₁	—	2	4	no	—
205	Q₁₆×C₃₁	—	2	4	no	—
206	SD₁₆×C₃₁	—	2	16	no	—
207	SD₁₆×C₃₁	—	2	16	no	—
208	SD₁₆×C₃₁	—	2	8	no	—
209	SD₁₆×C₃₁	—	2	8	no	—
210	D₈×C₃₁	—	2	16	no	—
211	D₈×C₃₁	—	2	4	no	—
212	D₈×C₃₁	—	2	4	no	—
213	M₁₆×C₃₁	—	2	2	no	—
214	M₁₆×C₃₁	—	2	4	no	—
215	M₁₆×C₃₁	—	2	4	no	—
216	M₁₆×C₃₁	—	2	8	no	—
217	M₁₆×C₃₁	—	2	8	no	—
218	M₁₆×C₃₁	—	2	8	no	—
219	M₁₆×C₃₁	C₂⁴	1	4	yes	—
220	M₁₆×C₃₁	C₂⁴	1	8	yes	—
221	C₂×C₈×C₃₁	—	2	2	no	—
222	C₂×C₈×C₃₁	—	2	4	no	—
223	C₂×C₈×C₃₁	—	2	4	no	—
224	C₂×C₈×C₃₁	—	2	8	no	—
225	C₂×C₈×C₃₁	—	2	8	no	—
226	C₂×C₈×C₃₁	—	2	8	no	—
227	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
228	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
229	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
230	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
231	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
232	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
233	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
234	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
235	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
236	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
237	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
238	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
239	C₂×#3067	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂³×C₃₁	—	2	4	no	—
241	C₂³×C₃₁	—	2	2	no	—
242	C₂³×C₃₁	—	2	4	no	—
243	C₂³×C₃₁	—	4	2	no	—
244	C₂³×C₃₁	—	4	4	no	—
245	C₂³×C₃₁	—	4	2	no	—
246	C₂³×C₃₁	—	4	2	no	—
247	C₂³×C₃₁	—	8	4	no	—
248	C₂³×C₃₁	#121	1	1	yes	characteristic
249	C₂³×C₃₁	#121	1	1	yes	characteristic
250	C₂³×C₃₁	#121	1	2	yes	—
251	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
252	C₂³×C₃₁	C₂×#41	1	2	yes	—
253	Q₈×C₃₁	—	2	4	no	—
254	Q₈×C₃₁	—	2	2	no	—
255	Q₈×C₃₁	—	2	4	no	—
256	Q₈×C₃₁	—	2	4	no	—
257	Q₈×C₃₁	—	4	2	no	—
258	Q₈×C₃₁	—	4	2	no	—
259	Q₈×C₃₁	C₂²×D₄	1	4	yes	—
260	D₄×C₃₁	—	2	4	no	—
261	D₄×C₃₁	—	2	8	no	—
262	D₄×C₃₁	—	2	4	no	—
263	D₄×C₃₁	—	2	4	no	—
264	D₄×C₃₁	—	2	2	no	—
265	D₄×C₃₁	—	2	4	no	—
266	D₄×C₃₁	—	2	4	no	—
267	D₄×C₃₁	—	4	8	no	—
268	D₄×C₃₁	—	4	8	no	—
269	D₄×C₃₁	—	4	2	no	—
270	D₄×C₃₁	—	4	2	no	—
271	D₄×C₃₁	C₂²×D₄	1	4	yes	—
272	C₂×C₄×C₃₁	—	2	4	no	—
273	C₂×C₄×C₃₁	—	2	2	no	—
274	C₂×C₄×C₃₁	—	2	8	no	—
275	C₂×C₄×C₃₁	—	2	4	no	—
276	C₂×C₄×C₃₁	—	2	4	no	—
277	C₂×C₄×C₃₁	—	2	4	no	—
278	C₂×C₄×C₃₁	—	4	2	no	—
279	C₂×C₄×C₃₁	—	4	2	no	—
280	C₂×C₄×C₃₁	—	4	8	no	—
281	C₂×C₄×C₃₁	—	4	8	no	—
282	C₂×C₄×C₃₁	#121	1	2	yes	—
283	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
284	C₂×C₄×C₃₁	#121	1	2	yes	—
285	C₂×C₄×C₃₁	#121	1	2	yes	—
286	C₂×C₄×C₃₁	#121	1	2	yes	—
287	C₂×C₄×C₃₁	#121	1	2	yes	—
288	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
289	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
291	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
293	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
297	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
298	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
299	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
300	C₈×C₃₁	—	2	4	no	—
301	C₈×C₃₁	—	2	8	no	—
302	C₈×C₃₁	C₂²×D₄	1	2	yes	—
303	C₈×C₃₁	C₂²×D₄	1	2	yes	—
304	C₈×C₃₁	C₂²×D₄	1	2	yes	—
305	C₈×C₃₁	C₂²×D₄	1	2	yes	—
306	C₂×#575	C₂×C₃₁	1	2	yes	complemented
307	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
308	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
309	#3067	C₂×C₃₁	1	16	yes	direct product factor
310	C₂×#473	C₂×C₃₁	1	2	yes	complemented
311	C₂×#472	C₂×C₃₁	1	2	yes	complemented
312	C₂×#471	C₂×C₃₁	1	4	yes	complemented
313	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
314	C₂×#413	C₂×C₃₁	1	2	yes	complemented
315	C₂²×C₃₁	—	2	2	no	—
316	C₂²×C₃₁	—	2	2	no	—
317	C₂²×C₃₁	—	2	2	no	—
318	C₂²×C₃₁	—	2	4	no	—
319	C₂²×C₃₁	—	4	2	no	—
320	C₂²×C₃₁	—	4	2	no	—
321	C₂²×C₃₁	—	4	8	no	—
322	C₂²×C₃₁	—	4	4	no	—
323	C₂²×C₃₁	—	4	4	no	—
324	C₂²×C₃₁	—	8	4	no	—
325	C₂²×C₃₁	—	8	4	no	—
326	C₂²×C₃₁	—	8	4	no	—
327	C₂²×C₃₁	C₂×#121	1	2	yes	—
328	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
329	C₂²×C₃₁	C₂×#121	1	2	yes	—
330	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
331	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
332	C₄×C₃₁	—	2	4	no	—
333	C₄×C₃₁	—	4	4	no	—
334	C₄×C₃₁	C₂×#121	1	2	yes	—
335	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
336	C₄×C₃₁	C₂×#121	1	2	yes	—
337	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
338	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
339	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
340	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
341	C₂²×#41	C₂²×C₃₁	1	2	yes	—
342	#575	—	2	8	no	—
343	#575	C₂²×C₃₁	1	16	yes	complemented
344	C₂×#137	—	2	2	no	—
345	C₂×#137	C₂²×C₃₁	1	4	yes	—
346	C₂×AGL(1,ℤ/8)	—	2	2	no	—
347	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
348	C₂×#135	—	2	4	no	—
349	C₂×#135	—	2	1	no	—
350	C₂×#135	—	2	1	no	—
351	C₂×#135	—	2	1	no	—
352	C₂×#135	—	2	1	no	—
353	C₂×#135	—	2	4	no	—
354	C₂×#135	C₂²×C₃₁	1	8	yes	complemented
355	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
356	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
357	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
358	C₂×#131	—	2	1	no	—
359	C₂×#131	—	2	1	no	—
360	C₂×#131	—	2	4	no	—
361	C₂×#131	C₂²×C₃₁	1	2	yes	—
362	C₂×#131	C₂²×C₃₁	1	2	yes	—
363	C₂×#131	C₂²×C₃₁	1	8	yes	complemented
364	C₂²×M₁₆	—	2	1	no	—
365	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
366	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
367	C₂³×C₈	—	2	1	no	—
368	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
369	#473	C₂²×C₃₁	1	16	yes	complemented
370	#472	C₂²×C₃₁	1	16	yes	complemented
371	#471	C₂²×C₃₁	1	32	yes	complemented
372	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
373	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
374	C₂×#108	C₂²×C₃₁	1	8	yes	complemented
375	#413	C₂²×C₃₁	1	16	yes	complemented
376	C₂×#101	C₂²×C₃₁	1	4	yes	complemented
377	C₂×#100	C₂²×C₃₁	1	4	yes	complemented
378	C₂×C₃₁	—	2	2	no	—
379	C₂×C₃₁	—	2	1	no	—
380	C₂×C₃₁	—	2	1	no	—
381	C₂×C₃₁	—	2	2	no	—
382	C₂×C₃₁	—	4	4	no	—
383	C₂×C₃₁	—	8	4	no	—
384	C₂×C₃₁	#3067	1	2	yes	direct product factor
385	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
386	C₂×#41	—	2	1	no	—
387	C₂×#41	—	2	2	no	—
388	C₂×#41	—	2	1	no	—
389	C₂×#41	—	2	2	no	—
390	C₂×#41	—	2	4	no	—
391	C₂×#41	—	2	2	no	—
392	C₂×#41	—	2	2	no	—
393	C₂×#41	—	2	2	no	—
394	C₂×#41	—	2	4	no	—
395	C₂×#41	—	2	2	no	—
396	C₂×#41	—	2	4	no	—
397	C₂×#41	D₄×C₃₁	1	2	yes	—
398	C₂×#41	D₄×C₃₁	1	4	yes	—
399	C₂×#41	C₂³×C₃₁	1	2	yes	—
400	C₂×#41	C₂³×C₃₁	1	4	yes	—
401	C₂×#41	C₂³×C₃₁	1	8	yes	—
402	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
403	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
404	C₂²×D₄	—	2	1	no	—
405	C₂²×D₄	—	2	2	no	—
406	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
407	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
408	C₂³×C₄	—	2	1	no	—
409	C₂³×C₄	—	2	2	no	—
410	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
411	#137	—	2	8	no	—
412	#137	—	2	16	no	—
413	#137	C₂³×C₃₁	1	16	yes	—
414	AGL(1,ℤ/8)	—	2	8	no	—
415	AGL(1,ℤ/8)	—	2	16	no	—
416	AGL(1,ℤ/8)	C₂³×C₃₁	1	16	yes	—
417	#135	—	2	32	no	—
418	#135	—	2	8	no	—
419	#135	—	2	8	no	—
420	#135	—	2	8	no	—
421	#135	—	2	8	no	—
422	C₂×Q₁₆	—	2	4	no	—
423	C₂×Q₁₆	—	2	1	no	—
424	C₂×Q₁₆	—	2	1	no	—
425	C₂×Q₁₆	—	2	2	no	—
426	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	complemented
427	C₂×SD₁₆	—	2	4	no	—
428	C₂×SD₁₆	—	2	4	no	—
429	C₂×SD₁₆	—	2	2	no	—
430	C₂×SD₁₆	—	2	2	no	—
431	C₂×SD₁₆	—	2	4	no	—
432	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	complemented
433	C₂×D₈	—	2	4	no	—
434	C₂×D₈	—	2	1	no	—
435	C₂×D₈	—	2	1	no	—
436	C₂×D₈	—	2	2	no	—
437	C₂×D₈	C₂³×C₃₁	1	4	yes	complemented
438	#131	—	2	4	no	—
439	#131	—	2	4	no	—
440	#131	—	2	8	no	—
441	#131	—	2	8	no	—
442	#131	C₂³×C₃₁	1	8	yes	—
443	#131	C₂³×C₃₁	1	8	yes	—
444	C₂×M₁₆	—	2	1	no	—
445	C₂×M₁₆	—	2	2	no	—
446	C₂×M₁₆	—	2	2	no	—
447	C₂×M₁₆	—	2	4	no	—
448	C₂×M₁₆	—	2	2	no	—
449	C₂×M₁₆	—	2	2	no	—
450	C₂×M₁₆	—	2	8	no	—
451	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
452	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
453	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
454	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
455	C₂²×C₈	—	2	1	no	—
456	C₂²×C₈	—	2	2	no	—
457	C₂²×C₈	—	2	2	no	—
458	C₂²×C₈	—	2	2	no	—
459	C₂²×C₈	—	2	8	no	—
460	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
462	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
464	C₂²×C₈	D₄×C₃₁	1	4	yes	—
465	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
466	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
467	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
468	#108	C₂³×C₃₁	1	8	yes	complemented
469	#108	C₂³×C₃₁	1	8	yes	complemented
470	#101	C₂³×C₃₁	1	16	yes	complemented
471	#100	C₂³×C₃₁	1	16	yes	complemented
472	C₃₁	C₂×#3067	1	1	yes	characteristic, direct product factor, Sylow
473	C₂⁴	—	2	1	no	—
474	C₂⁴	—	2	2	no	—
475	C₂⁴	#41×C₃₁	1	1	yes	characteristic
476	#41	—	2	4	no	—
477	#41	—	2	4	no	—
478	#41	—	2	4	no	—
479	#41	—	2	2	no	—
480	#41	—	2	4	no	—
481	#41	—	2	4	no	—
482	#41	—	2	2	no	—
483	#41	—	2	8	no	—
484	#41	—	2	8	no	—
485	#41	—	2	8	no	—
486	#41	—	2	8	no	—
487	#41	—	4	4	no	—
488	#41	—	4	4	no	—
489	#41	—	4	4	no	—
490	#41	—	4	4	no	—
491	#41	C₂×D₄×C₃₁	1	8	yes	—
492	#41	C₂×D₄×C₃₁	1	4	yes	—
493	#41	C₂⁴×C₃₁	1	4	yes	—
494	C₂×Q₈	—	2	1	no	—
495	C₂×Q₈	—	2	2	no	—
496	C₂×Q₈	—	2	2	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	2	no	—
500	C₂×Q₈	—	2	2	no	—
501	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
502	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
503	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
504	C₂×D₄	—	2	2	no	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	2	2	no	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	2	8	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	—	2	2	no	—
511	C₂×D₄	—	2	2	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	2	no	—
514	C₂×D₄	—	2	2	no	—
515	C₂×D₄	—	2	2	no	—
516	C₂×D₄	—	4	2	no	—
517	C₂×D₄	—	4	2	no	—
518	C₂×D₄	—	4	4	no	—
519	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
520	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
521	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	4	no	—
524	C₂²×C₄	—	2	2	no	—
525	C₂²×C₄	—	2	2	no	—
526	C₂²×C₄	—	2	4	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	8	no	—
529	C₂²×C₄	—	4	2	no	—
530	C₂²×C₄	—	4	4	no	—
531	C₂²×C₄	—	4	2	no	—
532	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
533	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
534	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
535	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
536	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
537	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
538	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
539	C₂²×C₄	#41×C₃₁	1	2	yes	—
540	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
541	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
542	Q₁₆	—	2	16	no	—
543	Q₁₆	—	2	4	no	—
544	Q₁₆	—	2	4	no	—
545	SD₁₆	—	2	16	no	—
546	SD₁₆	—	2	16	no	—
547	SD₁₆	—	2	8	no	—
548	SD₁₆	—	2	8	no	—
549	D₈	—	2	16	no	—
550	D₈	—	2	4	no	—
551	D₈	—	2	4	no	—
552	M₁₆	—	2	2	no	—
553	M₁₆	—	2	4	no	—
554	M₁₆	—	2	4	no	—
555	M₁₆	—	2	8	no	—
556	M₁₆	—	2	8	no	—
557	M₁₆	—	2	8	no	—
558	M₁₆	C₂⁴×C₃₁	1	4	yes	—
559	M₁₆	C₂⁴×C₃₁	1	8	yes	—
560	C₂×C₈	—	2	2	no	—
561	C₂×C₈	—	2	4	no	—
562	C₂×C₈	—	2	4	no	—
563	C₂×C₈	—	2	8	no	—
564	C₂×C₈	—	2	8	no	—
565	C₂×C₈	—	2	8	no	—
566	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
567	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
568	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
569	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
570	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
571	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
572	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
573	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
574	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
575	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
576	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
577	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
578	C₂³	—	2	4	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	2	4	no	—
581	C₂³	—	4	2	no	—
582	C₂³	—	4	4	no	—
583	C₂³	—	4	2	no	—
584	C₂³	—	4	2	no	—
585	C₂³	—	8	4	no	—
586	C₂³	C₃₁×#121	1	1	yes	characteristic
587	C₂³	C₃₁×#121	1	1	yes	characteristic
588	C₂³	C₃₁×#121	1	2	yes	—
589	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
590	C₂³	C₂×#41×C₃₁	1	2	yes	—
591	Q₈	—	2	4	no	—
592	Q₈	—	2	2	no	—
593	Q₈	—	2	4	no	—
594	Q₈	—	2	4	no	—
595	Q₈	—	4	2	no	—
596	Q₈	—	4	2	no	—
597	Q₈	C₂²×D₄×C₃₁	1	4	yes	—
598	D₄	—	2	4	no	—
599	D₄	—	2	8	no	—
600	D₄	—	2	4	no	—
601	D₄	—	2	4	no	—
602	D₄	—	2	2	no	—
603	D₄	—	2	4	no	—
604	D₄	—	2	4	no	—
605	D₄	—	4	8	no	—
606	D₄	—	4	8	no	—
607	D₄	—	4	2	no	—
608	D₄	—	4	2	no	—
609	D₄	C₂²×D₄×C₃₁	1	4	yes	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	8	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	4	8	no	—
619	C₂×C₄	—	4	8	no	—
620	C₂×C₄	C₃₁×#121	1	2	yes	—
621	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
622	C₂×C₄	C₃₁×#121	1	2	yes	—
623	C₂×C₄	C₃₁×#121	1	2	yes	—
624	C₂×C₄	C₃₁×#121	1	2	yes	—
625	C₂×C₄	C₃₁×#121	1	2	yes	—
626	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
627	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
628	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
629	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
630	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
631	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
632	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
633	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
634	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
635	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
636	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
637	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
638	C₈	—	2	4	no	—
639	C₈	—	2	8	no	—
640	C₈	C₂²×D₄×C₃₁	1	2	yes	—
641	C₈	C₂²×D₄×C₃₁	1	2	yes	—
642	C₈	C₂²×D₄×C₃₁	1	2	yes	—
643	C₈	C₂²×D₄×C₃₁	1	2	yes	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	2	no	—
646	C₂²	—	2	2	no	—
647	C₂²	—	2	4	no	—
648	C₂²	—	4	2	no	—
649	C₂²	—	4	2	no	—
650	C₂²	—	4	8	no	—
651	C₂²	—	4	4	no	—
652	C₂²	—	4	4	no	—
653	C₂²	—	8	4	no	—
654	C₂²	—	8	4	no	—
655	C₂²	—	8	4	no	—
656	C₂²	C₂×C₃₁×#121	1	2	yes	—
657	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
658	C₂²	C₂×C₃₁×#121	1	2	yes	—
659	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
660	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
661	C₄	—	2	4	no	—
662	C₄	—	4	4	no	—
663	C₄	C₂×C₃₁×#121	1	2	yes	—
664	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
665	C₄	C₂×C₃₁×#121	1	2	yes	—
666	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
667	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
668	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
669	C₂	—	2	2	no	—
670	C₂	—	2	1	no	—
671	C₂	—	2	1	no	—
672	C₂	—	2	2	no	—
673	C₂	—	4	4	no	—
674	C₂	—	8	4	no	—
675	C₂	C₃₁×#3067	1	2	yes	direct product factor
676	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂×#121
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#575	C₂	semidirect
2. C₂²×C₃₁×#135	C₂	semidirect
3. C₂²×C₃₁×#131	C₂	semidirect
4. C₃₁×#3067	C₂	direct (N × H)
5. C₂×C₃₁×#473	C₂	semidirect
6. C₂×C₃₁×#472	C₂	semidirect
7. C₂×C₃₁×#471	C₂	semidirect
8. C₂²×C₃₁×#108	C₂	semidirect
9. C₂×C₃₁×#413	C₂	semidirect
13. C₃₁×#575	C₂²	semidirect
24. C₂×C₃₁×#135	C₂²	semidirect
25. C₂²×Q₁₆×C₃₁	C₂²	semidirect
26. C₂²×SD₁₆×C₃₁	C₂²	semidirect
27. C₂²×D₈×C₃₁	C₂²	semidirect
33. C₂×C₃₁×#131	C₂²	semidirect
39. C₃₁×#473	C₂²	semidirect
40. C₃₁×#472	C₂²	semidirect
41. C₃₁×#471	C₂²	semidirect
42. C₂×C₃₁×#108	C₂²	semidirect
43. C₂×C₃₁×#108	C₂²	semidirect
44. C₂×C₃₁×#108	C₂²	semidirect
45. C₃₁×#413	C₂²	semidirect
46. C₂×C₃₁×#101	C₂²	semidirect
47. C₂×C₃₁×#100	C₂²	semidirect
88. C₂×Q₁₆×C₃₁	C₂³	semidirect
94. C₂×SD₁₆×C₃₁	C₂³	semidirect
99. C₂×D₈×C₃₁	C₂³	semidirect
130. C₃₁×#108	C₂³	semidirect
131. C₃₁×#108	C₂³	semidirect
132. C₃₁×#101	C₂³	semidirect
133. C₃₁×#100	C₂³	semidirect
239. C₂×#3067	C₃₁	direct (N × H)
306. C₂×#575	C₂×C₃₁	semidirect
307. C₂²×#135	C₂×C₃₁	semidirect
308. C₂²×#131	C₂×C₃₁	semidirect
309. #3067	C₂×C₃₁	direct (N × H)
310. C₂×#473	C₂×C₃₁	semidirect
311. C₂×#472	C₂×C₃₁	semidirect
312. C₂×#471	C₂×C₃₁	semidirect
313. C₂²×#108	C₂×C₃₁	semidirect
314. C₂×#413	C₂×C₃₁	semidirect
343. #575	C₂²×C₃₁	semidirect
354. C₂×#135	C₂²×C₃₁	semidirect
355. C₂²×Q₁₆	C₂²×C₃₁	semidirect
356. C₂²×SD₁₆	C₂²×C₃₁	semidirect
357. C₂²×D₈	C₂²×C₃₁	semidirect
363. C₂×#131	C₂²×C₃₁	semidirect
369. #473	C₂²×C₃₁	semidirect
370. #472	C₂²×C₃₁	semidirect
371. #471	C₂²×C₃₁	semidirect
372. C₂×#108	C₂²×C₃₁	semidirect
373. C₂×#108	C₂²×C₃₁	semidirect
374. C₂×#108	C₂²×C₃₁	semidirect
375. #413	C₂²×C₃₁	semidirect
376. C₂×#101	C₂²×C₃₁	semidirect
377. C₂×#100	C₂²×C₃₁	semidirect
384. C₂×C₃₁	#3067	direct (N × H)
426. C₂×Q₁₆	C₂³×C₃₁	semidirect
432. C₂×SD₁₆	C₂³×C₃₁	semidirect
437. C₂×D₈	C₂³×C₃₁	semidirect
468. #108	C₂³×C₃₁	semidirect
469. #108	C₂³×C₃₁	semidirect
470. #101	C₂³×C₃₁	semidirect
471. #100	C₂³×C₃₁	semidirect
472. C₃₁	C₂×#3067	direct (N × H)
675. C₂	C₃₁×#3067	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#575	2
C₂²×C₃₁×#135	2
C₂²×C₃₁×#131	2
C₃₁×#3067	2
C₂×C₃₁×#473	2
C₂×C₃₁×#472	2
C₂×C₃₁×#471	2
C₂²×C₃₁×#108	2
C₂×C₃₁×#413	2
C₂×#3067	31

Subgroups — C2×C31×#3067 #21517225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3067 #21517225