Subgroups — C₂×D₄×M₁₆×C₃₁ #21516897

Order 7936.

3218 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	D₄×M₁₆×C₃₁	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#436	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#435	C₂	1	4	yes	maximal, complemented
6	C₂×C₈×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#407	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₃₁	C₄	1	1	yes	characteristic
11	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₃₁	—	2	16	no	—
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	complemented
17	C₂²×M₁₆×C₃₁	C₂²	1	16	yes	complemented
18	C₂³×C₈×C₃₁	C₂²	1	4	yes	complemented
19	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
20	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₄	1	8	yes	—
22	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
27	C₂²×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#436	C₂²	1	16	yes	complemented
31	C₃₁×#435	C₂²	1	32	yes	complemented
32	C₈×D₄×C₃₁	C₂²	1	16	yes	complemented
33	C₃₁×#423	C₂²	1	8	yes	complemented
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
35	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₃₁×#407	C₂²	1	16	yes	complemented
37	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	complemented
38	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	complemented
39	C₄×M₁₆×C₃₁	C₂²	1	8	yes	complemented
40	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
41	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
42	C₂⁵×C₃₁	C₂×C₄	1	2	yes	—
43	C₂²×D₄×C₃₁	—	2	8	no	—
44	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
45	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
46	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	—	2	8	no	—
49	C₂³×C₄×C₃₁	—	2	8	no	—
50	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	—
54	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	—	2	8	no	—
61	C₂×M₁₆×C₃₁	—	2	4	no	—
62	C₂×M₁₆×C₃₁	—	2	64	no	—
63	C₂×M₁₆×C₃₁	—	2	16	no	—
64	C₂×M₁₆×C₃₁	D₄	1	8	yes	direct product factor
65	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
66	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
67	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
68	C₂×M₁₆×C₃₁	C₂³	1	16	yes	complemented
69	C₂²×C₈×C₃₁	—	2	16	no	—
70	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
71	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
72	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
73	C₂²×C₈×C₃₁	C₂³	1	16	yes	complemented
74	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
75	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
76	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	—
77	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
78	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
79	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₃₁	C₂³	1	4	yes	—
81	C₄×D₄×C₃₁	C₂³	1	8	yes	—
82	C₄×D₄×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	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	C₂×C₄	1	2	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
92	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
95	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
98	C₃₁×C₄⋊C₈	C₂³	1	8	yes	complemented
99	C₃₁×C₄⋊C₈	C₂³	1	8	yes	complemented
100	C₃₁×C₂²⋊C₈	C₂³	1	16	yes	complemented
101	C₃₁×C₂²⋊C₈	C₂³	1	16	yes	complemented
102	C₃₁×#97	C₂³	1	8	yes	complemented
103	C₄×C₈×C₃₁	C₂³	1	8	yes	complemented
104	C₂⁴×C₃₁	—	2	8	no	—
105	C₂⁴×C₃₁	—	2	8	no	—
106	C₂⁴×C₃₁	—	4	4	no	—
107	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
109	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₃₁	—	2	8	no	—
112	C₂×D₄×C₃₁	—	2	32	no	—
113	C₂×D₄×C₃₁	—	2	4	no	—
114	C₂×D₄×C₃₁	M₁₆	1	8	yes	direct product factor
115	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
116	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
117	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
118	C₂²×C₄×C₃₁	—	2	8	no	—
119	C₂²×C₄×C₃₁	—	2	8	no	—
120	C₂²×C₄×C₃₁	—	2	32	no	—
121	C₂²×C₄×C₃₁	—	2	8	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	16	no	—
125	C₂²×C₄×C₃₁	—	2	4	no	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	16	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
135	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
138	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
139	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
141	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
142	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₃₁	#41	1	4	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
151	M₁₆×C₃₁	—	2	16	no	—
152	M₁₆×C₃₁	—	2	8	no	—
153	M₁₆×C₃₁	—	2	32	no	—
154	M₁₆×C₃₁	C₂×D₄	1	16	yes	direct product factor
155	C₂×C₈×C₃₁	—	2	8	no	—
156	C₂×C₈×C₃₁	—	2	4	no	—
157	C₂×C₈×C₃₁	—	2	32	no	—
158	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	complemented
159	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
160	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
161	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
162	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
163	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
164	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
165	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
166	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
167	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
168	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
169	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
170	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
171	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
172	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
173	C₄²×C₃₁	C₂⁴	1	2	yes	—
174	C₄²×C₃₁	C₂⁴	1	2	yes	—
175	C₂×D₄×M₁₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	4	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	4	no	—
180	C₂³×C₃₁	—	2	16	no	—
181	C₂³×C₃₁	—	2	4	no	—
182	C₂³×C₃₁	—	2	4	no	—
183	C₂³×C₃₁	—	2	16	no	—
184	C₂³×C₃₁	—	2	4	no	—
185	C₂³×C₃₁	—	4	8	no	—
186	C₂³×C₃₁	—	4	32	no	—
187	C₂³×C₃₁	C₄×D₄	1	4	yes	—
188	C₂³×C₃₁	C₂×M₁₆	1	8	yes	complemented
189	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
190	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
191	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
192	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
193	D₄×C₃₁	—	2	16	no	—
194	D₄×C₃₁	—	2	8	no	—
195	D₄×C₃₁	C₂×M₁₆	1	16	yes	direct product factor
196	C₂×C₄×C₃₁	—	2	16	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	8	no	—
203	C₂×C₄×C₃₁	—	2	16	no	—
204	C₂×C₄×C₃₁	—	2	16	no	—
205	C₂×C₄×C₃₁	—	2	8	no	—
206	C₂×C₄×C₃₁	—	2	8	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	C₄×D₄	1	8	yes	—
213	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
214	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	complemented
215	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
216	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
217	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
218	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
219	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
220	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
225	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
226	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
227	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
228	C₈×C₃₁	—	2	8	no	—
229	C₈×C₃₁	—	2	4	no	—
230	C₈×C₃₁	C₂²×D₄	1	8	yes	complemented
231	C₂³×M₁₆	C₂×C₃₁	1	2	yes	complemented
232	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
233	D₄×M₁₆	C₂×C₃₁	1	16	yes	direct product factor
234	C₂×#436	C₂×C₃₁	1	2	yes	complemented
235	C₂×#435	C₂×C₃₁	1	4	yes	complemented
236	C₂×C₈×D₄	C₂×C₃₁	1	2	yes	complemented
237	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂×#407	C₂×C₃₁	1	2	yes	complemented
239	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂²×C₃₁	—	2	4	no	—
241	C₂²×C₃₁	—	2	8	no	—
242	C₂²×C₃₁	—	2	16	no	—
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	2	4	no	—
246	C₂²×C₃₁	—	2	4	no	—
247	C₂²×C₃₁	—	2	4	no	—
248	C₂²×C₃₁	—	2	4	no	—
249	C₂²×C₃₁	—	4	32	no	—
250	C₂²×C₃₁	—	4	8	no	—
251	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
252	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
253	C₂²×C₃₁	C₂²×M₁₆	1	2	yes	—
254	C₂²×C₃₁	C₂²×M₁₆	1	8	yes	complemented
255	C₂²×C₃₁	#568	1	2	yes	—
256	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
257	C₄×C₃₁	—	2	4	no	—
258	C₄×C₃₁	—	2	4	no	—
259	C₄×C₃₁	—	2	2	no	—
260	C₄×C₃₁	—	2	2	no	—
261	C₄×C₃₁	—	2	2	no	—
262	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
263	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
264	C₄×C₃₁	C₂²×M₁₆	1	4	yes	complemented
265	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
266	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
267	C₂³×D₄	C₄×C₃₁	1	1	yes	characteristic
268	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
269	C₂²×M₁₆	—	2	16	no	—
270	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
271	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
272	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
273	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	complemented
274	C₂²×M₁₆	C₂²×C₃₁	1	16	yes	complemented
275	C₂³×C₈	C₂²×C₃₁	1	4	yes	complemented
276	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
277	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄×D₄	C₄×C₃₁	1	8	yes	—
279	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
282	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
284	C₂²×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
285	C₂²×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
286	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
287	#436	C₂²×C₃₁	1	16	yes	complemented
288	#435	C₂²×C₃₁	1	32	yes	complemented
289	C₈×D₄	C₂²×C₃₁	1	16	yes	complemented
290	#423	C₂²×C₃₁	1	8	yes	complemented
291	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
292	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
293	#407	C₂²×C₃₁	1	16	yes	complemented
294	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	complemented
295	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	complemented
296	C₄×M₁₆	C₂²×C₃₁	1	8	yes	complemented
297	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
298	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
299	C₂×C₃₁	—	2	4	no	—
300	C₂×C₃₁	—	2	8	no	—
301	C₂×C₃₁	—	4	4	no	—
302	C₂×C₃₁	D₄×M₁₆	1	4	yes	direct product factor
303	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
304	C₂×C₃₁	C₂³×M₁₆	1	1	yes	characteristic
305	C₂×C₃₁	C₂×#568	1	1	yes	characteristic
306	C₂⁵	C₂×C₄×C₃₁	1	2	yes	—
307	C₂²×D₄	—	2	8	no	—
308	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
309	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
310	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
311	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
312	C₂³×C₄	—	2	8	no	—
313	C₂³×C₄	—	2	8	no	—
314	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
315	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
316	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
317	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	—
318	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
319	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
321	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
322	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
323	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂×M₁₆	—	2	8	no	—
325	C₂×M₁₆	—	2	4	no	—
326	C₂×M₁₆	—	2	64	no	—
327	C₂×M₁₆	—	2	16	no	—
328	C₂×M₁₆	D₄×C₃₁	1	8	yes	direct product factor
329	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
330	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
331	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
332	C₂×M₁₆	C₂³×C₃₁	1	16	yes	complemented
333	C₂²×C₈	—	2	16	no	—
334	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
335	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
336	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
337	C₂²×C₈	C₂³×C₃₁	1	16	yes	complemented
338	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
339	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
340	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	—
341	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
342	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
343	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
344	C₄×D₄	C₂³×C₃₁	1	4	yes	—
345	C₄×D₄	C₂³×C₃₁	1	8	yes	—
346	C₄×D₄	C₂³×C₃₁	1	4	yes	—
347	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
348	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
349	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
351	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
352	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
353	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
354	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
355	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	—
356	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
357	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
358	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
359	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
360	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
362	C₄⋊C₈	C₂³×C₃₁	1	8	yes	complemented
363	C₄⋊C₈	C₂³×C₃₁	1	8	yes	complemented
364	C₂²⋊C₈	C₂³×C₃₁	1	16	yes	complemented
365	C₂²⋊C₈	C₂³×C₃₁	1	16	yes	complemented
366	#97	C₂³×C₃₁	1	8	yes	complemented
367	C₄×C₈	C₂³×C₃₁	1	8	yes	complemented
368	C₃₁	C₂×D₄×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
369	C₂⁴	—	2	8	no	—
370	C₂⁴	—	2	8	no	—
371	C₂⁴	—	4	4	no	—
372	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
373	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
374	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
375	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	2	32	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	M₁₆×C₃₁	1	8	yes	direct product factor
380	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
381	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
382	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	2	8	no	—
385	C₂²×C₄	—	2	32	no	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	16	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	16	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
399	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
400	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
402	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
403	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
404	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
405	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
406	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
407	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
408	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
409	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
410	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
411	C₂²×C₄	#41×C₃₁	1	4	yes	—
412	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
413	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
414	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
416	M₁₆	—	2	16	no	—
417	M₁₆	—	2	8	no	—
418	M₁₆	—	2	32	no	—
419	M₁₆	C₂×D₄×C₃₁	1	16	yes	direct product factor
420	C₂×C₈	—	2	8	no	—
421	C₂×C₈	—	2	4	no	—
422	C₂×C₈	—	2	32	no	—
423	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	complemented
424	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
425	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
426	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
427	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
428	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
429	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
430	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
431	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
432	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
433	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
434	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
435	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
436	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
437	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
438	C₄²	C₂⁴×C₃₁	1	2	yes	—
439	C₄²	C₂⁴×C₃₁	1	2	yes	—
440	C₂³	—	2	4	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	16	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	16	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	4	8	no	—
450	C₂³	—	4	32	no	—
451	C₂³	C₄×D₄×C₃₁	1	4	yes	—
452	C₂³	C₂×M₁₆×C₃₁	1	8	yes	complemented
453	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
454	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
455	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
456	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
457	D₄	—	2	16	no	—
458	D₄	—	2	8	no	—
459	D₄	C₂×M₁₆×C₃₁	1	16	yes	direct product factor
460	C₂×C₄	—	2	16	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	16	no	—
468	C₂×C₄	—	2	16	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
477	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
478	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	complemented
479	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
480	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
481	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
482	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
483	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
484	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
485	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
487	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
488	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
489	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
490	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
491	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
492	C₈	—	2	8	no	—
493	C₈	—	2	4	no	—
494	C₈	C₂²×D₄×C₃₁	1	8	yes	complemented
495	C₂²	—	2	4	no	—
496	C₂²	—	2	8	no	—
497	C₂²	—	2	16	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	2	4	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	4	32	no	—
505	C₂²	—	4	8	no	—
506	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
507	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
508	C₂²	C₂²×M₁₆×C₃₁	1	2	yes	—
509	C₂²	C₂²×M₁₆×C₃₁	1	8	yes	complemented
510	C₂²	C₃₁×#568	1	2	yes	—
511	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
512	C₄	—	2	4	no	—
513	C₄	—	2	4	no	—
514	C₄	—	2	2	no	—
515	C₄	—	2	2	no	—
516	C₄	—	2	2	no	—
517	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
518	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
519	C₄	C₂²×M₁₆×C₃₁	1	4	yes	complemented
520	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
521	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
522	C₂	—	2	4	no	—
523	C₂	—	2	8	no	—
524	C₂	—	4	4	no	—
525	C₂	D₄×M₁₆×C₃₁	1	4	yes	direct product factor
526	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
527	C₂	C₂³×M₁₆×C₃₁	1	1	yes	characteristic
528	C₂	C₂×C₃₁×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂³×D₄×C₃₁	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. C₂³×M₁₆×C₃₁	C₂	semidirect
3. D₄×M₁₆×C₃₁	C₂	direct (N × H)
4. C₂×C₃₁×#436	C₂	semidirect
5. C₂×C₃₁×#435	C₂	semidirect
6. C₂×C₈×D₄×C₃₁	C₂	semidirect
7. C₂×C₃₁×#423	C₂	semidirect
8. C₂×C₃₁×#407	C₂	semidirect
9. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
16. C₂²×M₁₆×C₃₁	C₂²	semidirect
17. C₂²×M₁₆×C₃₁	C₂²	semidirect
18. C₂³×C₈×C₃₁	C₂²	semidirect
30. C₃₁×#436	C₂²	semidirect
31. C₃₁×#435	C₂²	semidirect
32. C₈×D₄×C₃₁	C₂²	semidirect
33. C₃₁×#423	C₂²	semidirect
34. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
35. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
36. C₃₁×#407	C₂²	semidirect
37. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
38. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
39. C₄×M₁₆×C₃₁	C₂²	semidirect
40. C₂×C₃₁×#97	C₂²	semidirect
41. C₂×C₄×C₈×C₃₁	C₂²	semidirect
64. C₂×M₁₆×C₃₁	D₄	direct (N × H)
68. C₂×M₁₆×C₃₁	C₂³	semidirect
73. C₂²×C₈×C₃₁	C₂³	semidirect
98. C₃₁×C₄⋊C₈	C₂³	semidirect
99. C₃₁×C₄⋊C₈	C₂³	semidirect
100. C₃₁×C₂²⋊C₈	C₂³	semidirect
101. C₃₁×C₂²⋊C₈	C₂³	semidirect
102. C₃₁×#97	C₂³	semidirect
103. C₄×C₈×C₃₁	C₂³	semidirect
114. C₂×D₄×C₃₁	M₁₆	direct (N × H)
154. M₁₆×C₃₁	C₂×D₄	direct (N × H)
158. C₂×C₈×C₃₁	C₂×D₄	semidirect
175. C₂×D₄×M₁₆	C₃₁	direct (N × H)
188. C₂³×C₃₁	C₂×M₁₆	semidirect
195. D₄×C₃₁	C₂×M₁₆	direct (N × H)
214. C₂×C₄×C₃₁	C₂×M₁₆	semidirect
230. C₈×C₃₁	C₂²×D₄	semidirect
231. C₂³×M₁₆	C₂×C₃₁	semidirect
233. D₄×M₁₆	C₂×C₃₁	direct (N × H)
234. C₂×#436	C₂×C₃₁	semidirect
235. C₂×#435	C₂×C₃₁	semidirect
236. C₂×C₈×D₄	C₂×C₃₁	semidirect
237. C₂×#423	C₂×C₃₁	semidirect
238. C₂×#407	C₂×C₃₁	semidirect
239. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
254. C₂²×C₃₁	C₂²×M₁₆	semidirect
264. C₄×C₃₁	C₂²×M₁₆	semidirect
273. C₂²×M₁₆	C₂²×C₃₁	semidirect
274. C₂²×M₁₆	C₂²×C₃₁	semidirect
275. C₂³×C₈	C₂²×C₃₁	semidirect
287. #436	C₂²×C₃₁	semidirect
288. #435	C₂²×C₃₁	semidirect
289. C₈×D₄	C₂²×C₃₁	semidirect
290. #423	C₂²×C₃₁	semidirect
291. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
292. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
293. #407	C₂²×C₃₁	semidirect
294. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
295. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
296. C₄×M₁₆	C₂²×C₃₁	semidirect
297. C₂×#97	C₂²×C₃₁	semidirect
298. C₂×C₄×C₈	C₂²×C₃₁	semidirect
302. C₂×C₃₁	D₄×M₁₆	direct (N × H)
328. C₂×M₁₆	D₄×C₃₁	direct (N × H)
332. C₂×M₁₆	C₂³×C₃₁	semidirect
337. C₂²×C₈	C₂³×C₃₁	semidirect
362. C₄⋊C₈	C₂³×C₃₁	semidirect
363. C₄⋊C₈	C₂³×C₃₁	semidirect
364. C₂²⋊C₈	C₂³×C₃₁	semidirect
365. C₂²⋊C₈	C₂³×C₃₁	semidirect
366. #97	C₂³×C₃₁	semidirect
367. C₄×C₈	C₂³×C₃₁	semidirect
368. C₃₁	C₂×D₄×M₁₆	direct (N × H)
379. C₂×D₄	M₁₆×C₃₁	direct (N × H)
419. M₁₆	C₂×D₄×C₃₁	direct (N × H)
423. C₂×C₈	C₂×D₄×C₃₁	semidirect
452. C₂³	C₂×M₁₆×C₃₁	semidirect
459. D₄	C₂×M₁₆×C₃₁	direct (N × H)
478. C₂×C₄	C₂×M₁₆×C₃₁	semidirect
494. C₈	C₂²×D₄×C₃₁	semidirect
509. C₂²	C₂²×M₁₆×C₃₁	semidirect
519. C₄	C₂²×M₁₆×C₃₁	semidirect
525. C₂	D₄×M₁₆×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₃₁	2
C₂²×C₄×D₄×C₃₁	2
D₄×M₁₆×C₃₁	2
C₂×C₃₁×#436	2
C₂×C₃₁×#435	2
C₂×C₈×D₄×C₃₁	2
C₂×C₃₁×#423	2
C₂×C₃₁×#407	2
C₂×C₄×M₁₆×C₃₁	2
C₂×D₄×M₁₆	31

Subgroups — C2×D4×M16×C31 #21516897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×M₁₆×C₃₁ #21516897