Subgroups — C₄×C₃₁×#575 #21516917

Order 7936.

2842 conjugacy classes of subgroups.

The subgroup lattice has 482 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	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₁×#137	C₂	1	8	yes	maximal, complemented
4	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2943	C₂	1	4	yes	maximal, complemented
6	C₄×C₃₁×#135	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2896	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#575	C₄	1	16	yes	direct product factor
12	C₂×C₃₁×#137	C₂²	1	4	yes	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
14	C₂×C₃₁×#135	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₄×#41×C₃₁	—	2	4	no	—
18	C₄×#41×C₃₁	—	2	2	no	—
19	C₄×#41×C₃₁	—	2	2	no	—
20	C₄×#41×C₃₁	C₂²	1	8	yes	—
21	C₄×#41×C₃₁	C₂²	1	4	yes	complemented
22	C₄×#41×C₃₁	C₂²	1	4	yes	complemented
23	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
24	C₂×C₄×D₄×C₃₁	—	2	4	no	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
26	C₂×C₃₁×#117	—	2	2	no	—
27	C₂×C₃₁×#117	C₂²	1	2	yes	—
28	C₂²×C₄²×C₃₁	—	2	2	no	—
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#442	C₂²	1	8	yes	complemented
31	C₃₁×#441	C₂²	1	8	yes	complemented
32	C₃₁×#440	C₂²	1	16	yes	complemented
33	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
34	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
35	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
36	C₃₁×#428	C₂²	1	8	yes	complemented
37	C₃₁×#427	C₂²	1	4	yes	complemented
38	C₃₁×#419	C₂²	1	4	yes	complemented
39	C₃₁×#418	C₂²	1	4	yes	complemented
40	C₃₁×#417	C₂²	1	8	yes	—
41	C₃₁×#416	C₂²	1	8	yes	complemented
42	C₄×M₁₆×C₃₁	C₂²	1	8	yes	complemented
43	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
44	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂×#41×C₃₁	—	2	2	no	—
47	C₂×#41×C₃₁	—	2	8	no	—
48	C₂×#41×C₃₁	C₂×C₄	1	16	yes	complemented
49	C₂×#41×C₃₁	C₂³	1	4	yes	—
50	C₂×#41×C₃₁	C₂³	1	4	yes	—
51	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×D₄×C₃₁	—	2	2	no	—
53	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
54	C₂³×C₄×C₃₁	—	2	2	no	—
55	C₂³×C₄×C₃₁	—	2	4	no	—
56	C₂³×C₄×C₃₁	D₄	1	2	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#137	C₂×C₄	1	32	yes	complemented
59	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	32	yes	complemented
60	C₃₁×#135	C₂×C₄	1	32	yes	complemented
61	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
62	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
63	C₂×D₈×C₃₁	C₂³	1	4	yes	—
64	C₂×M₁₆×C₃₁	—	2	4	no	—
65	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
66	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
67	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
68	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
69	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
70	C₄×Q₈×C₃₁	—	2	4	no	—
71	C₄×Q₈×C₃₁	—	2	4	no	—
72	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
73	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
74	C₄×D₄×C₃₁	—	2	8	no	—
75	C₄×D₄×C₃₁	—	2	8	no	—
76	C₄×D₄×C₃₁	—	2	8	no	—
77	C₄×D₄×C₃₁	—	2	8	no	—
78	C₄×D₄×C₃₁	—	2	4	no	—
79	C₄×D₄×C₃₁	—	2	4	no	—
80	C₄×D₄×C₃₁	C₂³	1	8	yes	—
81	C₄×D₄×C₃₁	C₂³	1	8	yes	—
82	C₃₁×#117	—	2	2	no	—
83	C₃₁×#117	—	2	2	no	—
84	C₃₁×#117	—	2	8	no	—
85	C₃₁×#117	—	2	4	no	—
86	C₃₁×#117	—	2	4	no	—
87	C₃₁×#117	C₂³	1	4	yes	—
88	C₃₁×#117	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
93	C₂×C₄²×C₃₁	—	2	2	no	—
94	C₂×C₄²×C₃₁	—	2	4	no	—
95	C₂×C₄²×C₃₁	—	2	8	no	—
96	C₂×C₄²×C₃₁	—	2	4	no	—
97	C₂×C₄²×C₃₁	—	2	4	no	—
98	C₂×C₄²×C₃₁	D₄	1	2	yes	—
99	C₂×C₄²×C₃₁	D₄	1	2	yes	—
100	C₂×C₄²×C₃₁	D₄	1	2	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₃₁×#107	C₂³	1	8	yes	—
105	C₃₁×#106	C₂³	1	8	yes	—
106	C₃₁×#103	C₂³	1	16	yes	—
107	C₃₁×#102	C₂³	1	16	yes	—
108	C₃₁×#97	C₂³	1	8	yes	—
109	C₄×C₈×C₃₁	C₂³	1	8	yes	—
110	C₂⁴×C₃₁	—	2	2	no	—
111	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
112	#41×C₃₁	—	2	16	no	—
113	#41×C₃₁	—	2	16	no	—
114	#41×C₃₁	—	2	8	no	—
115	#41×C₃₁	—	2	8	no	—
116	#41×C₃₁	C₂²×C₄	1	16	yes	—
117	#41×C₃₁	C₂²×C₄	1	16	yes	complemented
118	C₂×Q₈×C₃₁	—	2	2	no	—
119	C₂×Q₈×C₃₁	—	2	4	no	—
120	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	complemented
121	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
122	C₂×D₄×C₃₁	—	2	16	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	2	4	no	—
127	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	complemented
128	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	16	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	8	no	—
135	C₂²×C₄×C₃₁	—	2	8	no	—
136	C₂²×C₄×C₃₁	—	4	4	no	—
137	C₂²×C₄×C₃₁	—	4	16	no	—
138	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	#41	1	4	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	Q₁₆×C₃₁	C₂²×C₄	1	16	yes	complemented
149	SD₁₆×C₃₁	C₂²×C₄	1	32	yes	complemented
150	D₈×C₃₁	C₂²×C₄	1	16	yes	complemented
151	M₁₆×C₃₁	—	2	8	no	—
152	M₁₆×C₃₁	C₂²×C₄	1	16	yes	complemented
153	C₂×C₈×C₃₁	—	2	4	no	—
154	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	complemented
155	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
156	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
157	C₄⋊C₄×C₃₁	—	2	8	no	—
158	C₄⋊C₄×C₃₁	—	2	8	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
161	C₂²⋊C₄×C₃₁	—	2	4	no	—
162	C₂²⋊C₄×C₃₁	—	2	4	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	8	no	—
165	C₄²×C₃₁	—	2	4	no	—
166	C₄²×C₃₁	—	2	4	no	—
167	C₄²×C₃₁	—	2	8	no	—
168	C₄²×C₃₁	—	2	8	no	—
169	C₄²×C₃₁	C₂×D₄	1	2	yes	—
170	C₄²×C₃₁	C₂×D₄	1	4	yes	—
171	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₃₁	C₂×D₄	1	4	yes	—
173	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₃₁	C₂⁴	1	2	yes	—
175	C₄²×C₃₁	C₂⁴	1	2	yes	—
176	C₄×#575	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₃₁	—	2	8	no	—
178	C₂³×C₃₁	—	2	4	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	8	no	—
181	C₂³×C₃₁	C₄×D₄	1	4	yes	—
182	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
184	Q₈×C₃₁	—	2	8	no	—
185	Q₈×C₃₁	—	2	4	no	—
186	Q₈×C₃₁	C₂³×C₄	1	8	yes	—
187	D₄×C₃₁	—	2	16	no	—
188	D₄×C₃₁	—	2	16	no	—
189	D₄×C₃₁	—	2	8	no	—
190	D₄×C₃₁	—	2	4	no	—
191	D₄×C₃₁	C₂³×C₄	1	8	yes	—
192	C₂×C₄×C₃₁	—	2	8	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	16	no	—
195	C₂×C₄×C₃₁	—	2	16	no	—
196	C₂×C₄×C₃₁	—	2	8	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	8	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	4	no	—
201	C₂×C₄×C₃₁	—	4	16	no	—
202	C₂×C₄×C₃₁	—	4	8	no	—
203	C₂×C₄×C₃₁	—	4	16	no	—
204	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
205	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
206	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
207	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
208	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
209	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
214	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
215	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
216	C₈×C₃₁	—	2	4	no	—
217	C₈×C₃₁	C₂³×C₄	1	8	yes	—
218	C₂×#575	C₂×C₃₁	1	1	yes	characteristic
219	C₂×C₄×#41	C₂×C₃₁	1	2	yes	complemented
220	C₄×#137	C₂×C₃₁	1	8	yes	complemented
221	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	8	yes	complemented
222	#2943	C₂×C₃₁	1	4	yes	complemented
223	C₄×#135	C₂×C₃₁	1	4	yes	complemented
224	#2911	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#2896	C₂×C₃₁	1	2	yes	complemented
226	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂²×C₃₁	—	2	4	no	—
228	C₂²×C₃₁	—	2	2	no	—
229	C₂²×C₃₁	—	2	8	no	—
230	C₂²×C₃₁	—	4	4	no	—
231	C₂²×C₃₁	—	4	16	no	—
232	C₂²×C₃₁	—	4	8	no	—
233	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
234	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
235	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
236	C₄×C₃₁	—	2	8	no	—
237	C₄×C₃₁	—	2	4	no	—
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	4	8	no	—
240	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
241	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
242	C₄×C₃₁	#575	1	4	yes	direct product factor
243	C₄×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
244	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
245	C₂²×#41	C₂²×C₃₁	1	2	yes	—
246	#575	C₄×C₃₁	1	16	yes	direct product factor
247	C₂×#137	C₂²×C₃₁	1	4	yes	—
248	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
249	C₂×#135	C₂²×C₃₁	1	4	yes	—
250	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
251	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
252	C₄×#41	—	2	4	no	—
253	C₄×#41	—	2	2	no	—
254	C₄×#41	—	2	2	no	—
255	C₄×#41	C₂²×C₃₁	1	8	yes	—
256	C₄×#41	C₂²×C₃₁	1	4	yes	complemented
257	C₄×#41	C₂²×C₃₁	1	4	yes	complemented
258	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
259	C₂×C₄×D₄	—	2	4	no	—
260	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
261	C₂×#117	—	2	2	no	—
262	C₂×#117	C₂²×C₃₁	1	2	yes	—
263	C₂²×C₄²	—	2	2	no	—
264	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
265	#442	C₂²×C₃₁	1	8	yes	complemented
266	#441	C₂²×C₃₁	1	8	yes	complemented
267	#440	C₂²×C₃₁	1	16	yes	complemented
268	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
269	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
270	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
271	#428	C₂²×C₃₁	1	8	yes	complemented
272	#427	C₂²×C₃₁	1	4	yes	complemented
273	#419	C₂²×C₃₁	1	4	yes	complemented
274	#418	C₂²×C₃₁	1	4	yes	complemented
275	#417	C₂²×C₃₁	1	8	yes	—
276	#416	C₂²×C₃₁	1	8	yes	complemented
277	C₄×M₁₆	C₂²×C₃₁	1	8	yes	complemented
278	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
279	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
280	C₂×C₃₁	—	2	4	no	—
281	C₂×C₃₁	—	2	2	no	—
282	C₂×C₃₁	—	4	8	no	—
283	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
284	C₂×C₃₁	C₂×#575	1	2	yes	—
285	C₂×#41	—	2	2	no	—
286	C₂×#41	—	2	2	no	—
287	C₂×#41	—	2	8	no	—
288	C₂×#41	C₂×C₄×C₃₁	1	16	yes	complemented
289	C₂×#41	C₂³×C₃₁	1	4	yes	—
290	C₂×#41	C₂³×C₃₁	1	4	yes	—
291	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
292	C₂²×D₄	—	2	2	no	—
293	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
294	C₂³×C₄	—	2	2	no	—
295	C₂³×C₄	—	2	4	no	—
296	C₂³×C₄	D₄×C₃₁	1	2	yes	—
297	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
298	#137	C₂×C₄×C₃₁	1	32	yes	complemented
299	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	32	yes	complemented
300	#135	C₂×C₄×C₃₁	1	32	yes	complemented
301	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
302	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
303	C₂×D₈	C₂³×C₃₁	1	4	yes	—
304	C₂×M₁₆	—	2	4	no	—
305	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
306	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
307	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
308	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
309	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
310	C₄×Q₈	—	2	4	no	—
311	C₄×Q₈	—	2	4	no	—
312	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
313	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
314	C₄×D₄	—	2	8	no	—
315	C₄×D₄	—	2	8	no	—
316	C₄×D₄	—	2	8	no	—
317	C₄×D₄	—	2	8	no	—
318	C₄×D₄	—	2	4	no	—
319	C₄×D₄	—	2	4	no	—
320	C₄×D₄	C₂³×C₃₁	1	8	yes	—
321	C₄×D₄	C₂³×C₃₁	1	8	yes	—
322	#117	—	2	2	no	—
323	#117	—	2	2	no	—
324	#117	—	2	8	no	—
325	#117	—	2	4	no	—
326	#117	—	2	4	no	—
327	#117	C₂³×C₃₁	1	4	yes	—
328	#117	C₂³×C₃₁	1	4	yes	—
329	C₂×C₄⋊C₄	—	2	4	no	—
330	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	—	2	4	no	—
333	C₂×C₄²	—	2	2	no	—
334	C₂×C₄²	—	2	4	no	—
335	C₂×C₄²	—	2	8	no	—
336	C₂×C₄²	—	2	4	no	—
337	C₂×C₄²	—	2	4	no	—
338	C₂×C₄²	D₄×C₃₁	1	2	yes	—
339	C₂×C₄²	D₄×C₃₁	1	2	yes	—
340	C₂×C₄²	D₄×C₃₁	1	2	yes	—
341	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
342	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
344	#107	C₂³×C₃₁	1	8	yes	—
345	#106	C₂³×C₃₁	1	8	yes	—
346	#103	C₂³×C₃₁	1	16	yes	—
347	#102	C₂³×C₃₁	1	16	yes	—
348	#97	C₂³×C₃₁	1	8	yes	—
349	C₄×C₈	C₂³×C₃₁	1	8	yes	—
350	C₃₁	C₄×#575	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	—	2	2	no	—
352	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
353	#41	—	2	16	no	—
354	#41	—	2	16	no	—
355	#41	—	2	8	no	—
356	#41	—	2	8	no	—
357	#41	C₂²×C₄×C₃₁	1	16	yes	—
358	#41	C₂²×C₄×C₃₁	1	16	yes	complemented
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	complemented
362	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
363	C₂×D₄	—	2	16	no	—
364	C₂×D₄	—	2	4	no	—
365	C₂×D₄	—	2	4	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	complemented
369	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	16	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	8	no	—
376	C₂²×C₄	—	2	8	no	—
377	C₂²×C₄	—	4	4	no	—
378	C₂²×C₄	—	4	16	no	—
379	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
380	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
381	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
382	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
383	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
384	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	#41×C₃₁	1	4	yes	—
387	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
389	Q₁₆	C₂²×C₄×C₃₁	1	16	yes	complemented
390	SD₁₆	C₂²×C₄×C₃₁	1	32	yes	complemented
391	D₈	C₂²×C₄×C₃₁	1	16	yes	complemented
392	M₁₆	—	2	8	no	—
393	M₁₆	C₂²×C₄×C₃₁	1	16	yes	complemented
394	C₂×C₈	—	2	4	no	—
395	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	complemented
396	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
397	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
398	C₄⋊C₄	—	2	8	no	—
399	C₄⋊C₄	—	2	8	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	2	8	no	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₄²	—	2	4	no	—
407	C₄²	—	2	4	no	—
408	C₄²	—	2	8	no	—
409	C₄²	—	2	8	no	—
410	C₄²	C₂×D₄×C₃₁	1	2	yes	—
411	C₄²	C₂×D₄×C₃₁	1	4	yes	—
412	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
413	C₄²	C₂×D₄×C₃₁	1	4	yes	—
414	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
415	C₄²	C₂⁴×C₃₁	1	2	yes	—
416	C₄²	C₂⁴×C₃₁	1	2	yes	—
417	C₂³	—	2	8	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	8	no	—
421	C₂³	C₄×D₄×C₃₁	1	4	yes	—
422	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
423	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
424	Q₈	—	2	8	no	—
425	Q₈	—	2	4	no	—
426	Q₈	C₂³×C₄×C₃₁	1	8	yes	—
427	D₄	—	2	16	no	—
428	D₄	—	2	16	no	—
429	D₄	—	2	8	no	—
430	D₄	—	2	4	no	—
431	D₄	C₂³×C₄×C₃₁	1	8	yes	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	16	no	—
435	C₂×C₄	—	2	16	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	16	no	—
442	C₂×C₄	—	4	8	no	—
443	C₂×C₄	—	4	16	no	—
444	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
445	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
446	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
447	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
448	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
449	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
450	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
453	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
454	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
455	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
456	C₈	—	2	4	no	—
457	C₈	C₂³×C₄×C₃₁	1	8	yes	—
458	C₂²	—	2	4	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	2	8	no	—
461	C₂²	—	4	4	no	—
462	C₂²	—	4	16	no	—
463	C₂²	—	4	8	no	—
464	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
465	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
466	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
467	C₄	—	2	8	no	—
468	C₄	—	2	4	no	—
469	C₄	—	2	2	no	—
470	C₄	—	4	8	no	—
471	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
472	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
473	C₄	C₃₁×#575	1	4	yes	direct product factor
474	C₄	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
475	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
476	C₂	—	2	4	no	—
477	C₂	—	2	2	no	—
478	C₂	—	4	8	no	—
479	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
480	C₂	C₂×C₃₁×#575	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₂×C₃₁×#575	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#41×C₃₁	C₂	semidirect
3. C₄×C₃₁×#137	C₂	semidirect
4. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
5. C₃₁×#2943	C₂	semidirect
6. C₄×C₃₁×#135	C₂	semidirect
7. C₃₁×#2911	C₂	semidirect
8. C₃₁×#2896	C₂	semidirect
9. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
11. C₃₁×#575	C₄	direct (N × H)
21. C₄×#41×C₃₁	C₂²	semidirect
22. C₄×#41×C₃₁	C₂²	semidirect
23. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
25. C₂×C₄×D₄×C₃₁	C₂²	semidirect
30. C₃₁×#442	C₂²	semidirect
31. C₃₁×#441	C₂²	semidirect
32. C₃₁×#440	C₂²	semidirect
33. C₄×Q₁₆×C₃₁	C₂²	semidirect
34. C₄×SD₁₆×C₃₁	C₂²	semidirect
35. C₄×D₈×C₃₁	C₂²	semidirect
36. C₃₁×#428	C₂²	semidirect
37. C₃₁×#427	C₂²	semidirect
38. C₃₁×#419	C₂²	semidirect
39. C₃₁×#418	C₂²	semidirect
41. C₃₁×#416	C₂²	semidirect
42. C₄×M₁₆×C₃₁	C₂²	semidirect
43. C₂×C₃₁×#97	C₂²	semidirect
44. C₂×C₄×C₈×C₃₁	C₂²	semidirect
48. C₂×#41×C₃₁	C₂×C₄	semidirect
58. C₃₁×#137	C₂×C₄	semidirect
59. C₃₁×AGL(1,ℤ/8)	C₂×C₄	semidirect
60. C₃₁×#135	C₂×C₄	semidirect
65. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
117. #41×C₃₁	C₂²×C₄	semidirect
120. C₂×Q₈×C₃₁	C₂²×C₄	semidirect
127. C₂×D₄×C₃₁	C₂²×C₄	semidirect
148. Q₁₆×C₃₁	C₂²×C₄	semidirect
149. SD₁₆×C₃₁	C₂²×C₄	semidirect
150. D₈×C₃₁	C₂²×C₄	semidirect
152. M₁₆×C₃₁	C₂²×C₄	semidirect
154. C₂×C₈×C₃₁	C₂²×C₄	semidirect
176. C₄×#575	C₃₁	direct (N × H)
219. C₂×C₄×#41	C₂×C₃₁	semidirect
220. C₄×#137	C₂×C₃₁	semidirect
221. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
222. #2943	C₂×C₃₁	semidirect
223. C₄×#135	C₂×C₃₁	semidirect
224. #2911	C₂×C₃₁	semidirect
225. #2896	C₂×C₃₁	semidirect
226. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
242. C₄×C₃₁	#575	direct (N × H)
246. #575	C₄×C₃₁	direct (N × H)
256. C₄×#41	C₂²×C₃₁	semidirect
257. C₄×#41	C₂²×C₃₁	semidirect
258. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
260. C₂×C₄×D₄	C₂²×C₃₁	semidirect
265. #442	C₂²×C₃₁	semidirect
266. #441	C₂²×C₃₁	semidirect
267. #440	C₂²×C₃₁	semidirect
268. C₄×Q₁₆	C₂²×C₃₁	semidirect
269. C₄×SD₁₆	C₂²×C₃₁	semidirect
270. C₄×D₈	C₂²×C₃₁	semidirect
271. #428	C₂²×C₃₁	semidirect
272. #427	C₂²×C₃₁	semidirect
273. #419	C₂²×C₃₁	semidirect
274. #418	C₂²×C₃₁	semidirect
276. #416	C₂²×C₃₁	semidirect
277. C₄×M₁₆	C₂²×C₃₁	semidirect
278. C₂×#97	C₂²×C₃₁	semidirect
279. C₂×C₄×C₈	C₂²×C₃₁	semidirect
288. C₂×#41	C₂×C₄×C₃₁	semidirect
298. #137	C₂×C₄×C₃₁	semidirect
299. AGL(1,ℤ/8)	C₂×C₄×C₃₁	semidirect
300. #135	C₂×C₄×C₃₁	semidirect
305. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
350. C₃₁	C₄×#575	direct (N × H)
358. #41	C₂²×C₄×C₃₁	semidirect
361. C₂×Q₈	C₂²×C₄×C₃₁	semidirect
368. C₂×D₄	C₂²×C₄×C₃₁	semidirect
389. Q₁₆	C₂²×C₄×C₃₁	semidirect
390. SD₁₆	C₂²×C₄×C₃₁	semidirect
391. D₈	C₂²×C₄×C₃₁	semidirect
393. M₁₆	C₂²×C₄×C₃₁	semidirect
395. C₂×C₈	C₂²×C₄×C₃₁	semidirect
473. C₄	C₃₁×#575	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
C₂×C₄×#41×C₃₁	2
C₄×C₃₁×#137	2
C₄×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#2943	2
C₄×C₃₁×#135	2
C₃₁×#2911	2
C₃₁×#2896	2
C₂×C₄×M₁₆×C₃₁	2
C₄×#575	31

Subgroups — C4×C31×#575 #21516917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#575 #21516917