Subgroups — C₂×C₃₁×#3254 #21517663

Order 7936.

2146 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3254	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#477	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#474	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#468	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×#460	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#461	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₃₁	—	2	1	no	—
11	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	—
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#533	—	2	4	no	—
14	C₃₁×#533	C₂²	1	8	yes	complemented
15	C₃₁×#532	—	2	4	no	—
16	C₃₁×#532	C₂²	1	8	yes	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#125	—	2	1	no	—
22	C₂×C₃₁×#124	—	2	1	no	—
23	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#121	—	2	1	no	—
26	C₂×C₃₁×#121	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	—	2	1	no	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₃₁×#117	—	2	1	no	—
32	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#477	C₂²	1	16	yes	complemented
34	C₃₁×#474	C₂²	1	16	yes	complemented
35	C₃₁×#468	C₂²	1	32	yes	complemented
36	C₃₁×#461	C₂²	1	16	yes	complemented
37	C₃₁×#460	C₂²	1	16	yes	complemented
38	C₂×C₃₁×#107	C₂²	1	4	yes	—
39	C₃₁×#423	C₂²	1	8	yes	complemented
40	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
41	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
42	C₂×C₃₁×#103	C₂²	1	4	yes	—
43	C₂×C₃₁×#102	C₂²	1	4	yes	—
44	C₂×#41×C₃₁	—	2	8	no	—
45	C₂×#41×C₃₁	—	4	8	no	—
46	C₂²×Q₈×C₃₁	—	2	1	no	—
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×D₄×C₃₁	—	2	2	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
53	C₂³×C₄×C₃₁	—	2	2	no	—
54	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×SD₁₆×C₃₁	—	4	8	no	—
58	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	—
59	C₂×M₁₆×C₃₁	—	4	4	no	—
60	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
61	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
62	C₃₁×C₄⋊Q₈	—	2	2	no	—
63	C₃₁×C₄⋊Q₈	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	—	2	2	no	—
65	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
66	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
67	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
68	C₃₁×#127	—	2	2	no	—
69	C₃₁×#127	C₂³	1	4	yes	—
70	C₃₁×#125	—	2	8	no	—
71	C₃₁×#124	—	2	8	no	—
72	C₃₁×C₂²⋊Q₈	—	2	8	no	—
73	C₃₁×C₂²⋊Q₈	—	2	4	no	—
74	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
75	C₃₁×#121	—	2	8	no	—
76	C₃₁×#121	—	2	4	no	—
77	C₃₁×#121	C₂³	1	8	yes	—
78	C₄×Q₈×C₃₁	—	2	8	no	—
79	C₄×Q₈×C₃₁	—	2	4	no	—
80	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
81	C₄×D₄×C₃₁	—	2	8	no	—
82	C₄×D₄×C₃₁	—	2	4	no	—
83	C₄×D₄×C₃₁	C₂³	1	8	yes	—
84	C₃₁×#117	—	2	8	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
89	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₂×C₄²×C₃₁	—	2	2	no	—
97	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	D₄	1	4	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
103	C₃₁×#107	C₂³	1	16	yes	—
104	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
105	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
106	C₃₁×#103	C₂³	1	16	yes	—
107	C₃₁×#102	C₂³	1	16	yes	—
108	C₂⁴×C₃₁	—	2	2	no	—
109	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
110	#41×C₃₁	—	4	32	no	—
111	C₂×Q₈×C₃₁	—	2	2	no	—
112	C₂×Q₈×C₃₁	—	2	4	no	—
113	C₂×Q₈×C₃₁	—	2	4	no	—
114	C₂×Q₈×C₃₁	—	2	4	no	—
115	C₂×Q₈×C₃₁	—	2	4	no	—
116	C₂×Q₈×C₃₁	—	4	4	no	—
117	C₂×Q₈×C₃₁	—	4	4	no	—
118	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
119	C₂×D₄×C₃₁	—	2	8	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	4	4	no	—
126	C₂×D₄×C₃₁	—	4	8	no	—
127	C₂×D₄×C₃₁	—	4	4	no	—
128	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	2	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	8	no	—
137	C₂²×C₄×C₃₁	—	4	8	no	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	SD₁₆×C₃₁	—	4	16	no	—
149	M₁₆×C₃₁	—	4	8	no	—
150	C₂×C₈×C₃₁	—	2	8	no	—
151	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	8	no	—
157	C₄⋊C₄×C₃₁	—	2	8	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
161	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	8	no	—
165	C₄²×C₃₁	—	2	8	no	—
166	C₄²×C₃₁	C₂×D₄	1	4	yes	—
167	C₄²×C₃₁	C₂×D₄	1	2	yes	—
168	C₄²×C₃₁	C₂×D₄	1	2	yes	—
169	C₄²×C₃₁	C₂×D₄	1	4	yes	—
170	C₄²×C₃₁	C₂⁴	1	2	yes	—
171	C₄²×C₃₁	C₂⁴	1	2	yes	—
172	C₂×#3254	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	2	2	no	—
176	C₂³×C₃₁	—	2	4	no	—
177	C₂³×C₃₁	—	4	4	no	—
178	C₂³×C₃₁	—	4	4	no	—
179	C₂³×C₃₁	—	4	4	no	—
180	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
181	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₃₁	—	2	4	no	—
183	Q₈×C₃₁	—	2	4	no	—
184	Q₈×C₃₁	—	4	8	no	—
185	Q₈×C₃₁	—	4	4	no	—
186	Q₈×C₃₁	—	4	8	no	—
187	D₄×C₃₁	—	2	4	no	—
188	D₄×C₃₁	—	2	4	no	—
189	D₄×C₃₁	—	4	8	no	—
190	D₄×C₃₁	—	4	16	no	—
191	D₄×C₃₁	—	4	4	no	—
192	D₄×C₃₁	—	4	8	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	8	no	—
198	C₂×C₄×C₃₁	—	2	8	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	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	—	4	4	no	—
205	C₂×C₄×C₃₁	—	4	4	no	—
206	C₂×C₄×C₃₁	—	4	4	no	—
207	C₂×C₄×C₃₁	—	4	16	no	—
208	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
209	C₂×C₄×C₃₁	#137	1	4	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
217	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
218	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
219	C₈×C₃₁	—	2	8	no	—
220	C₂×#533	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
222	#3254	C₂×C₃₁	1	16	yes	direct product factor
223	C₂×#477	C₂×C₃₁	1	2	yes	complemented
224	C₂×#474	C₂×C₃₁	1	2	yes	complemented
225	C₂×#468	C₂×C₃₁	1	4	yes	complemented
226	C₂×#460	C₂×C₃₁	1	2	yes	complemented
227	C₂×#461	C₂×C₃₁	1	2	yes	complemented
228	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₂²×C₃₁	—	2	2	no	—
230	C₂²×C₃₁	—	2	2	no	—
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	4	4	no	—
233	C₂²×C₃₁	—	4	4	no	—
234	C₂²×C₃₁	—	4	4	no	—
235	C₂²×C₃₁	—	4	4	no	—
236	C₂²×C₃₁	—	8	8	no	—
237	C₂²×C₃₁	#537	1	2	yes	—
238	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
239	C₂²×C₃₁	C₂×#137	1	2	yes	—
240	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	2	2	no	—
244	C₄×C₃₁	—	4	4	no	—
245	C₄×C₃₁	—	4	4	no	—
246	C₄×C₃₁	—	4	4	no	—
247	C₄×C₃₁	#537	1	2	yes	—
248	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
249	C₄×C₃₁	C₂×#137	1	4	yes	—
250	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
251	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
252	C₂²×#41	—	2	1	no	—
253	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	—
254	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
255	#533	—	2	4	no	—
256	#533	C₂²×C₃₁	1	8	yes	complemented
257	#532	—	2	4	no	—
258	#532	C₂²×C₃₁	1	8	yes	—
259	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×#125	—	2	1	no	—
264	C₂×#124	—	2	1	no	—
265	C₂×C₂²⋊Q₈	—	2	1	no	—
266	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
267	C₂×#121	—	2	1	no	—
268	C₂×#121	C₂²×C₃₁	1	2	yes	—
269	C₂×C₄×Q₈	—	2	1	no	—
270	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
271	C₂×C₄×D₄	—	2	1	no	—
272	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
273	C₂×#117	—	2	1	no	—
274	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
275	#477	C₂²×C₃₁	1	16	yes	complemented
276	#474	C₂²×C₃₁	1	16	yes	complemented
277	#468	C₂²×C₃₁	1	32	yes	complemented
278	#461	C₂²×C₃₁	1	16	yes	complemented
279	#460	C₂²×C₃₁	1	16	yes	complemented
280	C₂×#107	C₂²×C₃₁	1	4	yes	—
281	#423	C₂²×C₃₁	1	8	yes	complemented
282	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
283	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
284	C₂×#103	C₂²×C₃₁	1	4	yes	—
285	C₂×#102	C₂²×C₃₁	1	4	yes	—
286	C₂×C₃₁	—	4	2	no	—
287	C₂×C₃₁	—	8	4	no	—
288	C₂×C₃₁	#3254	1	4	yes	direct product factor
289	C₂×C₃₁	C₂×#537	1	1	yes	characteristic
290	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
292	C₂×#41	—	2	8	no	—
293	C₂×#41	—	4	8	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	—	2	1	no	—
296	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
297	C₂²×D₄	—	2	2	no	—
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	—	2	1	no	—
300	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
301	C₂³×C₄	—	2	2	no	—
302	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
303	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
304	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₂×SD₁₆	—	4	8	no	—
306	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	—
307	C₂×M₁₆	—	4	4	no	—
308	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
309	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
314	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
315	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
316	#127	—	2	2	no	—
317	#127	C₂³×C₃₁	1	4	yes	—
318	#125	—	2	8	no	—
319	#124	—	2	8	no	—
320	C₂²⋊Q₈	—	2	8	no	—
321	C₂²⋊Q₈	—	2	4	no	—
322	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
323	#121	—	2	8	no	—
324	#121	—	2	4	no	—
325	#121	C₂³×C₃₁	1	8	yes	—
326	C₄×Q₈	—	2	8	no	—
327	C₄×Q₈	—	2	4	no	—
328	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
329	C₄×D₄	—	2	8	no	—
330	C₄×D₄	—	2	4	no	—
331	C₄×D₄	C₂³×C₃₁	1	8	yes	—
332	#117	—	2	8	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	2	no	—
338	C₂×C₄⋊C₄	—	2	2	no	—
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₄²	—	2	2	no	—
345	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
346	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
347	C₂×C₄²	D₄×C₃₁	1	4	yes	—
348	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
349	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
351	#107	C₂³×C₃₁	1	16	yes	—
352	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
353	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
354	#103	C₂³×C₃₁	1	16	yes	—
355	#102	C₂³×C₃₁	1	16	yes	—
356	C₃₁	C₂×#3254	1	1	yes	characteristic, direct product factor, Sylow
357	C₂⁴	—	2	2	no	—
358	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
359	#41	—	4	32	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	4	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	4	4	no	—
366	C₂×Q₈	—	4	4	no	—
367	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	8	no	—
386	C₂²×C₄	—	4	8	no	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
392	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
393	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
397	SD₁₆	—	4	16	no	—
398	M₁₆	—	4	8	no	—
399	C₂×C₈	—	2	8	no	—
400	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	8	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
410	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	8	no	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₄²	—	2	8	no	—
415	C₄²	C₂×D₄×C₃₁	1	4	yes	—
416	C₄²	C₂×D₄×C₃₁	1	2	yes	—
417	C₄²	C₂×D₄×C₃₁	1	2	yes	—
418	C₄²	C₂×D₄×C₃₁	1	4	yes	—
419	C₄²	C₂⁴×C₃₁	1	2	yes	—
420	C₄²	C₂⁴×C₃₁	1	2	yes	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	4	no	—
425	C₂³	—	4	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	—	4	4	no	—
428	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
429	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	Q₈	—	2	4	no	—
431	Q₈	—	2	4	no	—
432	Q₈	—	4	8	no	—
433	Q₈	—	4	4	no	—
434	Q₈	—	4	8	no	—
435	D₄	—	2	4	no	—
436	D₄	—	2	4	no	—
437	D₄	—	4	8	no	—
438	D₄	—	4	16	no	—
439	D₄	—	4	4	no	—
440	D₄	—	4	8	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	8	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	16	no	—
456	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
457	C₂×C₄	C₃₁×#137	1	4	yes	—
458	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
464	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
465	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
466	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
467	C₈	—	2	8	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	8	8	no	—
476	C₂²	C₃₁×#537	1	2	yes	—
477	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
478	C₂²	C₂×C₃₁×#137	1	2	yes	—
479	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	4	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	C₃₁×#537	1	2	yes	—
487	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
488	C₄	C₂×C₃₁×#137	1	4	yes	—
489	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
490	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
491	C₂	—	4	2	no	—
492	C₂	—	8	4	no	—
493	C₂	C₃₁×#3254	1	4	yes	direct product factor
494	C₂	C₂×C₃₁×#537	1	1	yes	characteristic
495	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
496	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#533	C₂	semidirect
3. C₃₁×#3254	C₂	direct (N × H)
4. C₂×C₃₁×#477	C₂	semidirect
5. C₂×C₃₁×#474	C₂	semidirect
6. C₂×C₃₁×#468	C₂	semidirect
7. C₂×C₃₁×#460	C₂	semidirect
8. C₂×C₃₁×#461	C₂	semidirect
9. C₂×C₃₁×#423	C₂	semidirect
14. C₃₁×#533	C₂²	semidirect
33. C₃₁×#477	C₂²	semidirect
34. C₃₁×#474	C₂²	semidirect
35. C₃₁×#468	C₂²	semidirect
36. C₃₁×#461	C₂²	semidirect
37. C₃₁×#460	C₂²	semidirect
39. C₃₁×#423	C₂²	semidirect
172. C₂×#3254	C₃₁	direct (N × H)
220. C₂×#533	C₂×C₃₁	semidirect
222. #3254	C₂×C₃₁	direct (N × H)
223. C₂×#477	C₂×C₃₁	semidirect
224. C₂×#474	C₂×C₃₁	semidirect
225. C₂×#468	C₂×C₃₁	semidirect
226. C₂×#460	C₂×C₃₁	semidirect
227. C₂×#461	C₂×C₃₁	semidirect
228. C₂×#423	C₂×C₃₁	semidirect
256. #533	C₂²×C₃₁	semidirect
275. #477	C₂²×C₃₁	semidirect
276. #474	C₂²×C₃₁	semidirect
277. #468	C₂²×C₃₁	semidirect
278. #461	C₂²×C₃₁	semidirect
279. #460	C₂²×C₃₁	semidirect
281. #423	C₂²×C₃₁	semidirect
288. C₂×C₃₁	#3254	direct (N × H)
356. C₃₁	C₂×#3254	direct (N × H)
493. C₂	C₃₁×#3254	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×#533	2
C₂×C₃₁×#532	2
C₃₁×#3254	2
C₂×C₃₁×#477	2
C₂×C₃₁×#474	2
C₂×C₃₁×#468	2
C₂×C₃₁×#460	2
C₂×C₃₁×#461	2
C₂×C₃₁×#423	2
C₂×#3254	31

Subgroups — C2×C31×#3254 #21517663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3254 #21517663