Subgroups — C₂×C₃₁×#3399 #21518155

Order 7936.

2314 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3399	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#496	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#463	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#459	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#442	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	—	2	1	no	—
14	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
17	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
18	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#559	—	2	4	no	—
20	C₃₁×#559	C₂²	1	8	yes	complemented
21	C₃₁×#550	—	2	4	no	—
22	C₃₁×#550	C₂²	1	8	yes	complemented
23	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#127	C₂²	1	2	yes	—
26	C₂×C₃₁×#124	—	2	1	no	—
27	C₂×C₃₁×#124	C₂²	1	2	yes	—
28	C₂×C₃₁×#123	—	2	2	no	—
29	C₂×C₃₁×#121	—	2	1	no	—
30	C₂×C₃₁×#121	—	2	2	no	—
31	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₃₁	—	2	1	no	—
34	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
37	C₃₁×#496	C₂²	1	16	yes	complemented
38	C₃₁×#493	C₂²	1	8	yes	complemented
39	C₃₁×#463	C₂²	1	16	yes	complemented
40	C₃₁×#460	C₂²	1	8	yes	complemented
41	C₃₁×#459	C₂²	1	16	yes	complemented
42	C₃₁×#459	C₂²	1	8	yes	complemented
43	C₃₁×#446	C₂²	1	8	yes	complemented
44	C₃₁×#442	C₂²	1	16	yes	complemented
45	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
46	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
47	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
48	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
49	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×#102	C₂²	1	2	yes	—
52	C₂×C₃₁×#102	C₂²	1	2	yes	—
53	C₂×C₃₁×#97	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₂×#41×C₃₁	—	2	8	no	—
56	C₂×#41×C₃₁	—	4	8	no	—
57	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₃₁	—	2	1	no	—
60	C₂²×D₄×C₃₁	—	2	2	no	—
61	C₂²×D₄×C₃₁	—	2	1	no	—
62	C₂²×D₄×C₃₁	—	2	2	no	—
63	C₂²×D₄×C₃₁	—	2	1	no	—
64	C₂²×D₄×C₃₁	—	2	1	no	—
65	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₃₁	—	2	2	no	—
69	C₂³×C₄×C₃₁	—	2	1	no	—
70	C₂³×C₄×C₃₁	—	2	2	no	—
71	C₂×SD₁₆×C₃₁	—	2	4	no	—
72	C₂×SD₁₆×C₃₁	—	4	4	no	—
73	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
74	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
75	C₂×D₈×C₃₁	—	2	8	no	—
76	C₂×D₈×C₃₁	—	4	4	no	—
77	C₂×D₈×C₃₁	—	4	4	no	—
78	C₂×D₈×C₃₁	C₂³	1	8	yes	—
79	C₂×D₈×C₃₁	C₂³	1	8	yes	—
80	C₂×D₈×C₃₁	C₂³	1	8	yes	—
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
83	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#127	—	2	4	no	—
85	C₃₁×#127	—	2	2	no	—
86	C₃₁×#127	—	2	2	no	—
87	C₃₁×#127	C₂³	1	4	yes	—
88	C₃₁×#127	C₂³	1	8	yes	—
89	C₃₁×#127	C₂³	1	4	yes	—
90	C₃₁×#124	—	2	8	no	—
91	C₃₁×#124	—	2	4	no	—
92	C₃₁×#124	C₂³	1	8	yes	—
93	C₃₁×#123	—	2	16	no	—
94	C₃₁×#121	—	2	8	no	—
95	C₃₁×#121	—	2	16	no	—
96	C₄×Q₈×C₃₁	—	2	2	no	—
97	C₄×Q₈×C₃₁	D₄	1	4	yes	—
98	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
99	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
100	C₄×D₄×C₃₁	—	2	8	no	—
101	C₄×D₄×C₃₁	—	2	4	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	4	no	—
104	C₄×D₄×C₃₁	C₂³	1	4	yes	—
105	C₄×D₄×C₃₁	C₂³	1	8	yes	—
106	C₄×D₄×C₃₁	C₂³	1	8	yes	—
107	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₄²×C₃₁	—	2	1	no	—
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
120	C₃₁×#106	C₂³	1	4	yes	—
121	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
122	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
123	C₃₁×#103	C₂³	1	4	yes	—
124	C₃₁×#102	C₂³	1	4	yes	—
125	C₃₁×#102	C₂³	1	8	yes	—
126	C₃₁×#102	C₂³	1	8	yes	—
127	C₃₁×#97	C₂³	1	8	yes	—
128	C₄×C₈×C₃₁	C₂³	1	4	yes	—
129	C₂⁴×C₃₁	—	2	2	no	—
130	C₂⁴×C₃₁	—	2	2	no	—
131	C₂⁴×C₃₁	—	2	1	no	—
132	#41×C₃₁	—	4	32	no	—
133	C₂×Q₈×C₃₁	—	2	4	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	—	2	2	no	—
136	C₂×Q₈×C₃₁	—	2	2	no	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
138	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	8	no	—
144	C₂×D₄×C₃₁	—	2	4	no	—
145	C₂×D₄×C₃₁	—	2	8	no	—
146	C₂×D₄×C₃₁	—	2	4	no	—
147	C₂×D₄×C₃₁	—	2	4	no	—
148	C₂×D₄×C₃₁	—	2	4	no	—
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	2	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	—	2	4	no	—
154	C₂×D₄×C₃₁	—	4	8	no	—
155	C₂×D₄×C₃₁	—	4	4	no	—
156	C₂×D₄×C₃₁	—	4	4	no	—
157	C₂×D₄×C₃₁	—	8	4	no	—
158	C₂×D₄×C₃₁	—	8	2	no	—
159	C₂×D₄×C₃₁	—	8	2	no	—
160	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
161	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	2	1	no	—
166	C₂²×C₄×C₃₁	—	2	8	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	8	no	—
169	C₂²×C₄×C₃₁	—	4	8	no	—
170	C₂²×C₄×C₃₁	—	4	4	no	—
171	C₂²×C₄×C₃₁	—	8	4	no	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	SD₁₆×C₃₁	—	2	8	no	—
178	SD₁₆×C₃₁	—	4	8	no	—
179	D₈×C₃₁	—	2	16	no	—
180	D₈×C₃₁	—	4	8	no	—
181	D₈×C₃₁	—	4	8	no	—
182	C₂×C₈×C₃₁	—	2	2	no	—
183	C₂×C₈×C₃₁	—	2	4	no	—
184	C₂×C₈×C₃₁	#41	1	4	yes	complemented
185	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
186	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
187	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
192	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
193	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
194	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
195	C₂²⋊C₄×C₃₁	—	2	8	no	—
196	C₂²⋊C₄×C₃₁	—	2	8	no	—
197	C₂²⋊C₄×C₃₁	—	2	4	no	—
198	C₂²⋊C₄×C₃₁	—	2	8	no	—
199	C₂²⋊C₄×C₃₁	—	2	8	no	—
200	C₄²×C₃₁	—	2	4	no	—
201	C₄²×C₃₁	C₂×D₄	1	4	yes	—
202	C₄²×C₃₁	C₂×D₄	1	2	yes	—
203	C₄²×C₃₁	C₂⁴	1	2	yes	—
204	C₄²×C₃₁	C₂⁴	1	4	yes	—
205	C₂×#3399	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂³×C₃₁	—	2	4	no	—
207	C₂³×C₃₁	—	2	4	no	—
208	C₂³×C₃₁	—	2	2	no	—
209	C₂³×C₃₁	—	4	4	no	—
210	C₂³×C₃₁	—	4	4	no	—
211	C₂³×C₃₁	—	4	2	no	—
212	C₂³×C₃₁	—	4	4	no	—
213	C₂³×C₃₁	—	4	2	no	—
214	C₂³×C₃₁	—	4	4	no	—
215	C₂³×C₃₁	—	4	4	no	—
216	C₂³×C₃₁	—	4	2	no	—
217	C₂³×C₃₁	—	4	4	no	—
218	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₃₁	—	2	8	no	—
220	Q₈×C₃₁	—	2	2	no	—
221	Q₈×C₃₁	—	2	2	no	—
222	Q₈×C₃₁	—	4	2	no	—
223	D₄×C₃₁	—	2	4	no	—
224	D₄×C₃₁	—	2	4	no	—
225	D₄×C₃₁	—	2	2	no	—
226	D₄×C₃₁	—	2	4	no	—
227	D₄×C₃₁	—	2	2	no	—
228	D₄×C₃₁	—	2	4	no	—
229	D₄×C₃₁	—	4	16	no	—
230	D₄×C₃₁	—	4	4	no	—
231	D₄×C₃₁	—	4	4	no	—
232	D₄×C₃₁	—	4	2	no	—
233	D₄×C₃₁	—	4	8	no	—
234	D₄×C₃₁	—	4	8	no	—
235	D₄×C₃₁	—	8	8	no	—
236	D₄×C₃₁	—	8	4	no	—
237	D₄×C₃₁	—	8	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	4	16	no	—
245	C₂×C₄×C₃₁	—	4	2	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	4	2	no	—
248	C₂×C₄×C₃₁	—	4	8	no	—
249	C₂×C₄×C₃₁	—	8	8	no	—
250	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
257	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
258	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
259	C₈×C₃₁	—	2	2	no	—
260	C₈×C₃₁	—	2	4	no	—
261	C₈×C₃₁	C₂×#41	1	4	yes	complemented
262	C₂×#559	C₂×C₃₁	1	1	yes	characteristic, complemented
263	C₂×#550	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#3399	C₂×C₃₁	1	16	yes	direct product factor
265	C₂×#496	C₂×C₃₁	1	2	yes	complemented
266	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₂×#460	C₂×C₃₁	1	1	yes	characteristic, complemented
268	C₂×#463	C₂×C₃₁	1	2	yes	complemented
269	C₂×#459	C₂×C₃₁	1	1	yes	characteristic, complemented
270	C₂×#459	C₂×C₃₁	1	2	yes	complemented
271	C₂×#446	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂×#442	C₂×C₃₁	1	2	yes	complemented
273	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
274	C₂²×C₃₁	—	4	4	no	—
275	C₂²×C₃₁	—	4	4	no	—
276	C₂²×C₃₁	—	4	2	no	—
277	C₂²×C₃₁	—	4	4	no	—
278	C₂²×C₃₁	—	4	2	no	—
279	C₂²×C₃₁	—	4	4	no	—
280	C₂²×C₃₁	—	4	4	no	—
281	C₂²×C₃₁	—	4	2	no	—
282	C₂²×C₃₁	—	4	4	no	—
283	C₂²×C₃₁	—	8	8	no	—
284	C₂²×C₃₁	—	8	4	no	—
285	C₂²×C₃₁	—	8	8	no	—
286	C₂²×C₃₁	#550	1	2	yes	—
287	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
288	C₂²×C₃₁	#576	1	2	yes	—
289	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
290	C₄×C₃₁	—	2	4	no	—
291	C₄×C₃₁	—	2	4	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	4	2	no	—
294	C₄×C₃₁	—	4	2	no	—
295	C₄×C₃₁	—	4	2	no	—
296	C₄×C₃₁	#550	1	2	yes	—
297	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
298	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
299	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
300	C₂²×#41	—	2	1	no	—
301	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
302	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
303	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
304	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
305	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
306	#559	—	2	4	no	—
307	#559	C₂²×C₃₁	1	8	yes	complemented
308	#550	—	2	4	no	—
309	#550	C₂²×C₃₁	1	8	yes	complemented
310	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×#127	C₂²×C₃₁	1	2	yes	—
313	C₂×#124	—	2	1	no	—
314	C₂×#124	C₂²×C₃₁	1	2	yes	—
315	C₂×#123	—	2	2	no	—
316	C₂×#121	—	2	1	no	—
317	C₂×#121	—	2	2	no	—
318	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×C₄×D₄	—	2	1	no	—
321	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
323	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
324	#496	C₂²×C₃₁	1	16	yes	complemented
325	#493	C₂²×C₃₁	1	8	yes	complemented
326	#463	C₂²×C₃₁	1	16	yes	complemented
327	#460	C₂²×C₃₁	1	8	yes	complemented
328	#459	C₂²×C₃₁	1	16	yes	complemented
329	#459	C₂²×C₃₁	1	8	yes	complemented
330	#446	C₂²×C₃₁	1	8	yes	complemented
331	#442	C₂²×C₃₁	1	16	yes	complemented
332	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
333	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
335	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
336	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×#102	C₂²×C₃₁	1	2	yes	—
339	C₂×#102	C₂²×C₃₁	1	2	yes	—
340	C₂×#97	C₂²×C₃₁	1	2	yes	—
341	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×C₃₁	—	8	4	no	—
343	C₂×C₃₁	—	8	2	no	—
344	C₂×C₃₁	—	8	4	no	—
345	C₂×C₃₁	#3399	1	4	yes	direct product factor
346	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
347	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
348	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
349	C₂×#41	—	2	8	no	—
350	C₂×#41	—	4	8	no	—
351	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
352	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
353	C₂²×D₄	—	2	1	no	—
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	—	2	2	no	—
357	C₂²×D₄	—	2	1	no	—
358	C₂²×D₄	—	2	1	no	—
359	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
360	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
361	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
362	C₂³×C₄	—	2	2	no	—
363	C₂³×C₄	—	2	1	no	—
364	C₂³×C₄	—	2	2	no	—
365	C₂×SD₁₆	—	2	4	no	—
366	C₂×SD₁₆	—	4	4	no	—
367	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
368	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
369	C₂×D₈	—	2	8	no	—
370	C₂×D₈	—	4	4	no	—
371	C₂×D₈	—	4	4	no	—
372	C₂×D₈	C₂³×C₃₁	1	8	yes	—
373	C₂×D₈	C₂³×C₃₁	1	8	yes	—
374	C₂×D₈	C₂³×C₃₁	1	8	yes	—
375	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
377	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
378	#127	—	2	4	no	—
379	#127	—	2	2	no	—
380	#127	—	2	2	no	—
381	#127	C₂³×C₃₁	1	8	yes	—
382	#127	C₂³×C₃₁	1	4	yes	—
383	#127	C₂³×C₃₁	1	4	yes	—
384	#124	—	2	8	no	—
385	#124	—	2	4	no	—
386	#124	C₂³×C₃₁	1	8	yes	—
387	#123	—	2	16	no	—
388	#121	—	2	8	no	—
389	#121	—	2	16	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	D₄×C₃₁	1	4	yes	—
392	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
393	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	4	no	—
398	C₄×D₄	C₂³×C₃₁	1	4	yes	—
399	C₄×D₄	C₂³×C₃₁	1	8	yes	—
400	C₄×D₄	C₂³×C₃₁	1	8	yes	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
403	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
406	C₂×C₂²⋊C₄	—	2	2	no	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	—	2	1	no	—
409	C₂×C₂²⋊C₄	—	2	2	no	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₄²	—	2	1	no	—
412	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
413	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
414	#106	C₂³×C₃₁	1	4	yes	—
415	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
416	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
417	#103	C₂³×C₃₁	1	4	yes	—
418	#102	C₂³×C₃₁	1	4	yes	—
419	#102	C₂³×C₃₁	1	8	yes	—
420	#102	C₂³×C₃₁	1	8	yes	—
421	#97	C₂³×C₃₁	1	8	yes	—
422	C₄×C₈	C₂³×C₃₁	1	4	yes	—
423	C₃₁	C₂×#3399	1	1	yes	characteristic, direct product factor, Sylow
424	C₂⁴	—	2	2	no	—
425	C₂⁴	—	2	2	no	—
426	C₂⁴	—	2	1	no	—
427	#41	—	4	32	no	—
428	C₂×Q₈	—	2	4	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	2	2	no	—
432	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
433	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	8	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	8	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	4	8	no	—
450	C₂×D₄	—	4	4	no	—
451	C₂×D₄	—	4	4	no	—
452	C₂×D₄	—	8	4	no	—
453	C₂×D₄	—	8	2	no	—
454	C₂×D₄	—	8	2	no	—
455	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
456	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
457	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	8	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	—	4	8	no	—
465	C₂²×C₄	—	4	4	no	—
466	C₂²×C₄	—	8	4	no	—
467	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
468	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
472	SD₁₆	—	2	8	no	—
473	SD₁₆	—	4	8	no	—
474	D₈	—	2	16	no	—
475	D₈	—	4	8	no	—
476	D₈	—	4	8	no	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	2	4	no	—
479	C₂×C₈	#41×C₃₁	1	4	yes	complemented
480	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
481	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
482	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
487	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
488	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
489	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	—	2	8	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	8	no	—
494	C₂²⋊C₄	—	2	8	no	—
495	C₄²	—	2	4	no	—
496	C₄²	C₂×D₄×C₃₁	1	4	yes	—
497	C₄²	C₂×D₄×C₃₁	1	2	yes	—
498	C₄²	C₂⁴×C₃₁	1	2	yes	—
499	C₄²	C₂⁴×C₃₁	1	4	yes	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	4	4	no	—
504	C₂³	—	4	4	no	—
505	C₂³	—	4	2	no	—
506	C₂³	—	4	4	no	—
507	C₂³	—	4	2	no	—
508	C₂³	—	4	4	no	—
509	C₂³	—	4	4	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	4	no	—
512	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
513	Q₈	—	2	8	no	—
514	Q₈	—	2	2	no	—
515	Q₈	—	2	2	no	—
516	Q₈	—	4	2	no	—
517	D₄	—	2	4	no	—
518	D₄	—	2	4	no	—
519	D₄	—	2	2	no	—
520	D₄	—	2	4	no	—
521	D₄	—	2	2	no	—
522	D₄	—	2	4	no	—
523	D₄	—	4	16	no	—
524	D₄	—	4	4	no	—
525	D₄	—	4	4	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	8	no	—
528	D₄	—	4	8	no	—
529	D₄	—	8	8	no	—
530	D₄	—	8	4	no	—
531	D₄	—	8	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	4	16	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	8	no	—
543	C₂×C₄	—	8	8	no	—
544	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
545	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
546	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
547	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
548	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
550	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
551	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
552	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
553	C₈	—	2	2	no	—
554	C₈	—	2	4	no	—
555	C₈	C₂×#41×C₃₁	1	4	yes	complemented
556	C₂²	—	4	4	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	4	no	—
562	C₂²	—	4	4	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	8	8	no	—
566	C₂²	—	8	4	no	—
567	C₂²	—	8	8	no	—
568	C₂²	C₃₁×#550	1	2	yes	—
569	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
570	C₂²	C₃₁×#576	1	2	yes	—
571	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
572	C₄	—	2	4	no	—
573	C₄	—	2	4	no	—
574	C₄	—	2	2	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	2	no	—
578	C₄	C₃₁×#550	1	2	yes	—
579	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
580	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
581	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
582	C₂	—	8	4	no	—
583	C₂	—	8	2	no	—
584	C₂	—	8	4	no	—
585	C₂	C₃₁×#3399	1	4	yes	direct product factor
586	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
587	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
588	C₂	C₂×C₃₁×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#559	C₂	semidirect
2. C₂×C₃₁×#550	C₂	semidirect
3. C₃₁×#3399	C₂	direct (N × H)
4. C₂×C₃₁×#496	C₂	semidirect
5. C₂×C₃₁×#493	C₂	semidirect
6. C₂×C₃₁×#460	C₂	semidirect
7. C₂×C₃₁×#463	C₂	semidirect
8. C₂×C₃₁×#459	C₂	semidirect
9. C₂×C₃₁×#459	C₂	semidirect
10. C₂×C₃₁×#446	C₂	semidirect
11. C₂×C₃₁×#442	C₂	semidirect
12. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
20. C₃₁×#559	C₂²	semidirect
22. C₃₁×#550	C₂²	semidirect
37. C₃₁×#496	C₂²	semidirect
38. C₃₁×#493	C₂²	semidirect
39. C₃₁×#463	C₂²	semidirect
40. C₃₁×#460	C₂²	semidirect
41. C₃₁×#459	C₂²	semidirect
42. C₃₁×#459	C₂²	semidirect
43. C₃₁×#446	C₂²	semidirect
44. C₃₁×#442	C₂²	semidirect
45. C₄×SD₁₆×C₃₁	C₂²	semidirect
184. C₂×C₈×C₃₁	#41	semidirect
205. C₂×#3399	C₃₁	direct (N × H)
261. C₈×C₃₁	C₂×#41	semidirect
262. C₂×#559	C₂×C₃₁	semidirect
263. C₂×#550	C₂×C₃₁	semidirect
264. #3399	C₂×C₃₁	direct (N × H)
265. C₂×#496	C₂×C₃₁	semidirect
266. C₂×#493	C₂×C₃₁	semidirect
267. C₂×#460	C₂×C₃₁	semidirect
268. C₂×#463	C₂×C₃₁	semidirect
269. C₂×#459	C₂×C₃₁	semidirect
270. C₂×#459	C₂×C₃₁	semidirect
271. C₂×#446	C₂×C₃₁	semidirect
272. C₂×#442	C₂×C₃₁	semidirect
273. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
307. #559	C₂²×C₃₁	semidirect
309. #550	C₂²×C₃₁	semidirect
324. #496	C₂²×C₃₁	semidirect
325. #493	C₂²×C₃₁	semidirect
326. #463	C₂²×C₃₁	semidirect
327. #460	C₂²×C₃₁	semidirect
328. #459	C₂²×C₃₁	semidirect
329. #459	C₂²×C₃₁	semidirect
330. #446	C₂²×C₃₁	semidirect
331. #442	C₂²×C₃₁	semidirect
332. C₄×SD₁₆	C₂²×C₃₁	semidirect
345. C₂×C₃₁	#3399	direct (N × H)
423. C₃₁	C₂×#3399	direct (N × H)
479. C₂×C₈	#41×C₃₁	semidirect
555. C₈	C₂×#41×C₃₁	semidirect
585. C₂	C₃₁×#3399	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×#559	2
C₂×C₃₁×#550	2
C₃₁×#3399	2
C₂×C₃₁×#496	2
C₂×C₃₁×#493	2
C₂×C₃₁×#460	2
C₂×C₃₁×#463	2
C₂×C₃₁×#459	2
C₂×C₃₁×#446	2
C₂×C₃₁×#442	2
C₂×C₄×SD₁₆×C₃₁	2
C₂×#3399	31

Subgroups — C2×C31×#3399 #21518155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3399 #21518155