Subgroups — C₂×C₃₁×#3220 #21517550

Order 7936.

2342 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#535	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3220	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#463	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#459	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#449	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#447	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
16	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
17	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
18	C₂²×D₈×C₃₁	C₂²	1	2	yes	complemented
19	C₃₁×#540	—	2	4	no	—
20	C₃₁×#540	C₂²	1	8	yes	complemented
21	C₃₁×#535	—	2	4	no	—
22	C₃₁×#535	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	1	yes	characteristic, complemented
26	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#123	—	2	2	no	—
30	C₂×C₃₁×#121	—	2	2	no	—
31	C₂×C₃₁×#121	—	2	2	no	—
32	C₂×C₃₁×#121	C₂²	1	2	yes	—
33	C₂×C₃₁×#120	—	2	2	no	—
34	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
35	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
36	C₃₁×#496	C₂²	1	8	yes	complemented
37	C₃₁×#493	C₂²	1	8	yes	complemented
38	C₃₁×#490	C₂²	1	8	yes	complemented
39	C₃₁×#488	C₂²	1	8	yes	complemented
40	C₃₁×#463	C₂²	1	16	yes	complemented
41	C₃₁×#459	C₂²	1	16	yes	complemented
42	C₃₁×#449	C₂²	1	16	yes	complemented
43	C₃₁×#447	C₂²	1	16	yes	complemented
44	C₃₁×#433	C₂²	1	8	yes	complemented
45	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
46	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
47	C₂×C₃₁×#102	C₂²	1	2	yes	—
48	C₂×C₃₁×#102	C₂²	1	2	yes	—
49	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
50	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
51	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
52	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
53	C₂⁵×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	—	2	2	no	—
55	C₂×#41×C₃₁	—	4	4	no	—
56	C₂×#41×C₃₁	C₂³	1	4	yes	—
57	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₃₁	—	2	1	no	—
59	C₂²×D₄×C₃₁	—	2	2	no	—
60	C₂²×D₄×C₃₁	—	2	1	no	—
61	C₂²×D₄×C₃₁	—	2	2	no	—
62	C₂²×D₄×C₃₁	—	2	2	no	—
63	C₂²×D₄×C₃₁	—	2	2	no	—
64	C₂²×D₄×C₃₁	—	2	2	no	—
65	C₂²×D₄×C₃₁	—	4	4	no	—
66	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
70	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₃₁	—	2	1	no	—
72	C₂³×C₄×C₃₁	—	2	2	no	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×SD₁₆×C₃₁	—	4	4	no	—
75	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	complemented
76	C₂×D₈×C₃₁	—	4	4	no	—
77	C₂×D₈×C₃₁	—	4	4	no	—
78	C₂×D₈×C₃₁	—	4	4	no	—
79	C₂×D₈×C₃₁	C₂³	1	8	yes	—
80	C₂×D₈×C₃₁	C₂³	1	8	yes	—
81	C₂×D₈×C₃₁	C₂³	1	8	yes	complemented
82	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
83	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
84	C₃₁×#127	—	2	2	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	4	yes	—
89	C₃₁×#127	C₂³	1	4	yes	complemented
90	C₃₁×#125	—	2	2	no	—
91	C₃₁×#125	C₂³	1	4	yes	complemented
92	C₃₁×#124	—	2	2	no	—
93	C₃₁×#124	—	2	2	no	—
94	C₃₁×#124	C₂³	1	4	yes	—
95	C₃₁×#124	C₂³	1	4	yes	—
96	C₃₁×#123	—	2	16	no	—
97	C₃₁×#121	—	2	4	no	—
98	C₃₁×#121	—	2	16	no	—
99	C₃₁×#121	—	2	16	no	—
100	C₃₁×#121	C₂³	1	8	yes	—
101	C₃₁×#120	—	2	16	no	—
102	C₄×D₄×C₃₁	—	2	4	no	—
103	C₄×D₄×C₃₁	C₂³	1	8	yes	complemented
104	C₃₁×#117	D₄	1	4	yes	—
105	C₃₁×#117	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
114	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	C₂³	1	8	yes	complemented
117	C₃₁×#103	C₂³	1	8	yes	complemented
118	C₃₁×#102	C₂³	1	8	yes	—
119	C₃₁×#102	C₂³	1	8	yes	—
120	C₃₁×#102	C₂³	1	8	yes	complemented
121	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
122	C₃₁×#97	C₂³	1	4	yes	complemented
123	C₄×C₈×C₃₁	C₂³	1	4	yes	complemented
124	C₂⁴×C₃₁	—	2	2	no	—
125	C₂⁴×C₃₁	—	2	1	no	—
126	C₂⁴×C₃₁	—	2	1	no	—
127	C₂⁴×C₃₁	—	2	1	no	—
128	C₂⁴×C₃₁	—	2	4	no	—
129	C₂⁴×C₃₁	—	4	4	no	—
130	C₂⁴×C₃₁	—	8	4	no	—
131	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
132	#41×C₃₁	—	4	8	no	—
133	#41×C₃₁	—	4	8	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	—	2	4	no	—
136	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	8	no	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	8	no	—
145	C₂×D₄×C₃₁	—	2	8	no	—
146	C₂×D₄×C₃₁	—	2	8	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	4	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	4	8	no	—
153	C₂×D₄×C₃₁	—	4	4	no	—
154	C₂×D₄×C₃₁	—	4	8	no	—
155	C₂×D₄×C₃₁	—	4	4	no	—
156	C₂×D₄×C₃₁	—	8	4	no	—
157	C₂×D₄×C₃₁	—	8	4	no	—
158	C₂×D₄×C₃₁	—	8	4	no	—
159	C₂×D₄×C₃₁	—	8	4	no	—
160	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
162	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
163	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	1	no	—
166	C₂²×C₄×C₃₁	—	2	2	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	8	no	—
169	C₂²×C₄×C₃₁	—	4	4	no	—
170	C₂²×C₄×C₃₁	—	8	4	no	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
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	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	SD₁₆×C₃₁	—	4	8	no	—
178	D₈×C₃₁	—	4	8	no	—
179	D₈×C₃₁	—	4	8	no	—
180	D₈×C₃₁	—	4	8	no	—
181	C₂×C₈×C₃₁	—	2	4	no	—
182	C₂×C₈×C₃₁	—	2	4	no	—
183	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
184	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₃₁	—	2	8	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
188	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
189	C₂²⋊C₄×C₃₁	—	2	8	no	—
190	C₂²⋊C₄×C₃₁	—	2	8	no	—
191	C₂²⋊C₄×C₃₁	—	2	8	no	—
192	C₂²⋊C₄×C₃₁	—	2	8	no	—
193	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
194	C₄²×C₃₁	C₂×D₄	1	2	yes	—
195	C₄²×C₃₁	C₂×D₄	1	2	yes	—
196	C₄²×C₃₁	C₂⁴	1	2	yes	—
197	C₄²×C₃₁	C₂⁴	1	2	yes	—
198	C₂×#3220	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂³×C₃₁	—	2	2	no	—
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	2	4	no	—
202	C₂³×C₃₁	—	2	2	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	2	2	no	—
205	C₂³×C₃₁	—	4	8	no	—
206	C₂³×C₃₁	—	4	2	no	—
207	C₂³×C₃₁	—	4	2	no	—
208	C₂³×C₃₁	—	4	2	no	—
209	C₂³×C₃₁	—	4	4	no	—
210	C₂³×C₃₁	—	4	4	no	—
211	C₂³×C₃₁	—	4	4	no	—
212	C₂³×C₃₁	—	4	4	no	—
213	C₂³×C₃₁	—	4	4	no	—
214	C₂³×C₃₁	—	4	4	no	—
215	C₂³×C₃₁	—	8	8	no	—
216	C₂³×C₃₁	—	8	16	no	—
217	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
218	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₃₁	—	2	4	no	—
220	Q₈×C₃₁	—	4	2	no	—
221	D₄×C₃₁	—	2	4	no	—
222	D₄×C₃₁	—	2	4	no	—
223	D₄×C₃₁	—	2	4	no	—
224	D₄×C₃₁	—	2	4	no	—
225	D₄×C₃₁	—	2	4	no	—
226	D₄×C₃₁	—	4	8	no	—
227	D₄×C₃₁	—	4	8	no	—
228	D₄×C₃₁	—	4	4	no	—
229	D₄×C₃₁	—	4	2	no	—
230	D₄×C₃₁	—	4	2	no	—
231	D₄×C₃₁	—	4	2	no	—
232	D₄×C₃₁	—	8	8	no	—
233	D₄×C₃₁	—	8	8	no	—
234	D₄×C₃₁	—	8	8	no	—
235	D₄×C₃₁	—	8	8	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	4	8	no	—
244	C₂×C₄×C₃₁	—	4	2	no	—
245	C₂×C₄×C₃₁	—	4	4	no	—
246	C₂×C₄×C₃₁	—	8	8	no	—
247	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
248	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
249	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
252	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
253	C₈×C₃₁	—	2	4	no	—
254	C₈×C₃₁	—	2	4	no	—
255	C₂×#540	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₂×#535	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3220	C₂×C₃₁	1	16	yes	direct product factor
258	C₂×#496	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
260	C₂×#490	C₂×C₃₁	1	1	yes	characteristic, complemented
261	C₂×#488	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂×#463	C₂×C₃₁	1	2	yes	complemented
263	C₂×#459	C₂×C₃₁	1	2	yes	complemented
264	C₂×#449	C₂×C₃₁	1	2	yes	complemented
265	C₂×#447	C₂×C₃₁	1	2	yes	complemented
266	C₂×#433	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₂²×C₃₁	—	2	2	no	—
268	C₂²×C₃₁	—	2	4	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	4	2	no	—
271	C₂²×C₃₁	—	4	2	no	—
272	C₂²×C₃₁	—	4	2	no	—
273	C₂²×C₃₁	—	4	4	no	—
274	C₂²×C₃₁	—	4	4	no	—
275	C₂²×C₃₁	—	4	4	no	—
276	C₂²×C₃₁	—	4	4	no	—
277	C₂²×C₃₁	—	4	4	no	—
278	C₂²×C₃₁	—	4	4	no	—
279	C₂²×C₃₁	—	8	16	no	—
280	C₂²×C₃₁	—	8	4	no	—
281	C₂²×C₃₁	—	8	8	no	—
282	C₂²×C₃₁	—	8	4	no	—
283	C₂²×C₃₁	—	8	4	no	—
284	C₂²×C₃₁	#535	1	2	yes	—
285	C₂²×C₃₁	#576	1	4	yes	—
286	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
287	C₄×C₃₁	—	2	2	no	—
288	C₄×C₃₁	—	2	4	no	—
289	C₄×C₃₁	—	2	4	no	—
290	C₄×C₃₁	—	4	2	no	—
291	C₄×C₃₁	—	4	4	no	—
292	C₄×C₃₁	#535	1	2	yes	—
293	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
294	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
295	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
296	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
297	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
298	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
299	C₂²×D₈	C₂²×C₃₁	1	2	yes	complemented
300	#540	—	2	4	no	—
301	#540	C₂²×C₃₁	1	8	yes	complemented
302	#535	—	2	4	no	—
303	#535	C₂²×C₃₁	1	8	yes	complemented
304	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
307	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×#123	—	2	2	no	—
311	C₂×#121	—	2	2	no	—
312	C₂×#121	—	2	2	no	—
313	C₂×#121	C₂²×C₃₁	1	2	yes	—
314	C₂×#120	—	2	2	no	—
315	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
316	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
317	#496	C₂²×C₃₁	1	8	yes	complemented
318	#493	C₂²×C₃₁	1	8	yes	complemented
319	#490	C₂²×C₃₁	1	8	yes	complemented
320	#488	C₂²×C₃₁	1	8	yes	complemented
321	#463	C₂²×C₃₁	1	16	yes	complemented
322	#459	C₂²×C₃₁	1	16	yes	complemented
323	#449	C₂²×C₃₁	1	16	yes	complemented
324	#447	C₂²×C₃₁	1	16	yes	complemented
325	#433	C₂²×C₃₁	1	8	yes	complemented
326	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
327	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
328	C₂×#102	C₂²×C₃₁	1	2	yes	—
329	C₂×#102	C₂²×C₃₁	1	2	yes	—
330	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
331	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
332	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
333	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
334	C₂×C₃₁	—	4	2	no	—
335	C₂×C₃₁	—	8	2	no	—
336	C₂×C₃₁	—	8	4	no	—
337	C₂×C₃₁	—	8	2	no	—
338	C₂×C₃₁	—	8	2	no	—
339	C₂×C₃₁	#3220	1	4	yes	direct product factor
340	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
341	C₂×C₃₁	C₂×#576	1	2	yes	—
342	C₂⁵	—	2	1	no	—
343	C₂×#41	—	2	2	no	—
344	C₂×#41	—	4	4	no	—
345	C₂×#41	C₂³×C₃₁	1	4	yes	—
346	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
347	C₂²×D₄	—	2	1	no	—
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	—	2	1	no	—
350	C₂²×D₄	—	2	2	no	—
351	C₂²×D₄	—	2	2	no	—
352	C₂²×D₄	—	2	2	no	—
353	C₂²×D₄	—	2	2	no	—
354	C₂²×D₄	—	4	4	no	—
355	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
356	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
357	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
359	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
360	C₂³×C₄	—	2	1	no	—
361	C₂³×C₄	—	2	2	no	—
362	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
363	C₂×SD₁₆	—	4	4	no	—
364	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	complemented
365	C₂×D₈	—	4	4	no	—
366	C₂×D₈	—	4	4	no	—
367	C₂×D₈	—	4	4	no	—
368	C₂×D₈	C₂³×C₃₁	1	8	yes	—
369	C₂×D₈	C₂³×C₃₁	1	8	yes	—
370	C₂×D₈	C₂³×C₃₁	1	8	yes	complemented
371	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
372	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
373	#127	—	2	2	no	—
374	#127	—	2	2	no	—
375	#127	—	2	2	no	—
376	#127	C₂³×C₃₁	1	4	yes	—
377	#127	C₂³×C₃₁	1	4	yes	—
378	#127	C₂³×C₃₁	1	4	yes	complemented
379	#125	—	2	2	no	—
380	#125	C₂³×C₃₁	1	4	yes	complemented
381	#124	—	2	2	no	—
382	#124	—	2	2	no	—
383	#124	C₂³×C₃₁	1	4	yes	—
384	#124	C₂³×C₃₁	1	4	yes	—
385	#123	—	2	16	no	—
386	#121	—	2	4	no	—
387	#121	—	2	16	no	—
388	#121	—	2	16	no	—
389	#121	C₂³×C₃₁	1	8	yes	—
390	#120	—	2	16	no	—
391	C₄×D₄	—	2	4	no	—
392	C₄×D₄	C₂³×C₃₁	1	8	yes	complemented
393	#117	D₄×C₃₁	1	4	yes	—
394	#117	C₂³×C₃₁	1	4	yes	—
395	C₂×C₄⋊C₄	—	2	2	no	—
396	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
398	C₂×C₂²⋊C₄	—	2	2	no	—
399	C₂×C₂²⋊C₄	—	2	2	no	—
400	C₂×C₂²⋊C₄	—	2	2	no	—
401	C₂×C₂²⋊C₄	—	2	2	no	—
402	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
403	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₃₁	1	8	yes	complemented
406	#103	C₂³×C₃₁	1	8	yes	complemented
407	#102	C₂³×C₃₁	1	8	yes	—
408	#102	C₂³×C₃₁	1	8	yes	—
409	#102	C₂³×C₃₁	1	8	yes	complemented
410	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
411	#97	C₂³×C₃₁	1	4	yes	complemented
412	C₄×C₈	C₂³×C₃₁	1	4	yes	complemented
413	C₃₁	C₂×#3220	1	1	yes	characteristic, direct product factor, Sylow
414	C₂⁴	—	2	2	no	—
415	C₂⁴	—	2	1	no	—
416	C₂⁴	—	2	1	no	—
417	C₂⁴	—	2	1	no	—
418	C₂⁴	—	2	4	no	—
419	C₂⁴	—	4	4	no	—
420	C₂⁴	—	8	4	no	—
421	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
422	#41	—	4	8	no	—
423	#41	—	4	8	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	4	no	—
426	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	8	no	—
435	C₂×D₄	—	2	8	no	—
436	C₂×D₄	—	2	8	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	4	8	no	—
443	C₂×D₄	—	4	4	no	—
444	C₂×D₄	—	4	8	no	—
445	C₂×D₄	—	4	4	no	—
446	C₂×D₄	—	8	4	no	—
447	C₂×D₄	—	8	4	no	—
448	C₂×D₄	—	8	4	no	—
449	C₂×D₄	—	8	4	no	—
450	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
451	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
452	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
453	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	8	no	—
459	C₂²×C₄	—	4	4	no	—
460	C₂²×C₄	—	8	4	no	—
461	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
462	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
463	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
466	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
467	SD₁₆	—	4	8	no	—
468	D₈	—	4	8	no	—
469	D₈	—	4	8	no	—
470	D₈	—	4	8	no	—
471	C₂×C₈	—	2	4	no	—
472	C₂×C₈	—	2	4	no	—
473	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
474	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
475	C₄⋊C₄	—	2	8	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
478	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
479	C₂²⋊C₄	—	2	8	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
484	C₄²	C₂×D₄×C₃₁	1	2	yes	—
485	C₄²	C₂×D₄×C₃₁	1	2	yes	—
486	C₄²	C₂⁴×C₃₁	1	2	yes	—
487	C₄²	C₂⁴×C₃₁	1	2	yes	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	4	no	—
490	C₂³	—	2	4	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	4	8	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	4	no	—
499	C₂³	—	4	4	no	—
500	C₂³	—	4	4	no	—
501	C₂³	—	4	4	no	—
502	C₂³	—	4	4	no	—
503	C₂³	—	4	4	no	—
504	C₂³	—	8	8	no	—
505	C₂³	—	8	16	no	—
506	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
507	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	Q₈	—	2	4	no	—
509	Q₈	—	4	2	no	—
510	D₄	—	2	4	no	—
511	D₄	—	2	4	no	—
512	D₄	—	2	4	no	—
513	D₄	—	2	4	no	—
514	D₄	—	2	4	no	—
515	D₄	—	4	8	no	—
516	D₄	—	4	8	no	—
517	D₄	—	4	4	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	2	no	—
521	D₄	—	8	8	no	—
522	D₄	—	8	8	no	—
523	D₄	—	8	8	no	—
524	D₄	—	8	8	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	4	8	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	4	no	—
535	C₂×C₄	—	8	8	no	—
536	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
537	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
538	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
539	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
540	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
541	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
542	C₈	—	2	4	no	—
543	C₈	—	2	4	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	4	no	—
546	C₂²	—	4	4	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	4	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	4	no	—
554	C₂²	—	4	4	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	8	16	no	—
557	C₂²	—	8	4	no	—
558	C₂²	—	8	4	no	—
559	C₂²	—	8	4	no	—
560	C₂²	—	8	8	no	—
561	C₂²	C₃₁×#535	1	2	yes	—
562	C₂²	C₃₁×#576	1	4	yes	—
563	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
564	C₄	—	2	2	no	—
565	C₄	—	2	4	no	—
566	C₄	—	2	4	no	—
567	C₄	—	4	2	no	—
568	C₄	—	4	4	no	—
569	C₄	C₃₁×#535	1	2	yes	—
570	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
571	C₂	—	4	2	no	—
572	C₂	—	8	2	no	—
573	C₂	—	8	2	no	—
574	C₂	—	8	2	no	—
575	C₂	—	8	4	no	—
576	C₂	C₃₁×#3220	1	4	yes	direct product factor
577	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
578	C₂	C₂×C₃₁×#576	1	2	yes	—

	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×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#540	C₂	semidirect
2. C₂×C₃₁×#535	C₂	semidirect
3. C₃₁×#3220	C₂	direct (N × H)
4. C₂×C₃₁×#496	C₂	semidirect
5. C₂×C₃₁×#493	C₂	semidirect
6. C₂×C₃₁×#490	C₂	semidirect
7. C₂×C₃₁×#488	C₂	semidirect
8. C₂×C₃₁×#463	C₂	semidirect
9. C₂×C₃₁×#459	C₂	semidirect
10. C₂×C₃₁×#449	C₂	semidirect
11. C₂×C₃₁×#447	C₂	semidirect
12. C₂×C₃₁×#433	C₂	semidirect
15. C₂²×SD₁₆×C₃₁	C₂²	semidirect
18. C₂²×D₈×C₃₁	C₂²	semidirect
20. C₃₁×#540	C₂²	semidirect
22. C₃₁×#535	C₂²	semidirect
25. C₂×C₃₁×#127	C₂²	semidirect
26. C₂×C₃₁×#125	C₂²	semidirect
34. C₂×C₄×D₄×C₃₁	C₂²	semidirect
36. C₃₁×#496	C₂²	semidirect
37. C₃₁×#493	C₂²	semidirect
38. C₃₁×#490	C₂²	semidirect
39. C₃₁×#488	C₂²	semidirect
40. C₃₁×#463	C₂²	semidirect
41. C₃₁×#459	C₂²	semidirect
42. C₃₁×#449	C₂²	semidirect
43. C₃₁×#447	C₂²	semidirect
44. C₃₁×#433	C₂²	semidirect
45. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
46. C₂×C₃₁×#103	C₂²	semidirect
49. C₂×C₃₁×#102	C₂²	semidirect
51. C₂×C₃₁×#97	C₂²	semidirect
52. C₂×C₄×C₈×C₃₁	C₂²	semidirect
75. C₂×SD₁₆×C₃₁	C₂³	semidirect
81. C₂×D₈×C₃₁	C₂³	semidirect
89. C₃₁×#127	C₂³	semidirect
91. C₃₁×#125	C₂³	semidirect
103. C₄×D₄×C₃₁	C₂³	semidirect
116. C₃₁×C₄⋊C₈	C₂³	semidirect
117. C₃₁×#103	C₂³	semidirect
120. C₃₁×#102	C₂³	semidirect
122. C₃₁×#97	C₂³	semidirect
123. C₄×C₈×C₃₁	C₂³	semidirect
198. C₂×#3220	C₃₁	direct (N × H)
255. C₂×#540	C₂×C₃₁	semidirect
256. C₂×#535	C₂×C₃₁	semidirect
257. #3220	C₂×C₃₁	direct (N × H)
258. C₂×#496	C₂×C₃₁	semidirect
259. C₂×#493	C₂×C₃₁	semidirect
260. C₂×#490	C₂×C₃₁	semidirect
261. C₂×#488	C₂×C₃₁	semidirect
262. C₂×#463	C₂×C₃₁	semidirect
263. C₂×#459	C₂×C₃₁	semidirect
264. C₂×#449	C₂×C₃₁	semidirect
265. C₂×#447	C₂×C₃₁	semidirect
266. C₂×#433	C₂×C₃₁	semidirect
296. C₂²×SD₁₆	C₂²×C₃₁	semidirect
299. C₂²×D₈	C₂²×C₃₁	semidirect
301. #540	C₂²×C₃₁	semidirect
303. #535	C₂²×C₃₁	semidirect
306. C₂×#127	C₂²×C₃₁	semidirect
307. C₂×#125	C₂²×C₃₁	semidirect
315. C₂×C₄×D₄	C₂²×C₃₁	semidirect
317. #496	C₂²×C₃₁	semidirect
318. #493	C₂²×C₃₁	semidirect
319. #490	C₂²×C₃₁	semidirect
320. #488	C₂²×C₃₁	semidirect
321. #463	C₂²×C₃₁	semidirect
322. #459	C₂²×C₃₁	semidirect
323. #449	C₂²×C₃₁	semidirect
324. #447	C₂²×C₃₁	semidirect
325. #433	C₂²×C₃₁	semidirect
326. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
327. C₂×#103	C₂²×C₃₁	semidirect
330. C₂×#102	C₂²×C₃₁	semidirect
332. C₂×#97	C₂²×C₃₁	semidirect
333. C₂×C₄×C₈	C₂²×C₃₁	semidirect
339. C₂×C₃₁	#3220	direct (N × H)
364. C₂×SD₁₆	C₂³×C₃₁	semidirect
370. C₂×D₈	C₂³×C₃₁	semidirect
378. #127	C₂³×C₃₁	semidirect
380. #125	C₂³×C₃₁	semidirect
392. C₄×D₄	C₂³×C₃₁	semidirect
405. C₄⋊C₈	C₂³×C₃₁	semidirect
406. #103	C₂³×C₃₁	semidirect
409. #102	C₂³×C₃₁	semidirect
411. #97	C₂³×C₃₁	semidirect
412. C₄×C₈	C₂³×C₃₁	semidirect
413. C₃₁	C₂×#3220	direct (N × H)
576. C₂	C₃₁×#3220	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×#540	2
C₂×C₃₁×#535	2
C₃₁×#3220	2
C₂×C₃₁×#496	2
C₂×C₃₁×#493	2
C₂×C₃₁×#490	2
C₂×C₃₁×#488	2
C₂×C₃₁×#463	2
C₂×C₃₁×#459	2
C₂×C₃₁×#449	2
C₂×C₃₁×#447	2
C₂×C₃₁×#433	2
C₂×#3220	31

Subgroups — C2×C31×#3220 #21517550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3220 #21517550