Subgroups — C₃₁×#60484 #21517243

Order 7936.

2130 conjugacy classes of subgroups.

The subgroup lattice has 606 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×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3080	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3079	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3078	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3077	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3076	C₂	1	8	yes	maximal, complemented
8	C₃₁×#2911	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2896	C₂	1	2	yes	maximal
11	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#533	—	2	1	no	—
14	C₃₁×#533	—	2	1	no	—
15	C₃₁×#533	—	2	1	no	—
16	C₃₁×#533	—	2	1	no	—
17	C₃₁×#533	C₂²	1	4	yes	—
18	C₃₁×#533	C₂²	1	4	yes	complemented
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
20	C₂×C₃₁×#125	C₂²	1	2	yes	complemented
21	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
22	C₄×#41×C₃₁	—	2	2	no	—
23	C₄×#41×C₃₁	—	2	1	no	—
24	C₄×#41×C₃₁	—	2	1	no	—
25	C₄×#41×C₃₁	C₂²	1	4	yes	—
26	C₄×#41×C₃₁	C₂²	1	2	yes	—
27	C₄×#41×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	—	2	2	no	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₃₁×#117	—	2	1	no	—
33	C₂×C₃₁×#117	C₂²	1	2	yes	—
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₃₁	—	2	1	no	—
36	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#479	C₂²	1	16	yes	complemented
38	C₃₁×#478	C₂²	1	16	yes	complemented
39	C₃₁×#477	C₂²	1	8	yes	complemented
40	C₃₁×#476	C₂²	1	8	yes	complemented
41	C₃₁×#475	C₂²	1	8	yes	complemented
42	C₃₁×#474	C₂²	1	8	yes	complemented
43	C₃₁×#428	C₂²	1	8	yes	—
44	C₃₁×#428	C₂²	1	8	yes	complemented
45	C₃₁×#427	C₂²	1	8	yes	complemented
46	C₃₁×#423	C₂²	1	4	yes	—
47	C₃₁×#423	C₂²	1	2	yes	complemented
48	C₃₁×#423	C₂²	1	2	yes	complemented
49	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
50	C₃₁×#419	C₂²	1	4	yes	—
51	C₃₁×#418	C₂²	1	4	yes	—
52	C₃₁×#417	C₂²	1	8	yes	—
53	C₃₁×#416	C₂²	1	8	yes	—
54	C₂×#41×C₃₁	—	2	1	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂×#41×C₃₁	—	2	4	no	—
57	C₂×#41×C₃₁	Q₈	1	8	yes	complemented
58	C₂×#41×C₃₁	C₂³	1	2	yes	—
59	C₂×#41×C₃₁	C₂³	1	2	yes	—
60	C₂²×Q₈×C₃₁	—	2	2	no	—
61	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₃₁	—	2	1	no	—
63	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₃₁	—	2	1	no	—
65	C₂³×C₄×C₃₁	—	2	2	no	—
66	C₂³×C₄×C₃₁	D₄	1	2	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₃₁	—	2	8	no	—
69	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
71	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
72	C₃₁×C₄⋊Q₈	—	2	2	no	—
73	C₃₁×C₄⋊Q₈	—	2	2	no	—
74	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
75	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
76	C₃₁×#125	—	2	2	no	—
77	C₃₁×#125	—	2	2	no	—
78	C₃₁×#125	C₂³	1	8	yes	—
79	C₃₁×C₂²⋊Q₈	—	2	4	no	—
80	C₃₁×C₂²⋊Q₈	—	2	4	no	—
81	C₃₁×C₂²⋊Q₈	—	2	4	no	—
82	C₃₁×C₂²⋊Q₈	—	2	4	no	—
83	C₄×Q₈×C₃₁	—	2	4	no	—
84	C₄×Q₈×C₃₁	—	2	4	no	—
85	C₄×Q₈×C₃₁	—	2	4	no	—
86	C₄×Q₈×C₃₁	—	2	4	no	—
87	C₄×Q₈×C₃₁	—	2	2	no	—
88	C₄×Q₈×C₃₁	—	2	2	no	—
89	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
90	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
91	C₄×D₄×C₃₁	—	2	4	no	—
92	C₄×D₄×C₃₁	—	2	4	no	—
93	C₄×D₄×C₃₁	—	2	4	no	—
94	C₄×D₄×C₃₁	—	2	4	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	2	no	—
97	C₄×D₄×C₃₁	C₂³	1	4	yes	—
98	C₄×D₄×C₃₁	C₂³	1	4	yes	—
99	C₃₁×#117	—	2	1	no	—
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	—	2	2	no	—
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	—	2	2	no	—
105	C₃₁×#117	—	2	4	no	—
106	C₃₁×#117	C₂³	1	4	yes	—
107	C₃₁×#117	C₂³	1	4	yes	—
108	C₃₁×#117	C₂³	1	2	yes	—
109	C₃₁×#117	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
112	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₄²×C₃₁	—	2	1	no	—
119	C₂×C₄²×C₃₁	—	2	2	no	—
120	C₂×C₄²×C₃₁	—	2	2	no	—
121	C₂×C₄²×C₃₁	—	2	2	no	—
122	C₂×C₄²×C₃₁	—	2	2	no	—
123	C₂×C₄²×C₃₁	—	2	4	no	—
124	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₃₁	D₄	1	2	yes	—
127	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
129	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
130	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
131	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
133	C₃₁×#107	C₂³	1	16	yes	—
134	C₃₁×#106	C₂³	1	16	yes	—
135	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
136	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
137	C₃₁×#103	C₂³	1	16	yes	—
138	C₃₁×#102	C₂³	1	16	yes	—
139	C₂⁴×C₃₁	—	2	1	no	—
140	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
141	#41×C₃₁	—	2	8	no	—
142	#41×C₃₁	—	2	8	no	—
143	#41×C₃₁	—	2	4	no	—
144	#41×C₃₁	—	2	4	no	—
145	#41×C₃₁	C₂×Q₈	1	8	yes	—
146	#41×C₃₁	C₂×Q₈	1	8	yes	complemented
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	—	2	4	no	—
149	C₂×Q₈×C₃₁	—	2	4	no	—
150	C₂×Q₈×C₃₁	—	2	2	no	—
151	C₂×Q₈×C₃₁	—	4	8	no	—
152	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	complemented
153	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
154	C₂×D₄×C₃₁	—	2	8	no	—
155	C₂×D₄×C₃₁	—	2	2	no	—
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	—	2	1	no	—
158	C₂×D₄×C₃₁	—	2	2	no	—
159	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	complemented
160	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	—	2	1	no	—
162	C₂²×C₄×C₃₁	—	2	8	no	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	4	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	4	no	—
169	C₂²×C₄×C₃₁	—	4	4	no	—
170	C₂²×C₄×C₃₁	—	4	8	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	2	yes	—
175	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
181	C₂²×C₄×C₃₁	#41	1	4	yes	—
182	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	M₁₆×C₃₁	—	2	8	no	—
186	M₁₆×C₃₁	—	2	8	no	—
187	C₂×C₈×C₃₁	—	2	8	no	—
188	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	4	no	—
191	C₄⋊C₄×C₃₁	—	2	8	no	—
192	C₄⋊C₄×C₃₁	—	2	8	no	—
193	C₄⋊C₄×C₃₁	—	2	4	no	—
194	C₄⋊C₄×C₃₁	—	2	4	no	—
195	C₄⋊C₄×C₃₁	—	2	2	no	—
196	C₄⋊C₄×C₃₁	—	2	4	no	—
197	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
198	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
199	C₂²⋊C₄×C₃₁	—	2	2	no	—
200	C₂²⋊C₄×C₃₁	—	2	2	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	—	2	4	no	—
203	C₂²⋊C₄×C₃₁	—	2	4	no	—
204	C₂²⋊C₄×C₃₁	—	2	4	no	—
205	C₄²×C₃₁	—	2	2	no	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	—	2	4	no	—
208	C₄²×C₃₁	—	2	4	no	—
209	C₄²×C₃₁	—	2	2	no	—
210	C₄²×C₃₁	—	2	4	no	—
211	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₃₁	C₂×D₄	1	2	yes	—
216	C₄²×C₃₁	C₂×D₄	1	2	yes	—
217	C₄²×C₃₁	C₂×D₄	1	2	yes	—
218	C₄²×C₃₁	C₂×D₄	1	2	yes	—
219	C₄²×C₃₁	C₂⁴	1	2	yes	—
220	C₄²×C₃₁	C₂⁴	1	2	yes	—
221	#60484	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂³×C₃₁	—	2	4	no	—
223	C₂³×C₃₁	—	2	2	no	—
224	C₂³×C₃₁	—	4	2	no	—
225	C₂³×C₃₁	—	4	4	no	—
226	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
227	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
230	Q₈×C₃₁	—	2	4	no	—
231	Q₈×C₃₁	—	2	2	no	—
232	Q₈×C₃₁	—	4	8	no	—
233	Q₈×C₃₁	—	4	8	no	—
234	Q₈×C₃₁	C₂²×Q₈	1	4	yes	—
235	D₄×C₃₁	—	2	8	no	—
236	D₄×C₃₁	—	2	8	no	—
237	D₄×C₃₁	—	2	4	no	—
238	D₄×C₃₁	—	2	2	no	—
239	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	8	no	—
243	C₂×C₄×C₃₁	—	2	8	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	2	4	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	8	no	—
249	C₂×C₄×C₃₁	—	4	2	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	2	no	—
252	C₂×C₄×C₃₁	—	4	8	no	—
253	C₂×C₄×C₃₁	—	4	4	no	—
254	C₂×C₄×C₃₁	—	4	8	no	—
255	C₂×C₄×C₃₁	—	4	8	no	—
256	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
257	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
258	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
259	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
267	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
268	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
269	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
270	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
271	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
272	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
273	C₈×C₃₁	—	2	8	no	—
274	C₂×#533	C₂×C₃₁	1	1	yes	characteristic, complemented
275	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
276	#3080	C₂×C₃₁	1	4	yes	complemented
277	#3079	C₂×C₃₁	1	4	yes	complemented
278	#3078	C₂×C₃₁	1	4	yes	complemented
279	#3077	C₂×C₃₁	1	4	yes	complemented
280	#3076	C₂×C₃₁	1	8	yes	complemented
281	#2911	C₂×C₃₁	1	2	yes	complemented
282	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
283	#2896	C₂×C₃₁	1	2	yes	—
284	C₂²×C₃₁	—	2	2	no	—
285	C₂²×C₃₁	—	2	2	no	—
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	4	no	—
288	C₂²×C₃₁	—	4	4	no	—
289	C₂²×C₃₁	—	4	8	no	—
290	C₂²×C₃₁	—	4	4	no	—
291	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
292	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
293	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
294	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
295	C₄×C₃₁	—	2	4	no	—
296	C₄×C₃₁	—	2	2	no	—
297	C₄×C₃₁	—	2	2	no	—
298	C₄×C₃₁	—	2	2	no	—
299	C₄×C₃₁	—	4	4	no	—
300	C₄×C₃₁	—	4	8	no	—
301	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
302	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
303	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
304	C₄×C₃₁	#575	1	4	yes	—
305	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
306	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
307	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
308	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
309	#533	—	2	1	no	—
310	#533	—	2	1	no	—
311	#533	—	2	1	no	—
312	#533	—	2	1	no	—
313	#533	C₂²×C₃₁	1	4	yes	—
314	#533	C₂²×C₃₁	1	4	yes	complemented
315	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
316	C₂×#125	C₂²×C₃₁	1	2	yes	complemented
317	C₂×C₂²⋊Q₈	—	2	2	no	—
318	C₄×#41	—	2	2	no	—
319	C₄×#41	—	2	1	no	—
320	C₄×#41	—	2	1	no	—
321	C₄×#41	C₂²×C₃₁	1	4	yes	—
322	C₄×#41	C₂²×C₃₁	1	2	yes	—
323	C₄×#41	C₂²×C₃₁	1	2	yes	—
324	C₂×C₄×Q₈	—	2	2	no	—
325	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
326	C₂×C₄×D₄	—	2	2	no	—
327	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
328	C₂×#117	—	2	1	no	—
329	C₂×#117	C₂²×C₃₁	1	2	yes	—
330	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
331	C₂²×C₄²	—	2	1	no	—
332	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
333	#479	C₂²×C₃₁	1	16	yes	complemented
334	#478	C₂²×C₃₁	1	16	yes	complemented
335	#477	C₂²×C₃₁	1	8	yes	complemented
336	#476	C₂²×C₃₁	1	8	yes	complemented
337	#475	C₂²×C₃₁	1	8	yes	complemented
338	#474	C₂²×C₃₁	1	8	yes	complemented
339	#428	C₂²×C₃₁	1	8	yes	—
340	#428	C₂²×C₃₁	1	8	yes	complemented
341	#427	C₂²×C₃₁	1	8	yes	complemented
342	#423	C₂²×C₃₁	1	4	yes	—
343	#423	C₂²×C₃₁	1	2	yes	complemented
344	#423	C₂²×C₃₁	1	2	yes	complemented
345	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	complemented
346	#419	C₂²×C₃₁	1	4	yes	—
347	#418	C₂²×C₃₁	1	4	yes	—
348	#417	C₂²×C₃₁	1	8	yes	—
349	#416	C₂²×C₃₁	1	8	yes	—
350	C₂×C₃₁	—	2	2	no	—
351	C₂×C₃₁	—	2	2	no	—
352	C₂×C₃₁	—	2	2	no	—
353	C₂×C₃₁	—	4	4	no	—
354	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
355	C₂×C₃₁	C₂×#575	1	2	yes	—
356	C₂×#41	—	2	1	no	—
357	C₂×#41	—	2	1	no	—
358	C₂×#41	—	2	4	no	—
359	C₂×#41	Q₈×C₃₁	1	8	yes	complemented
360	C₂×#41	C₂³×C₃₁	1	2	yes	—
361	C₂×#41	C₂³×C₃₁	1	2	yes	—
362	C₂²×Q₈	—	2	2	no	—
363	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₂²×D₄	—	2	1	no	—
365	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	—	2	2	no	—
368	C₂³×C₄	D₄×C₃₁	1	2	yes	—
369	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×M₁₆	—	2	8	no	—
371	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
372	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
373	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
374	C₄⋊Q₈	—	2	2	no	—
375	C₄⋊Q₈	—	2	2	no	—
376	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
377	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
378	#125	—	2	2	no	—
379	#125	—	2	2	no	—
380	#125	C₂³×C₃₁	1	8	yes	—
381	C₂²⋊Q₈	—	2	4	no	—
382	C₂²⋊Q₈	—	2	4	no	—
383	C₂²⋊Q₈	—	2	4	no	—
384	C₂²⋊Q₈	—	2	4	no	—
385	C₄×Q₈	—	2	4	no	—
386	C₄×Q₈	—	2	4	no	—
387	C₄×Q₈	—	2	4	no	—
388	C₄×Q₈	—	2	4	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
392	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
393	C₄×D₄	—	2	4	no	—
394	C₄×D₄	—	2	4	no	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	4	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	C₂³×C₃₁	1	4	yes	—
400	C₄×D₄	C₂³×C₃₁	1	4	yes	—
401	#117	—	2	1	no	—
402	#117	—	2	1	no	—
403	#117	—	2	2	no	—
404	#117	—	2	2	no	—
405	#117	—	2	4	no	—
406	#117	—	2	2	no	—
407	#117	—	2	2	no	—
408	#117	C₂³×C₃₁	1	2	yes	—
409	#117	C₂³×C₃₁	1	2	yes	—
410	#117	C₂³×C₃₁	1	4	yes	—
411	#117	C₂³×C₃₁	1	4	yes	—
412	C₂×C₄⋊C₄	—	2	2	no	—
413	C₂×C₄⋊C₄	—	2	4	no	—
414	C₂×C₄⋊C₄	—	2	2	no	—
415	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
417	C₂×C₂²⋊C₄	—	2	1	no	—
418	C₂×C₂²⋊C₄	—	2	2	no	—
419	C₂×C₂²⋊C₄	—	2	2	no	—
420	C₂×C₄²	—	2	1	no	—
421	C₂×C₄²	—	2	2	no	—
422	C₂×C₄²	—	2	2	no	—
423	C₂×C₄²	—	2	4	no	—
424	C₂×C₄²	—	2	2	no	—
425	C₂×C₄²	—	2	2	no	—
426	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄²	D₄×C₃₁	1	2	yes	—
428	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
432	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
435	#107	C₂³×C₃₁	1	16	yes	—
436	#106	C₂³×C₃₁	1	16	yes	—
437	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
438	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
439	#103	C₂³×C₃₁	1	16	yes	—
440	#102	C₂³×C₃₁	1	16	yes	—
441	C₃₁	#60484	1	1	yes	characteristic, direct product factor, Sylow
442	C₂⁴	—	2	1	no	—
443	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
444	#41	—	2	8	no	—
445	#41	—	2	8	no	—
446	#41	—	2	4	no	—
447	#41	—	2	4	no	—
448	#41	C₂×Q₈×C₃₁	1	8	yes	—
449	#41	C₂×Q₈×C₃₁	1	8	yes	complemented
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	4	no	—
452	C₂×Q₈	—	2	4	no	—
453	C₂×Q₈	—	2	2	no	—
454	C₂×Q₈	—	4	8	no	—
455	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	complemented
456	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
457	C₂×D₄	—	2	8	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	complemented
463	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	8	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	—	4	8	no	—
474	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
477	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
479	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
480	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
484	C₂²×C₄	#41×C₃₁	1	4	yes	—
485	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
488	M₁₆	—	2	8	no	—
489	M₁₆	—	2	8	no	—
490	C₂×C₈	—	2	8	no	—
491	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	8	no	—
495	C₄⋊C₄	—	2	8	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
501	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₄²	—	2	2	no	—
509	C₄²	—	2	2	no	—
510	C₄²	—	2	4	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	—	2	4	no	—
514	C₄²	C₂×D₄×C₃₁	1	2	yes	—
515	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
516	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
517	C₄²	C₂×D₄×C₃₁	1	2	yes	—
518	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
519	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
520	C₄²	C₂×D₄×C₃₁	1	2	yes	—
521	C₄²	C₂×D₄×C₃₁	1	2	yes	—
522	C₄²	C₂⁴×C₃₁	1	2	yes	—
523	C₄²	C₂⁴×C₃₁	1	2	yes	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	4	2	no	—
527	C₂³	—	4	4	no	—
528	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
529	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
530	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
531	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
532	Q₈	—	2	4	no	—
533	Q₈	—	2	2	no	—
534	Q₈	—	4	8	no	—
535	Q₈	—	4	8	no	—
536	Q₈	C₂²×Q₈×C₃₁	1	4	yes	—
537	D₄	—	2	8	no	—
538	D₄	—	2	8	no	—
539	D₄	—	2	4	no	—
540	D₄	—	2	2	no	—
541	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	8	no	—
545	C₂×C₄	—	2	8	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	8	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	8	no	—
555	C₂×C₄	—	4	4	no	—
556	C₂×C₄	—	4	8	no	—
557	C₂×C₄	—	4	8	no	—
558	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
559	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
560	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
561	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
562	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
563	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
564	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
568	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
569	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
570	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
571	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
572	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
573	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
574	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
575	C₈	—	2	8	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	8	no	—
582	C₂²	—	4	4	no	—
583	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
584	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
585	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
586	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
587	C₄	—	2	4	no	—
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	4	4	no	—
592	C₄	—	4	8	no	—
593	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
594	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
595	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
596	C₄	C₃₁×#575	1	4	yes	—
597	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
598	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
599	C₂	—	2	2	no	—
600	C₂	—	2	2	no	—
601	C₂	—	2	2	no	—
602	C₂	—	4	4	no	—
603	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
604	C₂	C₂×C₃₁×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#3080	C₂	semidirect
4. C₃₁×#3079	C₂	semidirect
5. C₃₁×#3078	C₂	semidirect
6. C₃₁×#3077	C₂	semidirect
7. C₃₁×#3076	C₂	semidirect
8. C₃₁×#2911	C₂	semidirect
9. C₂×C₃₁×#423	C₂	semidirect
18. C₃₁×#533	C₂²	semidirect
19. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
20. C₂×C₃₁×#125	C₂²	semidirect
37. C₃₁×#479	C₂²	semidirect
38. C₃₁×#478	C₂²	semidirect
39. C₃₁×#477	C₂²	semidirect
40. C₃₁×#476	C₂²	semidirect
41. C₃₁×#475	C₂²	semidirect
42. C₃₁×#474	C₂²	semidirect
44. C₃₁×#428	C₂²	semidirect
45. C₃₁×#427	C₂²	semidirect
47. C₃₁×#423	C₂²	semidirect
48. C₃₁×#423	C₂²	semidirect
49. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
57. C₂×#41×C₃₁	Q₈	semidirect
146. #41×C₃₁	C₂×Q₈	semidirect
152. C₂×Q₈×C₃₁	C₂×Q₈	semidirect
159. C₂×D₄×C₃₁	C₂×Q₈	semidirect
221. #60484	C₃₁	direct (N × H)
274. C₂×#533	C₂×C₃₁	semidirect
276. #3080	C₂×C₃₁	semidirect
277. #3079	C₂×C₃₁	semidirect
278. #3078	C₂×C₃₁	semidirect
279. #3077	C₂×C₃₁	semidirect
280. #3076	C₂×C₃₁	semidirect
281. #2911	C₂×C₃₁	semidirect
282. C₂×#423	C₂×C₃₁	semidirect
314. #533	C₂²×C₃₁	semidirect
315. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
316. C₂×#125	C₂²×C₃₁	semidirect
333. #479	C₂²×C₃₁	semidirect
334. #478	C₂²×C₃₁	semidirect
335. #477	C₂²×C₃₁	semidirect
336. #476	C₂²×C₃₁	semidirect
337. #475	C₂²×C₃₁	semidirect
338. #474	C₂²×C₃₁	semidirect
340. #428	C₂²×C₃₁	semidirect
341. #427	C₂²×C₃₁	semidirect
343. #423	C₂²×C₃₁	semidirect
344. #423	C₂²×C₃₁	semidirect
345. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
359. C₂×#41	Q₈×C₃₁	semidirect
441. C₃₁	#60484	direct (N × H)
449. #41	C₂×Q₈×C₃₁	semidirect
455. C₂×Q₈	C₂×Q₈×C₃₁	semidirect
462. C₂×D₄	C₂×Q₈×C₃₁	semidirect

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×C₄×#41×C₃₁	2
C₃₁×#3080	2
C₃₁×#3079	2
C₃₁×#3078	2
C₃₁×#3077	2
C₃₁×#3076	2
C₃₁×#2911	2
C₂×C₃₁×#423	2
C₃₁×#2896	2
#60484	31

Subgroups — C31×#60484 #21517243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60484 #21517243