Subgroups — C₃₁×#60618 #21517377

Order 7936.

2170 conjugacy classes of subgroups.

The subgroup lattice has 538 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×#3142	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3141	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3140	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3139	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3138	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3135	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3131	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2896	C₂	1	4	yes	maximal, complemented
11	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#533	—	2	2	no	—
15	C₃₁×#533	—	2	2	no	—
16	C₃₁×#533	C₂²	1	2	yes	—
17	C₃₁×#533	C₂²	1	2	yes	—
18	C₃₁×#533	C₂²	1	2	yes	complemented
19	C₃₁×#533	C₂²	1	2	yes	complemented
20	C₃₁×#532	—	2	2	no	—
21	C₃₁×#532	—	2	2	no	—
22	C₃₁×#532	C₂²	1	2	yes	—
23	C₃₁×#532	C₂²	1	2	yes	—
24	C₃₁×#532	C₂²	1	2	yes	—
25	C₃₁×#532	C₂²	1	2	yes	—
26	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
28	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#125	C₂²	1	2	yes	complemented
30	C₂×C₃₁×#124	C₂²	1	2	yes	—
31	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
32	C₂×C₃₁×#121	—	2	2	no	—
33	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
34	C₂×C₄×D₄×C₃₁	—	2	2	no	—
35	C₂×C₃₁×#117	C₂²	1	2	yes	—
36	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#491	C₂²	1	8	yes	complemented
38	C₃₁×#490	C₂²	1	8	yes	complemented
39	C₃₁×#489	C₂²	1	16	yes	complemented
40	C₃₁×#488	C₂²	1	16	yes	complemented
41	C₃₁×#487	C₂²	1	8	yes	complemented
42	C₃₁×#486	C₂²	1	8	yes	complemented
43	C₃₁×#419	C₂²	1	8	yes	—
44	C₃₁×#418	C₂²	1	8	yes	—
45	C₃₁×#417	C₂²	1	16	yes	complemented
46	C₃₁×#416	C₂²	1	16	yes	complemented
47	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
48	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
49	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
50	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
51	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
52	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
53	C₂×#41×C₃₁	—	2	2	no	—
54	C₂×#41×C₃₁	—	2	2	no	—
55	C₂×#41×C₃₁	—	2	16	no	—
56	C₂×#41×C₃₁	C₂³	1	4	yes	—
57	C₂×#41×C₃₁	C₂³	1	4	yes	—
58	C₂²×Q₈×C₃₁	—	2	2	no	—
59	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×D₄×C₃₁	—	2	2	no	—
61	C₂²×D₄×C₃₁	—	2	2	no	—
62	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	—	2	2	no	—
64	C₂³×C₄×C₃₁	D₄	1	2	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₃₁	—	2	8	no	—
67	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
68	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
69	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
70	C₃₁×C₄⋊Q₈	—	2	2	no	—
71	C₃₁×C₄⋊Q₈	—	2	2	no	—
72	C₃₁×C₄⋊Q₈	—	2	2	no	—
73	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
76	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
77	C₃₁×#127	—	2	2	no	—
78	C₃₁×#127	C₂³	1	2	yes	—
79	C₃₁×#127	C₂³	1	2	yes	—
80	C₃₁×#125	—	2	4	no	—
81	C₃₁×#125	C₂³	1	4	yes	—
82	C₃₁×#125	C₂³	1	4	yes	—
83	C₃₁×#124	—	2	4	no	—
84	C₃₁×#124	C₂³	1	4	yes	—
85	C₃₁×#124	C₂³	1	4	yes	—
86	C₃₁×C₂²⋊Q₈	—	2	4	no	—
87	C₃₁×C₂²⋊Q₈	—	2	4	no	—
88	C₃₁×C₂²⋊Q₈	—	2	4	no	—
89	C₃₁×C₂²⋊Q₈	—	2	4	no	—
90	C₃₁×#121	—	2	4	no	—
91	C₃₁×#121	—	2	4	no	—
92	C₃₁×#121	—	2	4	no	—
93	C₃₁×#121	—	2	4	no	—
94	C₄×Q₈×C₃₁	—	2	4	no	—
95	C₄×Q₈×C₃₁	—	2	4	no	—
96	C₄×Q₈×C₃₁	—	2	4	no	—
97	C₄×Q₈×C₃₁	—	2	4	no	—
98	C₄×D₄×C₃₁	—	2	4	no	—
99	C₄×D₄×C₃₁	—	2	4	no	—
100	C₄×D₄×C₃₁	—	2	4	no	—
101	C₄×D₄×C₃₁	—	2	4	no	—
102	C₃₁×#117	—	2	2	no	—
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	C₂³	1	4	yes	—
105	C₃₁×#117	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
108	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
112	C₂×C₄²×C₃₁	—	2	2	no	—
113	C₂×C₄²×C₃₁	D₄	1	2	yes	—
114	C₂×C₄²×C₃₁	D₄	1	2	yes	—
115	C₂×C₄²×C₃₁	D₄	1	2	yes	—
116	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₃₁×#103	C₂³	1	32	yes	—
123	C₃₁×#102	C₂³	1	32	yes	—
124	C₃₁×#97	C₂³	1	4	yes	—
125	C₃₁×#97	C₂³	1	4	yes	—
126	C₄×C₈×C₃₁	C₂³	1	4	yes	—
127	C₄×C₈×C₃₁	C₂³	1	4	yes	—
128	C₂⁴×C₃₁	—	2	2	no	—
129	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
130	#41×C₃₁	—	2	16	no	—
131	#41×C₃₁	—	2	16	no	—
132	#41×C₃₁	—	4	8	no	—
133	#41×C₃₁	—	4	8	no	—
134	C₂×Q₈×C₃₁	—	2	4	no	—
135	C₂×Q₈×C₃₁	—	2	4	no	—
136	C₂×Q₈×C₃₁	—	2	2	no	—
137	C₂×Q₈×C₃₁	—	2	8	no	—
138	C₂×Q₈×C₃₁	—	4	8	no	—
139	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	4	no	—
142	C₂×D₄×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	2	4	no	—
144	C₂×D₄×C₃₁	—	2	2	no	—
145	C₂×D₄×C₃₁	—	2	8	no	—
146	C₂×D₄×C₃₁	—	4	8	no	—
147	C₂×D₄×C₃₁	—	4	8	no	—
148	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	4	4	no	—
154	C₂²×C₄×C₃₁	—	4	8	no	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	#41	1	8	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	M₁₆×C₃₁	—	2	8	no	—
169	M₁₆×C₃₁	—	2	8	no	—
170	C₂×C₈×C₃₁	—	2	8	no	—
171	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	4	no	—
174	C₄⋊C₄×C₃₁	—	2	8	no	—
175	C₄⋊C₄×C₃₁	—	2	8	no	—
176	C₄⋊C₄×C₃₁	—	2	4	no	—
177	C₄⋊C₄×C₃₁	—	2	4	no	—
178	C₄⋊C₄×C₃₁	—	2	4	no	—
179	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
180	C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₂²⋊C₄×C₃₁	—	2	4	no	—
182	C₂²⋊C₄×C₃₁	—	2	8	no	—
183	C₂²⋊C₄×C₃₁	—	2	8	no	—
184	C₄²×C₃₁	—	2	4	no	—
185	C₄²×C₃₁	—	2	4	no	—
186	C₄²×C₃₁	C₂×D₄	1	2	yes	—
187	C₄²×C₃₁	C₂×D₄	1	2	yes	—
188	C₄²×C₃₁	C₂×D₄	1	2	yes	—
189	C₄²×C₃₁	C₂×D₄	1	2	yes	—
190	C₄²×C₃₁	C₂×D₄	1	2	yes	—
191	C₄²×C₃₁	C₂×D₄	1	2	yes	—
192	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
193	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
194	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
196	#60618	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	4	no	—
198	C₂³×C₃₁	—	4	2	no	—
199	C₂³×C₃₁	—	4	8	no	—
200	C₂³×C₃₁	—	4	4	no	—
201	C₂³×C₃₁	#124	1	4	yes	—
202	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₃₁	—	2	8	no	—
206	Q₈×C₃₁	—	4	8	no	—
207	Q₈×C₃₁	—	4	4	no	—
208	Q₈×C₃₁	—	4	8	no	—
209	D₄×C₃₁	—	2	8	no	—
210	D₄×C₃₁	—	4	8	no	—
211	D₄×C₃₁	—	4	8	no	—
212	D₄×C₃₁	—	4	8	no	—
213	D₄×C₃₁	—	4	4	no	—
214	D₄×C₃₁	—	4	8	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	8	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	8	no	—
224	C₂×C₄×C₃₁	—	4	4	no	—
225	C₂×C₄×C₃₁	—	4	8	no	—
226	C₂×C₄×C₃₁	—	4	8	no	—
227	C₂×C₄×C₃₁	#124	1	4	yes	—
228	C₂×C₄×C₃₁	#124	1	4	yes	—
229	C₂×C₄×C₃₁	#124	1	4	yes	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
238	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
239	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
240	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
241	C₈×C₃₁	—	2	8	no	—
242	C₂×#533	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
244	#3142	C₂×C₃₁	1	4	yes	complemented
245	#3141	C₂×C₃₁	1	4	yes	complemented
246	#3140	C₂×C₃₁	1	4	yes	complemented
247	#3139	C₂×C₃₁	1	2	yes	complemented
248	#3138	C₂×C₃₁	1	2	yes	complemented
249	#3135	C₂×C₃₁	1	4	yes	complemented
250	#3131	C₂×C₃₁	1	4	yes	complemented
251	#2896	C₂×C₃₁	1	4	yes	complemented
252	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	4	4	no	—
257	C₂²×C₃₁	—	4	4	no	—
258	C₂²×C₃₁	—	4	8	no	—
259	C₂²×C₃₁	—	8	8	no	—
260	C₂²×C₃₁	C₂×#124	1	2	yes	—
261	C₂²×C₃₁	C₂×#124	1	2	yes	—
262	C₂²×C₃₁	C₂×#124	1	2	yes	—
263	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
264	C₄×C₃₁	—	2	2	no	—
265	C₄×C₃₁	—	2	2	no	—
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	4	4	no	—
268	C₄×C₃₁	—	4	8	no	—
269	C₄×C₃₁	C₂×#124	1	2	yes	—
270	C₄×C₃₁	C₂×#124	1	2	yes	—
271	C₄×C₃₁	C₂×#124	1	2	yes	—
272	C₄×C₃₁	#575	1	4	yes	—
273	C₄×C₃₁	C₂²×#41	1	2	yes	—
274	C₂²×#41	C₂²×C₃₁	1	2	yes	—
275	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
276	#533	—	2	2	no	—
277	#533	—	2	2	no	—
278	#533	C₂²×C₃₁	1	2	yes	—
279	#533	C₂²×C₃₁	1	2	yes	—
280	#533	C₂²×C₃₁	1	2	yes	complemented
281	#533	C₂²×C₃₁	1	2	yes	complemented
282	#532	—	2	2	no	—
283	#532	—	2	2	no	—
284	#532	C₂²×C₃₁	1	2	yes	—
285	#532	C₂²×C₃₁	1	2	yes	—
286	#532	C₂²×C₃₁	1	2	yes	—
287	#532	C₂²×C₃₁	1	2	yes	—
288	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
290	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×#125	C₂²×C₃₁	1	2	yes	complemented
292	C₂×#124	C₂²×C₃₁	1	2	yes	—
293	C₂×C₂²⋊Q₈	—	2	2	no	—
294	C₂×#121	—	2	2	no	—
295	C₂×C₄×Q₈	—	2	2	no	—
296	C₂×C₄×D₄	—	2	2	no	—
297	C₂×#117	C₂²×C₃₁	1	2	yes	—
298	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
299	#491	C₂²×C₃₁	1	8	yes	complemented
300	#490	C₂²×C₃₁	1	8	yes	complemented
301	#489	C₂²×C₃₁	1	16	yes	complemented
302	#488	C₂²×C₃₁	1	16	yes	complemented
303	#487	C₂²×C₃₁	1	8	yes	complemented
304	#486	C₂²×C₃₁	1	8	yes	complemented
305	#419	C₂²×C₃₁	1	8	yes	—
306	#418	C₂²×C₃₁	1	8	yes	—
307	#417	C₂²×C₃₁	1	16	yes	complemented
308	#416	C₂²×C₃₁	1	16	yes	complemented
309	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
310	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
311	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
312	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
313	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
314	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
315	C₂×C₃₁	—	2	2	no	—
316	C₂×C₃₁	—	2	2	no	—
317	C₂×C₃₁	—	2	2	no	—
318	C₂×C₃₁	—	8	4	no	—
319	C₂×C₃₁	C₂²×#124	1	1	yes	characteristic
320	C₂×C₃₁	C₂×#575	1	2	yes	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	16	no	—
324	C₂×#41	C₂³×C₃₁	1	4	yes	—
325	C₂×#41	C₂³×C₃₁	1	4	yes	—
326	C₂²×Q₈	—	2	2	no	—
327	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
328	C₂²×D₄	—	2	2	no	—
329	C₂²×D₄	—	2	2	no	—
330	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
331	C₂³×C₄	—	2	2	no	—
332	C₂³×C₄	D₄×C₃₁	1	2	yes	—
333	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
334	C₂×M₁₆	—	2	8	no	—
335	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
336	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
337	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
338	C₄⋊Q₈	—	2	2	no	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	—	2	2	no	—
341	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
342	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
343	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
344	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
345	#127	—	2	2	no	—
346	#127	C₂³×C₃₁	1	2	yes	—
347	#127	C₂³×C₃₁	1	2	yes	—
348	#125	—	2	4	no	—
349	#125	C₂³×C₃₁	1	4	yes	—
350	#125	C₂³×C₃₁	1	4	yes	—
351	#124	—	2	4	no	—
352	#124	C₂³×C₃₁	1	4	yes	—
353	#124	C₂³×C₃₁	1	4	yes	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	#121	—	2	4	no	—
359	#121	—	2	4	no	—
360	#121	—	2	4	no	—
361	#121	—	2	4	no	—
362	C₄×Q₈	—	2	4	no	—
363	C₄×Q₈	—	2	4	no	—
364	C₄×Q₈	—	2	4	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×D₄	—	2	4	no	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	4	no	—
369	C₄×D₄	—	2	4	no	—
370	#117	—	2	2	no	—
371	#117	—	2	2	no	—
372	#117	C₂³×C₃₁	1	4	yes	—
373	#117	C₂³×C₃₁	1	4	yes	—
374	C₂×C₄⋊C₄	—	2	2	no	—
375	C₂×C₄⋊C₄	—	2	4	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
378	C₂×C₂²⋊C₄	—	2	2	no	—
379	C₂×C₂²⋊C₄	—	2	4	no	—
380	C₂×C₄²	—	2	2	no	—
381	C₂×C₄²	D₄×C₃₁	1	2	yes	—
382	C₂×C₄²	D₄×C₃₁	1	2	yes	—
383	C₂×C₄²	D₄×C₃₁	1	2	yes	—
384	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
385	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
390	#103	C₂³×C₃₁	1	32	yes	—
391	#102	C₂³×C₃₁	1	32	yes	—
392	#97	C₂³×C₃₁	1	4	yes	—
393	#97	C₂³×C₃₁	1	4	yes	—
394	C₄×C₈	C₂³×C₃₁	1	4	yes	—
395	C₄×C₈	C₂³×C₃₁	1	4	yes	—
396	C₃₁	#60618	1	1	yes	characteristic, direct product factor, Sylow
397	C₂⁴	—	2	2	no	—
398	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
399	#41	—	2	16	no	—
400	#41	—	2	16	no	—
401	#41	—	4	8	no	—
402	#41	—	4	8	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	8	no	—
407	C₂×Q₈	—	4	8	no	—
408	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	4	8	no	—
416	C₂×D₄	—	4	8	no	—
417	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	—	4	8	no	—
424	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
425	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
428	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
430	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	#41×C₃₁	1	8	yes	—
434	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
437	M₁₆	—	2	8	no	—
438	M₁₆	—	2	8	no	—
439	C₂×C₈	—	2	8	no	—
440	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	8	no	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	4	no	—
455	C₄²	C₂×D₄×C₃₁	1	2	yes	—
456	C₄²	C₂×D₄×C₃₁	1	2	yes	—
457	C₄²	C₂×D₄×C₃₁	1	2	yes	—
458	C₄²	C₂×D₄×C₃₁	1	2	yes	—
459	C₄²	C₂×D₄×C₃₁	1	2	yes	—
460	C₄²	C₂×D₄×C₃₁	1	2	yes	—
461	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
462	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
463	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
464	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
465	C₂³	—	2	4	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	8	no	—
468	C₂³	—	4	4	no	—
469	C₂³	C₃₁×#124	1	4	yes	—
470	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
471	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
472	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
473	Q₈	—	2	8	no	—
474	Q₈	—	4	8	no	—
475	Q₈	—	4	4	no	—
476	Q₈	—	4	8	no	—
477	D₄	—	2	8	no	—
478	D₄	—	4	8	no	—
479	D₄	—	4	8	no	—
480	D₄	—	4	8	no	—
481	D₄	—	4	4	no	—
482	D₄	—	4	8	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	8	no	—
492	C₂×C₄	—	4	4	no	—
493	C₂×C₄	—	4	8	no	—
494	C₂×C₄	—	4	8	no	—
495	C₂×C₄	C₃₁×#124	1	4	yes	—
496	C₂×C₄	C₃₁×#124	1	4	yes	—
497	C₂×C₄	C₃₁×#124	1	4	yes	—
498	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
499	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
500	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
501	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
505	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
506	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
507	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
508	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
509	C₈	—	2	8	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	8	no	—
516	C₂²	—	8	8	no	—
517	C₂²	C₂×C₃₁×#124	1	2	yes	—
518	C₂²	C₂×C₃₁×#124	1	2	yes	—
519	C₂²	C₂×C₃₁×#124	1	2	yes	—
520	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	4	4	no	—
525	C₄	—	4	8	no	—
526	C₄	C₂×C₃₁×#124	1	2	yes	—
527	C₄	C₂×C₃₁×#124	1	2	yes	—
528	C₄	C₂×C₃₁×#124	1	2	yes	—
529	C₄	C₃₁×#575	1	4	yes	—
530	C₄	C₂²×#41×C₃₁	1	2	yes	—
531	C₂	—	2	2	no	—
532	C₂	—	2	2	no	—
533	C₂	—	2	2	no	—
534	C₂	—	8	4	no	—
535	C₂	C₂²×C₃₁×#124	1	1	yes	characteristic
536	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	C₂×C₃₁×#532	C₂
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×#3142	C₂	semidirect
4. C₃₁×#3141	C₂	semidirect
5. C₃₁×#3140	C₂	semidirect
6. C₃₁×#3139	C₂	semidirect
7. C₃₁×#3138	C₂	semidirect
8. C₃₁×#3135	C₂	semidirect
9. C₃₁×#3131	C₂	semidirect
10. C₃₁×#2896	C₂	semidirect
11. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
18. C₃₁×#533	C₂²	semidirect
19. C₃₁×#533	C₂²	semidirect
27. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
29. C₂×C₃₁×#125	C₂²	semidirect
37. C₃₁×#491	C₂²	semidirect
38. C₃₁×#490	C₂²	semidirect
39. C₃₁×#489	C₂²	semidirect
40. C₃₁×#488	C₂²	semidirect
41. C₃₁×#487	C₂²	semidirect
42. C₃₁×#486	C₂²	semidirect
45. C₃₁×#417	C₂²	semidirect
46. C₃₁×#416	C₂²	semidirect
49. C₄×M₁₆×C₃₁	C₂²	semidirect
50. C₄×M₁₆×C₃₁	C₂²	semidirect
51. C₂×C₃₁×#97	C₂²	semidirect
52. C₂×C₄×C₈×C₃₁	C₂²	semidirect
196. #60618	C₃₁	direct (N × H)
242. C₂×#533	C₂×C₃₁	semidirect
244. #3142	C₂×C₃₁	semidirect
245. #3141	C₂×C₃₁	semidirect
246. #3140	C₂×C₃₁	semidirect
247. #3139	C₂×C₃₁	semidirect
248. #3138	C₂×C₃₁	semidirect
249. #3135	C₂×C₃₁	semidirect
250. #3131	C₂×C₃₁	semidirect
251. #2896	C₂×C₃₁	semidirect
252. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
280. #533	C₂²×C₃₁	semidirect
281. #533	C₂²×C₃₁	semidirect
289. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
291. C₂×#125	C₂²×C₃₁	semidirect
299. #491	C₂²×C₃₁	semidirect
300. #490	C₂²×C₃₁	semidirect
301. #489	C₂²×C₃₁	semidirect
302. #488	C₂²×C₃₁	semidirect
303. #487	C₂²×C₃₁	semidirect
304. #486	C₂²×C₃₁	semidirect
307. #417	C₂²×C₃₁	semidirect
308. #416	C₂²×C₃₁	semidirect
311. C₄×M₁₆	C₂²×C₃₁	semidirect
312. C₄×M₁₆	C₂²×C₃₁	semidirect
313. C₂×#97	C₂²×C₃₁	semidirect
314. C₂×C₄×C₈	C₂²×C₃₁	semidirect
396. C₃₁	#60618	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×#3142	2
C₃₁×#3141	2
C₃₁×#3140	2
C₃₁×#3139	2
C₃₁×#3138	2
C₃₁×#3135	2
C₃₁×#3131	2
C₃₁×#2896	2
C₂×C₄×M₁₆×C₃₁	2
#60618	31

Subgroups — C31×#60618 #21517377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60618 #21517377