Subgroups — C₃₁×#17517 #21474276

Order 7936.

1096 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3481	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1494	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1490	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1489	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1484	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1483	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#559	C₄	1	1	yes	characteristic
14	C₃₁×#554	C₄	1	1	yes	characteristic
15	C₃₁×#554	C₄	1	1	yes	characteristic
16	C₃₁×#553	—	2	1	no	—
17	C₃₁×#553	C₄	1	2	yes	—
18	C₃₁×#551	—	2	2	no	—
19	C₃₁×#546	—	2	1	no	—
20	C₃₁×#546	C₄	1	2	yes	—
21	D₄²×C₃₁	C₄	1	1	yes	characteristic
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₃₁×#529	—	2	1	no	—
24	C₃₁×#529	C₂²	1	2	yes	—
25	C₃₁×#525	—	2	2	no	—
26	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₃₁	—	2	1	no	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#446	—	2	1	no	—
31	C₃₁×#435	—	2	2	no	—
32	C₈×D₄×C₃₁	—	2	1	no	—
33	C₃₁×#432	C₂²	1	2	yes	complemented
34	C₃₁×#431	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₃₁×#405	—	2	2	no	—
38	C₃₁×#405	—	2	1	no	—
39	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
40	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
41	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
46	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
47	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
48	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
51	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
52	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×D₄×C₃₁	—	2	1	no	—
57	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂³×C₄×C₃₁	—	2	2	no	—
59	C₃₁×#131	—	2	8	no	—
60	C₂×M₁₆×C₃₁	—	2	2	no	—
61	C₂×M₁₆×C₃₁	—	2	1	no	—
62	C₂×M₁₆×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	—	2	1	no	—
65	C₂²×C₈×C₃₁	—	2	2	no	—
66	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
67	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
68	C₃₁×#126	—	2	2	no	—
69	C₃₁×#126	—	2	2	no	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#125	C₂×C₄	1	2	yes	—
72	C₃₁×#124	—	2	1	no	—
73	C₃₁×#124	—	2	1	no	—
74	C₃₁×#124	C₂×C₄	1	2	yes	—
75	C₃₁×#123	—	2	2	no	—
76	C₃₁×#123	—	2	2	no	—
77	C₃₁×#123	—	2	2	no	—
78	C₃₁×#123	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	—	2	2	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
83	C₃₁×#121	—	2	2	no	—
84	C₃₁×#121	—	2	1	no	—
85	C₃₁×#121	—	2	1	no	—
86	C₃₁×#121	—	2	2	no	—
87	C₃₁×#121	C₂×C₄	1	2	yes	—
88	C₃₁×#121	C₂×C₄	1	2	yes	—
89	C₃₁×#120	—	2	2	no	—
90	C₃₁×#120	—	2	2	no	—
91	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
92	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	—	2	2	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	2	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	2	no	—
104	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
105	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
106	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
109	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
110	C₄×D₄×C₃₁	D₄	1	2	yes	—
111	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₄×D₄×C₃₁	C₂³	1	2	yes	—
113	C₃₁×#117	—	2	1	no	—
114	C₃₁×#117	—	2	2	no	—
115	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
116	C₃₁×#117	D₄	1	2	yes	—
117	C₃₁×#117	D₄	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
120	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
122	C₂×C₄²×C₃₁	—	2	1	no	—
123	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₃₁×C₄⋊C₈	—	2	2	no	—
126	C₃₁×C₄⋊C₈	—	2	1	no	—
127	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
128	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
129	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
130	C₃₁×C₂²⋊C₈	—	2	1	no	—
131	C₃₁×C₂²⋊C₈	—	2	2	no	—
132	C₃₁×#97	—	2	1	no	—
133	C₃₁×#97	—	2	2	no	—
134	C₃₁×#97	—	2	2	no	—
135	C₄×C₈×C₃₁	—	2	1	no	—
136	C₄×C₈×C₃₁	—	2	2	no	—
137	C₄×C₈×C₃₁	—	2	1	no	—
138	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₃₁	C₂³	1	2	yes	—
140	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
141	C₂⁴×C₃₁	—	2	2	no	—
142	#41×C₃₁	—	2	4	no	—
143	#41×C₃₁	—	2	2	no	—
144	#41×C₃₁	—	4	1	no	—
145	#41×C₃₁	—	8	1	no	—
146	#41×C₃₁	—	8	1	no	—
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂×D₄×C₃₁	—	2	4	no	—
151	C₂×D₄×C₃₁	—	2	1	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	—	2	2	no	—
154	C₂×D₄×C₃₁	—	2	2	no	—
155	C₂×D₄×C₃₁	—	2	1	no	—
156	C₂×D₄×C₃₁	—	2	1	no	—
157	C₂×D₄×C₃₁	—	4	2	no	—
158	C₂×D₄×C₃₁	—	4	4	no	—
159	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
160	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
161	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
162	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	2	1	no	—
166	C₂²×C₄×C₃₁	—	2	1	no	—
167	C₂²×C₄×C₃₁	—	2	2	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	2	no	—
170	C₂²×C₄×C₃₁	—	8	2	no	—
171	C₂²×C₄×C₃₁	—	8	2	no	—
172	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	M₁₆×C₃₁	—	2	8	no	—
179	M₁₆×C₃₁	—	2	4	no	—
180	M₁₆×C₃₁	—	8	1	no	—
181	C₂×C₈×C₃₁	—	2	8	no	—
182	C₂×C₈×C₃₁	—	2	4	no	—
183	C₂×C₈×C₃₁	—	2	1	no	—
184	C₂×C₈×C₃₁	—	2	1	no	—
185	C₂×C₈×C₃₁	—	2	2	no	—
186	C₂×C₈×C₃₁	—	2	1	no	—
187	C₂×C₈×C₃₁	—	2	1	no	—
188	C₂×C₈×C₃₁	—	2	2	no	—
189	C₂×C₈×C₃₁	—	8	2	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	2	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	2	2	no	—
195	C₄⋊C₄×C₃₁	—	2	2	no	—
196	C₄⋊C₄×C₃₁	—	2	1	no	—
197	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
200	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
201	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
202	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
203	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
204	C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂²⋊C₄×C₃₁	—	2	1	no	—
206	C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	2	no	—
208	C₂²⋊C₄×C₃₁	—	2	2	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	—	2	1	no	—
211	C₂²⋊C₄×C₃₁	—	2	2	no	—
212	C₂²⋊C₄×C₃₁	—	2	2	no	—
213	C₂²⋊C₄×C₃₁	—	2	1	no	—
214	C₂²⋊C₄×C₃₁	—	2	2	no	—
215	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
216	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
217	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
218	C₄²×C₃₁	—	2	1	no	—
219	C₄²×C₃₁	—	2	1	no	—
220	C₄²×C₃₁	—	2	1	no	—
221	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
222	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
223	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
224	C₄²×C₃₁	C₂×D₄	1	2	yes	—
225	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₃₁	C₂×D₄	1	2	yes	—
227	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
228	#17517	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂³×C₃₁	—	2	1	no	—
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	4	4	no	—
232	C₂³×C₃₁	—	4	4	no	—
233	C₂³×C₃₁	—	8	2	no	—
234	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
236	Q₈×C₃₁	—	2	2	no	—
237	Q₈×C₃₁	—	2	1	no	—
238	Q₈×C₃₁	—	2	1	no	—
239	Q₈×C₃₁	—	8	1	no	—
240	D₄×C₃₁	—	2	4	no	—
241	D₄×C₃₁	—	2	2	no	—
242	D₄×C₃₁	—	2	1	no	—
243	D₄×C₃₁	—	2	2	no	—
244	D₄×C₃₁	—	4	4	no	—
245	D₄×C₃₁	—	4	2	no	—
246	D₄×C₃₁	—	4	4	no	—
247	D₄×C₃₁	—	4	2	no	—
248	D₄×C₃₁	—	8	1	no	—
249	D₄×C₃₁	—	8	2	no	—
250	D₄×C₃₁	#104	1	4	yes	—
251	D₄×C₃₁	#131	1	2	yes	complemented
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	1	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	1	no	—
259	C₂×C₄×C₃₁	—	2	2	no	—
260	C₂×C₄×C₃₁	—	2	1	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	1	no	—
263	C₂×C₄×C₃₁	—	2	1	no	—
264	C₂×C₄×C₃₁	—	2	1	no	—
265	C₂×C₄×C₃₁	—	2	2	no	—
266	C₂×C₄×C₃₁	—	4	2	no	—
267	C₂×C₄×C₃₁	—	4	1	no	—
268	C₂×C₄×C₃₁	—	4	2	no	—
269	C₂×C₄×C₃₁	—	4	1	no	—
270	C₂×C₄×C₃₁	—	4	2	no	—
271	C₂×C₄×C₃₁	—	4	2	no	—
272	C₂×C₄×C₃₁	—	8	2	no	—
273	C₂×C₄×C₃₁	—	8	2	no	—
274	C₂×C₄×C₃₁	—	8	2	no	—
275	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
277	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
278	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
280	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
282	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
284	C₈×C₃₁	—	2	2	no	—
285	C₈×C₃₁	—	2	4	no	—
286	C₈×C₃₁	—	2	2	no	—
287	C₈×C₃₁	—	4	1	no	—
288	C₈×C₃₁	—	4	2	no	—
289	C₈×C₃₁	—	4	1	no	—
290	#3481	C₂×C₃₁	1	1	yes	characteristic
291	#2990	C₂×C₃₁	1	1	yes	characteristic, complemented
292	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#1494	C₂×C₃₁	1	2	yes	complemented
294	#1492	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#1490	C₂×C₃₁	1	2	yes	complemented
296	#1489	C₂×C₃₁	1	1	yes	characteristic, complemented
297	#1484	C₂×C₃₁	1	1	yes	characteristic, complemented
298	#1483	C₂×C₃₁	1	2	yes	complemented
299	#1476	C₂×C₃₁	1	1	yes	characteristic, complemented
300	C₂×D₄⋊C₈	C₂×C₃₁	1	2	yes	complemented
301	C₂²×C₃₁	—	2	2	no	—
302	C₂²×C₃₁	—	2	2	no	—
303	C₂²×C₃₁	—	2	1	no	—
304	C₂²×C₃₁	—	2	2	no	—
305	C₂²×C₃₁	—	2	1	no	—
306	C₂²×C₃₁	—	2	1	no	—
307	C₂²×C₃₁	—	4	1	no	—
308	C₂²×C₃₁	—	4	1	no	—
309	C₂²×C₃₁	—	4	2	no	—
310	C₂²×C₃₁	—	4	2	no	—
311	C₂²×C₃₁	—	4	2	no	—
312	C₂²×C₃₁	—	8	8	no	—
313	C₂²×C₃₁	C₂×#104	1	4	yes	—
314	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
315	C₄×C₃₁	—	2	1	no	—
316	C₄×C₃₁	—	2	2	no	—
317	C₄×C₃₁	—	2	2	no	—
318	C₄×C₃₁	—	2	1	no	—
319	C₄×C₃₁	—	2	1	no	—
320	C₄×C₃₁	—	2	2	no	—
321	C₄×C₃₁	—	4	1	no	—
322	C₄×C₃₁	—	4	1	no	—
323	C₄×C₃₁	—	4	2	no	—
324	C₄×C₃₁	—	4	1	no	—
325	C₄×C₃₁	C₂×#104	1	2	yes	—
326	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
327	C₄×C₃₁	#568	1	1	yes	characteristic
328	C₄×C₃₁	#576	1	1	yes	characteristic
329	C₄×C₃₁	#577	1	1	yes	characteristic
330	C₂×#131	—	2	1	no	—
331	#559	C₄×C₃₁	1	1	yes	characteristic
332	#554	C₄×C₃₁	1	1	yes	characteristic
333	#554	C₄×C₃₁	1	1	yes	characteristic
334	#553	—	2	1	no	—
335	#553	C₄×C₃₁	1	2	yes	—
336	#551	—	2	2	no	—
337	#546	—	2	1	no	—
338	#546	C₄×C₃₁	1	2	yes	—
339	D₄²	C₄×C₃₁	1	1	yes	characteristic
340	#532	C₂²×C₃₁	1	1	yes	characteristic
341	#529	—	2	1	no	—
342	#529	C₂²×C₃₁	1	2	yes	—
343	#525	—	2	2	no	—
344	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
345	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
346	C₂×C₄×D₄	—	2	1	no	—
347	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
348	#446	—	2	1	no	—
349	#435	—	2	2	no	—
350	C₈×D₄	—	2	1	no	—
351	#432	C₂²×C₃₁	1	2	yes	complemented
352	#431	C₂²×C₃₁	1	1	yes	characteristic, complemented
353	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
354	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
355	#405	—	2	2	no	—
356	#405	—	2	1	no	—
357	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
358	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
359	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
360	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
361	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
362	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
363	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
364	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
365	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
366	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
367	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
368	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
369	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
370	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
371	C₂×C₃₁	—	2	4	no	—
372	C₂×C₃₁	—	4	1	no	—
373	C₂×C₃₁	—	4	2	no	—
374	C₂×C₃₁	—	8	2	no	—
375	C₂×C₃₁	#2880	1	1	yes	characteristic
376	C₂×C₃₁	#2897	1	1	yes	characteristic
377	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
378	C₂×#41	—	2	1	no	—
379	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
380	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
381	C₂²×D₄	—	2	1	no	—
382	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
383	C₂³×C₄	—	2	2	no	—
384	#131	—	2	8	no	—
385	C₂×M₁₆	—	2	2	no	—
386	C₂×M₁₆	—	2	1	no	—
387	C₂×M₁₆	—	2	1	no	—
388	C₂²×C₈	—	2	2	no	—
389	C₂²×C₈	—	2	1	no	—
390	C₂²×C₈	—	2	2	no	—
391	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
392	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
393	#126	—	2	2	no	—
394	#126	—	2	2	no	—
395	#125	—	2	1	no	—
396	#125	C₂×C₄×C₃₁	1	2	yes	—
397	#124	—	2	1	no	—
398	#124	—	2	1	no	—
399	#124	C₂×C₄×C₃₁	1	2	yes	—
400	#123	—	2	2	no	—
401	#123	—	2	2	no	—
402	#123	—	2	2	no	—
403	#123	—	2	2	no	—
404	C₂²⋊Q₈	—	2	1	no	—
405	C₂²⋊Q₈	—	2	2	no	—
406	C₂²⋊Q₈	—	2	2	no	—
407	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
408	#121	—	2	2	no	—
409	#121	—	2	1	no	—
410	#121	—	2	1	no	—
411	#121	—	2	2	no	—
412	#121	C₂×C₄×C₃₁	1	2	yes	—
413	#121	C₂×C₄×C₃₁	1	2	yes	—
414	#120	—	2	2	no	—
415	#120	—	2	2	no	—
416	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
417	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
418	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
419	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
420	C₄×D₄	—	2	1	no	—
421	C₄×D₄	—	2	2	no	—
422	C₄×D₄	—	2	1	no	—
423	C₄×D₄	—	2	1	no	—
424	C₄×D₄	—	2	1	no	—
425	C₄×D₄	—	2	2	no	—
426	C₄×D₄	—	2	1	no	—
427	C₄×D₄	—	2	2	no	—
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
430	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
431	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
432	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
433	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
434	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
435	C₄×D₄	D₄×C₃₁	1	2	yes	—
436	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
437	C₄×D₄	C₂³×C₃₁	1	2	yes	—
438	#117	—	2	1	no	—
439	#117	—	2	2	no	—
440	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
441	#117	D₄×C₃₁	1	2	yes	—
442	#117	D₄×C₃₁	1	1	yes	characteristic
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
445	C₂×C₂²⋊C₄	—	2	2	no	—
446	C₂×C₂²⋊C₄	—	2	2	no	—
447	C₂×C₄²	—	2	1	no	—
448	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
449	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
450	C₄⋊C₈	—	2	2	no	—
451	C₄⋊C₈	—	2	1	no	—
452	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
453	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
454	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
455	C₂²⋊C₈	—	2	1	no	—
456	C₂²⋊C₈	—	2	2	no	—
457	#97	—	2	1	no	—
458	#97	—	2	2	no	—
459	#97	—	2	2	no	—
460	C₄×C₈	—	2	1	no	—
461	C₄×C₈	—	2	2	no	—
462	C₄×C₈	—	2	1	no	—
463	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
464	C₄×C₈	C₂³×C₃₁	1	2	yes	—
465	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
466	C₃₁	#17517	1	1	yes	characteristic, direct product factor, Sylow
467	C₂⁴	—	2	2	no	—
468	#41	—	2	4	no	—
469	#41	—	2	2	no	—
470	#41	—	4	1	no	—
471	#41	—	8	1	no	—
472	#41	—	8	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
475	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	4	2	no	—
484	C₂×D₄	—	4	4	no	—
485	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
486	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
487	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
488	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	2	no	—
494	C₂²×C₄	—	2	2	no	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	8	2	no	—
497	C₂²×C₄	—	8	2	no	—
498	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
499	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
501	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
503	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
504	M₁₆	—	2	8	no	—
505	M₁₆	—	2	4	no	—
506	M₁₆	—	8	1	no	—
507	C₂×C₈	—	2	8	no	—
508	C₂×C₈	—	2	4	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	2	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	2	no	—
515	C₂×C₈	—	8	2	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
524	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
525	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
526	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
527	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
528	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
529	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	1	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	2	no	—
534	C₂²⋊C₄	—	2	2	no	—
535	C₂²⋊C₄	—	2	2	no	—
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	2	1	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
542	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
543	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
548	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
549	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
550	C₄²	C₂×D₄×C₃₁	1	2	yes	—
551	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
552	C₄²	C₂×D₄×C₃₁	1	2	yes	—
553	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
554	C₂³	—	2	1	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	4	4	no	—
557	C₂³	—	4	4	no	—
558	C₂³	—	8	2	no	—
559	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
560	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
561	Q₈	—	2	2	no	—
562	Q₈	—	2	1	no	—
563	Q₈	—	2	1	no	—
564	Q₈	—	8	1	no	—
565	D₄	—	2	4	no	—
566	D₄	—	2	2	no	—
567	D₄	—	2	1	no	—
568	D₄	—	2	2	no	—
569	D₄	—	4	4	no	—
570	D₄	—	4	2	no	—
571	D₄	—	4	4	no	—
572	D₄	—	4	2	no	—
573	D₄	—	8	1	no	—
574	D₄	—	8	2	no	—
575	D₄	C₃₁×#104	1	4	yes	—
576	D₄	C₃₁×#131	1	2	yes	complemented
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	8	2	no	—
598	C₂×C₄	—	8	2	no	—
599	C₂×C₄	—	8	2	no	—
600	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
601	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
602	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
603	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
604	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
605	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
606	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
607	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
608	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
609	C₈	—	2	2	no	—
610	C₈	—	2	4	no	—
611	C₈	—	2	2	no	—
612	C₈	—	4	1	no	—
613	C₈	—	4	2	no	—
614	C₈	—	4	1	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	1	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	2	no	—
626	C₂²	—	8	8	no	—
627	C₂²	C₂×C₃₁×#104	1	4	yes	—
628	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
629	C₄	—	2	1	no	—
630	C₄	—	2	2	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	1	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	2	no	—
635	C₄	—	4	1	no	—
636	C₄	—	4	1	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	1	no	—
639	C₄	C₂×C₃₁×#104	1	2	yes	—
640	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
641	C₄	C₃₁×#568	1	1	yes	characteristic
642	C₄	C₃₁×#576	1	1	yes	characteristic
643	C₄	C₃₁×#577	1	1	yes	characteristic
644	C₂	—	2	4	no	—
645	C₂	—	4	1	no	—
646	C₂	—	4	2	no	—
647	C₂	—	8	2	no	—
648	C₂	C₃₁×#2880	1	1	yes	characteristic
649	C₂	C₃₁×#2897	1	1	yes	characteristic
650	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3481	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2990	C₂	semidirect
3. C₄×C₃₁×#131	C₂	semidirect
4. C₃₁×#1494	C₂	semidirect
5. C₃₁×#1492	C₂	semidirect
6. C₃₁×#1490	C₂	semidirect
7. C₃₁×#1489	C₂	semidirect
8. C₃₁×#1484	C₂	semidirect
9. C₃₁×#1483	C₂	semidirect
10. C₃₁×#1476	C₂	semidirect
11. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
33. C₃₁×#432	C₂²	semidirect
34. C₃₁×#431	C₂²	semidirect
35. C₃₁×#423	C₂²	semidirect
36. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
39. C₄×M₁₆×C₃₁	C₂²	semidirect
40. C₄×M₁₆×C₃₁	C₂²	semidirect
41. C₂×C₄×C₈×C₃₁	C₂²	semidirect
42. C₂×C₄×C₈×C₃₁	C₂²	semidirect
43. C₃₁×Q₈⋊C₈	C₂²	semidirect
44. C₃₁×Q₈⋊C₈	C₂²	semidirect
45. C₃₁×Q₈⋊C₈	C₂²	semidirect
49. C₃₁×D₄⋊C₈	C₂²	semidirect
50. C₃₁×D₄⋊C₈	C₂²	semidirect
51. C₃₁×D₄⋊C₈	C₂²	semidirect
52. C₃₁×D₄⋊C₈	C₂²	semidirect
228. #17517	C₃₁	direct (N × H)
251. D₄×C₃₁	#131	semidirect
291. #2990	C₂×C₃₁	semidirect
292. C₄×#131	C₂×C₃₁	semidirect
293. #1494	C₂×C₃₁	semidirect
294. #1492	C₂×C₃₁	semidirect
295. #1490	C₂×C₃₁	semidirect
296. #1489	C₂×C₃₁	semidirect
297. #1484	C₂×C₃₁	semidirect
298. #1483	C₂×C₃₁	semidirect
299. #1476	C₂×C₃₁	semidirect
300. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
351. #432	C₂²×C₃₁	semidirect
352. #431	C₂²×C₃₁	semidirect
353. #423	C₂²×C₃₁	semidirect
354. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
357. C₄×M₁₆	C₂²×C₃₁	semidirect
358. C₄×M₁₆	C₂²×C₃₁	semidirect
359. C₂×C₄×C₈	C₂²×C₃₁	semidirect
360. C₂×C₄×C₈	C₂²×C₃₁	semidirect
361. Q₈⋊C₈	C₂²×C₃₁	semidirect
362. Q₈⋊C₈	C₂²×C₃₁	semidirect
363. Q₈⋊C₈	C₂²×C₃₁	semidirect
367. D₄⋊C₈	C₂²×C₃₁	semidirect
368. D₄⋊C₈	C₂²×C₃₁	semidirect
369. D₄⋊C₈	C₂²×C₃₁	semidirect
370. D₄⋊C₈	C₂²×C₃₁	semidirect
466. C₃₁	#17517	direct (N × H)
576. D₄	C₃₁×#131	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3481	2
C₃₁×#2990	2
C₄×C₃₁×#131	2
C₃₁×#1494	2
C₃₁×#1492	2
C₃₁×#1490	2
C₃₁×#1489	2
C₃₁×#1484	2
C₃₁×#1483	2
C₃₁×#1476	2
C₂×C₃₁×D₄⋊C₈	2
#17517	31

Subgroups — C31×#17517 #21474276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17517 #21474276