Subgroups — C₃₁×#60915 #21517674

Order 7936.

1790 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3455	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3263	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3262	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3261	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3260	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3259	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3258	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3257	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3076	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3054	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3050	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3035	C₂	1	4	yes	maximal, complemented
13	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×#135	C₂²	1	2	yes	—
15	C₂×C₃₁×#135	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#542	C₂²	1	4	yes	—
19	C₃₁×#542	C₂²	1	4	yes	—
20	C₃₁×#541	C₂²	1	2	yes	—
21	C₃₁×#541	C₂²	1	2	yes	—
22	C₃₁×#540	C₂²	1	2	yes	—
23	C₃₁×#540	C₂²	1	2	yes	—
24	C₃₁×#539	—	2	4	no	—
25	C₃₁×#538	—	2	4	no	—
26	C₃₁×#533	C₂²	1	2	yes	—
27	C₃₁×#532	C₂²	1	2	yes	—
28	C₃₁×#529	—	2	4	no	—
29	C₃₁×#529	—	2	4	no	—
30	C₃₁×#525	C₂²	1	4	yes	—
31	C₄×#41×C₃₁	C₂²	1	4	yes	—
32	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
33	C₃₁×#479	C₂²	1	4	yes	complemented
34	C₃₁×#479	C₂²	1	4	yes	complemented
35	C₃₁×#478	C₂²	1	4	yes	complemented
36	C₃₁×#478	C₂²	1	4	yes	complemented
37	C₃₁×#477	C₂²	1	4	yes	complemented
38	C₃₁×#476	C₂²	1	4	yes	complemented
39	C₃₁×#475	C₂²	1	4	yes	complemented
40	C₃₁×#474	C₂²	1	4	yes	complemented
41	C₃₁×#470	C₂²	1	2	yes	—
42	C₃₁×#470	C₂²	1	2	yes	—
43	C₃₁×#469	C₂²	1	2	yes	—
44	C₃₁×#469	C₂²	1	2	yes	—
45	C₃₁×#468	C₂²	1	4	yes	—
46	C₃₁×#468	C₂²	1	4	yes	—
47	C₃₁×#467	C₂²	1	2	yes	—
48	C₃₁×#467	C₂²	1	2	yes	—
49	C₃₁×#466	C₂²	1	2	yes	—
50	C₃₁×#466	C₂²	1	2	yes	—
51	C₃₁×#465	C₂²	1	4	yes	—
52	C₃₁×#465	C₂²	1	4	yes	—
53	C₃₁×#464	C₂²	1	4	yes	complemented
54	C₃₁×#464	C₂²	1	4	yes	complemented
55	C₃₁×#463	C₂²	1	4	yes	complemented
56	C₃₁×#463	C₂²	1	4	yes	complemented
57	C₃₁×#462	C₂²	1	4	yes	complemented
58	C₃₁×#461	C₂²	1	4	yes	complemented
59	C₃₁×#460	C₂²	1	4	yes	complemented
60	C₃₁×#459	C₂²	1	4	yes	complemented
61	C₃₁×#427	C₂²	1	2	yes	—
62	C₃₁×#427	C₂²	1	2	yes	—
63	C₃₁×#424	C₂²	1	4	yes	—
64	C₃₁×#424	C₂²	1	1	yes	characteristic
65	C₃₁×#424	C₂²	1	1	yes	characteristic
66	C₃₁×#424	C₂²	1	1	yes	characteristic
67	C₃₁×#424	C₂²	1	1	yes	characteristic
68	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
69	C₃₁×#416	C₂²	1	4	yes	—
70	C₃₁×#416	C₂²	1	4	yes	—
71	C₂×#41×C₃₁	C₂³	1	4	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#135	—	4	4	no	—
74	C₃₁×#135	—	4	4	no	—
75	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
76	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
77	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
78	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
79	C₂×D₈×C₃₁	C₂³	1	2	yes	—
80	C₂×D₈×C₃₁	C₂³	1	2	yes	—
81	C₂×M₁₆×C₃₁	—	4	2	no	—
82	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
84	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	—	4	2	no	—
86	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
87	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
88	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
90	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
92	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
93	C₃₁×#127	C₂³	1	2	yes	—
94	C₃₁×#126	—	2	4	no	—
95	C₃₁×#126	—	2	4	no	—
96	C₃₁×#126	—	2	4	no	—
97	C₃₁×#126	—	2	4	no	—
98	C₃₁×#125	—	2	4	no	—
99	C₃₁×#125	—	2	4	no	—
100	C₃₁×#125	C₂³	1	2	yes	—
101	C₃₁×#125	C₂³	1	2	yes	—
102	C₃₁×#124	—	2	4	no	—
103	C₃₁×#124	—	2	4	no	—
104	C₃₁×#124	C₂³	1	4	yes	—
105	C₃₁×#123	—	2	4	no	—
106	C₃₁×#123	—	2	4	no	—
107	C₃₁×#123	—	2	4	no	—
108	C₃₁×#123	—	2	4	no	—
109	C₃₁×#123	—	2	4	no	—
110	C₃₁×C₂²⋊Q₈	—	2	4	no	—
111	C₃₁×C₂²⋊Q₈	—	2	4	no	—
112	C₃₁×C₂²⋊Q₈	—	2	4	no	—
113	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
114	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
115	C₃₁×#121	—	2	4	no	—
116	C₃₁×#121	—	2	4	no	—
117	C₃₁×#121	—	2	4	no	—
118	C₃₁×#121	C₂³	1	4	yes	—
119	C₃₁×#121	C₂³	1	4	yes	—
120	C₃₁×#120	—	2	4	no	—
121	C₄×Q₈×C₃₁	—	2	4	no	—
122	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
123	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
124	C₄×D₄×C₃₁	—	2	4	no	—
125	C₄×D₄×C₃₁	—	2	4	no	—
126	C₄×D₄×C₃₁	—	2	4	no	—
127	C₄×D₄×C₃₁	—	2	4	no	—
128	C₄×D₄×C₃₁	—	2	4	no	—
129	C₄×D₄×C₃₁	C₂³	1	4	yes	—
130	C₄×D₄×C₃₁	C₂³	1	4	yes	—
131	C₃₁×#117	—	2	4	no	—
132	C₃₁×#117	D₄	1	2	yes	—
133	C₃₁×#117	D₄	1	2	yes	—
134	C₃₁×#117	C₂³	1	4	yes	—
135	C₃₁×#117	C₂³	1	1	yes	characteristic
136	C₃₁×#117	C₂³	1	2	yes	—
137	C₃₁×#117	C₂³	1	1	yes	characteristic
138	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
139	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
140	C₂×C₄²×C₃₁	—	2	4	no	—
141	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
142	C₃₁×#107	C₂³	1	2	yes	—
143	C₃₁×#107	C₂³	1	2	yes	—
144	C₃₁×#107	C₂³	1	4	yes	—
145	C₃₁×#106	C₂³	1	4	yes	—
146	C₃₁×#106	C₂³	1	2	yes	—
147	C₃₁×#106	C₂³	1	2	yes	—
148	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
150	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
151	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
152	C₃₁×#103	C₂³	1	4	yes	—
153	C₃₁×#103	C₂³	1	4	yes	—
154	C₃₁×#103	C₂³	1	4	yes	—
155	C₃₁×#103	C₂³	1	4	yes	—
156	C₃₁×#102	C₂³	1	4	yes	—
157	C₃₁×#102	C₂³	1	4	yes	—
158	C₃₁×#102	C₂³	1	4	yes	—
159	C₃₁×#102	C₂³	1	4	yes	—
160	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
161	#41×C₃₁	—	2	8	no	—
162	#41×C₃₁	—	2	8	no	—
163	C₂×Q₈×C₃₁	—	2	4	no	—
164	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
165	C₂×D₄×C₃₁	—	2	4	no	—
166	C₂×D₄×C₃₁	—	2	4	no	—
167	C₂×D₄×C₃₁	—	2	4	no	—
168	C₂×D₄×C₃₁	—	2	4	no	—
169	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₃₁	—	2	2	no	—
171	C₂²×C₄×C₃₁	—	2	2	no	—
172	C₂²×C₄×C₃₁	—	2	4	no	—
173	C₂²×C₄×C₃₁	—	2	4	no	—
174	C₂²×C₄×C₃₁	—	2	4	no	—
175	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	Q₁₆×C₃₁	—	4	2	no	—
183	Q₁₆×C₃₁	—	4	2	no	—
184	SD₁₆×C₃₁	—	4	4	no	—
185	SD₁₆×C₃₁	—	4	4	no	—
186	D₈×C₃₁	—	4	2	no	—
187	D₈×C₃₁	—	4	2	no	—
188	M₁₆×C₃₁	—	4	1	no	—
189	M₁₆×C₃₁	—	4	2	no	—
190	M₁₆×C₃₁	—	4	1	no	—
191	C₂×C₈×C₃₁	—	2	4	no	—
192	C₂×C₈×C₃₁	—	2	4	no	—
193	C₂×C₈×C₃₁	—	4	1	no	—
194	C₂×C₈×C₃₁	—	4	2	no	—
195	C₂×C₈×C₃₁	—	4	1	no	—
196	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
197	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
198	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
199	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
200	C₄⋊C₄×C₃₁	—	2	4	no	—
201	C₄⋊C₄×C₃₁	—	2	4	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	4	no	—
204	C₄⋊C₄×C₃₁	—	2	4	no	—
205	C₄⋊C₄×C₃₁	—	2	4	no	—
206	C₄⋊C₄×C₃₁	—	2	4	no	—
207	C₄⋊C₄×C₃₁	—	2	4	no	—
208	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
209	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
210	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
211	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
212	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
213	C₂²⋊C₄×C₃₁	—	2	4	no	—
214	C₂²⋊C₄×C₃₁	—	2	4	no	—
215	C₂²⋊C₄×C₃₁	—	2	4	no	—
216	C₂²⋊C₄×C₃₁	—	2	4	no	—
217	C₂²⋊C₄×C₃₁	—	2	4	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	—	2	4	no	—
221	C₂²⋊C₄×C₃₁	—	2	4	no	—
222	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
223	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
224	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
225	C₄²×C₃₁	—	2	4	no	—
226	C₄²×C₃₁	—	2	4	no	—
227	C₄²×C₃₁	—	2	4	no	—
228	C₄²×C₃₁	C₂×D₄	1	4	yes	—
229	C₄²×C₃₁	C₂⁴	1	2	yes	—
230	C₄²×C₃₁	C₂⁴	1	2	yes	—
231	#60915	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	4	no	—
234	C₂³×C₃₁	—	4	2	no	—
235	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	Q₈×C₃₁	—	2	4	no	—
239	Q₈×C₃₁	—	2	4	no	—
240	Q₈×C₃₁	—	8	4	no	—
241	D₄×C₃₁	—	2	4	no	—
242	D₄×C₃₁	—	2	4	no	—
243	D₄×C₃₁	—	4	8	no	—
244	D₄×C₃₁	—	8	4	no	—
245	D₄×C₃₁	—	8	4	no	—
246	D₄×C₃₁	—	8	4	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	1	no	—
252	C₂×C₄×C₃₁	—	2	1	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	4	4	no	—
258	C₂×C₄×C₃₁	—	4	8	no	—
259	C₂×C₄×C₃₁	—	8	4	no	—
260	C₂×C₄×C₃₁	—	8	4	no	—
261	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
262	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
267	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
268	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
269	C₈×C₃₁	—	2	2	no	—
270	C₈×C₃₁	—	2	2	no	—
271	C₈×C₃₁	—	2	2	no	—
272	C₈×C₃₁	—	2	2	no	—
273	#3455	C₂×C₃₁	1	2	yes	complemented
274	#3263	C₂×C₃₁	1	2	yes	complemented
275	#3262	C₂×C₃₁	1	2	yes	complemented
276	#3261	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#3260	C₂×C₃₁	1	2	yes	complemented
278	#3259	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#3258	C₂×C₃₁	1	4	yes	complemented
280	#3257	C₂×C₃₁	1	4	yes	complemented
281	#3076	C₂×C₃₁	1	4	yes	complemented
282	#3054	C₂×C₃₁	1	2	yes	complemented
283	#3050	C₂×C₃₁	1	2	yes	complemented
284	#3035	C₂×C₃₁	1	4	yes	complemented
285	C₂×#424	C₂×C₃₁	1	1	yes	characteristic, complemented
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	2	1	no	—
290	C₂²×C₃₁	—	2	1	no	—
291	C₂²×C₃₁	—	4	4	no	—
292	C₂²×C₃₁	—	4	4	no	—
293	C₂²×C₃₁	—	4	4	no	—
294	C₂²×C₃₁	—	4	4	no	—
295	C₂²×C₃₁	#537	1	2	yes	—
296	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
297	C₄×C₃₁	—	2	4	no	—
298	C₄×C₃₁	—	2	4	no	—
299	C₄×C₃₁	—	2	1	no	—
300	C₄×C₃₁	—	2	1	no	—
301	C₄×C₃₁	—	4	4	no	—
302	C₄×C₃₁	—	4	4	no	—
303	C₄×C₃₁	—	4	4	no	—
304	C₄×C₃₁	#537	1	2	yes	—
305	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
306	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
307	C₂×#135	C₂²×C₃₁	1	2	yes	—
308	C₂×#135	C₂²×C₃₁	1	2	yes	—
309	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
310	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
311	#542	C₂²×C₃₁	1	4	yes	—
312	#542	C₂²×C₃₁	1	4	yes	—
313	#541	C₂²×C₃₁	1	2	yes	—
314	#541	C₂²×C₃₁	1	2	yes	—
315	#540	C₂²×C₃₁	1	2	yes	—
316	#540	C₂²×C₃₁	1	2	yes	—
317	#539	—	2	4	no	—
318	#538	—	2	4	no	—
319	#533	C₂²×C₃₁	1	2	yes	—
320	#532	C₂²×C₃₁	1	2	yes	—
321	#529	—	2	4	no	—
322	#529	—	2	4	no	—
323	#525	C₂²×C₃₁	1	4	yes	—
324	C₄×#41	C₂²×C₃₁	1	4	yes	—
325	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
326	#479	C₂²×C₃₁	1	4	yes	complemented
327	#479	C₂²×C₃₁	1	4	yes	complemented
328	#478	C₂²×C₃₁	1	4	yes	complemented
329	#478	C₂²×C₃₁	1	4	yes	complemented
330	#477	C₂²×C₃₁	1	4	yes	complemented
331	#476	C₂²×C₃₁	1	4	yes	complemented
332	#475	C₂²×C₃₁	1	4	yes	complemented
333	#474	C₂²×C₃₁	1	4	yes	complemented
334	#470	C₂²×C₃₁	1	2	yes	—
335	#470	C₂²×C₃₁	1	2	yes	—
336	#469	C₂²×C₃₁	1	2	yes	—
337	#469	C₂²×C₃₁	1	2	yes	—
338	#468	C₂²×C₃₁	1	4	yes	—
339	#468	C₂²×C₃₁	1	4	yes	—
340	#467	C₂²×C₃₁	1	2	yes	—
341	#467	C₂²×C₃₁	1	2	yes	—
342	#466	C₂²×C₃₁	1	2	yes	—
343	#466	C₂²×C₃₁	1	2	yes	—
344	#465	C₂²×C₃₁	1	4	yes	—
345	#465	C₂²×C₃₁	1	4	yes	—
346	#464	C₂²×C₃₁	1	4	yes	complemented
347	#464	C₂²×C₃₁	1	4	yes	complemented
348	#463	C₂²×C₃₁	1	4	yes	complemented
349	#463	C₂²×C₃₁	1	4	yes	complemented
350	#462	C₂²×C₃₁	1	4	yes	complemented
351	#461	C₂²×C₃₁	1	4	yes	complemented
352	#460	C₂²×C₃₁	1	4	yes	complemented
353	#459	C₂²×C₃₁	1	4	yes	complemented
354	#427	C₂²×C₃₁	1	2	yes	—
355	#427	C₂²×C₃₁	1	2	yes	—
356	#424	C₂²×C₃₁	1	4	yes	—
357	#424	C₂²×C₃₁	1	1	yes	characteristic
358	#424	C₂²×C₃₁	1	1	yes	characteristic
359	#424	C₂²×C₃₁	1	1	yes	characteristic
360	#424	C₂²×C₃₁	1	1	yes	characteristic
361	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
362	#416	C₂²×C₃₁	1	4	yes	—
363	#416	C₂²×C₃₁	1	4	yes	—
364	C₂×C₃₁	—	2	2	no	—
365	C₂×C₃₁	—	4	1	no	—
366	C₂×C₃₁	—	4	1	no	—
367	C₂×C₃₁	—	8	4	no	—
368	C₂×C₃₁	C₂×#537	1	1	yes	characteristic
369	C₂×C₃₁	#3584	1	2	yes	—
370	C₂×#41	C₂³×C₃₁	1	4	yes	—
371	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
372	#135	—	4	4	no	—
373	#135	—	4	4	no	—
374	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
375	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
376	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
377	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
378	C₂×D₈	C₂³×C₃₁	1	2	yes	—
379	C₂×D₈	C₂³×C₃₁	1	2	yes	—
380	C₂×M₁₆	—	4	2	no	—
381	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
382	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
383	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
384	C₂²×C₈	—	4	2	no	—
385	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
386	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
387	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
388	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
389	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
390	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
391	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
392	#127	C₂³×C₃₁	1	2	yes	—
393	#126	—	2	4	no	—
394	#126	—	2	4	no	—
395	#126	—	2	4	no	—
396	#126	—	2	4	no	—
397	#125	—	2	4	no	—
398	#125	—	2	4	no	—
399	#125	C₂³×C₃₁	1	2	yes	—
400	#125	C₂³×C₃₁	1	2	yes	—
401	#124	—	2	4	no	—
402	#124	—	2	4	no	—
403	#124	C₂³×C₃₁	1	4	yes	—
404	#123	—	2	4	no	—
405	#123	—	2	4	no	—
406	#123	—	2	4	no	—
407	#123	—	2	4	no	—
408	#123	—	2	4	no	—
409	C₂²⋊Q₈	—	2	4	no	—
410	C₂²⋊Q₈	—	2	4	no	—
411	C₂²⋊Q₈	—	2	4	no	—
412	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
413	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
414	#121	—	2	4	no	—
415	#121	—	2	4	no	—
416	#121	—	2	4	no	—
417	#121	C₂³×C₃₁	1	4	yes	—
418	#121	C₂³×C₃₁	1	4	yes	—
419	#120	—	2	4	no	—
420	C₄×Q₈	—	2	4	no	—
421	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
422	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
423	C₄×D₄	—	2	4	no	—
424	C₄×D₄	—	2	4	no	—
425	C₄×D₄	—	2	4	no	—
426	C₄×D₄	—	2	4	no	—
427	C₄×D₄	—	2	4	no	—
428	C₄×D₄	C₂³×C₃₁	1	4	yes	—
429	C₄×D₄	C₂³×C₃₁	1	4	yes	—
430	#117	—	2	4	no	—
431	#117	D₄×C₃₁	1	2	yes	—
432	#117	D₄×C₃₁	1	2	yes	—
433	#117	C₂³×C₃₁	1	4	yes	—
434	#117	C₂³×C₃₁	1	1	yes	characteristic
435	#117	C₂³×C₃₁	1	2	yes	—
436	#117	C₂³×C₃₁	1	1	yes	characteristic
437	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
438	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
439	C₂×C₄²	—	2	4	no	—
440	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
441	#107	C₂³×C₃₁	1	2	yes	—
442	#107	C₂³×C₃₁	1	2	yes	—
443	#107	C₂³×C₃₁	1	4	yes	—
444	#106	C₂³×C₃₁	1	4	yes	—
445	#106	C₂³×C₃₁	1	2	yes	—
446	#106	C₂³×C₃₁	1	2	yes	—
447	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
448	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
449	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
450	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
451	#103	C₂³×C₃₁	1	4	yes	—
452	#103	C₂³×C₃₁	1	4	yes	—
453	#103	C₂³×C₃₁	1	4	yes	—
454	#103	C₂³×C₃₁	1	4	yes	—
455	#102	C₂³×C₃₁	1	4	yes	—
456	#102	C₂³×C₃₁	1	4	yes	—
457	#102	C₂³×C₃₁	1	4	yes	—
458	#102	C₂³×C₃₁	1	4	yes	—
459	C₃₁	#60915	1	1	yes	characteristic, direct product factor, Sylow
460	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
461	#41	—	2	8	no	—
462	#41	—	2	8	no	—
463	C₂×Q₈	—	2	4	no	—
464	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
482	Q₁₆	—	4	2	no	—
483	Q₁₆	—	4	2	no	—
484	SD₁₆	—	4	4	no	—
485	SD₁₆	—	4	4	no	—
486	D₈	—	4	2	no	—
487	D₈	—	4	2	no	—
488	M₁₆	—	4	1	no	—
489	M₁₆	—	4	2	no	—
490	M₁₆	—	4	1	no	—
491	C₂×C₈	—	2	4	no	—
492	C₂×C₈	—	2	4	no	—
493	C₂×C₈	—	4	1	no	—
494	C₂×C₈	—	4	2	no	—
495	C₂×C₈	—	4	1	no	—
496	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
497	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
498	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
499	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	4	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
509	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
510	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
511	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
512	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	2	4	no	—
522	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
523	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
524	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
525	C₄²	—	2	4	no	—
526	C₄²	—	2	4	no	—
527	C₄²	—	2	4	no	—
528	C₄²	C₂×D₄×C₃₁	1	4	yes	—
529	C₄²	C₂⁴×C₃₁	1	2	yes	—
530	C₄²	C₂⁴×C₃₁	1	2	yes	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	4	2	no	—
534	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
535	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
536	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
537	Q₈	—	2	4	no	—
538	Q₈	—	2	4	no	—
539	Q₈	—	8	4	no	—
540	D₄	—	2	4	no	—
541	D₄	—	2	4	no	—
542	D₄	—	4	8	no	—
543	D₄	—	8	4	no	—
544	D₄	—	8	4	no	—
545	D₄	—	8	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	4	4	no	—
557	C₂×C₄	—	4	8	no	—
558	C₂×C₄	—	8	4	no	—
559	C₂×C₄	—	8	4	no	—
560	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
561	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
562	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
563	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
564	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
565	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
566	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
567	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
568	C₈	—	2	2	no	—
569	C₈	—	2	2	no	—
570	C₈	—	2	2	no	—
571	C₈	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	C₃₁×#537	1	2	yes	—
582	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	2	4	no	—
586	C₄	—	2	4	no	—
587	C₄	—	4	4	no	—
588	C₄	—	4	4	no	—
589	C₄	—	4	4	no	—
590	C₄	C₃₁×#537	1	2	yes	—
591	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
592	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
593	C₂	—	2	2	no	—
594	C₂	—	4	1	no	—
595	C₂	—	4	1	no	—
596	C₂	—	8	4	no	—
597	C₂	C₂×C₃₁×#537	1	1	yes	characteristic
598	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3455	C₂	semidirect
2. C₃₁×#3263	C₂	semidirect
3. C₃₁×#3262	C₂	semidirect
4. C₃₁×#3261	C₂	semidirect
5. C₃₁×#3260	C₂	semidirect
6. C₃₁×#3259	C₂	semidirect
7. C₃₁×#3258	C₂	semidirect
8. C₃₁×#3257	C₂	semidirect
9. C₃₁×#3076	C₂	semidirect
10. C₃₁×#3054	C₂	semidirect
11. C₃₁×#3050	C₂	semidirect
12. C₃₁×#3035	C₂	semidirect
13. C₂×C₃₁×#424	C₂	semidirect
33. C₃₁×#479	C₂²	semidirect
34. C₃₁×#479	C₂²	semidirect
35. C₃₁×#478	C₂²	semidirect
36. C₃₁×#478	C₂²	semidirect
37. C₃₁×#477	C₂²	semidirect
38. C₃₁×#476	C₂²	semidirect
39. C₃₁×#475	C₂²	semidirect
40. C₃₁×#474	C₂²	semidirect
53. C₃₁×#464	C₂²	semidirect
54. C₃₁×#464	C₂²	semidirect
55. C₃₁×#463	C₂²	semidirect
56. C₃₁×#463	C₂²	semidirect
57. C₃₁×#462	C₂²	semidirect
58. C₃₁×#461	C₂²	semidirect
59. C₃₁×#460	C₂²	semidirect
60. C₃₁×#459	C₂²	semidirect
231. #60915	C₃₁	direct (N × H)
273. #3455	C₂×C₃₁	semidirect
274. #3263	C₂×C₃₁	semidirect
275. #3262	C₂×C₃₁	semidirect
276. #3261	C₂×C₃₁	semidirect
277. #3260	C₂×C₃₁	semidirect
278. #3259	C₂×C₃₁	semidirect
279. #3258	C₂×C₃₁	semidirect
280. #3257	C₂×C₃₁	semidirect
281. #3076	C₂×C₃₁	semidirect
282. #3054	C₂×C₃₁	semidirect
283. #3050	C₂×C₃₁	semidirect
284. #3035	C₂×C₃₁	semidirect
285. C₂×#424	C₂×C₃₁	semidirect
326. #479	C₂²×C₃₁	semidirect
327. #479	C₂²×C₃₁	semidirect
328. #478	C₂²×C₃₁	semidirect
329. #478	C₂²×C₃₁	semidirect
330. #477	C₂²×C₃₁	semidirect
331. #476	C₂²×C₃₁	semidirect
332. #475	C₂²×C₃₁	semidirect
333. #474	C₂²×C₃₁	semidirect
346. #464	C₂²×C₃₁	semidirect
347. #464	C₂²×C₃₁	semidirect
348. #463	C₂²×C₃₁	semidirect
349. #463	C₂²×C₃₁	semidirect
350. #462	C₂²×C₃₁	semidirect
351. #461	C₂²×C₃₁	semidirect
352. #460	C₂²×C₃₁	semidirect
353. #459	C₂²×C₃₁	semidirect
459. C₃₁	#60915	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3455	2
C₃₁×#3263	2
C₃₁×#3262	2
C₃₁×#3261	2
C₃₁×#3260	2
C₃₁×#3259	2
C₃₁×#3258	2
C₃₁×#3257	2
C₃₁×#3076	2
C₃₁×#3054	2
C₃₁×#3050	2
C₃₁×#3035	2
C₂×C₃₁×#424	2
#60915	31

Subgroups — C31×#60915 #21517674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60915 #21517674