Subgroups — C₃₁×#60831 #21517590

Order 7936.

2014 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3512	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3223	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3221	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3219	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3218	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3142	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3046	C₂	1	4	yes	maximal, complemented
12	C₃₁×#3013	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2923	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
15	C₂×C₃₁×#137	C₂²	1	4	yes	—
16	C₂×C₃₁×#135	C₂²	1	4	yes	—
17	C₃₁×#553	—	2	4	no	—
18	C₃₁×#553	—	2	2	no	—
19	C₃₁×#552	—	2	2	no	—
20	D₄×Q₈×C₃₁	—	2	2	no	—
21	C₃₁×#547	—	2	4	no	—
22	C₃₁×#547	—	2	2	no	—
23	C₃₁×#541	C₂²	1	2	yes	—
24	C₃₁×#540	C₂²	1	2	yes	—
25	C₃₁×#537	C₂²	1	2	yes	—
26	C₃₁×#536	C₂²	1	4	yes	—
27	C₃₁×#536	C₂²	1	2	yes	—
28	C₃₁×#532	C₂²	1	1	yes	characteristic
29	C₃₁×#532	C₂²	1	1	yes	characteristic
30	C₃₁×#532	C₂²	1	2	yes	—
31	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
33	C₃₁×#529	C₂²	1	4	yes	—
34	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
35	C₃₁×#520	C₂²	1	2	yes	—
36	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#497	C₂²	1	2	yes	—
38	C₃₁×#497	C₂²	1	2	yes	—
39	C₃₁×#497	C₂²	1	4	yes	—
40	C₃₁×#496	C₂²	1	2	yes	—
41	C₃₁×#496	C₂²	1	2	yes	—
42	C₃₁×#494	C₂²	1	2	yes	—
43	C₃₁×#492	C₂²	1	2	yes	—
44	C₃₁×#491	C₂²	1	4	yes	—
45	C₃₁×#491	C₂²	1	4	yes	—
46	C₃₁×#491	C₂²	1	2	yes	—
47	C₃₁×#490	C₂²	1	2	yes	—
48	C₃₁×#488	C₂²	1	4	yes	—
49	C₃₁×#464	C₂²	1	8	yes	—
50	C₃₁×#463	C₂²	1	8	yes	—
51	C₃₁×#462	C₂²	1	8	yes	—
52	C₃₁×#461	C₂²	1	8	yes	—
53	C₃₁×#452	C₂²	1	8	yes	—
54	C₃₁×#451	C₂²	1	8	yes	—
55	C₃₁×#449	C₂²	1	8	yes	—
56	C₃₁×#448	C₂²	1	8	yes	—
57	C₃₁×#433	C₂²	1	4	yes	—
58	C₃₁×#432	C₂²	1	4	yes	—
59	C₃₁×#424	C₂²	1	2	yes	—
60	C₃₁×#417	C₂²	1	4	yes	—
61	C₂×C₃₁×#103	C₂²	1	4	yes	—
62	C₃₁×#408	C₂²	1	2	yes	—
63	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
64	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
65	C₃₁×2₋¹⁺⁴	—	4	8	no	—
66	C₂×#41×C₃₁	—	2	2	no	—
67	C₂×#41×C₃₁	—	2	4	no	—
68	C₂×#41×C₃₁	C₂³	1	2	yes	—
69	C₂×#41×C₃₁	C₂³	1	2	yes	—
70	C₂×#41×C₃₁	C₂³	1	4	yes	—
71	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₃₁	—	2	2	no	—
73	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
74	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
75	C₃₁×#137	—	4	8	no	—
76	C₃₁×#135	—	4	8	no	—
77	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
78	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
79	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
80	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
81	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
82	C₂×D₈×C₃₁	C₂³	1	4	yes	—
83	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
84	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
85	C₃₁×C₄⋊Q₈	—	2	4	no	—
86	C₃₁×C₄⋊Q₈	—	2	2	no	—
87	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
88	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
89	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
90	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₃₁×#127	C₂³	1	1	yes	characteristic
93	C₃₁×#127	C₂³	1	1	yes	characteristic
94	C₃₁×#126	—	2	4	no	—
95	C₃₁×#126	—	2	4	no	—
96	C₃₁×#125	—	2	4	no	—
97	C₃₁×#125	—	2	2	no	—
98	C₃₁×#125	C₂³	1	2	yes	—
99	C₃₁×#125	C₂³	1	2	yes	—
100	C₃₁×#124	C₂³	1	4	yes	—
101	C₃₁×#124	C₂³	1	2	yes	—
102	C₃₁×#124	C₂³	1	2	yes	—
103	C₃₁×#123	—	2	4	no	—
104	C₃₁×#123	—	2	4	no	—
105	C₃₁×#123	—	2	4	no	—
106	C₃₁×#123	—	2	8	no	—
107	C₃₁×C₂²⋊Q₈	—	2	2	no	—
108	C₃₁×C₂²⋊Q₈	—	2	4	no	—
109	C₃₁×C₂²⋊Q₈	—	2	8	no	—
110	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
111	C₃₁×#121	—	2	2	no	—
112	C₃₁×#121	—	2	4	no	—
113	C₃₁×#121	—	2	2	no	—
114	C₃₁×#121	—	2	4	no	—
115	C₃₁×#121	—	2	2	no	—
116	C₃₁×#121	—	2	4	no	—
117	C₃₁×#121	C₂³	1	4	yes	—
118	C₄×Q₈×C₃₁	D₄	1	2	yes	—
119	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
120	C₄×D₄×C₃₁	—	2	4	no	—
121	C₄×D₄×C₃₁	—	2	4	no	—
122	C₄×D₄×C₃₁	—	2	4	no	—
123	C₄×D₄×C₃₁	—	2	2	no	—
124	C₄×D₄×C₃₁	—	2	2	no	—
125	C₄×D₄×C₃₁	D₄	1	4	yes	—
126	C₄×D₄×C₃₁	D₄	1	2	yes	—
127	C₄×D₄×C₃₁	C₂³	1	4	yes	—
128	C₃₁×#117	—	2	2	no	—
129	C₃₁×#117	—	2	4	no	—
130	C₃₁×#117	C₂³	1	1	yes	characteristic
131	C₃₁×#117	C₂³	1	2	yes	—
132	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
135	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
137	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
139	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
140	C₃₁×#103	C₂³	1	8	yes	—
141	C₃₁×#103	C₂³	1	8	yes	—
142	C₃₁×#103	C₂³	1	8	yes	—
143	C₃₁×#102	C₂³	1	8	yes	—
144	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
145	C₃₁×#97	C₂³	1	2	yes	—
146	C₃₁×#97	C₂³	1	2	yes	—
147	C₃₁×#97	C₂³	1	2	yes	—
148	C₄×C₈×C₃₁	C₂³	1	2	yes	—
149	#41×C₃₁	—	2	4	no	—
150	#41×C₃₁	—	2	8	no	—
151	#41×C₃₁	—	2	8	no	—
152	#41×C₃₁	—	4	8	no	—
153	#41×C₃₁	—	4	16	no	—
154	#41×C₃₁	—	4	8	no	—
155	C₂×Q₈×C₃₁	—	2	4	no	—
156	C₂×Q₈×C₃₁	—	2	2	no	—
157	C₂×Q₈×C₃₁	—	2	8	no	—
158	C₂×Q₈×C₃₁	—	4	8	no	—
159	C₂×Q₈×C₃₁	—	4	8	no	—
160	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
162	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
163	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
164	C₂×D₄×C₃₁	—	2	2	no	—
165	C₂×D₄×C₃₁	—	2	4	no	—
166	C₂×D₄×C₃₁	—	2	2	no	—
167	C₂×D₄×C₃₁	—	2	4	no	—
168	C₂×D₄×C₃₁	—	2	2	no	—
169	C₂×D₄×C₃₁	—	2	2	no	—
170	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
173	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₃₁	—	2	4	no	—
175	C₂²×C₄×C₃₁	—	2	2	no	—
176	C₂²×C₄×C₃₁	—	2	2	no	—
177	C₂²×C₄×C₃₁	—	2	2	no	—
178	C₂²×C₄×C₃₁	—	2	4	no	—
179	C₂²×C₄×C₃₁	—	2	2	no	—
180	C₂²×C₄×C₃₁	—	2	4	no	—
181	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	Q₁₆×C₃₁	—	4	4	no	—
186	Q₁₆×C₃₁	—	4	8	no	—
187	SD₁₆×C₃₁	—	4	4	no	—
188	SD₁₆×C₃₁	—	4	4	no	—
189	SD₁₆×C₃₁	—	4	8	no	—
190	D₈×C₃₁	—	4	4	no	—
191	M₁₆×C₃₁	—	2	4	no	—
192	C₂×C₈×C₃₁	—	2	4	no	—
193	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
194	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
195	C₄⋊C₄×C₃₁	—	2	4	no	—
196	C₄⋊C₄×C₃₁	—	2	2	no	—
197	C₄⋊C₄×C₃₁	—	2	2	no	—
198	C₄⋊C₄×C₃₁	—	2	4	no	—
199	C₄⋊C₄×C₃₁	—	2	4	no	—
200	C₄⋊C₄×C₃₁	—	2	4	no	—
201	C₄⋊C₄×C₃₁	—	2	8	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	4	no	—
204	C₄⋊C₄×C₃₁	—	2	2	no	—
205	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
206	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
207	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
208	C₂²⋊C₄×C₃₁	—	2	4	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	—	2	4	no	—
211	C₂²⋊C₄×C₃₁	—	2	4	no	—
212	C₂²⋊C₄×C₃₁	—	2	2	no	—
213	C₂²⋊C₄×C₃₁	—	2	8	no	—
214	C₂²⋊C₄×C₃₁	—	2	4	no	—
215	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
216	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
217	C₄²×C₃₁	—	2	4	no	—
218	C₄²×C₃₁	C₂×D₄	1	4	yes	—
219	C₄²×C₃₁	C₂×D₄	1	2	yes	—
220	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
222	C₄²×C₃₁	C₂⁴	1	2	yes	—
223	#60831	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂³×C₃₁	—	2	2	no	—
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
227	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	Q₈×C₃₁	—	2	2	no	—
229	Q₈×C₃₁	—	2	4	no	—
230	Q₈×C₃₁	—	2	8	no	—
231	Q₈×C₃₁	—	2	4	no	—
232	Q₈×C₃₁	—	4	8	no	—
233	Q₈×C₃₁	—	4	4	no	—
234	D₄×C₃₁	—	2	4	no	—
235	D₄×C₃₁	—	2	2	no	—
236	D₄×C₃₁	—	2	4	no	—
237	D₄×C₃₁	—	2	4	no	—
238	D₄×C₃₁	—	4	4	no	—
239	D₄×C₃₁	—	4	8	no	—
240	D₄×C₃₁	—	4	4	no	—
241	D₄×C₃₁	—	4	4	no	—
242	D₄×C₃₁	—	8	2	no	—
243	D₄×C₃₁	—	8	4	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	4	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	8	no	—
252	C₂×C₄×C₃₁	—	4	8	no	—
253	C₂×C₄×C₃₁	—	4	4	no	—
254	C₂×C₄×C₃₁	—	4	4	no	—
255	C₂×C₄×C₃₁	—	4	4	no	—
256	C₂×C₄×C₃₁	—	8	4	no	—
257	C₂×C₄×C₃₁	—	8	2	no	—
258	C₂×C₄×C₃₁	C₂²×D₄	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	H(5,𝔽₂)	1	4	yes	—
264	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
265	C₈×C₃₁	—	2	4	no	—
266	C₈×C₃₁	—	2	4	no	—
267	#3512	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#3499	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#3223	C₂×C₃₁	1	4	yes	complemented
270	#3221	C₂×C₃₁	1	4	yes	complemented
271	#3219	C₂×C₃₁	1	4	yes	complemented
272	#3218	C₂×C₃₁	1	4	yes	complemented
273	#3152	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#3150	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#3142	C₂×C₃₁	1	1	yes	characteristic, complemented
276	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#3046	C₂×C₃₁	1	4	yes	complemented
278	#3013	C₂×C₃₁	1	4	yes	complemented
279	#2923	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂²×C₃₁	—	2	2	no	—
281	C₂²×C₃₁	—	2	4	no	—
282	C₂²×C₃₁	—	2	2	no	—
283	C₂²×C₃₁	—	4	2	no	—
284	C₂²×C₃₁	—	4	4	no	—
285	C₂²×C₃₁	—	4	2	no	—
286	C₂²×C₃₁	—	4	2	no	—
287	C₂²×C₃₁	—	4	2	no	—
288	C₂²×C₃₁	#535	1	2	yes	—
289	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	4	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	4	4	no	—
295	C₄×C₃₁	—	4	4	no	—
296	C₄×C₃₁	—	4	2	no	—
297	C₄×C₃₁	—	4	2	no	—
298	C₄×C₃₁	#535	1	2	yes	—
299	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
300	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
301	C₂×#137	C₂²×C₃₁	1	4	yes	—
302	C₂×#135	C₂²×C₃₁	1	4	yes	—
303	#553	—	2	4	no	—
304	#553	—	2	2	no	—
305	#552	—	2	2	no	—
306	D₄×Q₈	—	2	2	no	—
307	#547	—	2	4	no	—
308	#547	—	2	2	no	—
309	#541	C₂²×C₃₁	1	2	yes	—
310	#540	C₂²×C₃₁	1	2	yes	—
311	#537	C₂²×C₃₁	1	2	yes	—
312	#536	C₂²×C₃₁	1	4	yes	—
313	#536	C₂²×C₃₁	1	2	yes	—
314	#532	C₂²×C₃₁	1	1	yes	characteristic
315	#532	C₂²×C₃₁	1	1	yes	characteristic
316	#532	C₂²×C₃₁	1	2	yes	—
317	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
319	#529	C₂²×C₃₁	1	4	yes	—
320	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
321	#520	C₂²×C₃₁	1	2	yes	—
322	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
323	#497	C₂²×C₃₁	1	2	yes	—
324	#497	C₂²×C₃₁	1	2	yes	—
325	#497	C₂²×C₃₁	1	4	yes	—
326	#496	C₂²×C₃₁	1	2	yes	—
327	#496	C₂²×C₃₁	1	2	yes	—
328	#494	C₂²×C₃₁	1	2	yes	—
329	#492	C₂²×C₃₁	1	2	yes	—
330	#491	C₂²×C₃₁	1	4	yes	—
331	#491	C₂²×C₃₁	1	4	yes	—
332	#491	C₂²×C₃₁	1	2	yes	—
333	#490	C₂²×C₃₁	1	2	yes	—
334	#488	C₂²×C₃₁	1	4	yes	—
335	#464	C₂²×C₃₁	1	8	yes	—
336	#463	C₂²×C₃₁	1	8	yes	—
337	#462	C₂²×C₃₁	1	8	yes	—
338	#461	C₂²×C₃₁	1	8	yes	—
339	#452	C₂²×C₃₁	1	8	yes	—
340	#451	C₂²×C₃₁	1	8	yes	—
341	#449	C₂²×C₃₁	1	8	yes	—
342	#448	C₂²×C₃₁	1	8	yes	—
343	#433	C₂²×C₃₁	1	4	yes	—
344	#432	C₂²×C₃₁	1	4	yes	—
345	#424	C₂²×C₃₁	1	2	yes	—
346	#417	C₂²×C₃₁	1	4	yes	—
347	C₂×#103	C₂²×C₃₁	1	4	yes	—
348	#408	C₂²×C₃₁	1	2	yes	—
349	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×C₃₁	—	2	2	no	—
352	C₂×C₃₁	—	4	2	no	—
353	C₂×C₃₁	—	8	2	no	—
354	C₂×C₃₁	—	8	2	no	—
355	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
356	C₂×C₃₁	#3586	1	2	yes	—
357	2₋¹⁺⁴	—	4	8	no	—
358	C₂×#41	—	2	2	no	—
359	C₂×#41	—	2	4	no	—
360	C₂×#41	C₂³×C₃₁	1	2	yes	—
361	C₂×#41	C₂³×C₃₁	1	2	yes	—
362	C₂×#41	C₂³×C₃₁	1	4	yes	—
363	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
364	C₂²×Q₈	—	2	2	no	—
365	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
366	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
367	#137	—	4	8	no	—
368	#135	—	4	8	no	—
369	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
370	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
371	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
372	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
373	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
374	C₂×D₈	C₂³×C₃₁	1	4	yes	—
375	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
376	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
377	C₄⋊Q₈	—	2	4	no	—
378	C₄⋊Q₈	—	2	2	no	—
379	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
380	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
381	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
382	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
383	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
384	#127	C₂³×C₃₁	1	1	yes	characteristic
385	#127	C₂³×C₃₁	1	1	yes	characteristic
386	#126	—	2	4	no	—
387	#126	—	2	4	no	—
388	#125	—	2	4	no	—
389	#125	—	2	2	no	—
390	#125	C₂³×C₃₁	1	2	yes	—
391	#125	C₂³×C₃₁	1	2	yes	—
392	#124	C₂³×C₃₁	1	2	yes	—
393	#124	C₂³×C₃₁	1	4	yes	—
394	#124	C₂³×C₃₁	1	2	yes	—
395	#123	—	2	4	no	—
396	#123	—	2	4	no	—
397	#123	—	2	4	no	—
398	#123	—	2	8	no	—
399	C₂²⋊Q₈	—	2	2	no	—
400	C₂²⋊Q₈	—	2	4	no	—
401	C₂²⋊Q₈	—	2	8	no	—
402	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
403	#121	—	2	2	no	—
404	#121	—	2	4	no	—
405	#121	—	2	2	no	—
406	#121	—	2	4	no	—
407	#121	—	2	2	no	—
408	#121	—	2	4	no	—
409	#121	C₂³×C₃₁	1	4	yes	—
410	C₄×Q₈	D₄×C₃₁	1	2	yes	—
411	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
412	C₄×D₄	—	2	4	no	—
413	C₄×D₄	—	2	4	no	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	2	no	—
417	C₄×D₄	D₄×C₃₁	1	4	yes	—
418	C₄×D₄	D₄×C₃₁	1	2	yes	—
419	C₄×D₄	C₂³×C₃₁	1	4	yes	—
420	#117	—	2	2	no	—
421	#117	—	2	4	no	—
422	#117	C₂³×C₃₁	1	1	yes	characteristic
423	#117	C₂³×C₃₁	1	2	yes	—
424	C₂×C₄⋊C₄	—	2	2	no	—
425	C₂×C₄⋊C₄	—	2	2	no	—
426	C₂×C₄⋊C₄	—	2	4	no	—
427	C₂×C₄⋊C₄	—	2	2	no	—
428	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
429	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
431	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
432	#103	C₂³×C₃₁	1	8	yes	—
433	#103	C₂³×C₃₁	1	8	yes	—
434	#103	C₂³×C₃₁	1	8	yes	—
435	#102	C₂³×C₃₁	1	8	yes	—
436	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
437	#97	C₂³×C₃₁	1	2	yes	—
438	#97	C₂³×C₃₁	1	2	yes	—
439	#97	C₂³×C₃₁	1	2	yes	—
440	C₄×C₈	C₂³×C₃₁	1	2	yes	—
441	C₃₁	#60831	1	1	yes	characteristic, direct product factor, Sylow
442	#41	—	2	4	no	—
443	#41	—	2	8	no	—
444	#41	—	2	8	no	—
445	#41	—	4	8	no	—
446	#41	—	4	16	no	—
447	#41	—	4	8	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	8	no	—
451	C₂×Q₈	—	4	8	no	—
452	C₂×Q₈	—	4	8	no	—
453	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
454	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
455	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
456	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
464	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
466	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
476	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
478	Q₁₆	—	4	4	no	—
479	Q₁₆	—	4	8	no	—
480	SD₁₆	—	4	4	no	—
481	SD₁₆	—	4	4	no	—
482	SD₁₆	—	4	8	no	—
483	D₈	—	4	4	no	—
484	M₁₆	—	2	4	no	—
485	C₂×C₈	—	2	4	no	—
486	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
487	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	8	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
499	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
500	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	8	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
509	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
510	C₄²	—	2	4	no	—
511	C₄²	C₂×D₄×C₃₁	1	4	yes	—
512	C₄²	C₂×D₄×C₃₁	1	2	yes	—
513	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
514	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
515	C₄²	C₂⁴×C₃₁	1	2	yes	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	2	no	—
518	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
519	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
520	Q₈	—	2	2	no	—
521	Q₈	—	2	4	no	—
522	Q₈	—	2	8	no	—
523	Q₈	—	2	4	no	—
524	Q₈	—	4	8	no	—
525	Q₈	—	4	4	no	—
526	D₄	—	2	4	no	—
527	D₄	—	2	2	no	—
528	D₄	—	2	4	no	—
529	D₄	—	2	4	no	—
530	D₄	—	4	4	no	—
531	D₄	—	4	8	no	—
532	D₄	—	4	4	no	—
533	D₄	—	4	4	no	—
534	D₄	—	8	2	no	—
535	D₄	—	8	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	8	no	—
544	C₂×C₄	—	4	8	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	4	no	—
548	C₂×C₄	—	8	4	no	—
549	C₂×C₄	—	8	2	no	—
550	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
551	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
555	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
556	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
557	C₈	—	2	4	no	—
558	C₈	—	2	4	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	2	no	—
567	C₂²	C₃₁×#535	1	2	yes	—
568	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
569	C₄	—	2	2	no	—
570	C₄	—	2	2	no	—
571	C₄	—	2	4	no	—
572	C₄	—	2	2	no	—
573	C₄	—	4	4	no	—
574	C₄	—	4	2	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	4	no	—
577	C₄	C₃₁×#535	1	2	yes	—
578	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
579	C₂	—	2	2	no	—
580	C₂	—	4	2	no	—
581	C₂	—	8	2	no	—
582	C₂	—	8	2	no	—
583	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
584	C₂	C₃₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3512	C₂	semidirect
2. C₃₁×#3499	C₂	semidirect
3. C₃₁×#3223	C₂	semidirect
4. C₃₁×#3221	C₂	semidirect
5. C₃₁×#3219	C₂	semidirect
6. C₃₁×#3218	C₂	semidirect
7. C₃₁×#3152	C₂	semidirect
8. C₃₁×#3150	C₂	semidirect
9. C₃₁×#3142	C₂	semidirect
10. C₂×C₃₁×#491	C₂	semidirect
11. C₃₁×#3046	C₂	semidirect
12. C₃₁×#3013	C₂	semidirect
13. C₃₁×#2923	C₂	semidirect
223. #60831	C₃₁	direct (N × H)
267. #3512	C₂×C₃₁	semidirect
268. #3499	C₂×C₃₁	semidirect
269. #3223	C₂×C₃₁	semidirect
270. #3221	C₂×C₃₁	semidirect
271. #3219	C₂×C₃₁	semidirect
272. #3218	C₂×C₃₁	semidirect
273. #3152	C₂×C₃₁	semidirect
274. #3150	C₂×C₃₁	semidirect
275. #3142	C₂×C₃₁	semidirect
276. C₂×#491	C₂×C₃₁	semidirect
277. #3046	C₂×C₃₁	semidirect
278. #3013	C₂×C₃₁	semidirect
279. #2923	C₂×C₃₁	semidirect
441. C₃₁	#60831	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×#3512	2
C₃₁×#3499	2
C₃₁×#3223	2
C₃₁×#3221	2
C₃₁×#3219	2
C₃₁×#3218	2
C₃₁×#3152	2
C₃₁×#3150	2
C₃₁×#3142	2
C₂×C₃₁×#491	2
C₃₁×#3046	2
C₃₁×#3013	2
C₃₁×#2923	2
#60831	31

Subgroups — C31×#60831 #21517590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60831 #21517590