Subgroups — C₃₁×#60907 #21517666

Order 7936.

1910 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3256	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3255	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3254	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3253	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3252	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3251	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3076	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3054	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3035	C₂	1	4	yes	maximal, complemented
11	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#533	C₂²	1	2	yes	—
15	C₃₁×#533	C₂²	1	2	yes	—
16	C₃₁×#533	C₂²	1	2	yes	—
17	C₃₁×#532	C₂²	1	2	yes	—
18	C₃₁×#532	C₂²	1	2	yes	—
19	C₃₁×#532	C₂²	1	2	yes	—
20	C₃₁×#529	—	2	2	no	—
21	C₃₁×#529	—	2	2	no	—
22	C₃₁×#529	—	2	2	no	—
23	C₃₁×#529	—	2	2	no	—
24	C₃₁×#529	C₂²	1	4	yes	—
25	C₃₁×#529	C₂²	1	4	yes	—
26	C₃₁×#525	—	2	2	no	—
27	C₃₁×#525	—	2	2	no	—
28	C₃₁×#525	C₂²	1	4	yes	—
29	C₄×#41×C₃₁	—	2	2	no	—
30	C₄×#41×C₃₁	—	2	2	no	—
31	C₄×#41×C₃₁	C₂²	1	4	yes	—
32	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#479	C₂²	1	8	yes	complemented
34	C₃₁×#478	C₂²	1	8	yes	complemented
35	C₃₁×#477	C₂²	1	4	yes	complemented
36	C₃₁×#476	C₂²	1	4	yes	complemented
37	C₃₁×#475	C₂²	1	4	yes	complemented
38	C₃₁×#474	C₂²	1	4	yes	complemented
39	C₃₁×#470	C₂²	1	4	yes	—
40	C₃₁×#470	C₂²	1	4	yes	—
41	C₃₁×#469	C₂²	1	4	yes	—
42	C₃₁×#469	C₂²	1	4	yes	—
43	C₃₁×#468	C₂²	1	8	yes	—
44	C₃₁×#468	C₂²	1	8	yes	—
45	C₃₁×#464	C₂²	1	8	yes	complemented
46	C₃₁×#463	C₂²	1	8	yes	complemented
47	C₃₁×#462	C₂²	1	4	yes	complemented
48	C₃₁×#461	C₂²	1	4	yes	complemented
49	C₃₁×#460	C₂²	1	4	yes	complemented
50	C₃₁×#459	C₂²	1	4	yes	complemented
51	C₃₁×#427	C₂²	1	4	yes	—
52	C₃₁×#423	C₂²	1	4	yes	—
53	C₃₁×#423	C₂²	1	1	yes	characteristic
54	C₃₁×#423	C₂²	1	1	yes	characteristic
55	C₃₁×#423	C₂²	1	2	yes	—
56	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
57	C₃₁×#416	C₂²	1	8	yes	—
58	C₂×#41×C₃₁	—	2	2	no	—
59	C₂×#41×C₃₁	—	2	2	no	—
60	C₂×#41×C₃₁	C₂³	1	4	yes	—
61	C₂³×C₄×C₃₁	D₄	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#135	—	4	8	no	—
64	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
65	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
66	C₂×D₈×C₃₁	C₂³	1	4	yes	—
67	C₂×M₁₆×C₃₁	—	4	4	no	—
68	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
72	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
73	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₃₁×#127	C₂³	1	2	yes	—
76	C₃₁×#126	—	2	4	no	—
77	C₃₁×#126	—	2	4	no	—
78	C₃₁×#125	—	2	2	no	—
79	C₃₁×#125	—	2	2	no	—
80	C₃₁×#125	C₂³	1	4	yes	—
81	C₃₁×#124	—	2	2	no	—
82	C₃₁×#124	—	2	2	no	—
83	C₃₁×#124	C₂³	1	4	yes	—
84	C₃₁×#123	—	2	4	no	—
85	C₃₁×#123	—	2	4	no	—
86	C₃₁×#123	—	2	4	no	—
87	C₃₁×C₂²⋊Q₈	—	2	2	no	—
88	C₃₁×C₂²⋊Q₈	—	2	2	no	—
89	C₃₁×C₂²⋊Q₈	—	2	2	no	—
90	C₃₁×C₂²⋊Q₈	—	2	2	no	—
91	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
92	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
93	C₃₁×#121	—	2	2	no	—
94	C₃₁×#121	—	2	2	no	—
95	C₃₁×#121	—	2	2	no	—
96	C₃₁×#121	—	2	2	no	—
97	C₃₁×#121	C₂³	1	4	yes	—
98	C₃₁×#121	C₂³	1	4	yes	—
99	C₃₁×#120	—	2	4	no	—
100	C₄×Q₈×C₃₁	—	2	2	no	—
101	C₄×Q₈×C₃₁	—	2	2	no	—
102	C₄×Q₈×C₃₁	—	2	2	no	—
103	C₄×Q₈×C₃₁	—	2	2	no	—
104	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
105	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
106	C₄×D₄×C₃₁	—	2	4	no	—
107	C₄×D₄×C₃₁	—	2	4	no	—
108	C₄×D₄×C₃₁	—	2	4	no	—
109	C₄×D₄×C₃₁	—	2	4	no	—
110	C₄×D₄×C₃₁	—	2	4	no	—
111	C₄×D₄×C₃₁	—	2	4	no	—
112	C₄×D₄×C₃₁	—	2	2	no	—
113	C₄×D₄×C₃₁	—	2	2	no	—
114	C₄×D₄×C₃₁	—	2	2	no	—
115	C₄×D₄×C₃₁	—	2	2	no	—
116	C₄×D₄×C₃₁	C₂³	1	4	yes	—
117	C₄×D₄×C₃₁	C₂³	1	4	yes	—
118	C₃₁×#117	—	2	2	no	—
119	C₃₁×#117	—	2	2	no	—
120	C₃₁×#117	—	2	4	no	—
121	C₃₁×#117	—	2	4	no	—
122	C₃₁×#117	—	2	4	no	—
123	C₃₁×#117	C₂³	1	4	yes	—
124	C₂×C₄²×C₃₁	—	2	4	no	—
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	D₄	1	1	yes	characteristic
130	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
131	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₃₁×#107	C₂³	1	8	yes	—
135	C₃₁×#106	C₂³	1	4	yes	—
136	C₃₁×#106	C₂³	1	4	yes	—
137	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
138	C₃₁×C₄⋊C₈	C₂³	1	4	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×#102	C₂³	1	8	yes	—
143	C₃₁×#102	C₂³	1	8	yes	—
144	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
145	#41×C₃₁	—	2	8	no	—
146	#41×C₃₁	—	2	8	no	—
147	#41×C₃₁	—	4	8	no	—
148	#41×C₃₁	—	4	8	no	—
149	C₂×Q₈×C₃₁	—	2	2	no	—
150	C₂×Q₈×C₃₁	—	2	2	no	—
151	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	4	no	—
154	C₂×D₄×C₃₁	—	2	4	no	—
155	C₂×D₄×C₃₁	—	2	2	no	—
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	4	no	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	4	4	no	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	Q₁₆×C₃₁	—	4	4	no	—
177	SD₁₆×C₃₁	—	4	8	no	—
178	D₈×C₃₁	—	4	4	no	—
179	M₁₆×C₃₁	—	4	2	no	—
180	M₁₆×C₃₁	—	4	4	no	—
181	M₁₆×C₃₁	—	4	2	no	—
182	C₂×C₈×C₃₁	—	2	8	no	—
183	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
184	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	2	2	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	4	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	2	4	no	—
194	C₄⋊C₄×C₃₁	—	2	4	no	—
195	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₃₁	—	2	4	no	—
198	C₂²⋊C₄×C₃₁	—	2	4	no	—
199	C₂²⋊C₄×C₃₁	—	2	4	no	—
200	C₂²⋊C₄×C₃₁	—	2	4	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	C₂²⋊C₄×C₃₁	—	2	4	no	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	—	2	4	no	—
208	C₄²×C₃₁	—	2	4	no	—
209	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂×D₄	1	2	yes	—
211	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂×D₄	1	2	yes	—
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	C₂⁴	1	2	yes	—
218	C₄²×C₃₁	C₂⁴	1	2	yes	—
219	#60907	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₃₁	—	2	2	no	—
221	C₂³×C₃₁	—	2	4	no	—
222	C₂³×C₃₁	—	4	2	no	—
223	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₃₁	—	2	4	no	—
227	Q₈×C₃₁	—	2	4	no	—
228	Q₈×C₃₁	—	4	4	no	—
229	Q₈×C₃₁	—	4	4	no	—
230	D₄×C₃₁	—	2	4	no	—
231	D₄×C₃₁	—	2	4	no	—
232	D₄×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	4	8	no	—
234	D₄×C₃₁	—	4	8	no	—
235	D₄×C₃₁	—	4	8	no	—
236	D₄×C₃₁	—	4	4	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	1	no	—
244	C₂×C₄×C₃₁	—	2	1	no	—
245	C₂×C₄×C₃₁	—	2	1	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	2	no	—
252	C₂×C₄×C₃₁	—	4	2	no	—
253	C₂×C₄×C₃₁	—	4	8	no	—
254	C₂×C₄×C₃₁	—	4	8	no	—
255	C₂×C₄×C₃₁	—	4	8	no	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
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	2	yes	—
264	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
265	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
266	C₈×C₃₁	—	2	4	no	—
267	C₈×C₃₁	—	2	4	no	—
268	#3444	C₂×C₃₁	1	2	yes	complemented
269	#3256	C₂×C₃₁	1	4	yes	complemented
270	#3255	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#3254	C₂×C₃₁	1	2	yes	complemented
272	#3253	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#3252	C₂×C₃₁	1	4	yes	complemented
274	#3251	C₂×C₃₁	1	4	yes	complemented
275	#3076	C₂×C₃₁	1	4	yes	complemented
276	#3054	C₂×C₃₁	1	4	yes	complemented
277	#3035	C₂×C₃₁	1	4	yes	complemented
278	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
279	C₂²×C₃₁	—	2	2	no	—
280	C₂²×C₃₁	—	2	1	no	—
281	C₂²×C₃₁	—	2	1	no	—
282	C₂²×C₃₁	—	2	2	no	—
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	4	4	no	—
285	C₂²×C₃₁	—	4	4	no	—
286	C₂²×C₃₁	—	4	4	no	—
287	C₂²×C₃₁	—	4	4	no	—
288	C₂²×C₃₁	#537	1	2	yes	—
289	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
290	C₄×C₃₁	—	2	1	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	4	4	no	—
296	C₄×C₃₁	—	4	4	no	—
297	C₄×C₃₁	—	4	4	no	—
298	C₄×C₃₁	#537	1	2	yes	—
299	C₄×C₃₁	#575	1	4	yes	—
300	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
301	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
302	C₂×#135	C₂²×C₃₁	1	4	yes	—
303	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
304	#533	C₂²×C₃₁	1	2	yes	—
305	#533	C₂²×C₃₁	1	2	yes	—
306	#533	C₂²×C₃₁	1	2	yes	—
307	#532	C₂²×C₃₁	1	2	yes	—
308	#532	C₂²×C₃₁	1	2	yes	—
309	#532	C₂²×C₃₁	1	2	yes	—
310	#529	—	2	2	no	—
311	#529	—	2	2	no	—
312	#529	—	2	2	no	—
313	#529	—	2	2	no	—
314	#529	C₂²×C₃₁	1	4	yes	—
315	#529	C₂²×C₃₁	1	4	yes	—
316	#525	—	2	2	no	—
317	#525	—	2	2	no	—
318	#525	C₂²×C₃₁	1	4	yes	—
319	C₄×#41	—	2	2	no	—
320	C₄×#41	—	2	2	no	—
321	C₄×#41	C₂²×C₃₁	1	4	yes	—
322	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
323	#479	C₂²×C₃₁	1	8	yes	complemented
324	#478	C₂²×C₃₁	1	8	yes	complemented
325	#477	C₂²×C₃₁	1	4	yes	complemented
326	#476	C₂²×C₃₁	1	4	yes	complemented
327	#475	C₂²×C₃₁	1	4	yes	complemented
328	#474	C₂²×C₃₁	1	4	yes	complemented
329	#470	C₂²×C₃₁	1	4	yes	—
330	#470	C₂²×C₃₁	1	4	yes	—
331	#469	C₂²×C₃₁	1	4	yes	—
332	#469	C₂²×C₃₁	1	4	yes	—
333	#468	C₂²×C₃₁	1	8	yes	—
334	#468	C₂²×C₃₁	1	8	yes	—
335	#464	C₂²×C₃₁	1	8	yes	complemented
336	#463	C₂²×C₃₁	1	8	yes	complemented
337	#462	C₂²×C₃₁	1	4	yes	complemented
338	#461	C₂²×C₃₁	1	4	yes	complemented
339	#460	C₂²×C₃₁	1	4	yes	complemented
340	#459	C₂²×C₃₁	1	4	yes	complemented
341	#427	C₂²×C₃₁	1	4	yes	—
342	#423	C₂²×C₃₁	1	4	yes	—
343	#423	C₂²×C₃₁	1	1	yes	characteristic
344	#423	C₂²×C₃₁	1	1	yes	characteristic
345	#423	C₂²×C₃₁	1	2	yes	—
346	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
347	#416	C₂²×C₃₁	1	8	yes	—
348	C₂×C₃₁	—	2	2	no	—
349	C₂×C₃₁	—	4	1	no	—
350	C₂×C₃₁	—	4	1	no	—
351	C₂×C₃₁	—	8	4	no	—
352	C₂×C₃₁	C₂×#537	1	1	yes	characteristic
353	C₂×C₃₁	C₂×#575	1	2	yes	—
354	C₂×#41	—	2	2	no	—
355	C₂×#41	—	2	2	no	—
356	C₂×#41	C₂³×C₃₁	1	4	yes	—
357	C₂³×C₄	D₄×C₃₁	1	2	yes	—
358	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
359	#135	—	4	8	no	—
360	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
361	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
362	C₂×D₈	C₂³×C₃₁	1	4	yes	—
363	C₂×M₁₆	—	4	4	no	—
364	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
365	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
366	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
367	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
368	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
369	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
370	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
371	#127	C₂³×C₃₁	1	2	yes	—
372	#126	—	2	4	no	—
373	#126	—	2	4	no	—
374	#125	—	2	2	no	—
375	#125	—	2	2	no	—
376	#125	C₂³×C₃₁	1	4	yes	—
377	#124	—	2	2	no	—
378	#124	—	2	2	no	—
379	#124	C₂³×C₃₁	1	4	yes	—
380	#123	—	2	4	no	—
381	#123	—	2	4	no	—
382	#123	—	2	4	no	—
383	C₂²⋊Q₈	—	2	2	no	—
384	C₂²⋊Q₈	—	2	2	no	—
385	C₂²⋊Q₈	—	2	2	no	—
386	C₂²⋊Q₈	—	2	2	no	—
387	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
388	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
389	#121	—	2	2	no	—
390	#121	—	2	2	no	—
391	#121	—	2	2	no	—
392	#121	—	2	2	no	—
393	#121	C₂³×C₃₁	1	4	yes	—
394	#121	C₂³×C₃₁	1	4	yes	—
395	#120	—	2	4	no	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×Q₈	—	2	2	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
401	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
402	C₄×D₄	—	2	4	no	—
403	C₄×D₄	—	2	4	no	—
404	C₄×D₄	—	2	4	no	—
405	C₄×D₄	—	2	4	no	—
406	C₄×D₄	—	2	4	no	—
407	C₄×D₄	—	2	4	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	C₂³×C₃₁	1	4	yes	—
413	C₄×D₄	C₂³×C₃₁	1	4	yes	—
414	#117	—	2	2	no	—
415	#117	—	2	2	no	—
416	#117	—	2	4	no	—
417	#117	—	2	4	no	—
418	#117	—	2	4	no	—
419	#117	C₂³×C₃₁	1	4	yes	—
420	C₂×C₄²	—	2	4	no	—
421	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄²	D₄×C₃₁	1	2	yes	—
423	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
427	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
428	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
429	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
430	#107	C₂³×C₃₁	1	8	yes	—
431	#106	C₂³×C₃₁	1	4	yes	—
432	#106	C₂³×C₃₁	1	4	yes	—
433	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
434	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
435	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
436	#103	C₂³×C₃₁	1	8	yes	—
437	#103	C₂³×C₃₁	1	8	yes	—
438	#102	C₂³×C₃₁	1	8	yes	—
439	#102	C₂³×C₃₁	1	8	yes	—
440	C₃₁	#60907	1	1	yes	characteristic, direct product factor, Sylow
441	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
442	#41	—	2	8	no	—
443	#41	—	2	8	no	—
444	#41	—	4	8	no	—
445	#41	—	4	8	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
465	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
466	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
468	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
473	Q₁₆	—	4	4	no	—
474	SD₁₆	—	4	8	no	—
475	D₈	—	4	4	no	—
476	M₁₆	—	4	2	no	—
477	M₁₆	—	4	4	no	—
478	M₁₆	—	4	2	no	—
479	C₂×C₈	—	2	8	no	—
480	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
481	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
493	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₄²	—	2	4	no	—
504	C₄²	—	2	4	no	—
505	C₄²	—	2	4	no	—
506	C₄²	C₂×D₄×C₃₁	1	2	yes	—
507	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
508	C₄²	C₂×D₄×C₃₁	1	2	yes	—
509	C₄²	C₂×D₄×C₃₁	1	2	yes	—
510	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
511	C₄²	C₂×D₄×C₃₁	1	2	yes	—
512	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
514	C₄²	C₂⁴×C₃₁	1	2	yes	—
515	C₄²	C₂⁴×C₃₁	1	2	yes	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	4	2	no	—
519	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
520	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
521	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
522	Q₈	—	2	4	no	—
523	Q₈	—	2	4	no	—
524	Q₈	—	4	4	no	—
525	Q₈	—	4	4	no	—
526	D₄	—	2	4	no	—
527	D₄	—	2	4	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	8	no	—
530	D₄	—	4	8	no	—
531	D₄	—	4	8	no	—
532	D₄	—	4	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	8	no	—
550	C₂×C₄	—	4	8	no	—
551	C₂×C₄	—	4	8	no	—
552	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
555	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
559	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
560	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
561	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
562	C₈	—	2	4	no	—
563	C₈	—	2	4	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	4	4	no	—
573	C₂²	C₃₁×#537	1	2	yes	—
574	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
575	C₄	—	2	1	no	—
576	C₄	—	2	1	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	4	no	—
583	C₄	C₃₁×#537	1	2	yes	—
584	C₄	C₃₁×#575	1	4	yes	—
585	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
586	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
587	C₂	—	2	2	no	—
588	C₂	—	4	1	no	—
589	C₂	—	4	1	no	—
590	C₂	—	8	4	no	—
591	C₂	C₂×C₃₁×#537	1	1	yes	characteristic
592	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
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. C₃₁×#3444	C₂	semidirect
2. C₃₁×#3256	C₂	semidirect
3. C₃₁×#3255	C₂	semidirect
4. C₃₁×#3254	C₂	semidirect
5. C₃₁×#3253	C₂	semidirect
6. C₃₁×#3252	C₂	semidirect
7. C₃₁×#3251	C₂	semidirect
8. C₃₁×#3076	C₂	semidirect
9. C₃₁×#3054	C₂	semidirect
10. C₃₁×#3035	C₂	semidirect
11. C₂×C₃₁×#423	C₂	semidirect
33. C₃₁×#479	C₂²	semidirect
34. C₃₁×#478	C₂²	semidirect
35. C₃₁×#477	C₂²	semidirect
36. C₃₁×#476	C₂²	semidirect
37. C₃₁×#475	C₂²	semidirect
38. C₃₁×#474	C₂²	semidirect
45. C₃₁×#464	C₂²	semidirect
46. C₃₁×#463	C₂²	semidirect
47. C₃₁×#462	C₂²	semidirect
48. C₃₁×#461	C₂²	semidirect
49. C₃₁×#460	C₂²	semidirect
50. C₃₁×#459	C₂²	semidirect
219. #60907	C₃₁	direct (N × H)
268. #3444	C₂×C₃₁	semidirect
269. #3256	C₂×C₃₁	semidirect
270. #3255	C₂×C₃₁	semidirect
271. #3254	C₂×C₃₁	semidirect
272. #3253	C₂×C₃₁	semidirect
273. #3252	C₂×C₃₁	semidirect
274. #3251	C₂×C₃₁	semidirect
275. #3076	C₂×C₃₁	semidirect
276. #3054	C₂×C₃₁	semidirect
277. #3035	C₂×C₃₁	semidirect
278. C₂×#423	C₂×C₃₁	semidirect
323. #479	C₂²×C₃₁	semidirect
324. #478	C₂²×C₃₁	semidirect
325. #477	C₂²×C₃₁	semidirect
326. #476	C₂²×C₃₁	semidirect
327. #475	C₂²×C₃₁	semidirect
328. #474	C₂²×C₃₁	semidirect
335. #464	C₂²×C₃₁	semidirect
336. #463	C₂²×C₃₁	semidirect
337. #462	C₂²×C₃₁	semidirect
338. #461	C₂²×C₃₁	semidirect
339. #460	C₂²×C₃₁	semidirect
340. #459	C₂²×C₃₁	semidirect
440. C₃₁	#60907	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×#3444	2
C₃₁×#3256	2
C₃₁×#3255	2
C₃₁×#3254	2
C₃₁×#3253	2
C₃₁×#3252	2
C₃₁×#3251	2
C₃₁×#3076	2
C₃₁×#3054	2
C₃₁×#3035	2
C₂×C₃₁×#423	2
#60907	31

Subgroups — C31×#60907 #21517666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60907 #21517666