Subgroups — C₃₁×#23019 #21479778

Order 7936.

880 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2933	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1929	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1871	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1866	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#525	C₂²	1	2	yes	—
17	C₃₁×#467	C₄	1	1	yes	characteristic, complemented
18	C₃₁×#467	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#466	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#466	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#465	C₄	1	2	yes	complemented
22	C₃₁×#465	C₄	1	2	yes	complemented
23	C₃₁×#444	—	2	4	no	—
24	C₃₁×#436	—	2	2	no	—
25	C₃₁×#435	—	2	2	no	—
26	C₃₁×#435	—	2	2	no	—
27	C₈×D₄×C₃₁	—	2	2	no	—
28	C₂×C₃₁×#108	—	2	4	no	—
29	C₂×C₃₁×#108	C₂²	1	2	yes	—
30	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
32	C₃₁×#427	C₂²	1	1	yes	characteristic
33	C₃₁×#424	C₂²	1	2	yes	—
34	C₂×C₃₁×#104	C₂²	1	2	yes	—
35	C₂×C₃₁×#104	C₂²	1	2	yes	—
36	C₃₁×#416	C₂²	1	2	yes	—
37	C₃₁×#408	C₂²	1	2	yes	—
38	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₃₁×#405	C₂²	1	2	yes	—
40	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#339	C₂²	1	2	yes	—
44	C₃₁×#339	C₂²	1	2	yes	—
45	C₃₁×#336	C₂²	1	1	yes	characteristic
46	C₃₁×#336	C₂²	1	1	yes	characteristic
47	C₂×#41×C₃₁	C₂³	1	2	yes	—
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×#135	—	2	2	no	—
50	C₃₁×#135	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₃₁×#131	—	2	4	no	—
55	C₃₁×#131	—	4	2	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	2	4	no	—
59	C₂×M₁₆×C₃₁	—	4	2	no	—
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂²×C₈×C₃₁	—	2	2	no	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	—	2	2	no	—
65	C₂²×C₈×C₃₁	—	2	2	no	—
66	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#123	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
76	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
77	C₃₁×#121	C₂×C₄	1	2	yes	—
78	C₃₁×#121	C₂×C₄	1	2	yes	—
79	C₃₁×#120	—	2	2	no	—
80	C₄×D₄×C₃₁	—	2	2	no	—
81	C₄×D₄×C₃₁	—	2	2	no	—
82	C₃₁×#117	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#108	—	2	4	no	—
87	C₃₁×#108	—	2	2	no	—
88	C₃₁×#108	D₄	1	4	yes	complemented
89	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
90	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
91	C₃₁×#106	C₂×C₄	1	2	yes	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×C₄⋊C₈	—	2	2	no	—
94	C₃₁×#104	—	2	4	no	—
95	C₃₁×#104	—	4	2	no	—
96	C₃₁×#103	C₂×C₄	1	2	yes	—
97	C₃₁×#103	C₂×C₄	1	2	yes	—
98	C₃₁×#102	C₂×C₄	1	2	yes	—
99	C₃₁×#102	C₂×C₄	1	2	yes	—
100	C₃₁×C₂²⋊C₈	—	2	2	no	—
101	C₃₁×C₂²⋊C₈	—	2	2	no	—
102	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
103	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
104	C₃₁×#97	—	2	2	no	—
105	C₃₁×#97	—	2	2	no	—
106	C₃₁×#97	—	2	2	no	—
107	C₄×C₈×C₃₁	—	2	2	no	—
108	C₃₁×#95	—	2	1	no	—
109	C₂⁴×C₃₁	#41	1	1	yes	characteristic
110	#41×C₃₁	—	2	2	no	—
111	#41×C₃₁	—	2	2	no	—
112	#41×C₃₁	—	2	2	no	—
113	#41×C₃₁	—	2	2	no	—
114	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₃₁	—	2	2	no	—
133	SD₁₆×C₃₁	—	2	4	no	—
134	D₈×C₃₁	—	2	2	no	—
135	M₁₆×C₃₁	—	2	2	no	—
136	M₁₆×C₃₁	—	2	2	no	—
137	M₁₆×C₃₁	—	2	2	no	—
138	M₁₆×C₃₁	—	4	4	no	—
139	M₁₆×C₃₁	—	4	4	no	—
140	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	4	4	no	—
154	C₂×C₈×C₃₁	—	4	4	no	—
155	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
162	C₂×C₈×C₃₁	#41	1	2	yes	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
166	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	—	2	2	no	—
171	C₂²⋊C₄×C₃₁	—	2	2	no	—
172	C₂²⋊C₄×C₃₁	—	4	2	no	—
173	C₄²×C₃₁	—	2	2	no	—
174	C₄²×C₃₁	—	2	1	no	—
175	C₄²×C₃₁	—	2	1	no	—
176	C₄²×C₃₁	—	2	1	no	—
177	C₄²×C₃₁	—	2	1	no	—
178	#23019	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₂³×C₃₁	—	2	1	no	—
180	C₂³×C₃₁	—	2	1	no	—
181	C₂³×C₃₁	—	2	2	no	—
182	C₂³×C₃₁	—	2	2	no	—
183	C₂³×C₃₁	—	4	1	no	—
184	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
185	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
186	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
187	Q₈×C₃₁	—	2	2	no	—
188	Q₈×C₃₁	—	2	2	no	—
189	D₄×C₃₁	—	2	2	no	—
190	D₄×C₃₁	—	2	2	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	4	2	no	—
193	D₄×C₃₁	—	4	4	no	—
194	D₄×C₃₁	—	8	2	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
217	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
223	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
224	C₈×C₃₁	—	2	1	no	—
225	C₈×C₃₁	—	2	1	no	—
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	2	4	no	—
228	C₈×C₃₁	—	2	2	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₈×C₃₁	C₄×D₄	1	2	yes	—
231	C₈×C₃₁	C₄×D₄	1	2	yes	—
232	#3050	C₂×C₃₁	1	1	yes	characteristic
233	C₂×#444	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#2933	C₂×C₃₁	1	2	yes	complemented
235	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1975	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1946	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1929	C₂×C₃₁	1	2	yes	complemented
239	#1871	C₂×C₃₁	1	2	yes	complemented
240	#1866	C₂×C₃₁	1	2	yes	complemented
241	#1773	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#1752	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	—	2	1	no	—
248	C₂²×C₃₁	—	2	1	no	—
249	C₂²×C₃₁	—	2	1	no	—
250	C₂²×C₃₁	—	4	2	no	—
251	C₂²×C₃₁	—	4	2	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	#444	1	2	yes	complemented
256	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
257	C₂²×C₃₁	#548	1	1	yes	characteristic
258	C₂²×C₃₁	#553	1	1	yes	characteristic
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	2	1	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	—	4	2	no	—
264	C₄×C₃₁	—	8	1	no	—
265	C₄×C₃₁	—	8	1	no	—
266	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
267	C₄×C₃₁	#520	1	1	yes	characteristic
268	C₄×C₃₁	D₄²	1	1	yes	characteristic
269	C₄×C₃₁	#547	1	1	yes	characteristic
270	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#131	C₂²×C₃₁	1	2	yes	—
272	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
273	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
274	#525	C₂²×C₃₁	1	2	yes	—
275	#467	C₄×C₃₁	1	1	yes	characteristic, complemented
276	#467	C₄×C₃₁	1	1	yes	characteristic, complemented
277	#466	C₄×C₃₁	1	1	yes	characteristic, complemented
278	#466	C₄×C₃₁	1	1	yes	characteristic, complemented
279	#465	C₄×C₃₁	1	2	yes	complemented
280	#465	C₄×C₃₁	1	2	yes	complemented
281	#444	—	2	4	no	—
282	#436	—	2	2	no	—
283	#435	—	2	2	no	—
284	#435	—	2	2	no	—
285	C₈×D₄	—	2	2	no	—
286	C₂×#108	—	2	4	no	—
287	C₂×#108	C₂²×C₃₁	1	2	yes	—
288	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
290	#427	C₂²×C₃₁	1	1	yes	characteristic
291	#424	C₂²×C₃₁	1	2	yes	—
292	C₂×#104	C₂²×C₃₁	1	2	yes	—
293	C₂×#104	C₂²×C₃₁	1	2	yes	—
294	#416	C₂²×C₃₁	1	2	yes	—
295	#408	C₂²×C₃₁	1	2	yes	—
296	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
297	#405	C₂²×C₃₁	1	2	yes	—
298	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
300	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
301	#339	C₂²×C₃₁	1	2	yes	—
302	#339	C₂²×C₃₁	1	2	yes	—
303	#336	C₂²×C₃₁	1	1	yes	characteristic
304	#336	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×C₃₁	—	2	1	no	—
306	C₂×C₃₁	—	2	1	no	—
307	C₂×C₃₁	—	2	2	no	—
308	C₂×C₃₁	—	4	1	no	—
309	C₂×C₃₁	—	8	2	no	—
310	C₂×C₃₁	#2361	1	1	yes	characteristic
311	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
312	C₂×C₃₁	#2957	1	1	yes	characteristic
313	C₂×#41	C₂³×C₃₁	1	2	yes	—
314	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
315	#135	—	2	2	no	—
316	#135	—	2	2	no	—
317	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
318	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
319	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
320	#131	—	2	4	no	—
321	#131	—	4	2	no	—
322	C₂×M₁₆	—	2	2	no	—
323	C₂×M₁₆	—	2	2	no	—
324	C₂×M₁₆	—	2	4	no	—
325	C₂×M₁₆	—	4	2	no	—
326	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
327	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
328	C₂²×C₈	—	2	2	no	—
329	C₂²×C₈	—	2	2	no	—
330	C₂²×C₈	—	2	2	no	—
331	C₂²×C₈	—	2	2	no	—
332	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
335	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
339	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
340	#123	—	2	2	no	—
341	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
342	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
343	#121	C₂×C₄×C₃₁	1	2	yes	—
344	#121	C₂×C₄×C₃₁	1	2	yes	—
345	#120	—	2	2	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	—	2	2	no	—
348	#117	C₂³×C₃₁	1	2	yes	—
349	C₂×C₂²⋊C₄	—	2	1	no	—
350	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
352	#108	—	2	4	no	—
353	#108	—	2	2	no	—
354	#108	D₄×C₃₁	1	4	yes	complemented
355	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
356	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
357	#106	C₂×C₄×C₃₁	1	2	yes	—
358	C₄⋊C₈	—	2	2	no	—
359	C₄⋊C₈	—	2	2	no	—
360	#104	—	2	4	no	—
361	#104	—	4	2	no	—
362	#103	C₂×C₄×C₃₁	1	2	yes	—
363	#103	C₂×C₄×C₃₁	1	2	yes	—
364	#102	C₂×C₄×C₃₁	1	2	yes	—
365	#102	C₂×C₄×C₃₁	1	2	yes	—
366	C₂²⋊C₈	—	2	2	no	—
367	C₂²⋊C₈	—	2	2	no	—
368	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
369	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
370	#97	—	2	2	no	—
371	#97	—	2	2	no	—
372	#97	—	2	2	no	—
373	C₄×C₈	—	2	2	no	—
374	#95	—	2	1	no	—
375	C₃₁	#23019	1	1	yes	characteristic, direct product factor, Sylow
376	C₂⁴	#41×C₃₁	1	1	yes	characteristic
377	#41	—	2	2	no	—
378	#41	—	2	2	no	—
379	#41	—	2	2	no	—
380	#41	—	2	2	no	—
381	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
399	Q₁₆	—	2	2	no	—
400	SD₁₆	—	2	4	no	—
401	D₈	—	2	2	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	4	4	no	—
406	M₁₆	—	4	4	no	—
407	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	4	4	no	—
421	C₂×C₈	—	4	4	no	—
422	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
423	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
424	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
425	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
426	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
427	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
428	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
429	C₂×C₈	#41×C₃₁	1	2	yes	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
433	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
434	C₂²⋊C₄	—	2	2	no	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	4	2	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	2	1	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	1	no	—
447	C₂³	—	2	1	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	4	1	no	—
450	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
451	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
452	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	Q₈	—	2	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	2	no	—
457	D₄	—	4	4	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	2	no	—
460	D₄	—	8	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
489	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
490	C₈	—	2	1	no	—
491	C₈	—	2	1	no	—
492	C₈	—	2	2	no	—
493	C₈	—	2	4	no	—
494	C₈	—	2	2	no	—
495	C₈	—	4	2	no	—
496	C₈	C₄×D₄×C₃₁	1	2	yes	—
497	C₈	C₄×D₄×C₃₁	1	2	yes	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	1	no	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	C₃₁×#444	1	2	yes	complemented
511	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
512	C₂²	C₃₁×#548	1	1	yes	characteristic
513	C₂²	C₃₁×#553	1	1	yes	characteristic
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	8	1	no	—
520	C₄	—	8	1	no	—
521	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
522	C₄	C₃₁×#520	1	1	yes	characteristic
523	C₄	D₄²×C₃₁	1	1	yes	characteristic
524	C₄	C₃₁×#547	1	1	yes	characteristic
525	C₂	—	2	1	no	—
526	C₂	—	2	1	no	—
527	C₂	—	2	2	no	—
528	C₂	—	4	1	no	—
529	C₂	—	8	2	no	—
530	C₂	C₃₁×#2361	1	1	yes	characteristic
531	C₂	C₂×C₃₁×#444	1	1	yes	characteristic
532	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3050	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×C₂²⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#444	C₂	semidirect
3. C₃₁×#2933	C₂	semidirect
4. C₂²×C₃₁×#108	C₂	semidirect
5. C₃₁×#1975	C₂	semidirect
6. C₃₁×#1946	C₂	semidirect
7. C₃₁×#1929	C₂	semidirect
8. C₃₁×#1871	C₂	semidirect
9. C₃₁×#1866	C₂	semidirect
10. C₃₁×#1773	C₂	semidirect
11. C₃₁×#1752	C₂	semidirect
17. C₃₁×#467	C₄	semidirect
18. C₃₁×#467	C₄	semidirect
19. C₃₁×#466	C₄	semidirect
20. C₃₁×#466	C₄	semidirect
21. C₃₁×#465	C₄	semidirect
22. C₃₁×#465	C₄	semidirect
88. C₃₁×#108	D₄	semidirect
178. #23019	C₃₁	direct (N × H)
233. C₂×#444	C₂×C₃₁	semidirect
234. #2933	C₂×C₃₁	semidirect
235. C₂²×#108	C₂×C₃₁	semidirect
236. #1975	C₂×C₃₁	semidirect
237. #1946	C₂×C₃₁	semidirect
238. #1929	C₂×C₃₁	semidirect
239. #1871	C₂×C₃₁	semidirect
240. #1866	C₂×C₃₁	semidirect
241. #1773	C₂×C₃₁	semidirect
242. #1752	C₂×C₃₁	semidirect
255. C₂²×C₃₁	#444	semidirect
275. #467	C₄×C₃₁	semidirect
276. #467	C₄×C₃₁	semidirect
277. #466	C₄×C₃₁	semidirect
278. #466	C₄×C₃₁	semidirect
279. #465	C₄×C₃₁	semidirect
280. #465	C₄×C₃₁	semidirect
354. #108	D₄×C₃₁	semidirect
375. C₃₁	#23019	direct (N × H)
510. C₂²	C₃₁×#444	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3050	2
C₂×C₃₁×#444	2
C₃₁×#2933	2
C₂²×C₃₁×#108	2
C₃₁×#1975	2
C₃₁×#1946	2
C₃₁×#1929	2
C₃₁×#1871	2
C₃₁×#1866	2
C₃₁×#1773	2
C₃₁×#1752	2
#23019	31

Subgroups — C31×#23019 #21479778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23019 #21479778