Subgroups — C₃₁×#22463 #21479222

Order 7936.

1380 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#576	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2952	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2897	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1907	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1883	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1815	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1792	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	—
13	C₃₁×#576	—	2	4	no	—
14	C₃₁×#576	C₄	1	8	yes	complemented
15	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
16	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
17	C₂×C₃₁×#135	—	2	1	no	—
18	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
19	C₂²×D₈×C₃₁	—	2	1	no	—
20	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#131	C₂²	1	2	yes	complemented
23	C₃₁×#520	C₂²	1	2	yes	complemented
24	C₃₁×#444	—	2	4	no	—
25	C₃₁×#442	—	2	1	no	—
26	C₃₁×#441	—	2	1	no	—
27	C₄×SD₁₆×C₃₁	—	2	2	no	—
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#104	C₂²	1	2	yes	complemented
32	C₃₁×#418	—	2	2	no	—
33	C₃₁×#418	C₂²	1	2	yes	complemented
34	C₃₁×#417	—	2	2	no	—
35	C₃₁×#417	C₂²	1	2	yes	—
36	C₃₁×#417	C₂²	1	2	yes	complemented
37	C₃₁×#416	—	2	2	no	—
38	C₃₁×#416	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#102	—	2	2	no	—
40	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
41	C₂×C₃₁×#100	C₂²	1	2	yes	complemented
42	C₃₁×#405	C₂²	1	1	yes	characteristic, complemented
43	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#390	C₂²	1	2	yes	complemented
45	C₃₁×#343	C₂²	1	2	yes	complemented
46	C₃₁×#341	C₂²	1	1	yes	characteristic, complemented
47	C₃₁×#339	C₂²	1	1	yes	characteristic, complemented
48	C₃₁×H(5,𝔽₂)	—	2	4	no	—
49	C₃₁×H(5,𝔽₂)	—	2	4	no	—
50	C₃₁×H(5,𝔽₂)	—	4	2	no	—
51	C₃₁×H(5,𝔽₂)	C₂×C₄	1	8	yes	complemented
52	C₂×#41×C₃₁	—	2	2	no	—
53	C₂×#41×C₃₁	—	2	2	no	—
54	C₂×#41×C₃₁	C₂³	1	2	yes	—
55	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₃₁	—	2	2	no	—
57	C₂²×D₄×C₃₁	—	2	2	no	—
58	C₂²×D₄×C₃₁	—	4	2	no	—
59	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
60	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
61	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
62	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	8	yes	complemented
63	C₃₁×#135	—	2	8	no	—
64	C₃₁×#135	C₂×C₄	1	4	yes	—
65	C₃₁×#135	D₄	1	4	yes	—
66	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
67	C₂×SD₁₆×C₃₁	—	2	2	no	—
68	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
69	C₂×D₈×C₃₁	—	2	2	no	—
70	C₂×D₈×C₃₁	—	2	8	no	—
71	C₂×D₈×C₃₁	—	2	1	no	—
72	C₂×D₈×C₃₁	—	2	1	no	—
73	C₂×D₈×C₃₁	—	2	2	no	—
74	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
75	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
76	C₃₁×#131	—	2	2	no	—
77	C₃₁×#131	—	2	4	no	—
78	C₃₁×#131	—	4	2	no	—
79	C₃₁×#131	C₂×C₄	1	4	yes	complemented
80	C₂×M₁₆×C₃₁	—	2	1	no	—
81	C₂×M₁₆×C₃₁	—	2	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	—	2	1	no	—
86	C₂²×C₈×C₃₁	—	2	2	no	—
87	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	4	2	no	—
92	C₃₁×#117	—	2	2	no	—
93	C₃₁×#117	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
97	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
98	C₃₁×#108	—	2	2	no	—
99	C₃₁×#106	—	2	1	no	—
100	C₃₁×#106	—	2	1	no	—
101	C₃₁×#104	—	2	4	no	—
102	C₃₁×#104	—	2	2	no	—
103	C₃₁×#103	—	2	2	no	—
104	C₃₁×#103	—	2	2	no	—
105	C₃₁×#103	D₄	1	2	yes	—
106	C₃₁×#102	—	2	2	no	—
107	C₃₁×#102	—	2	2	no	—
108	C₃₁×#102	—	2	2	no	—
109	C₃₁×#102	—	2	2	no	—
110	C₃₁×#102	—	2	2	no	—
111	C₃₁×#102	—	2	2	no	—
112	C₃₁×#102	D₄	1	2	yes	—
113	C₃₁×#100	—	4	2	no	—
114	C₃₁×#97	—	2	2	no	—
115	C₃₁×#97	—	2	1	no	—
116	C₄×C₈×C₃₁	—	2	1	no	—
117	C₂⁴×C₃₁	—	2	2	no	—
118	C₂⁴×C₃₁	—	4	2	no	—
119	#41×C₃₁	—	2	1	no	—
120	#41×C₃₁	—	2	2	no	—
121	#41×C₃₁	—	2	2	no	—
122	#41×C₃₁	—	2	1	no	—
123	#41×C₃₁	—	2	8	no	—
124	#41×C₃₁	—	2	8	no	—
125	#41×C₃₁	—	4	4	no	—
126	#41×C₃₁	—	4	4	no	—
127	#41×C₃₁	C₂²×C₄	1	4	yes	—
128	#41×C₃₁	C₂²×C₄	1	2	yes	—
129	#41×C₃₁	C₂×D₄	1	4	yes	—
130	#41×C₃₁	#41	1	2	yes	—
131	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
132	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	1	no	—
135	C₂×D₄×C₃₁	—	2	8	no	—
136	C₂×D₄×C₃₁	—	2	8	no	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	4	4	no	—
143	C₂×D₄×C₃₁	—	4	4	no	—
144	C₂×D₄×C₃₁	—	4	2	no	—
145	C₂×D₄×C₃₁	—	4	2	no	—
146	C₂×D₄×C₃₁	—	4	8	no	—
147	C₂×D₄×C₃₁	—	4	8	no	—
148	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
149	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	2	1	no	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	—	4	2	no	—
156	C₂²×C₄×C₃₁	—	4	2	no	—
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₃₁	C₂²⋊C₄	1	4	yes	complemented
159	SD₁₆×C₃₁	—	2	8	no	—
160	SD₁₆×C₃₁	C₂²⋊C₄	1	8	yes	complemented
161	D₈×C₃₁	—	2	8	no	—
162	D₈×C₃₁	—	2	4	no	—
163	D₈×C₃₁	—	2	4	no	—
164	D₈×C₃₁	C₂²⋊C₄	1	4	yes	complemented
165	M₁₆×C₃₁	—	2	1	no	—
166	M₁₆×C₃₁	—	2	4	no	—
167	M₁₆×C₃₁	—	2	2	no	—
168	M₁₆×C₃₁	—	2	1	no	—
169	M₁₆×C₃₁	—	2	1	no	—
170	M₁₆×C₃₁	—	2	2	no	—
171	M₁₆×C₃₁	—	4	4	no	—
172	M₁₆×C₃₁	C₂²×C₄	1	2	yes	—
173	C₂×C₈×C₃₁	—	2	1	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	1	no	—
176	C₂×C₈×C₃₁	—	2	4	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	2	no	—
179	C₂×C₈×C₃₁	—	4	4	no	—
180	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
181	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	4	2	no	—
188	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
189	C₂²⋊C₄×C₃₁	—	2	1	no	—
190	C₂²⋊C₄×C₃₁	—	4	2	no	—
191	C₂²⋊C₄×C₃₁	—	8	4	no	—
192	C₄²×C₃₁	—	2	1	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	2	1	no	—
195	C₄²×C₃₁	—	4	2	no	—
196	#22463	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	1	no	—
199	C₂³×C₃₁	—	2	4	no	—
200	C₂³×C₃₁	—	4	8	no	—
201	C₂³×C₃₁	—	4	2	no	—
202	C₂³×C₃₁	—	4	2	no	—
203	C₂³×C₃₁	—	8	2	no	—
204	C₂³×C₃₁	—	8	2	no	—
205	C₂³×C₃₁	—	16	2	no	—
206	C₂³×C₃₁	—	16	2	no	—
207	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
208	Q₈×C₃₁	—	2	1	no	—
209	Q₈×C₃₁	—	2	2	no	—
210	Q₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
211	Q₈×C₃₁	C₄×D₄	1	2	yes	—
212	D₄×C₃₁	—	2	2	no	—
213	D₄×C₃₁	—	2	4	no	—
214	D₄×C₃₁	—	2	2	no	—
215	D₄×C₃₁	—	2	4	no	—
216	D₄×C₃₁	—	2	4	no	—
217	D₄×C₃₁	—	2	4	no	—
218	D₄×C₃₁	—	2	1	no	—
219	D₄×C₃₁	—	2	2	no	—
220	D₄×C₃₁	—	4	8	no	—
221	D₄×C₃₁	—	4	8	no	—
222	D₄×C₃₁	—	4	2	no	—
223	D₄×C₃₁	—	4	2	no	—
224	D₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
225	D₄×C₃₁	C₄×D₄	1	2	yes	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	4	8	no	—
234	C₂×C₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	4	2	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	8	2	no	—
238	C₂×C₄×C₃₁	—	16	2	no	—
239	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
244	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
245	C₈×C₃₁	—	2	2	no	—
246	C₈×C₃₁	—	2	2	no	—
247	C₈×C₃₁	—	4	2	no	—
248	C₈×C₃₁	—	4	1	no	—
249	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
250	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
251	C₂×#576	C₂×C₃₁	1	1	yes	characteristic
252	C₂×#444	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#2952	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#2897	C₂×C₃₁	1	2	yes	complemented
255	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1907	C₂×C₃₁	1	2	yes	complemented
258	#1883	C₂×C₃₁	1	2	yes	complemented
259	#1815	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1792	C₂×C₃₁	1	2	yes	complemented
261	#1765	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	2	4	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	2	no	—
268	C₂²×C₃₁	—	4	4	no	—
269	C₂²×C₃₁	—	8	4	no	—
270	C₂²×C₃₁	—	8	2	no	—
271	C₂²×C₃₁	—	8	2	no	—
272	C₂²×C₃₁	—	16	2	no	—
273	C₂²×C₃₁	—	16	8	no	—
274	C₂²×C₃₁	#518	1	1	yes	characteristic
275	C₂²×C₃₁	D₄²	1	1	yes	characteristic
276	C₂²×C₃₁	#546	1	1	yes	characteristic
277	C₄×C₃₁	—	2	2	no	—
278	C₄×C₃₁	—	2	4	no	—
279	C₄×C₃₁	—	4	2	no	—
280	C₄×C₃₁	—	8	1	no	—
281	C₄×C₃₁	—	8	2	no	—
282	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
283	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
284	C₄×C₃₁	D₄²	1	1	yes	characteristic
285	C₄×C₃₁	#546	1	1	yes	characteristic
286	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	—
287	#576	—	2	4	no	—
288	#576	C₄×C₃₁	1	8	yes	complemented
289	C₂×AGL(1,ℤ/8)	—	2	2	no	—
290	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
291	C₂×#135	—	2	1	no	—
292	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
293	C₂²×D₈	—	2	1	no	—
294	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#131	C₂²×C₃₁	1	2	yes	complemented
297	#520	C₂²×C₃₁	1	2	yes	complemented
298	#444	—	2	4	no	—
299	#442	—	2	1	no	—
300	#441	—	2	1	no	—
301	C₄×SD₁₆	—	2	2	no	—
302	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic, complemented
303	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic, complemented
304	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×#104	C₂²×C₃₁	1	2	yes	complemented
306	#418	—	2	2	no	—
307	#418	C₂²×C₃₁	1	2	yes	complemented
308	#417	—	2	2	no	—
309	#417	C₂²×C₃₁	1	2	yes	—
310	#417	C₂²×C₃₁	1	2	yes	complemented
311	#416	—	2	2	no	—
312	#416	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×#102	—	2	2	no	—
314	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
315	C₂×#100	C₂²×C₃₁	1	2	yes	complemented
316	#405	C₂²×C₃₁	1	1	yes	characteristic, complemented
317	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
318	#390	C₂²×C₃₁	1	2	yes	complemented
319	#343	C₂²×C₃₁	1	2	yes	complemented
320	#341	C₂²×C₃₁	1	1	yes	characteristic, complemented
321	#339	C₂²×C₃₁	1	1	yes	characteristic, complemented
322	C₂×C₃₁	—	2	1	no	—
323	C₂×C₃₁	—	2	1	no	—
324	C₂×C₃₁	—	4	4	no	—
325	C₂×C₃₁	—	4	2	no	—
326	C₂×C₃₁	—	8	4	no	—
327	C₂×C₃₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
328	C₂×C₃₁	#3288	1	2	yes	—
329	H(5,𝔽₂)	—	2	4	no	—
330	H(5,𝔽₂)	—	2	4	no	—
331	H(5,𝔽₂)	—	4	2	no	—
332	H(5,𝔽₂)	C₂×C₄×C₃₁	1	8	yes	complemented
333	C₂×#41	—	2	2	no	—
334	C₂×#41	—	2	2	no	—
335	C₂×#41	C₂³×C₃₁	1	2	yes	—
336	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
337	C₂²×D₄	—	2	2	no	—
338	C₂²×D₄	—	2	2	no	—
339	C₂²×D₄	—	4	2	no	—
340	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
341	AGL(1,ℤ/8)	—	2	4	no	—
342	AGL(1,ℤ/8)	—	2	16	no	—
343	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	8	yes	complemented
344	#135	—	2	8	no	—
345	#135	C₂×C₄×C₃₁	1	4	yes	—
346	#135	D₄×C₃₁	1	4	yes	—
347	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
348	C₂×SD₁₆	—	2	2	no	—
349	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
350	C₂×D₈	—	2	2	no	—
351	C₂×D₈	—	2	8	no	—
352	C₂×D₈	—	2	1	no	—
353	C₂×D₈	—	2	1	no	—
354	C₂×D₈	—	2	2	no	—
355	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
356	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
357	#131	—	2	2	no	—
358	#131	—	2	4	no	—
359	#131	—	4	2	no	—
360	#131	C₂×C₄×C₃₁	1	4	yes	complemented
361	C₂×M₁₆	—	2	1	no	—
362	C₂×M₁₆	—	2	2	no	—
363	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
364	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
365	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
366	C₂²×C₈	—	2	1	no	—
367	C₂²×C₈	—	2	2	no	—
368	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
370	C₄×Q₈	—	2	2	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	—	4	2	no	—
373	#117	—	2	2	no	—
374	#117	C₂³×C₃₁	1	1	yes	characteristic
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
377	C₂×C₂²⋊C₄	—	4	2	no	—
378	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	#108	—	2	2	no	—
380	#106	—	2	1	no	—
381	#106	—	2	1	no	—
382	#104	—	2	4	no	—
383	#104	—	2	2	no	—
384	#103	—	2	2	no	—
385	#103	—	2	2	no	—
386	#103	D₄×C₃₁	1	2	yes	—
387	#102	—	2	2	no	—
388	#102	—	2	2	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	#102	D₄×C₃₁	1	2	yes	—
394	#100	—	4	2	no	—
395	#97	—	2	2	no	—
396	#97	—	2	1	no	—
397	C₄×C₈	—	2	1	no	—
398	C₃₁	#22463	1	1	yes	characteristic, direct product factor, Sylow
399	C₂⁴	—	2	2	no	—
400	C₂⁴	—	4	2	no	—
401	#41	—	2	1	no	—
402	#41	—	2	2	no	—
403	#41	—	2	2	no	—
404	#41	—	2	1	no	—
405	#41	—	2	8	no	—
406	#41	—	2	8	no	—
407	#41	—	4	4	no	—
408	#41	—	4	4	no	—
409	#41	C₂²×C₄×C₃₁	1	4	yes	—
410	#41	C₂²×C₄×C₃₁	1	2	yes	—
411	#41	C₂×D₄×C₃₁	1	4	yes	—
412	#41	#41×C₃₁	1	2	yes	—
413	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
414	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	4	no	—
426	C₂×D₄	—	4	2	no	—
427	C₂×D₄	—	4	2	no	—
428	C₂×D₄	—	4	8	no	—
429	C₂×D₄	—	4	8	no	—
430	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
431	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
432	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
440	Q₁₆	C₂²⋊C₄×C₃₁	1	4	yes	complemented
441	SD₁₆	—	2	8	no	—
442	SD₁₆	C₂²⋊C₄×C₃₁	1	8	yes	complemented
443	D₈	—	2	8	no	—
444	D₈	—	2	4	no	—
445	D₈	—	2	4	no	—
446	D₈	C₂²⋊C₄×C₃₁	1	4	yes	complemented
447	M₁₆	—	2	1	no	—
448	M₁₆	—	2	4	no	—
449	M₁₆	—	2	2	no	—
450	M₁₆	—	2	1	no	—
451	M₁₆	—	2	1	no	—
452	M₁₆	—	2	2	no	—
453	M₁₆	—	4	4	no	—
454	M₁₆	C₂²×C₄×C₃₁	1	2	yes	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	4	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	4	4	no	—
462	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
463	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
464	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	4	2	no	—
470	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
471	C₂²⋊C₄	—	2	1	no	—
472	C₂²⋊C₄	—	4	2	no	—
473	C₂²⋊C₄	—	8	4	no	—
474	C₄²	—	2	1	no	—
475	C₄²	—	2	1	no	—
476	C₄²	—	2	1	no	—
477	C₄²	—	4	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	1	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	4	8	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	8	2	no	—
485	C₂³	—	8	2	no	—
486	C₂³	—	16	2	no	—
487	C₂³	—	16	2	no	—
488	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
489	Q₈	—	2	1	no	—
490	Q₈	—	2	2	no	—
491	Q₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
492	Q₈	C₄×D₄×C₃₁	1	2	yes	—
493	D₄	—	2	2	no	—
494	D₄	—	2	4	no	—
495	D₄	—	2	2	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	4	no	—
498	D₄	—	2	4	no	—
499	D₄	—	2	1	no	—
500	D₄	—	2	2	no	—
501	D₄	—	4	8	no	—
502	D₄	—	4	8	no	—
503	D₄	—	4	2	no	—
504	D₄	—	4	2	no	—
505	D₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
506	D₄	C₄×D₄×C₃₁	1	2	yes	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	4	8	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	8	2	no	—
519	C₂×C₄	—	16	2	no	—
520	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
524	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
525	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
526	C₈	—	2	2	no	—
527	C₈	—	2	2	no	—
528	C₈	—	4	2	no	—
529	C₈	—	4	1	no	—
530	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
531	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	4	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	4	no	—
539	C₂²	—	8	4	no	—
540	C₂²	—	8	2	no	—
541	C₂²	—	8	2	no	—
542	C₂²	—	16	2	no	—
543	C₂²	—	16	8	no	—
544	C₂²	C₃₁×#518	1	1	yes	characteristic
545	C₂²	D₄²×C₃₁	1	1	yes	characteristic
546	C₂²	C₃₁×#546	1	1	yes	characteristic
547	C₄	—	2	2	no	—
548	C₄	—	2	4	no	—
549	C₄	—	4	2	no	—
550	C₄	—	8	1	no	—
551	C₄	—	8	2	no	—
552	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
553	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
554	C₄	D₄²×C₃₁	1	1	yes	characteristic
555	C₄	C₃₁×#546	1	1	yes	characteristic
556	C₂	—	2	1	no	—
557	C₂	—	2	1	no	—
558	C₂	—	4	2	no	—
559	C₂	—	4	4	no	—
560	C₂	—	8	4	no	—
561	C₂	D₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
562	C₂	C₃₁×#3288	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#576	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ 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×#2952	C₂	semidirect
4. C₃₁×#2897	C₂	semidirect
5. C₃₁×#1978	C₂	semidirect
6. C₃₁×#1972	C₂	semidirect
7. C₃₁×#1907	C₂	semidirect
8. C₃₁×#1883	C₂	semidirect
9. C₃₁×#1815	C₂	semidirect
10. C₃₁×#1792	C₂	semidirect
11. C₃₁×#1765	C₂	semidirect
14. C₃₁×#576	C₄	semidirect
22. C₂×C₃₁×#131	C₂²	semidirect
23. C₃₁×#520	C₂²	semidirect
28. C₂×C₃₁×#108	C₂²	semidirect
29. C₂×C₃₁×#106	C₂²	semidirect
31. C₂×C₃₁×#104	C₂²	semidirect
33. C₃₁×#418	C₂²	semidirect
36. C₃₁×#417	C₂²	semidirect
40. C₂×C₃₁×#102	C₂²	semidirect
41. C₂×C₃₁×#100	C₂²	semidirect
42. C₃₁×#405	C₂²	semidirect
43. C₂×C₃₁×#97	C₂²	semidirect
44. C₃₁×#390	C₂²	semidirect
45. C₃₁×#343	C₂²	semidirect
46. C₃₁×#341	C₂²	semidirect
47. C₃₁×#339	C₂²	semidirect
51. C₃₁×H(5,𝔽₂)	C₂×C₄	semidirect
62. C₃₁×AGL(1,ℤ/8)	C₂×C₄	semidirect
74. C₂×D₈×C₃₁	C₂×C₄	semidirect
79. C₃₁×#131	C₂×C₄	semidirect
158. Q₁₆×C₃₁	C₂²⋊C₄	semidirect
160. SD₁₆×C₃₁	C₂²⋊C₄	semidirect
164. D₈×C₃₁	C₂²⋊C₄	semidirect
196. #22463	C₃₁	direct (N × H)
252. C₂×#444	C₂×C₃₁	semidirect
253. #2952	C₂×C₃₁	semidirect
254. #2897	C₂×C₃₁	semidirect
255. #1978	C₂×C₃₁	semidirect
256. #1972	C₂×C₃₁	semidirect
257. #1907	C₂×C₃₁	semidirect
258. #1883	C₂×C₃₁	semidirect
259. #1815	C₂×C₃₁	semidirect
260. #1792	C₂×C₃₁	semidirect
261. #1765	C₂×C₃₁	semidirect
288. #576	C₄×C₃₁	semidirect
296. C₂×#131	C₂²×C₃₁	semidirect
297. #520	C₂²×C₃₁	semidirect
302. C₂×#108	C₂²×C₃₁	semidirect
303. C₂×#106	C₂²×C₃₁	semidirect
305. C₂×#104	C₂²×C₃₁	semidirect
307. #418	C₂²×C₃₁	semidirect
310. #417	C₂²×C₃₁	semidirect
314. C₂×#102	C₂²×C₃₁	semidirect
315. C₂×#100	C₂²×C₃₁	semidirect
316. #405	C₂²×C₃₁	semidirect
317. C₂×#97	C₂²×C₃₁	semidirect
318. #390	C₂²×C₃₁	semidirect
319. #343	C₂²×C₃₁	semidirect
320. #341	C₂²×C₃₁	semidirect
321. #339	C₂²×C₃₁	semidirect
332. H(5,𝔽₂)	C₂×C₄×C₃₁	semidirect
343. AGL(1,ℤ/8)	C₂×C₄×C₃₁	semidirect
355. C₂×D₈	C₂×C₄×C₃₁	semidirect
360. #131	C₂×C₄×C₃₁	semidirect
398. C₃₁	#22463	direct (N × H)
440. Q₁₆	C₂²⋊C₄×C₃₁	semidirect
442. SD₁₆	C₂²⋊C₄×C₃₁	semidirect
446. D₈	C₂²⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#576	2
C₂×C₃₁×#444	2
C₃₁×#2952	2
C₃₁×#2897	2
C₃₁×#1978	2
C₃₁×#1972	2
C₃₁×#1907	2
C₃₁×#1883	2
C₃₁×#1815	2
C₃₁×#1792	2
C₃₁×#1765	2
#22463	31

Subgroups — C31×#22463 #21479222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22463 #21479222