Subgroups — C₃₁×#22897 #21479656

Order 7936.

1074 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#453	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2946	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2897	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1931	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1907	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1887	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1859	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1764	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
12	C₂×C₃₁×#131	C₂²	1	2	yes	—
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₃₁×#520	C₂²	1	2	yes	—
15	C₃₁×#518	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#453	C₄	1	8	yes	complemented
17	C₃₁×#442	—	2	2	no	—
18	C₃₁×#440	—	2	2	no	—
19	C₄×SD₁₆×C₃₁	—	2	2	no	—
20	C₄×D₈×C₃₁	—	2	2	no	—
21	C₃₁×#428	C₂²	1	2	yes	complemented
22	C₂×C₃₁×#104	C₂²	1	2	yes	—
23	C₃₁×#418	—	2	2	no	—
24	C₃₁×#418	C₂²	1	2	yes	—
25	C₃₁×#417	—	2	2	no	—
26	C₃₁×#417	C₂²	1	2	yes	—
27	C₃₁×#416	—	2	2	no	—
28	C₃₁×#416	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	—	2	2	no	—
30	C₂×C₃₁×#102	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
32	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#100	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#405	C₂²	1	2	yes	complemented
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#384	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#343	C₂²	1	2	yes	complemented
38	C₃₁×#341	C₂²	1	2	yes	complemented
39	C₃₁×#339	C₂²	1	2	yes	complemented
40	C₃₁×H(5,𝔽₂)	—	2	4	no	—
41	C₃₁×H(5,𝔽₂)	—	4	1	no	—
42	C₃₁×H(5,𝔽₂)	C₂×C₄	1	4	yes	complemented
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₂×#41×C₃₁	C₂³	1	2	yes	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	4	1	no	—
47	C₂²×D₄×C₃₁	—	4	1	no	—
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
51	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	8	yes	complemented
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₃₁×#131	—	2	4	no	—
55	C₃₁×#131	—	4	2	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
61	C₃₁×#127	—	4	1	no	—
62	C₃₁×#127	C₂×C₄	1	4	yes	complemented
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	4	2	no	—
68	C₃₁×#117	—	2	2	no	—
69	C₃₁×#117	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
73	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#106	—	2	2	no	—
77	C₃₁×#104	C₂×C₄	1	8	yes	complemented
78	C₃₁×#103	—	2	2	no	—
79	C₃₁×#103	D₄	1	2	yes	complemented
80	C₃₁×#103	D₄	1	2	yes	complemented
81	C₃₁×#102	—	2	2	no	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×#102	D₄	1	2	yes	complemented
89	C₃₁×#102	D₄	1	2	yes	complemented
90	C₃₁×#100	—	4	1	no	—
91	C₃₁×#100	C₂×C₄	1	4	yes	complemented
92	C₃₁×#97	—	2	2	no	—
93	C₄×C₈×C₃₁	—	2	2	no	—
94	C₄×C₈×C₃₁	—	4	1	no	—
95	C₂⁴×C₃₁	—	2	1	no	—
96	C₂⁴×C₃₁	—	2	1	no	—
97	C₂⁴×C₃₁	—	4	1	no	—
98	#41×C₃₁	—	2	2	no	—
99	#41×C₃₁	—	2	2	no	—
100	#41×C₃₁	—	2	8	no	—
101	#41×C₃₁	—	4	4	no	—
102	#41×C₃₁	C₂²×C₄	1	4	yes	—
103	#41×C₃₁	#41	1	4	yes	—
104	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	8	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	4	4	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂×D₄×C₃₁	—	4	4	no	—
116	C₂×D₄×C₃₁	—	4	4	no	—
117	C₂×D₄×C₃₁	—	4	4	no	—
118	C₂×D₄×C₃₁	—	8	2	no	—
119	C₂×D₄×C₃₁	—	8	2	no	—
120	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
121	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
122	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	4	1	no	—
127	C₂²×C₄×C₃₁	—	4	1	no	—
128	C₂²×C₄×C₃₁	—	4	1	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
131	SD₁₆×C₃₁	—	2	8	no	—
132	D₈×C₃₁	—	2	8	no	—
133	M₁₆×C₃₁	—	2	2	no	—
134	M₁₆×C₃₁	—	2	2	no	—
135	M₁₆×C₃₁	—	2	2	no	—
136	M₁₆×C₃₁	—	4	4	no	—
137	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	4	4	no	—
143	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	—	8	2	no	—
149	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₃₁	—	2	2	no	—
152	C₂²⋊C₄×C₃₁	—	4	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	2	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	—	4	2	no	—
157	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
158	#22897	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	2	2	no	—
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	4	4	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	—	4	1	no	—
165	C₂³×C₃₁	—	4	1	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	8	1	no	—
168	C₂³×C₃₁	—	8	1	no	—
169	C₂³×C₃₁	—	8	1	no	—
170	C₂³×C₃₁	—	8	1	no	—
171	C₂³×C₃₁	—	16	1	no	—
172	C₂³×C₃₁	—	16	1	no	—
173	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
174	Q₈×C₃₁	—	2	2	no	—
175	Q₈×C₃₁	C₄×D₄	1	4	yes	—
176	D₄×C₃₁	—	2	4	no	—
177	D₄×C₃₁	—	2	4	no	—
178	D₄×C₃₁	—	2	4	no	—
179	D₄×C₃₁	—	2	4	no	—
180	D₄×C₃₁	—	2	2	no	—
181	D₄×C₃₁	—	4	8	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	—	8	8	no	—
185	D₄×C₃₁	C₄×D₄	1	4	yes	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	4	no	—
191	C₂×C₄×C₃₁	—	2	4	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	4	4	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	8	2	no	—
197	C₂×C₄×C₃₁	—	16	1	no	—
198	C₂×C₄×C₃₁	—	16	1	no	—
199	C₂×C₄×C₃₁	—	16	2	no	—
200	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
202	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
203	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
204	C₈×C₃₁	—	2	4	no	—
205	C₈×C₃₁	—	2	2	no	—
206	C₈×C₃₁	—	2	2	no	—
207	C₈×C₃₁	—	4	2	no	—
208	C₂×#453	C₂×C₃₁	1	1	yes	characteristic
209	#2946	C₂×C₃₁	1	2	yes	complemented
210	#2897	C₂×C₃₁	1	2	yes	complemented
211	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1931	C₂×C₃₁	1	2	yes	complemented
213	#1907	C₂×C₃₁	1	2	yes	complemented
214	#1887	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1859	C₂×C₃₁	1	2	yes	complemented
216	#1764	C₂×C₃₁	1	2	yes	complemented
217	C₂²×C₃₁	—	2	1	no	—
218	C₂²×C₃₁	—	2	1	no	—
219	C₂²×C₃₁	—	2	4	no	—
220	C₂²×C₃₁	—	4	2	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	8	2	no	—
225	C₂²×C₃₁	—	8	2	no	—
226	C₂²×C₃₁	—	8	1	no	—
227	C₂²×C₃₁	—	8	1	no	—
228	C₂²×C₃₁	—	8	1	no	—
229	C₂²×C₃₁	—	8	1	no	—
230	C₂²×C₃₁	—	16	1	no	—
231	C₂²×C₃₁	—	16	1	no	—
232	C₂²×C₃₁	—	16	4	no	—
233	C₂²×C₃₁	#518	1	1	yes	characteristic
234	C₂²×C₃₁	D₄²	1	1	yes	characteristic
235	C₂²×C₃₁	#551	1	1	yes	characteristic
236	C₄×C₃₁	—	2	4	no	—
237	C₄×C₃₁	—	2	4	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	—	8	2	no	—
241	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
242	C₄×C₃₁	#546	1	2	yes	—
243	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
245	C₂×#131	C₂²×C₃₁	1	2	yes	—
246	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
247	#520	C₂²×C₃₁	1	2	yes	—
248	#518	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	#453	C₄×C₃₁	1	8	yes	complemented
250	#442	—	2	2	no	—
251	#440	—	2	2	no	—
252	C₄×SD₁₆	—	2	2	no	—
253	C₄×D₈	—	2	2	no	—
254	#428	C₂²×C₃₁	1	2	yes	complemented
255	C₂×#104	C₂²×C₃₁	1	2	yes	—
256	#418	—	2	2	no	—
257	#418	C₂²×C₃₁	1	2	yes	—
258	#417	—	2	2	no	—
259	#417	C₂²×C₃₁	1	2	yes	—
260	#416	—	2	2	no	—
261	#416	C₂²×C₃₁	1	2	yes	—
262	C₂×#102	—	2	2	no	—
263	C₂×#102	C₂²×C₃₁	1	2	yes	—
264	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
265	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#100	C₂²×C₃₁	1	1	yes	characteristic, complemented
267	#405	C₂²×C₃₁	1	2	yes	complemented
268	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
269	#384	C₂²×C₃₁	1	1	yes	characteristic, complemented
270	#343	C₂²×C₃₁	1	2	yes	complemented
271	#341	C₂²×C₃₁	1	2	yes	complemented
272	#339	C₂²×C₃₁	1	2	yes	complemented
273	C₂×C₃₁	—	2	1	no	—
274	C₂×C₃₁	—	2	1	no	—
275	C₂×C₃₁	—	4	4	no	—
276	C₂×C₃₁	—	8	2	no	—
277	C₂×C₃₁	—	8	2	no	—
278	C₂×C₃₁	#2358	1	1	yes	characteristic
279	C₂×C₃₁	#3291	1	1	yes	characteristic
280	C₂×C₃₁	#3292	1	1	yes	characteristic
281	H(5,𝔽₂)	—	2	4	no	—
282	H(5,𝔽₂)	—	4	1	no	—
283	H(5,𝔽₂)	C₂×C₄×C₃₁	1	4	yes	complemented
284	C₂×#41	—	2	2	no	—
285	C₂×#41	C₂³×C₃₁	1	2	yes	—
286	C₂²×D₄	—	2	2	no	—
287	C₂²×D₄	—	4	1	no	—
288	C₂²×D₄	—	4	1	no	—
289	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
290	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
291	AGL(1,ℤ/8)	—	2	4	no	—
292	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	8	yes	complemented
293	C₂×SD₁₆	—	2	2	no	—
294	C₂×D₈	—	2	2	no	—
295	#131	—	2	4	no	—
296	#131	—	4	2	no	—
297	C₂×M₁₆	—	2	2	no	—
298	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
299	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
300	C₂²×C₈	—	2	2	no	—
301	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
302	#127	—	4	1	no	—
303	#127	C₂×C₄×C₃₁	1	4	yes	complemented
304	C₄×Q₈	—	2	2	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	2	2	no	—
307	C₄×D₄	—	2	2	no	—
308	C₄×D₄	—	4	2	no	—
309	#117	—	2	2	no	—
310	#117	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	4	1	no	—
313	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
314	C₂×C₂²⋊C₄	—	4	1	no	—
315	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
316	#107	—	2	2	no	—
317	#106	—	2	2	no	—
318	#104	C₂×C₄×C₃₁	1	8	yes	complemented
319	#103	—	2	2	no	—
320	#103	D₄×C₃₁	1	2	yes	complemented
321	#103	D₄×C₃₁	1	2	yes	complemented
322	#102	—	2	2	no	—
323	#102	—	2	2	no	—
324	#102	—	2	2	no	—
325	#102	—	2	2	no	—
326	#102	—	2	2	no	—
327	#102	—	2	2	no	—
328	#102	—	2	2	no	—
329	#102	D₄×C₃₁	1	2	yes	complemented
330	#102	D₄×C₃₁	1	2	yes	complemented
331	#100	—	4	1	no	—
332	#100	C₂×C₄×C₃₁	1	4	yes	complemented
333	#97	—	2	2	no	—
334	C₄×C₈	—	2	2	no	—
335	C₄×C₈	—	4	1	no	—
336	C₃₁	#22897	1	1	yes	characteristic, direct product factor, Sylow
337	C₂⁴	—	2	1	no	—
338	C₂⁴	—	2	1	no	—
339	C₂⁴	—	4	1	no	—
340	#41	—	2	2	no	—
341	#41	—	2	2	no	—
342	#41	—	2	8	no	—
343	#41	—	4	4	no	—
344	#41	C₂²×C₄×C₃₁	1	4	yes	—
345	#41	#41×C₃₁	1	4	yes	—
346	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	8	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	2	no	—
357	C₂×D₄	—	4	4	no	—
358	C₂×D₄	—	4	4	no	—
359	C₂×D₄	—	4	4	no	—
360	C₂×D₄	—	8	2	no	—
361	C₂×D₄	—	8	2	no	—
362	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
363	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
364	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	4	1	no	—
369	C₂²×C₄	—	4	1	no	—
370	C₂²×C₄	—	4	1	no	—
371	C₂²×C₄	—	4	2	no	—
372	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
373	SD₁₆	—	2	8	no	—
374	D₈	—	2	8	no	—
375	M₁₆	—	2	2	no	—
376	M₁₆	—	2	2	no	—
377	M₁₆	—	2	2	no	—
378	M₁₆	—	4	4	no	—
379	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	4	4	no	—
385	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
386	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	—	8	2	no	—
391	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
392	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
393	C₂²⋊C₄	—	2	2	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₂²⋊C₄	—	4	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	4	2	no	—
399	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	4	4	no	—
404	C₂³	—	4	1	no	—
405	C₂³	—	4	1	no	—
406	C₂³	—	4	1	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	8	1	no	—
409	C₂³	—	8	1	no	—
410	C₂³	—	8	1	no	—
411	C₂³	—	8	1	no	—
412	C₂³	—	16	1	no	—
413	C₂³	—	16	1	no	—
414	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
415	Q₈	—	2	2	no	—
416	Q₈	C₄×D₄×C₃₁	1	4	yes	—
417	D₄	—	2	4	no	—
418	D₄	—	2	4	no	—
419	D₄	—	2	4	no	—
420	D₄	—	2	4	no	—
421	D₄	—	2	2	no	—
422	D₄	—	4	8	no	—
423	D₄	—	4	2	no	—
424	D₄	—	4	2	no	—
425	D₄	—	8	8	no	—
426	D₄	C₄×D₄×C₃₁	1	4	yes	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	4	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	8	2	no	—
438	C₂×C₄	—	16	1	no	—
439	C₂×C₄	—	16	1	no	—
440	C₂×C₄	—	16	2	no	—
441	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
443	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
444	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
445	C₈	—	2	4	no	—
446	C₈	—	2	2	no	—
447	C₈	—	2	2	no	—
448	C₈	—	4	2	no	—
449	C₂²	—	2	1	no	—
450	C₂²	—	2	1	no	—
451	C₂²	—	2	4	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	8	2	no	—
457	C₂²	—	8	2	no	—
458	C₂²	—	8	1	no	—
459	C₂²	—	8	1	no	—
460	C₂²	—	8	1	no	—
461	C₂²	—	8	1	no	—
462	C₂²	—	16	1	no	—
463	C₂²	—	16	1	no	—
464	C₂²	—	16	4	no	—
465	C₂²	C₃₁×#518	1	1	yes	characteristic
466	C₂²	D₄²×C₃₁	1	1	yes	characteristic
467	C₂²	C₃₁×#551	1	1	yes	characteristic
468	C₄	—	2	4	no	—
469	C₄	—	2	4	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	2	no	—
472	C₄	—	8	2	no	—
473	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
474	C₄	C₃₁×#546	1	2	yes	—
475	C₂	—	2	1	no	—
476	C₂	—	2	1	no	—
477	C₂	—	4	4	no	—
478	C₂	—	8	2	no	—
479	C₂	—	8	2	no	—
480	C₂	C₃₁×#2358	1	1	yes	characteristic
481	C₂	C₃₁×#3291	1	1	yes	characteristic
482	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#453	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. C₃₁×#2946	C₂	semidirect
3. C₃₁×#2897	C₂	semidirect
4. C₃₁×#1957	C₂	semidirect
5. C₃₁×#1931	C₂	semidirect
6. C₃₁×#1907	C₂	semidirect
7. C₃₁×#1887	C₂	semidirect
8. C₃₁×#1859	C₂	semidirect
9. C₃₁×#1764	C₂	semidirect
15. C₃₁×#518	C₂²	semidirect
16. C₃₁×#453	C₄	semidirect
21. C₃₁×#428	C₂²	semidirect
31. C₂×C₃₁×#102	C₂²	semidirect
33. C₂×C₃₁×#100	C₂²	semidirect
34. C₃₁×#405	C₂²	semidirect
35. C₂×C₄×C₈×C₃₁	C₂²	semidirect
36. C₃₁×#384	C₂²	semidirect
37. C₃₁×#343	C₂²	semidirect
38. C₃₁×#341	C₂²	semidirect
39. C₃₁×#339	C₂²	semidirect
42. C₃₁×H(5,𝔽₂)	C₂×C₄	semidirect
51. C₃₁×AGL(1,ℤ/8)	C₂×C₄	semidirect
62. C₃₁×#127	C₂×C₄	semidirect
77. C₃₁×#104	C₂×C₄	semidirect
79. C₃₁×#103	D₄	semidirect
80. C₃₁×#103	D₄	semidirect
88. C₃₁×#102	D₄	semidirect
89. C₃₁×#102	D₄	semidirect
91. C₃₁×#100	C₂×C₄	semidirect
158. #22897	C₃₁	direct (N × H)
209. #2946	C₂×C₃₁	semidirect
210. #2897	C₂×C₃₁	semidirect
211. #1957	C₂×C₃₁	semidirect
212. #1931	C₂×C₃₁	semidirect
213. #1907	C₂×C₃₁	semidirect
214. #1887	C₂×C₃₁	semidirect
215. #1859	C₂×C₃₁	semidirect
216. #1764	C₂×C₃₁	semidirect
248. #518	C₂²×C₃₁	semidirect
249. #453	C₄×C₃₁	semidirect
254. #428	C₂²×C₃₁	semidirect
264. C₂×#102	C₂²×C₃₁	semidirect
266. C₂×#100	C₂²×C₃₁	semidirect
267. #405	C₂²×C₃₁	semidirect
268. C₂×C₄×C₈	C₂²×C₃₁	semidirect
269. #384	C₂²×C₃₁	semidirect
270. #343	C₂²×C₃₁	semidirect
271. #341	C₂²×C₃₁	semidirect
272. #339	C₂²×C₃₁	semidirect
283. H(5,𝔽₂)	C₂×C₄×C₃₁	semidirect
292. AGL(1,ℤ/8)	C₂×C₄×C₃₁	semidirect
303. #127	C₂×C₄×C₃₁	semidirect
318. #104	C₂×C₄×C₃₁	semidirect
320. #103	D₄×C₃₁	semidirect
321. #103	D₄×C₃₁	semidirect
329. #102	D₄×C₃₁	semidirect
330. #102	D₄×C₃₁	semidirect
332. #100	C₂×C₄×C₃₁	semidirect
336. C₃₁	#22897	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#453	2
C₃₁×#2946	2
C₃₁×#2897	2
C₃₁×#1957	2
C₃₁×#1931	2
C₃₁×#1907	2
C₃₁×#1887	2
C₃₁×#1859	2
C₃₁×#1764	2
#22897	31

Subgroups — C31×#22897 #21479656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22897 #21479656