Subgroups — C₃₁×#19900 #21476659

Order 7936.

710 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3393	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3324	C₂	1	2	yes	maximal, complemented
3	D₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1709	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1701	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1639	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1628	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	—	2	1	no	—
13	C₃₁×#556	C₂²	1	2	yes	complemented
14	D₄²×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#495	—	2	1	no	—
16	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#493	C₂²	1	1	yes	characteristic
18	C₃₁×#486	C₂²	1	2	yes	complemented
19	C₃₁×#486	C₂²	1	2	yes	complemented
20	C₃₁×#482	—	2	2	no	—
21	C₃₁×#474	C₂²	1	2	yes	—
22	C₃₁×#468	—	2	2	no	—
23	C₃₁×#466	—	2	1	no	—
24	C₃₁×#464	—	2	2	no	—
25	C₃₁×#461	C₂²	1	2	yes	complemented
26	C₃₁×#460	C₂²	1	2	yes	complemented
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#450	—	2	2	no	—
29	C₃₁×#447	—	2	2	no	—
30	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
31	C₄×Q₁₆×C₃₁	—	2	1	no	—
32	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#436	C₂²	1	2	yes	complemented
35	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#418	—	2	2	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#334	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#332	C₂²	1	2	yes	complemented
40	C₃₁×#331	C₂²	1	2	yes	complemented
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂×Q₁₆×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	—	2	1	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₂×D₈×C₃₁	—	2	1	no	—
55	C₂×D₈×C₃₁	—	4	4	no	—
56	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₃₁×#127	C₂³	1	2	yes	—
61	C₃₁×#126	—	4	2	no	—
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×C₂²⋊Q₈	—	2	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₃₁×#121	—	2	2	no	—
66	C₃₁×#120	—	4	2	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₄×D₄×C₃₁	C₂³	1	2	yes	—
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#107	—	2	2	no	—
73	C₃₁×#107	D₄	1	1	yes	characteristic
74	C₃₁×#106	D₄	1	1	yes	characteristic
75	C₃₁×C₄⋊C₈	—	2	1	no	—
76	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
78	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₃₁×#103	—	2	2	no	—
80	C₃₁×#103	D₄	1	1	yes	characteristic
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	—	4	2	no	—
87	C₃₁×#102	D₄	1	1	yes	characteristic
88	C₃₁×#102	D₄	1	2	yes	—
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	2	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₂⁴×C₃₁	—	4	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂×D₄×C₃₁	—	4	2	no	—
103	C₂×D₄×C₃₁	—	4	4	no	—
104	C₂×D₄×C₃₁	—	4	4	no	—
105	C₂×D₄×C₃₁	—	8	2	no	—
106	C₂×D₄×C₃₁	—	8	2	no	—
107	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	Q₁₆×C₃₁	—	4	2	no	—
112	Q₁₆×C₃₁	—	8	1	no	—
113	SD₁₆×C₃₁	—	2	2	no	—
114	SD₁₆×C₃₁	—	8	2	no	—
115	SD₁₆×C₃₁	—	8	2	no	—
116	D₈×C₃₁	—	2	2	no	—
117	D₈×C₃₁	—	4	4	no	—
118	D₈×C₃₁	—	4	2	no	—
119	D₈×C₃₁	—	8	1	no	—
120	M₁₆×C₃₁	—	8	2	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	4	2	no	—
126	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₃₁	—	2	1	no	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₂²⋊C₄×C₃₁	—	4	2	no	—
141	C₂²⋊C₄×C₃₁	—	4	2	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	4	2	no	—
144	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
145	#19900	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	—	8	4	no	—
150	C₂³×C₃₁	—	8	4	no	—
151	C₂³×C₃₁	—	16	2	no	—
152	Q₈×C₃₁	—	2	1	no	—
153	Q₈×C₃₁	—	2	1	no	—
154	Q₈×C₃₁	—	4	2	no	—
155	Q₈×C₃₁	—	4	2	no	—
156	D₄×C₃₁	—	2	2	no	—
157	D₄×C₃₁	—	2	1	no	—
158	D₄×C₃₁	—	2	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	8	2	no	—
164	D₄×C₃₁	—	8	4	no	—
165	D₄×C₃₁	—	8	4	no	—
166	D₄×C₃₁	#135	1	2	yes	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
179	C₈×C₃₁	—	2	1	no	—
180	C₈×C₃₁	—	2	1	no	—
181	C₈×C₃₁	—	2	1	no	—
182	C₈×C₃₁	—	4	1	no	—
183	C₈×C₃₁	—	4	1	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	—	4	1	no	—
186	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
187	#3393	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#3324	C₂×C₃₁	1	2	yes	complemented
189	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic
190	#1709	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1701	C₂×C₃₁	1	2	yes	complemented
192	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1668	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1639	C₂×C₃₁	1	2	yes	complemented
195	#1628	C₂×C₃₁	1	2	yes	complemented
196	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
197	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	4	1	no	—
200	C₂²×C₃₁	—	4	1	no	—
201	C₂²×C₃₁	—	4	2	no	—
202	C₂²×C₃₁	—	4	2	no	—
203	C₂²×C₃₁	—	4	2	no	—
204	C₂²×C₃₁	—	8	2	no	—
205	C₂²×C₃₁	—	8	2	no	—
206	C₂²×C₃₁	—	8	2	no	—
207	C₂²×C₃₁	—	16	8	no	—
208	C₂²×C₃₁	D₄²	1	1	yes	characteristic
209	C₄×C₃₁	—	2	1	no	—
210	C₄×C₃₁	—	4	1	no	—
211	C₄×C₃₁	—	4	2	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
216	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₄×C₃₁	#576	1	1	yes	characteristic
218	C₄×C₃₁	#576	1	1	yes	characteristic
219	C₂²×D₈	—	2	1	no	—
220	#556	C₂²×C₃₁	1	2	yes	complemented
221	D₄²	C₂²×C₃₁	1	2	yes	—
222	#495	—	2	1	no	—
223	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
224	#493	C₂²×C₃₁	1	1	yes	characteristic
225	#486	C₂²×C₃₁	1	2	yes	complemented
226	#486	C₂²×C₃₁	1	2	yes	complemented
227	#482	—	2	2	no	—
228	#474	C₂²×C₃₁	1	2	yes	—
229	#468	—	2	2	no	—
230	#466	—	2	1	no	—
231	#464	—	2	2	no	—
232	#461	C₂²×C₃₁	1	2	yes	complemented
233	#460	C₂²×C₃₁	1	2	yes	complemented
234	#459	C₂²×C₃₁	1	2	yes	—
235	#450	—	2	2	no	—
236	#447	—	2	2	no	—
237	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
238	C₄×Q₁₆	—	2	1	no	—
239	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
240	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
241	#436	C₂²×C₃₁	1	2	yes	complemented
242	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
243	#418	—	2	2	no	—
244	#335	C₂²×C₃₁	1	1	yes	characteristic, complemented
245	#334	C₂²×C₃₁	1	1	yes	characteristic, complemented
246	#332	C₂²×C₃₁	1	2	yes	complemented
247	#331	C₂²×C₃₁	1	2	yes	complemented
248	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
250	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
251	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
252	C₂×C₃₁	—	4	2	no	—
253	C₂×C₃₁	—	8	2	no	—
254	C₂×C₃₁	—	16	2	no	—
255	C₂×C₃₁	#3274	1	1	yes	characteristic
256	C₂×C₃₁	#3280	1	1	yes	characteristic
257	C₂×C₃₁	#3290	1	1	yes	characteristic
258	C₂²×D₄	—	2	2	no	—
259	C₂²×D₄	—	2	2	no	—
260	C₂×Q₁₆	—	2	1	no	—
261	C₂×Q₁₆	—	2	1	no	—
262	C₂×SD₁₆	—	2	2	no	—
263	C₂×SD₁₆	—	2	2	no	—
264	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	—	2	1	no	—
267	C₂×D₈	—	2	1	no	—
268	C₂×D₈	—	4	4	no	—
269	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
270	C₂×M₁₆	—	2	2	no	—
271	C₂²×C₈	—	2	1	no	—
272	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
273	#127	C₂³×C₃₁	1	2	yes	—
274	#126	—	4	2	no	—
275	#124	—	2	2	no	—
276	C₂²⋊Q₈	—	2	2	no	—
277	#121	—	2	2	no	—
278	#121	—	2	2	no	—
279	#120	—	4	2	no	—
280	C₄×Q₈	—	2	2	no	—
281	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
282	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₄×D₄	C₂³×C₃₁	1	2	yes	—
284	#117	—	2	2	no	—
285	#107	—	2	2	no	—
286	#107	D₄×C₃₁	1	1	yes	characteristic
287	#106	D₄×C₃₁	1	1	yes	characteristic
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
291	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
292	#103	—	2	2	no	—
293	#103	D₄×C₃₁	1	1	yes	characteristic
294	#102	—	2	2	no	—
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	4	2	no	—
300	#102	D₄×C₃₁	1	1	yes	characteristic
301	#102	D₄×C₃₁	1	2	yes	—
302	C₂²⋊C₈	—	2	1	no	—
303	C₂²⋊C₈	—	2	2	no	—
304	C₄×C₈	—	2	1	no	—
305	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
307	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₃₁	#19900	1	1	yes	characteristic, direct product factor, Sylow
309	C₂⁴	—	4	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	4	2	no	—
316	C₂×D₄	—	4	2	no	—
317	C₂×D₄	—	4	4	no	—
318	C₂×D₄	—	4	4	no	—
319	C₂×D₄	—	8	2	no	—
320	C₂×D₄	—	8	2	no	—
321	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
322	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	1	no	—
325	Q₁₆	—	4	2	no	—
326	Q₁₆	—	8	1	no	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	8	2	no	—
329	SD₁₆	—	8	2	no	—
330	D₈	—	2	2	no	—
331	D₈	—	4	4	no	—
332	D₈	—	4	2	no	—
333	D₈	—	8	1	no	—
334	M₁₆	—	8	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	4	2	no	—
340	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	4	2	no	—
348	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
350	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂²⋊C₄	—	2	1	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₂²⋊C₄	—	4	2	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₄²	—	2	1	no	—
357	C₄²	—	4	2	no	—
358	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
359	C₂³	—	2	1	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	8	4	no	—
363	C₂³	—	8	4	no	—
364	C₂³	—	16	2	no	—
365	Q₈	—	2	1	no	—
366	Q₈	—	2	1	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	4	2	no	—
369	D₄	—	2	2	no	—
370	D₄	—	2	1	no	—
371	D₄	—	2	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	2	no	—
375	D₄	—	4	2	no	—
376	D₄	—	8	2	no	—
377	D₄	—	8	4	no	—
378	D₄	—	8	4	no	—
379	D₄	C₃₁×#135	1	2	yes	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	1	no	—
397	C₈	—	4	2	no	—
398	C₈	—	4	1	no	—
399	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
400	C₂²	—	2	2	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	1	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	4	2	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	8	2	no	—
408	C₂²	—	8	2	no	—
409	C₂²	—	16	8	no	—
410	C₂²	D₄²×C₃₁	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	4	1	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	1	no	—
415	C₄	—	4	1	no	—
416	C₄	—	8	2	no	—
417	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
418	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
419	C₄	C₃₁×#576	1	1	yes	characteristic
420	C₄	C₃₁×#576	1	1	yes	characteristic
421	C₂	—	4	2	no	—
422	C₂	—	8	2	no	—
423	C₂	—	16	2	no	—
424	C₂	C₃₁×#3274	1	1	yes	characteristic
425	C₂	C₃₁×#3280	1	1	yes	characteristic
426	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	D₄×D₈×C₃₁	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3393	C₂	semidirect
2. C₃₁×#3324	C₂	semidirect
4. C₃₁×#1709	C₂	semidirect
5. C₃₁×#1701	C₂	semidirect
6. C₃₁×#1669	C₂	semidirect
7. C₃₁×#1668	C₂	semidirect
8. C₃₁×#1639	C₂	semidirect
9. C₃₁×#1628	C₂	semidirect
10. C₃₁×#1589	C₂	semidirect
11. C₈×SD₁₆×C₃₁	C₂	semidirect
13. C₃₁×#556	C₂²	semidirect
16. C₃₁×#494	C₂²	semidirect
18. C₃₁×#486	C₂²	semidirect
19. C₃₁×#486	C₂²	semidirect
25. C₃₁×#461	C₂²	semidirect
26. C₃₁×#460	C₂²	semidirect
30. C₈×Q₈×C₃₁	C₂²	semidirect
34. C₃₁×#436	C₂²	semidirect
37. C₃₁×#335	C₂²	semidirect
38. C₃₁×#334	C₂²	semidirect
39. C₃₁×#332	C₂²	semidirect
40. C₃₁×#331	C₂²	semidirect
41. C₃₁×Q₈⋊C₈	C₂²	semidirect
43. C₃₁×D₄⋊C₈	C₂²	semidirect
44. C₈²×C₃₁	C₂²	semidirect
145. #19900	C₃₁	direct (N × H)
187. #3393	C₂×C₃₁	semidirect
188. #3324	C₂×C₃₁	semidirect
190. #1709	C₂×C₃₁	semidirect
191. #1701	C₂×C₃₁	semidirect
192. #1669	C₂×C₃₁	semidirect
193. #1668	C₂×C₃₁	semidirect
194. #1639	C₂×C₃₁	semidirect
195. #1628	C₂×C₃₁	semidirect
196. #1589	C₂×C₃₁	semidirect
197. C₈×SD₁₆	C₂×C₃₁	semidirect
220. #556	C₂²×C₃₁	semidirect
223. #494	C₂²×C₃₁	semidirect
225. #486	C₂²×C₃₁	semidirect
226. #486	C₂²×C₃₁	semidirect
232. #461	C₂²×C₃₁	semidirect
233. #460	C₂²×C₃₁	semidirect
237. C₈×Q₈	C₂²×C₃₁	semidirect
241. #436	C₂²×C₃₁	semidirect
244. #335	C₂²×C₃₁	semidirect
245. #334	C₂²×C₃₁	semidirect
246. #332	C₂²×C₃₁	semidirect
247. #331	C₂²×C₃₁	semidirect
248. Q₈⋊C₈	C₂²×C₃₁	semidirect
250. D₄⋊C₈	C₂²×C₃₁	semidirect
251. C₈²	C₂²×C₃₁	semidirect
308. C₃₁	#19900	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×#3393	2
C₃₁×#3324	2
D₄×D₈×C₃₁	2
C₃₁×#1709	2
C₃₁×#1701	2
C₃₁×#1669	2
C₃₁×#1668	2
C₃₁×#1639	2
C₃₁×#1628	2
C₃₁×#1589	2
C₈×SD₁₆×C₃₁	2
#19900	31

Subgroups — C31×#19900 #21476659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19900 #21476659