Subgroups — C₃₁×#19898 #21476657

Order 7936.

724 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3341	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3325	C₂	1	2	yes	maximal, complemented
3	D₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1697	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1629	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1624	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	—	2	1	no	—
13	C₃₁×#555	C₂²	1	2	yes	complemented
14	D₄²×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#493	C₂²	1	1	yes	characteristic
17	C₃₁×#492	C₂²	1	2	yes	complemented
18	C₃₁×#486	C₂²	1	2	yes	complemented
19	C₃₁×#482	—	2	2	no	—
20	C₃₁×#482	—	2	2	no	—
21	C₃₁×#476	C₂²	1	2	yes	—
22	C₃₁×#476	C₂²	1	2	yes	complemented
23	C₃₁×#469	—	2	2	no	—
24	C₃₁×#466	—	2	1	no	—
25	C₃₁×#466	—	2	1	no	—
26	C₃₁×#459	C₂²	1	2	yes	—
27	C₃₁×#459	C₂²	1	2	yes	complemented
28	C₃₁×#447	—	2	2	no	—
29	C₃₁×#447	—	2	2	no	—
30	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#436	C₂²	1	2	yes	complemented
33	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#427	—	2	1	no	—
36	C₃₁×#418	—	2	2	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#335	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#333	C₂²	1	2	yes	complemented
40	C₃₁×#331	C₂²	1	2	yes	complemented
41	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
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×SD₁₆×C₃₁	—	4	2	no	—
48	C₂×D₈×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	1	no	—
51	C₂×D₈×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	—	4	4	no	—
53	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
54	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₃₁×#127	C₂³	1	2	yes	—
60	C₃₁×#124	—	4	2	no	—
61	C₃₁×#121	—	2	2	no	—
62	C₃₁×#121	—	2	2	no	—
63	C₃₁×#121	—	2	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₃₁×#120	—	4	2	no	—
66	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₃₁	C₂³	1	2	yes	—
69	C₃₁×#117	—	2	2	no	—
70	C₃₁×#117	—	2	2	no	—
71	C₃₁×#107	—	2	1	no	—
72	C₃₁×#107	—	2	1	no	—
73	C₃₁×#107	D₄	1	1	yes	characteristic
74	C₃₁×#107	D₄	1	1	yes	characteristic
75	C₃₁×#106	—	2	2	no	—
76	C₃₁×#106	—	2	2	no	—
77	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
80	C₃₁×#102	—	2	2	no	—
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	D₄	1	1	yes	characteristic
87	C₃₁×#102	D₄	1	2	yes	—
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	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×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	4	2	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂×D₄×C₃₁	—	4	4	no	—
103	C₂×D₄×C₃₁	—	4	4	no	—
104	C₂×D₄×C₃₁	—	4	2	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	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	SD₁₆×C₃₁	—	8	4	no	—
114	D₈×C₃₁	—	2	2	no	—
115	D₈×C₃₁	—	2	2	no	—
116	D₈×C₃₁	—	4	4	no	—
117	D₈×C₃₁	—	4	2	no	—
118	D₈×C₃₁	—	8	1	no	—
119	D₈×C₃₁	—	8	2	no	—
120	M₁₆×C₃₁	—	8	2	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	4	2	no	—
126	C₂×C₈×C₃₁	—	4	2	no	—
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	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²⋊C₄×C₃₁	—	2	1	no	—
137	C₂²⋊C₄×C₃₁	—	2	2	no	—
138	C₂²⋊C₄×C₃₁	—	2	1	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	—	4	2	no	—
143	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
144	#19898	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	1	no	—
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	—	8	2	no	—
153	D₄×C₃₁	—	2	1	no	—
154	D₄×C₃₁	—	2	2	no	—
155	D₄×C₃₁	—	2	1	no	—
156	D₄×C₃₁	—	2	1	no	—
157	D₄×C₃₁	—	2	2	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	8	2	no	—
163	D₄×C₃₁	—	8	4	no	—
164	D₄×C₃₁	—	8	4	no	—
165	D₄×C₃₁	—	16	2	no	—
166	D₄×C₃₁	#135	1	2	yes	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	1	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	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
180	C₈×C₃₁	—	2	2	no	—
181	C₈×C₃₁	—	2	1	no	—
182	C₈×C₃₁	—	2	1	no	—
183	C₈×C₃₁	—	2	1	no	—
184	C₈×C₃₁	—	4	1	no	—
185	C₈×C₃₁	—	4	1	no	—
186	C₈×C₃₁	—	4	2	no	—
187	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
188	#3341	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#3325	C₂×C₃₁	1	2	yes	complemented
190	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1697	C₂×C₃₁	1	2	yes	complemented
193	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1629	C₂×C₃₁	1	2	yes	complemented
196	#1624	C₂×C₃₁	1	2	yes	complemented
197	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂²×C₃₁	—	2	2	no	—
200	C₂²×C₃₁	—	4	1	no	—
201	C₂²×C₃₁	—	4	1	no	—
202	C₂²×C₃₁	—	4	1	no	—
203	C₂²×C₃₁	—	4	2	no	—
204	C₂²×C₃₁	—	4	2	no	—
205	C₂²×C₃₁	—	4	1	no	—
206	C₂²×C₃₁	—	4	1	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	8	2	no	—
209	C₂²×C₃₁	—	8	2	no	—
210	C₂²×C₃₁	—	8	2	no	—
211	C₂²×C₃₁	—	16	8	no	—
212	C₂²×C₃₁	D₄²	1	1	yes	characteristic
213	C₄×C₃₁	—	2	1	no	—
214	C₄×C₃₁	—	4	1	no	—
215	C₄×C₃₁	—	4	1	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	—	8	2	no	—
218	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
219	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₄×C₃₁	#576	1	1	yes	characteristic
221	C₄×C₃₁	#577	1	1	yes	characteristic
222	C₂²×D₈	—	2	1	no	—
223	#555	C₂²×C₃₁	1	2	yes	complemented
224	D₄²	C₂²×C₃₁	1	2	yes	—
225	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
226	#493	C₂²×C₃₁	1	1	yes	characteristic
227	#492	C₂²×C₃₁	1	2	yes	complemented
228	#486	C₂²×C₃₁	1	2	yes	complemented
229	#482	—	2	2	no	—
230	#482	—	2	2	no	—
231	#476	C₂²×C₃₁	1	2	yes	—
232	#476	C₂²×C₃₁	1	2	yes	complemented
233	#469	—	2	2	no	—
234	#466	—	2	1	no	—
235	#466	—	2	1	no	—
236	#459	C₂²×C₃₁	1	2	yes	—
237	#459	C₂²×C₃₁	1	2	yes	complemented
238	#447	—	2	2	no	—
239	#447	—	2	2	no	—
240	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
242	#436	C₂²×C₃₁	1	2	yes	complemented
243	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
244	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
245	#427	—	2	1	no	—
246	#418	—	2	2	no	—
247	#335	C₂²×C₃₁	1	1	yes	characteristic, complemented
248	#335	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	#333	C₂²×C₃₁	1	2	yes	complemented
250	#331	C₂²×C₃₁	1	2	yes	complemented
251	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
252	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
253	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
254	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
255	C₂×C₃₁	—	4	2	no	—
256	C₂×C₃₁	—	8	1	no	—
257	C₂×C₃₁	—	8	2	no	—
258	C₂×C₃₁	—	16	2	no	—
259	C₂×C₃₁	#3272	1	1	yes	characteristic
260	C₂×C₃₁	#3280	1	1	yes	characteristic
261	C₂×C₃₁	#3292	1	1	yes	characteristic
262	C₂²×D₄	—	2	2	no	—
263	C₂²×D₄	—	2	2	no	—
264	C₂×SD₁₆	—	4	2	no	—
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	—	2	2	no	—
267	C₂×D₈	—	2	1	no	—
268	C₂×D₈	—	2	1	no	—
269	C₂×D₈	—	4	4	no	—
270	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
271	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
272	C₂×M₁₆	—	2	2	no	—
273	C₂²×C₈	—	2	1	no	—
274	C₂²×C₈	—	2	1	no	—
275	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
276	#127	C₂³×C₃₁	1	2	yes	—
277	#124	—	4	2	no	—
278	#121	—	2	2	no	—
279	#121	—	2	2	no	—
280	#121	—	2	2	no	—
281	#121	—	2	2	no	—
282	#120	—	4	2	no	—
283	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
284	C₄×D₄	C₂³×C₃₁	1	2	yes	—
285	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
286	#117	—	2	2	no	—
287	#117	—	2	2	no	—
288	#107	—	2	1	no	—
289	#107	—	2	1	no	—
290	#107	D₄×C₃₁	1	1	yes	characteristic
291	#107	D₄×C₃₁	1	1	yes	characteristic
292	#106	—	2	2	no	—
293	#106	—	2	2	no	—
294	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
296	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	#102	—	2	2	no	—
301	#102	—	2	2	no	—
302	#102	—	2	2	no	—
303	#102	D₄×C₃₁	1	1	yes	characteristic
304	#102	D₄×C₃₁	1	2	yes	—
305	#102	D₄×C₃₁	1	1	yes	characteristic
306	C₂²⋊C₈	—	2	1	no	—
307	C₂²⋊C₈	—	2	1	no	—
308	C₂²⋊C₈	—	2	2	no	—
309	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
310	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₃₁	#19898	1	1	yes	characteristic, direct product factor, Sylow
313	C₂⁴	—	4	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	—	4	4	no	—
321	C₂×D₄	—	4	4	no	—
322	C₂×D₄	—	4	2	no	—
323	C₂×D₄	—	8	2	no	—
324	C₂×D₄	—	8	2	no	—
325	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
326	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	1	no	—
331	SD₁₆	—	8	4	no	—
332	D₈	—	2	2	no	—
333	D₈	—	2	2	no	—
334	D₈	—	4	4	no	—
335	D₈	—	4	2	no	—
336	D₈	—	8	1	no	—
337	D₈	—	8	2	no	—
338	M₁₆	—	8	2	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	4	2	no	—
344	C₂×C₈	—	4	2	no	—
345	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²⋊C₄	—	2	1	no	—
355	C₂²⋊C₄	—	2	2	no	—
356	C₂²⋊C₄	—	2	1	no	—
357	C₂²⋊C₄	—	4	2	no	—
358	C₂²⋊C₄	—	4	2	no	—
359	C₂²⋊C₄	—	4	2	no	—
360	C₄²	—	4	2	no	—
361	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
362	C₂³	—	2	1	no	—
363	C₂³	—	2	1	no	—
364	C₂³	—	2	2	no	—
365	C₂³	—	4	2	no	—
366	C₂³	—	8	4	no	—
367	C₂³	—	8	4	no	—
368	C₂³	—	16	2	no	—
369	Q₈	—	8	2	no	—
370	D₄	—	2	1	no	—
371	D₄	—	2	2	no	—
372	D₄	—	2	1	no	—
373	D₄	—	2	1	no	—
374	D₄	—	2	2	no	—
375	D₄	—	4	2	no	—
376	D₄	—	4	2	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	8	2	no	—
380	D₄	—	8	4	no	—
381	D₄	—	8	4	no	—
382	D₄	—	16	2	no	—
383	D₄	C₃₁×#135	1	2	yes	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	2	1	no	—
399	C₈	—	2	1	no	—
400	C₈	—	2	1	no	—
401	C₈	—	4	1	no	—
402	C₈	—	4	1	no	—
403	C₈	—	4	2	no	—
404	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
405	C₂²	—	2	2	no	—
406	C₂²	—	4	1	no	—
407	C₂²	—	4	1	no	—
408	C₂²	—	4	1	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	8	2	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	16	8	no	—
418	C₂²	D₄²×C₃₁	1	1	yes	characteristic
419	C₄	—	2	1	no	—
420	C₄	—	4	1	no	—
421	C₄	—	4	1	no	—
422	C₄	—	4	2	no	—
423	C₄	—	8	2	no	—
424	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
425	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
426	C₄	C₃₁×#576	1	1	yes	characteristic
427	C₄	C₃₁×#577	1	1	yes	characteristic
428	C₂	—	4	2	no	—
429	C₂	—	8	2	no	—
430	C₂	—	8	1	no	—
431	C₂	—	16	2	no	—
432	C₂	C₃₁×#3272	1	1	yes	characteristic
433	C₂	C₃₁×#3280	1	1	yes	characteristic
434	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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×#3341	C₂	semidirect
2. C₃₁×#3325	C₂	semidirect
3. D₄×D₈×C₃₁	C₂	semidirect
4. C₃₁×#1706	C₂	semidirect
5. C₃₁×#1697	C₂	semidirect
6. C₃₁×#1675	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1629	C₂	semidirect
9. C₃₁×#1624	C₂	semidirect
10. C₃₁×#1589	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
13. C₃₁×#555	C₂²	semidirect
15. C₃₁×#500	C₂²	semidirect
17. C₃₁×#492	C₂²	semidirect
18. C₃₁×#486	C₂²	semidirect
22. C₃₁×#476	C₂²	semidirect
27. C₃₁×#459	C₂²	semidirect
32. C₃₁×#436	C₂²	semidirect
34. C₈×D₄×C₃₁	C₂²	semidirect
37. C₃₁×#335	C₂²	semidirect
38. C₃₁×#335	C₂²	semidirect
39. C₃₁×#333	C₂²	semidirect
40. C₃₁×#331	C₂²	semidirect
42. C₃₁×D₄⋊C₈	C₂²	semidirect
43. C₃₁×D₄⋊C₈	C₂²	semidirect
44. C₈²×C₃₁	C₂²	semidirect
144. #19898	C₃₁	direct (N × H)
188. #3341	C₂×C₃₁	semidirect
189. #3325	C₂×C₃₁	semidirect
190. D₄×D₈	C₂×C₃₁	semidirect
191. #1706	C₂×C₃₁	semidirect
192. #1697	C₂×C₃₁	semidirect
193. #1675	C₂×C₃₁	semidirect
194. #1667	C₂×C₃₁	semidirect
195. #1629	C₂×C₃₁	semidirect
196. #1624	C₂×C₃₁	semidirect
197. #1589	C₂×C₃₁	semidirect
198. C₈×D₈	C₂×C₃₁	semidirect
223. #555	C₂²×C₃₁	semidirect
225. #500	C₂²×C₃₁	semidirect
227. #492	C₂²×C₃₁	semidirect
228. #486	C₂²×C₃₁	semidirect
232. #476	C₂²×C₃₁	semidirect
237. #459	C₂²×C₃₁	semidirect
242. #436	C₂²×C₃₁	semidirect
244. C₈×D₄	C₂²×C₃₁	semidirect
247. #335	C₂²×C₃₁	semidirect
248. #335	C₂²×C₃₁	semidirect
249. #333	C₂²×C₃₁	semidirect
250. #331	C₂²×C₃₁	semidirect
252. D₄⋊C₈	C₂²×C₃₁	semidirect
253. D₄⋊C₈	C₂²×C₃₁	semidirect
254. C₈²	C₂²×C₃₁	semidirect
312. C₃₁	#19898	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×#3341	2
C₃₁×#3325	2
D₄×D₈×C₃₁	2
C₃₁×#1706	2
C₃₁×#1697	2
C₃₁×#1675	2
C₃₁×#1667	2
C₃₁×#1629	2
C₃₁×#1624	2
C₃₁×#1589	2
C₈×D₈×C₃₁	2
#19898	31

Subgroups — C31×#19898 #21476657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19898 #21476657