Subgroups — C₃₁×#33859 #21490618

Order 7936.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3417	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3415	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3290	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2249	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2222	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2221	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2212	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2194	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2193	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2178	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2170	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#578	C₂²	1	2	yes	complemented
13	C₃₁×#577	—	2	2	no	—
14	C₃₁×#576	C₂²	1	2	yes	—
15	C₃₁×#510	—	2	2	no	—
16	C₃₁×#510	C₂²	1	2	yes	complemented
17	C₃₁×#509	—	2	2	no	—
18	C₃₁×#509	C₂²	1	2	yes	—
19	C₃₁×#508	—	2	1	no	—
20	C₃₁×#508	—	2	1	no	—
21	C₃₁×#508	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×Q₃₂	—	2	1	no	—
23	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
25	C₂×C₃₁×D₁₆	—	2	1	no	—
26	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
27	C₃₁×#504	C₂²	1	1	yes	characteristic
28	C₃₁×#504	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#495	C₂²	1	2	yes	complemented
30	C₃₁×#473	C₂²	1	2	yes	—
31	C₃₁×#472	C₂²	1	2	yes	complemented
32	C₃₁×#471	—	2	2	no	—
33	C₃₁×#455	—	2	2	no	—
34	C₃₁×#454	—	2	2	no	—
35	C₃₁×#443	C₂²	1	1	yes	characteristic
36	C₃₁×#443	C₂²	1	2	yes	complemented
37	C₃₁×#368	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#364	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#362	C₂²	1	2	yes	complemented
40	C₃₁×#361	C₂²	1	2	yes	complemented
41	C₃₁×#359	C₂²	1	2	yes	complemented
42	C₃₁×#350	C₂²	1	1	yes	characteristic
43	C₃₁×#350	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×2₋¹⁺⁴	—	2	2	no	—
46	C₃₁×H(5,𝔽₂)	—	2	2	no	—
47	C₃₁×#137	—	2	2	no	—
48	C₃₁×#137	—	2	2	no	—
49	C₃₁×#137	—	4	2	no	—
50	C₃₁×#137	—	4	2	no	—
51	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
52	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
53	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
54	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
55	C₃₁×#135	—	2	2	no	—
56	C₃₁×#135	—	2	2	no	—
57	C₃₁×#135	—	4	2	no	—
58	C₃₁×#135	D₄	1	1	yes	characteristic
59	C₃₁×#135	C₂³	1	2	yes	—
60	C₂×Q₁₆×C₃₁	—	2	2	no	—
61	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₃₁	—	2	2	no	—
63	C₂×SD₁₆×C₃₁	—	4	2	no	—
64	C₂×D₈×C₃₁	—	2	2	no	—
65	C₂×D₈×C₃₁	C₂³	1	2	yes	—
66	C₃₁×#131	—	2	2	no	—
67	C₃₁×#131	C₂³	1	1	yes	characteristic
68	C₃₁×#131	C₂³	1	1	yes	characteristic
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×Q₃₂	—	2	2	no	—
71	C₃₁×Q₃₂	—	2	1	no	—
72	C₃₁×Q₃₂	—	2	1	no	—
73	C₃₁×Q₃₂	D₄	1	1	yes	characteristic
74	C₃₁×SD₃₂	—	2	2	no	—
75	C₃₁×SD₃₂	—	2	2	no	—
76	C₃₁×SD₃₂	—	4	2	no	—
77	C₃₁×SD₃₂	D₄	1	2	yes	—
78	C₃₁×D₁₆	—	2	2	no	—
79	C₃₁×D₁₆	—	2	1	no	—
80	C₃₁×D₁₆	—	2	1	no	—
81	C₃₁×D₁₆	D₄	1	1	yes	characteristic
82	C₃₁×M₃₂	—	2	1	no	—
83	C₃₁×M₃₂	—	2	1	no	—
84	C₃₁×M₃₂	C₂³	1	1	yes	characteristic
85	C₃₁×M₃₂	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₃₁	—	2	1	no	—
87	C₂×C₁₆×C₃₁	—	2	1	no	—
88	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
90	C₃₁×#108	D₄	1	1	yes	characteristic
91	C₃₁×#108	C₂³	1	2	yes	—
92	C₃₁×#104	—	2	2	no	—
93	C₃₁×#104	D₄	1	1	yes	characteristic
94	C₃₁×#104	D₄	1	1	yes	characteristic
95	C₃₁×#101	—	2	2	no	—
96	C₃₁×#100	—	2	2	no	—
97	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
98	#41×C₃₁	—	2	2	no	—
99	#41×C₃₁	—	4	2	no	—
100	#41×C₃₁	—	4	2	no	—
101	#41×C₃₁	—	4	2	no	—
102	#41×C₃₁	—	4	2	no	—
103	#41×C₃₁	—	8	2	no	—
104	#41×C₃₁	C₂×D₄	1	1	yes	characteristic
105	#41×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	4	2	no	—
108	C₂×Q₈×C₃₁	—	4	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	4	2	no	—
111	C₂×D₄×C₃₁	—	4	2	no	—
112	C₂×D₄×C₃₁	—	8	2	no	—
113	Q₁₆×C₃₁	—	2	2	no	—
114	Q₁₆×C₃₁	—	2	2	no	—
115	Q₁₆×C₃₁	—	4	2	no	—
116	Q₁₆×C₃₁	—	4	2	no	—
117	Q₁₆×C₃₁	D₈	1	1	yes	characteristic
118	Q₁₆×C₃₁	C₂×D₄	1	2	yes	—
119	SD₁₆×C₃₁	—	2	2	no	—
120	SD₁₆×C₃₁	—	4	2	no	—
121	SD₁₆×C₃₁	—	4	2	no	—
122	SD₁₆×C₃₁	—	4	2	no	—
123	SD₁₆×C₃₁	—	4	2	no	—
124	SD₁₆×C₃₁	—	4	2	no	—
125	SD₁₆×C₃₁	—	4	2	no	—
126	SD₁₆×C₃₁	D₈	1	1	yes	characteristic
127	SD₁₆×C₃₁	D₈	1	1	yes	characteristic
128	D₈×C₃₁	—	2	2	no	—
129	D₈×C₃₁	—	2	2	no	—
130	D₈×C₃₁	—	4	2	no	—
131	D₈×C₃₁	—	4	2	no	—
132	D₈×C₃₁	D₈	1	1	yes	characteristic
133	D₈×C₃₁	C₂×D₄	1	2	yes	—
134	M₁₆×C₃₁	—	2	1	no	—
135	M₁₆×C₃₁	—	2	1	no	—
136	M₁₆×C₃₁	—	2	2	no	—
137	M₁₆×C₃₁	—	4	2	no	—
138	M₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
139	M₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	4	2	no	—
143	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
145	C₂²⋊C₄×C₃₁	—	8	2	no	—
146	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₁₆×C₃₁	—	2	1	no	—
148	C₁₆×C₃₁	—	2	1	no	—
149	C₁₆×C₃₁	—	2	1	no	—
150	C₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
152	#33859	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	4	2	no	—
154	C₂³×C₃₁	—	8	2	no	—
155	Q₈×C₃₁	—	2	2	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	Q₈×C₃₁	—	8	2	no	—
159	Q₈×C₃₁	C₂×D₈	1	1	yes	characteristic
160	Q₈×C₃₁	C₂×D₈	1	1	yes	characteristic
161	D₄×C₃₁	—	2	1	no	—
162	D₄×C₃₁	—	2	1	no	—
163	D₄×C₃₁	—	2	2	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	8	2	no	—
168	D₄×C₃₁	—	8	2	no	—
169	D₄×C₃₁	—	8	2	no	—
170	D₄×C₃₁	C₂×D₈	1	1	yes	characteristic
171	D₄×C₃₁	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	8	2	no	—
178	C₂×C₄×C₃₁	—	8	2	no	—
179	C₂×C₄×C₃₁	—	8	2	no	—
180	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	C₈×C₃₁	—	2	1	no	—
182	C₈×C₃₁	—	2	1	no	—
183	C₈×C₃₁	—	4	2	no	—
184	C₈×C₃₁	C₂×D₈	1	1	yes	characteristic
185	C₈×C₃₁	C₂×D₈	1	1	yes	characteristic
186	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
188	#3417	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#3415	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#3290	C₂×C₃₁	1	2	yes	complemented
191	#2249	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#2222	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#2221	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#2212	C₂×C₃₁	1	2	yes	complemented
195	#2194	C₂×C₃₁	1	2	yes	complemented
196	#2193	C₂×C₃₁	1	2	yes	complemented
197	#2178	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2170	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂²×C₃₁	—	2	1	no	—
200	C₂²×C₃₁	—	2	1	no	—
201	C₂²×C₃₁	—	4	2	no	—
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	—	16	2	no	—
204	C₂²×C₃₁	—	32	2	no	—
205	C₂²×C₃₁	#576	1	1	yes	characteristic
206	C₄×C₃₁	—	2	1	no	—
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	4	1	no	—
209	C₄×C₃₁	—	4	2	no	—
210	C₄×C₃₁	—	8	2	no	—
211	C₄×C₃₁	D₄²	1	1	yes	characteristic
212	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
213	#578	C₂²×C₃₁	1	2	yes	complemented
214	#577	—	2	2	no	—
215	#576	C₂²×C₃₁	1	2	yes	—
216	#510	—	2	2	no	—
217	#510	C₂²×C₃₁	1	2	yes	complemented
218	#509	—	2	2	no	—
219	#509	C₂²×C₃₁	1	2	yes	—
220	#508	—	2	1	no	—
221	#508	—	2	1	no	—
222	#508	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×Q₃₂	—	2	1	no	—
224	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
225	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
226	C₂×D₁₆	—	2	1	no	—
227	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
228	#504	C₂²×C₃₁	1	1	yes	characteristic
229	#504	C₂²×C₃₁	1	1	yes	characteristic, complemented
230	#495	C₂²×C₃₁	1	2	yes	complemented
231	#473	C₂²×C₃₁	1	2	yes	—
232	#472	C₂²×C₃₁	1	2	yes	complemented
233	#471	—	2	2	no	—
234	#455	—	2	2	no	—
235	#454	—	2	2	no	—
236	#443	C₂²×C₃₁	1	1	yes	characteristic
237	#443	C₂²×C₃₁	1	2	yes	complemented
238	#368	C₂²×C₃₁	1	1	yes	characteristic, complemented
239	#364	C₂²×C₃₁	1	1	yes	characteristic, complemented
240	#362	C₂²×C₃₁	1	2	yes	complemented
241	#361	C₂²×C₃₁	1	2	yes	complemented
242	#359	C₂²×C₃₁	1	2	yes	complemented
243	#350	C₂²×C₃₁	1	1	yes	characteristic
244	#350	C₂²×C₃₁	1	1	yes	characteristic, complemented
245	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
246	C₂×C₃₁	—	2	1	no	—
247	C₂×C₃₁	—	4	1	no	—
248	C₂×C₃₁	—	4	1	no	—
249	C₂×C₃₁	—	8	2	no	—
250	C₂×C₃₁	—	16	2	no	—
251	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
252	2₋¹⁺⁴	—	2	2	no	—
253	H(5,𝔽₂)	—	2	2	no	—
254	#137	—	2	2	no	—
255	#137	—	2	2	no	—
256	#137	—	4	2	no	—
257	#137	—	4	2	no	—
258	AGL(1,ℤ/8)	—	2	2	no	—
259	AGL(1,ℤ/8)	—	2	2	no	—
260	AGL(1,ℤ/8)	—	4	2	no	—
261	AGL(1,ℤ/8)	—	4	2	no	—
262	#135	—	2	2	no	—
263	#135	—	2	2	no	—
264	#135	—	4	2	no	—
265	#135	D₄×C₃₁	1	1	yes	characteristic
266	#135	C₂³×C₃₁	1	2	yes	—
267	C₂×Q₁₆	—	2	2	no	—
268	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
269	C₂×SD₁₆	—	2	2	no	—
270	C₂×SD₁₆	—	4	2	no	—
271	C₂×D₈	—	2	2	no	—
272	C₂×D₈	C₂³×C₃₁	1	2	yes	—
273	#131	—	2	2	no	—
274	#131	C₂³×C₃₁	1	1	yes	characteristic
275	#131	C₂³×C₃₁	1	1	yes	characteristic
276	#124	—	2	2	no	—
277	Q₃₂	—	2	2	no	—
278	Q₃₂	—	2	1	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	D₄×C₃₁	1	1	yes	characteristic
281	SD₃₂	—	2	2	no	—
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	4	2	no	—
284	SD₃₂	D₄×C₃₁	1	2	yes	—
285	D₁₆	—	2	2	no	—
286	D₁₆	—	2	1	no	—
287	D₁₆	—	2	1	no	—
288	D₁₆	D₄×C₃₁	1	1	yes	characteristic
289	M₃₂	—	2	1	no	—
290	M₃₂	—	2	1	no	—
291	M₃₂	C₂³×C₃₁	1	1	yes	characteristic
292	M₃₂	C₂³×C₃₁	1	1	yes	characteristic
293	C₂×C₁₆	—	2	1	no	—
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
297	#108	D₄×C₃₁	1	1	yes	characteristic
298	#108	C₂³×C₃₁	1	2	yes	—
299	#104	—	2	2	no	—
300	#104	D₄×C₃₁	1	1	yes	characteristic
301	#104	D₄×C₃₁	1	1	yes	characteristic
302	#101	—	2	2	no	—
303	#100	—	2	2	no	—
304	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₃₁	#33859	1	1	yes	characteristic, direct product factor, Sylow
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	4	2	no	—
310	#41	—	4	2	no	—
311	#41	—	8	2	no	—
312	#41	C₂×D₄×C₃₁	1	1	yes	characteristic
313	#41	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	—	8	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	4	2	no	—
325	Q₁₆	D₈×C₃₁	1	1	yes	characteristic
326	Q₁₆	C₂×D₄×C₃₁	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	4	2	no	—
330	SD₁₆	—	4	2	no	—
331	SD₁₆	—	4	2	no	—
332	SD₁₆	—	4	2	no	—
333	SD₁₆	—	4	2	no	—
334	SD₁₆	D₈×C₃₁	1	1	yes	characteristic
335	SD₁₆	D₈×C₃₁	1	1	yes	characteristic
336	D₈	—	2	2	no	—
337	D₈	—	2	2	no	—
338	D₈	—	4	2	no	—
339	D₈	—	4	2	no	—
340	D₈	D₈×C₃₁	1	1	yes	characteristic
341	D₈	C₂×D₄×C₃₁	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	2	1	no	—
344	M₁₆	—	2	2	no	—
345	M₁₆	—	4	2	no	—
346	M₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
347	M₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	4	2	no	—
351	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
353	C₂²⋊C₄	—	8	2	no	—
354	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₁₆	—	2	1	no	—
356	C₁₆	—	2	1	no	—
357	C₁₆	—	2	1	no	—
358	C₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂³	—	4	2	no	—
361	C₂³	—	8	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	Q₈	C₂×D₈×C₃₁	1	1	yes	characteristic
367	Q₈	C₂×D₈×C₃₁	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	1	no	—
370	D₄	—	2	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	8	2	no	—
375	D₄	—	8	2	no	—
376	D₄	—	8	2	no	—
377	D₄	C₂×D₈×C₃₁	1	1	yes	characteristic
378	D₄	C₂×D₈×C₃₁	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	8	2	no	—
385	C₂×C₄	—	8	2	no	—
386	C₂×C₄	—	8	2	no	—
387	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	2	1	no	—
390	C₈	—	4	2	no	—
391	C₈	C₂×D₈×C₃₁	1	1	yes	characteristic
392	C₈	C₂×D₈×C₃₁	1	1	yes	characteristic
393	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
394	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	4	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	16	2	no	—
400	C₂²	—	32	2	no	—
401	C₂²	C₃₁×#576	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	4	1	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	2	no	—
407	C₄	D₄²×C₃₁	1	1	yes	characteristic
408	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	4	1	no	—
412	C₂	—	8	2	no	—
413	C₂	—	16	2	no	—
414	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	D₄×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#443 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3417	C₂	semidirect
2. C₃₁×#3415	C₂	semidirect
3. C₃₁×#3290	C₂	semidirect
4. C₃₁×#2249	C₂	semidirect
5. C₃₁×#2222	C₂	semidirect
6. C₃₁×#2221	C₂	semidirect
7. C₃₁×#2212	C₂	semidirect
8. C₃₁×#2194	C₂	semidirect
9. C₃₁×#2193	C₂	semidirect
10. C₃₁×#2178	C₂	semidirect
11. C₃₁×#2170	C₂	semidirect
12. C₃₁×#578	C₂²	semidirect
16. C₃₁×#510	C₂²	semidirect
23. C₂×C₃₁×Q₃₂	C₂²	semidirect
24. C₂×C₃₁×SD₃₂	C₂²	semidirect
28. C₃₁×#504	C₂²	semidirect
29. C₃₁×#495	C₂²	semidirect
31. C₃₁×#472	C₂²	semidirect
36. C₃₁×#443	C₂²	semidirect
37. C₃₁×#368	C₂²	semidirect
38. C₃₁×#364	C₂²	semidirect
39. C₃₁×#362	C₂²	semidirect
40. C₃₁×#361	C₂²	semidirect
41. C₃₁×#359	C₂²	semidirect
43. C₃₁×#350	C₂²	semidirect
44. C₄×C₁₆×C₃₁	C₂²	semidirect
152. #33859	C₃₁	direct (N × H)
188. #3417	C₂×C₃₁	semidirect
189. #3415	C₂×C₃₁	semidirect
190. #3290	C₂×C₃₁	semidirect
191. #2249	C₂×C₃₁	semidirect
192. #2222	C₂×C₃₁	semidirect
193. #2221	C₂×C₃₁	semidirect
194. #2212	C₂×C₃₁	semidirect
195. #2194	C₂×C₃₁	semidirect
196. #2193	C₂×C₃₁	semidirect
197. #2178	C₂×C₃₁	semidirect
198. #2170	C₂×C₃₁	semidirect
213. #578	C₂²×C₃₁	semidirect
217. #510	C₂²×C₃₁	semidirect
224. C₂×Q₃₂	C₂²×C₃₁	semidirect
225. C₂×SD₃₂	C₂²×C₃₁	semidirect
229. #504	C₂²×C₃₁	semidirect
230. #495	C₂²×C₃₁	semidirect
232. #472	C₂²×C₃₁	semidirect
237. #443	C₂²×C₃₁	semidirect
238. #368	C₂²×C₃₁	semidirect
239. #364	C₂²×C₃₁	semidirect
240. #362	C₂²×C₃₁	semidirect
241. #361	C₂²×C₃₁	semidirect
242. #359	C₂²×C₃₁	semidirect
244. #350	C₂²×C₃₁	semidirect
245. C₄×C₁₆	C₂²×C₃₁	semidirect
305. C₃₁	#33859	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×#3417	2
C₃₁×#3415	2
C₃₁×#3290	2
C₃₁×#2249	2
C₃₁×#2222	2
C₃₁×#2221	2
C₃₁×#2212	2
C₃₁×#2194	2
C₃₁×#2193	2
C₃₁×#2178	2
C₃₁×#2170	2
#33859	31

Subgroups — C31×#33859 #21490618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33859 #21490618