Subgroups — C₃₁×#33840 #21490599

Order 7936.

748 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#509	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3287	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2246	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2218	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2207	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2186	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2184	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	2	yes	complemented
13	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#550	—	2	1	no	—
15	D₄²×C₃₁	—	2	1	no	—
16	C₃₁×#509	—	2	8	no	—
17	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
19	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
20	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
21	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
22	C₂×C₃₁×M₃₂	C₂²	1	2	yes	complemented
23	C₃₁×#493	C₂²	1	1	yes	characteristic
24	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#466	C₂²	1	2	yes	—
26	C₃₁×#466	C₂²	1	2	yes	complemented
27	C₃₁×#460	—	2	1	no	—
28	C₃₁×#459	—	2	1	no	—
29	C₃₁×#449	—	2	2	no	—
30	C₃₁×#447	—	2	2	no	—
31	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
32	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#367	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#358	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#357	C₂²	1	1	yes	characteristic
38	C₃₁×#357	C₂²	1	2	yes	complemented
39	C₃₁×#357	C₂²	1	2	yes	complemented
40	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
41	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
42	C₂×#41×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	—	2	2	no	—
44	C₂²×D₄×C₃₁	—	4	1	no	—
45	C₃₁×#135	—	2	4	no	—
46	C₃₁×#135	—	4	2	no	—
47	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	—	2	4	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂×D₈×C₃₁	—	4	2	no	—
54	C₂×D₈×C₃₁	C₂³	1	2	yes	—
55	C₂×D₈×C₃₁	C₂³	1	2	yes	—
56	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
58	C₃₁×#127	—	2	1	no	—
59	C₃₁×#127	—	2	1	no	—
60	C₃₁×#127	—	4	1	no	—
61	C₃₁×#121	—	2	2	no	—
62	C₃₁×#121	—	2	2	no	—
63	C₃₁×#121	—	4	1	no	—
64	C₃₁×#121	—	4	2	no	—
65	C₃₁×#120	—	4	2	no	—
66	C₄×Q₈×C₃₁	—	2	1	no	—
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	—	4	1	no	—
69	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
70	C₃₁×SD₃₂	—	2	4	no	—
71	C₃₁×SD₃₂	D₄	1	4	yes	complemented
72	C₃₁×D₁₆	—	2	4	no	—
73	C₃₁×D₁₆	—	2	4	no	—
74	C₃₁×D₁₆	—	4	2	no	—
75	C₃₁×M₃₂	—	2	4	no	—
76	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
77	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	C₂³	1	1	yes	characteristic
80	C₃₁×#107	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
82	C₃₁×#103	—	2	1	no	—
83	C₃₁×#102	—	2	1	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
87	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂⁴×C₃₁	—	4	2	no	—
89	#41×C₃₁	—	4	4	no	—
90	#41×C₃₁	—	8	2	no	—
91	C₂×Q₈×C₃₁	—	2	1	no	—
92	C₂×D₄×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	4	4	no	—
96	C₂×D₄×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	1	no	—
100	C₂×D₄×C₃₁	—	4	2	no	—
101	C₂×D₄×C₃₁	—	4	1	no	—
102	C₂×D₄×C₃₁	—	4	1	no	—
103	C₂×D₄×C₃₁	—	4	1	no	—
104	C₂×D₄×C₃₁	—	8	2	no	—
105	C₂×D₄×C₃₁	—	8	4	no	—
106	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	—	4	1	no	—
108	C₂²×C₄×C₃₁	—	4	2	no	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	Q₁₆×C₃₁	—	2	1	no	—
111	Q₁₆×C₃₁	C₂×D₄	1	2	yes	complemented
112	SD₁₆×C₃₁	—	4	4	no	—
113	D₈×C₃₁	—	2	2	no	—
114	D₈×C₃₁	—	2	2	no	—
115	D₈×C₃₁	—	2	1	no	—
116	D₈×C₃₁	—	2	2	no	—
117	D₈×C₃₁	—	2	2	no	—
118	D₈×C₃₁	—	4	4	no	—
119	D₈×C₃₁	—	8	1	no	—
120	D₈×C₃₁	—	8	1	no	—
121	D₈×C₃₁	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	1	no	—
129	C₄⋊C₄×C₃₁	—	4	1	no	—
130	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
131	C₂²⋊C₄×C₃₁	—	4	1	no	—
132	C₂²⋊C₄×C₃₁	—	4	2	no	—
133	C₂²⋊C₄×C₃₁	—	4	2	no	—
134	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
135	C₄²×C₃₁	—	4	1	no	—
136	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₁₆×C₃₁	—	2	2	no	—
138	C₁₆×C₃₁	—	2	1	no	—
139	C₁₆×C₃₁	C₂×D₄	1	2	yes	complemented
140	#33840	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	4	1	no	—
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	—	4	2	no	—
144	C₂³×C₃₁	—	8	4	no	—
145	C₂³×C₃₁	—	8	4	no	—
146	C₂³×C₃₁	—	16	2	no	—
147	C₂³×C₃₁	C₂×D₈	1	2	yes	—
148	Q₈×C₃₁	—	2	2	no	—
149	Q₈×C₃₁	—	4	1	no	—
150	D₄×C₃₁	—	2	2	no	—
151	D₄×C₃₁	—	2	2	no	—
152	D₄×C₃₁	—	4	2	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	1	no	—
155	D₄×C₃₁	—	4	2	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	8	4	no	—
158	D₄×C₃₁	—	8	4	no	—
159	D₄×C₃₁	—	8	4	no	—
160	D₄×C₃₁	—	8	2	no	—
161	D₄×C₃₁	—	8	2	no	—
162	D₄×C₃₁	—	16	1	no	—
163	D₄×C₃₁	—	16	1	no	—
164	D₄×C₃₁	—	16	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	8	4	no	—
171	C₂×C₄×C₃₁	—	8	2	no	—
172	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
174	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	C₈×C₃₁	—	2	2	no	—
177	C₈×C₃₁	—	2	1	no	—
178	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂×#509	C₂×C₃₁	1	2	yes	complemented
181	#3287	C₂×C₃₁	1	1	yes	characteristic, complemented
182	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#2246	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#2218	C₂×C₃₁	1	2	yes	complemented
185	#2207	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#2206	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#2186	C₂×C₃₁	1	2	yes	complemented
188	#2184	C₂×C₃₁	1	2	yes	complemented
189	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#2169	C₂×C₃₁	1	1	yes	characteristic, complemented
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	2	2	no	—
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	—	8	1	no	—
196	C₂²×C₃₁	—	8	1	no	—
197	C₂²×C₃₁	—	8	2	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	—	8	2	no	—
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	—	16	8	no	—
204	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
205	C₄×C₃₁	—	2	1	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	4	2	no	—
208	C₄×C₃₁	—	8	1	no	—
209	C₄×C₃₁	—	8	1	no	—
210	C₄×C₃₁	#509	1	2	yes	complemented
211	C₄×C₃₁	D₄²	1	1	yes	characteristic
212	C₄×C₃₁	#576	1	1	yes	characteristic
213	C₂×#135	C₂²×C₃₁	1	2	yes	complemented
214	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
215	#550	—	2	1	no	—
216	D₄²	—	2	1	no	—
217	#509	—	2	8	no	—
218	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
220	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
221	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
222	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
223	C₂×M₃₂	C₂²×C₃₁	1	2	yes	complemented
224	#493	C₂²×C₃₁	1	1	yes	characteristic
225	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
226	#466	C₂²×C₃₁	1	2	yes	—
227	#466	C₂²×C₃₁	1	2	yes	complemented
228	#460	—	2	1	no	—
229	#459	—	2	1	no	—
230	#449	—	2	2	no	—
231	#447	—	2	2	no	—
232	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
233	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
235	#367	C₂²×C₃₁	1	1	yes	characteristic, complemented
236	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
237	#358	C₂²×C₃₁	1	1	yes	characteristic, complemented
238	#357	C₂²×C₃₁	1	1	yes	characteristic
239	#357	C₂²×C₃₁	1	2	yes	complemented
240	#357	C₂²×C₃₁	1	2	yes	complemented
241	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
242	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
243	C₂×C₃₁	—	4	2	no	—
244	C₂×C₃₁	—	8	2	no	—
245	C₂×C₃₁	—	16	2	no	—
246	C₂×C₃₁	—	16	2	no	—
247	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
248	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
249	C₂×C₃₁	#3415	1	1	yes	characteristic
250	C₂×#41	—	2	2	no	—
251	C₂²×D₄	—	2	2	no	—
252	C₂²×D₄	—	4	1	no	—
253	#135	—	2	4	no	—
254	#135	—	4	2	no	—
255	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
256	C₂×SD₁₆	—	2	2	no	—
257	C₂×D₈	—	2	4	no	—
258	C₂×D₈	—	2	2	no	—
259	C₂×D₈	—	2	1	no	—
260	C₂×D₈	—	2	1	no	—
261	C₂×D₈	—	4	2	no	—
262	C₂×D₈	C₂³×C₃₁	1	2	yes	—
263	C₂×D₈	C₂³×C₃₁	1	2	yes	—
264	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
266	#127	—	2	1	no	—
267	#127	—	2	1	no	—
268	#127	—	4	1	no	—
269	#121	—	2	2	no	—
270	#121	—	2	2	no	—
271	#121	—	4	1	no	—
272	#121	—	4	2	no	—
273	#120	—	4	2	no	—
274	C₄×Q₈	—	2	1	no	—
275	C₄×D₄	—	2	1	no	—
276	C₄×D₄	—	4	1	no	—
277	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
278	SD₃₂	—	2	4	no	—
279	SD₃₂	D₄×C₃₁	1	4	yes	complemented
280	D₁₆	—	2	4	no	—
281	D₁₆	—	2	4	no	—
282	D₁₆	—	4	2	no	—
283	M₃₂	—	2	4	no	—
284	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
286	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
287	#107	C₂³×C₃₁	1	1	yes	characteristic
288	#107	C₂³×C₃₁	1	1	yes	characteristic
289	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
290	#103	—	2	1	no	—
291	#102	—	2	1	no	—
292	#102	—	2	2	no	—
293	#102	—	2	2	no	—
294	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
295	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₃₁	#33840	1	1	yes	characteristic, direct product factor, Sylow
297	C₂⁴	—	4	2	no	—
298	#41	—	4	4	no	—
299	#41	—	8	2	no	—
300	C₂×Q₈	—	2	1	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	4	4	no	—
305	C₂×D₄	—	4	2	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	—	4	1	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂×D₄	—	4	1	no	—
311	C₂×D₄	—	4	1	no	—
312	C₂×D₄	—	4	1	no	—
313	C₂×D₄	—	8	2	no	—
314	C₂×D₄	—	8	4	no	—
315	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
319	Q₁₆	—	2	1	no	—
320	Q₁₆	C₂×D₄×C₃₁	1	2	yes	complemented
321	SD₁₆	—	4	4	no	—
322	D₈	—	2	2	no	—
323	D₈	—	2	2	no	—
324	D₈	—	2	1	no	—
325	D₈	—	2	2	no	—
326	D₈	—	2	2	no	—
327	D₈	—	4	4	no	—
328	D₈	—	8	1	no	—
329	D₈	—	8	1	no	—
330	D₈	C₂×D₄×C₃₁	1	2	yes	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
335	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	4	1	no	—
339	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
340	C₂²⋊C₄	—	4	1	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
344	C₄²	—	4	1	no	—
345	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₁₆	—	2	2	no	—
347	C₁₆	—	2	1	no	—
348	C₁₆	C₂×D₄×C₃₁	1	2	yes	complemented
349	C₂³	—	4	1	no	—
350	C₂³	—	4	2	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	8	4	no	—
353	C₂³	—	8	4	no	—
354	C₂³	—	16	2	no	—
355	C₂³	C₂×D₈×C₃₁	1	2	yes	—
356	Q₈	—	2	2	no	—
357	Q₈	—	4	1	no	—
358	D₄	—	2	2	no	—
359	D₄	—	2	2	no	—
360	D₄	—	4	2	no	—
361	D₄	—	4	2	no	—
362	D₄	—	4	1	no	—
363	D₄	—	4	2	no	—
364	D₄	—	4	2	no	—
365	D₄	—	8	4	no	—
366	D₄	—	8	4	no	—
367	D₄	—	8	4	no	—
368	D₄	—	8	2	no	—
369	D₄	—	8	2	no	—
370	D₄	—	16	1	no	—
371	D₄	—	16	1	no	—
372	D₄	—	16	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	8	4	no	—
379	C₂×C₄	—	8	2	no	—
380	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
382	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
383	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
384	C₈	—	2	2	no	—
385	C₈	—	2	1	no	—
386	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	8	1	no	—
393	C₂²	—	8	1	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	2	no	—
396	C₂²	—	8	2	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	16	8	no	—
401	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	2	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	8	1	no	—
406	C₄	—	8	1	no	—
407	C₄	C₃₁×#509	1	2	yes	complemented
408	C₄	D₄²×C₃₁	1	1	yes	characteristic
409	C₄	C₃₁×#576	1	1	yes	characteristic
410	C₂	—	4	2	no	—
411	C₂	—	8	2	no	—
412	C₂	—	16	2	no	—
413	C₂	—	16	2	no	—
414	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
415	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
416	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#509	C₂	semidirect
2. C₃₁×#3287	C₂	semidirect
3. D₄×D₈×C₃₁	C₂	semidirect
4. C₃₁×#2246	C₂	semidirect
5. C₃₁×#2218	C₂	semidirect
6. C₃₁×#2207	C₂	semidirect
7. C₃₁×#2206	C₂	semidirect
8. C₃₁×#2186	C₂	semidirect
9. C₃₁×#2184	C₂	semidirect
10. C₄×C₃₁×SD₃₂	C₂	semidirect
11. C₃₁×#2169	C₂	semidirect
12. C₂×C₃₁×#135	C₂²	semidirect
18. C₂×C₃₁×SD₃₂	C₂²	semidirect
20. C₂×C₃₁×D₁₆	C₂²	semidirect
21. C₂×C₃₁×D₁₆	C₂²	semidirect
22. C₂×C₃₁×M₃₂	C₂²	semidirect
24. C₃₁×#493	C₂²	semidirect
26. C₃₁×#466	C₂²	semidirect
31. C₄×Q₁₆×C₃₁	C₂²	semidirect
34. C₃₁×#367	C₂²	semidirect
35. C₃₁×C₄⋊C₁₆	C₂²	semidirect
36. C₃₁×#358	C₂²	semidirect
38. C₃₁×#357	C₂²	semidirect
39. C₃₁×#357	C₂²	semidirect
40. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
41. C₄×C₁₆×C₃₁	C₂²	semidirect
71. C₃₁×SD₃₂	D₄	semidirect
111. Q₁₆×C₃₁	C₂×D₄	semidirect
139. C₁₆×C₃₁	C₂×D₄	semidirect
140. #33840	C₃₁	direct (N × H)
180. C₂×#509	C₂×C₃₁	semidirect
181. #3287	C₂×C₃₁	semidirect
182. D₄×D₈	C₂×C₃₁	semidirect
183. #2246	C₂×C₃₁	semidirect
184. #2218	C₂×C₃₁	semidirect
185. #2207	C₂×C₃₁	semidirect
186. #2206	C₂×C₃₁	semidirect
187. #2186	C₂×C₃₁	semidirect
188. #2184	C₂×C₃₁	semidirect
189. C₄×SD₃₂	C₂×C₃₁	semidirect
190. #2169	C₂×C₃₁	semidirect
210. C₄×C₃₁	#509	semidirect
213. C₂×#135	C₂²×C₃₁	semidirect
219. C₂×SD₃₂	C₂²×C₃₁	semidirect
221. C₂×D₁₆	C₂²×C₃₁	semidirect
222. C₂×D₁₆	C₂²×C₃₁	semidirect
223. C₂×M₃₂	C₂²×C₃₁	semidirect
225. #493	C₂²×C₃₁	semidirect
227. #466	C₂²×C₃₁	semidirect
232. C₄×Q₁₆	C₂²×C₃₁	semidirect
235. #367	C₂²×C₃₁	semidirect
236. C₄⋊C₁₆	C₂²×C₃₁	semidirect
237. #358	C₂²×C₃₁	semidirect
239. #357	C₂²×C₃₁	semidirect
240. #357	C₂²×C₃₁	semidirect
241. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
242. C₄×C₁₆	C₂²×C₃₁	semidirect
279. SD₃₂	D₄×C₃₁	semidirect
296. C₃₁	#33840	direct (N × H)
320. Q₁₆	C₂×D₄×C₃₁	semidirect
348. C₁₆	C₂×D₄×C₃₁	semidirect
407. C₄	C₃₁×#509	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#509	2
C₃₁×#3287	2
D₄×D₈×C₃₁	2
C₃₁×#2246	2
C₃₁×#2218	2
C₃₁×#2207	2
C₃₁×#2206	2
C₃₁×#2186	2
C₃₁×#2184	2
C₄×C₃₁×SD₃₂	2
C₃₁×#2169	2
#33840	31

Subgroups — C31×#33840 #21490599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33840 #21490599