Subgroups — C₃₁×#26501 #21483260

Order 7936.

724 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3081	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2063	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2027	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1856	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#544	C₂²	1	2	yes	—
14	C₃₁×#518	C₂²	1	2	yes	—
15	C₃₁×#501	—	2	2	no	—
16	C₃₁×#500	—	2	1	no	—
17	C₃₁×#499	—	2	1	no	—
18	C₃₁×#478	—	2	2	no	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#476	—	2	2	no	—
21	C₃₁×#474	—	2	2	no	—
22	C₃₁×#442	—	2	2	no	—
23	C₄×D₈×C₃₁	—	2	1	no	—
24	C₄×D₈×C₃₁	—	2	1	no	—
25	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
27	C₃₁×#428	C₂²	1	2	yes	—
28	C₃₁×#428	C₂²	1	2	yes	—
29	C₃₁×#427	C₂²	1	1	yes	characteristic
30	C₃₁×#427	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
32	C₃₁×#424	C₂²	1	2	yes	—
33	C₃₁×#418	C₂²	1	2	yes	—
34	C₃₁×#418	C₂²	1	2	yes	—
35	C₂×C₃₁×#102	C₂²	1	2	yes	—
36	C₂×C₃₁×#100	C₂²	1	2	yes	—
37	C₃₁×#405	C₂²	1	1	yes	characteristic
38	C₃₁×#405	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#390	C₂²	1	2	yes	—
41	C₃₁×#343	C₂²	1	2	yes	—
42	C₃₁×#339	C₂²	1	1	yes	characteristic
43	C₃₁×#339	C₂²	1	2	yes	—
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
46	C₂×D₈×C₃₁	—	2	1	no	—
47	C₂×D₈×C₃₁	—	2	1	no	—
48	C₂×D₈×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	Q₈	1	4	yes	complemented
50	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
52	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×#125	—	2	2	no	—
58	C₃₁×#125	—	2	2	no	—
59	C₃₁×C₂²⋊Q₈	—	4	2	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₃₁×#117	C₂³	1	1	yes	characteristic
65	C₃₁×#117	C₂³	1	1	yes	characteristic
66	C₃₁×#117	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#108	—	2	2	no	—
72	C₃₁×#108	—	2	2	no	—
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#107	—	2	2	no	—
77	C₃₁×#107	D₄	1	1	yes	characteristic
78	C₃₁×#107	D₄	1	1	yes	characteristic
79	C₃₁×#107	D₄	1	1	yes	characteristic
80	C₃₁×#107	D₄	1	1	yes	characteristic
81	C₃₁×#106	—	2	2	no	—
82	C₃₁×#106	—	2	2	no	—
83	C₃₁×#106	D₄	1	2	yes	—
84	C₃₁×#106	D₄	1	1	yes	characteristic
85	C₃₁×#106	D₄	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#102	—	2	2	no	—
90	C₃₁×#102	—	2	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#100	—	2	4	no	—
95	C₃₁×#97	—	2	1	no	—
96	C₃₁×#97	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	1	no	—
98	C₄×C₈×C₃₁	—	2	1	no	—
99	C₄×C₈×C₃₁	—	2	2	no	—
100	C₂⁴×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	4	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
113	D₈×C₃₁	—	2	4	no	—
114	D₈×C₃₁	—	2	4	no	—
115	M₁₆×C₃₁	—	2	1	no	—
116	M₁₆×C₃₁	—	2	2	no	—
117	M₁₆×C₃₁	—	2	2	no	—
118	M₁₆×C₃₁	—	2	1	no	—
119	M₁₆×C₃₁	—	2	1	no	—
120	M₁₆×C₃₁	#41	1	2	yes	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	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	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
131	C₂×C₈×C₃₁	#41	1	2	yes	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₃₁	—	2	1	no	—
144	C₂²⋊C₄×C₃₁	—	2	2	no	—
145	C₂²⋊C₄×C₃₁	—	2	1	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₂²⋊C₄×C₃₁	—	4	2	no	—
148	C₂²⋊C₄×C₃₁	—	8	4	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	2	2	no	—
152	C₄²×C₃₁	—	2	1	no	—
153	C₄²×C₃₁	—	2	1	no	—
154	#26501	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	8	2	no	—
159	C₂³×C₃₁	—	8	2	no	—
160	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
161	Q₈×C₃₁	—	16	2	no	—
162	D₄×C₃₁	—	2	4	no	—
163	D₄×C₃₁	—	2	4	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
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	2	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	8	2	no	—
177	C₂×C₄×C₃₁	—	16	2	no	—
178	C₂×C₄×C₃₁	—	16	2	no	—
179	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
185	C₈×C₃₁	—	2	1	no	—
186	C₈×C₃₁	—	2	1	no	—
187	C₈×C₃₁	—	2	2	no	—
188	C₈×C₃₁	—	4	2	no	—
189	C₈×C₃₁	—	4	1	no	—
190	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
191	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
192	#3164	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#3081	C₂×C₃₁	1	2	yes	complemented
194	#2948	C₂×C₃₁	1	1	yes	characteristic
195	#2063	C₂×C₃₁	1	2	yes	complemented
196	#2027	C₂×C₃₁	1	2	yes	complemented
197	#1978	C₂×C₃₁	1	1	yes	characteristic
198	#1970	C₂×C₃₁	1	1	yes	characteristic
199	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1856	C₂×C₃₁	1	2	yes	complemented
202	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	1	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	—	8	4	no	—
207	C₂²×C₃₁	—	8	2	no	—
208	C₂²×C₃₁	—	8	2	no	—
209	C₂²×C₃₁	—	16	2	no	—
210	C₂²×C₃₁	#542	1	1	yes	characteristic
211	C₂²×C₃₁	D₄²	1	1	yes	characteristic
212	C₂²×C₃₁	#554	1	1	yes	characteristic
213	C₄×C₃₁	—	4	2	no	—
214	C₄×C₃₁	—	4	1	no	—
215	C₄×C₃₁	—	4	2	no	—
216	C₄×C₃₁	—	4	1	no	—
217	C₄×C₃₁	—	4	1	no	—
218	C₄×C₃₁	—	4	2	no	—
219	C₄×C₃₁	—	4	1	no	—
220	C₄×C₃₁	—	8	2	no	—
221	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
222	C₄×C₃₁	#532	1	1	yes	characteristic
223	C₄×C₃₁	#548	1	1	yes	characteristic
224	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
225	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
226	#544	C₂²×C₃₁	1	2	yes	—
227	#518	C₂²×C₃₁	1	2	yes	—
228	#501	—	2	2	no	—
229	#500	—	2	1	no	—
230	#499	—	2	1	no	—
231	#478	—	2	2	no	—
232	#478	—	2	2	no	—
233	#476	—	2	2	no	—
234	#474	—	2	2	no	—
235	#442	—	2	2	no	—
236	C₄×D₈	—	2	1	no	—
237	C₄×D₈	—	2	1	no	—
238	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
240	#428	C₂²×C₃₁	1	2	yes	—
241	#428	C₂²×C₃₁	1	2	yes	—
242	#427	C₂²×C₃₁	1	1	yes	characteristic
243	#427	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
245	#424	C₂²×C₃₁	1	2	yes	—
246	#418	C₂²×C₃₁	1	2	yes	—
247	#418	C₂²×C₃₁	1	2	yes	—
248	C₂×#102	C₂²×C₃₁	1	2	yes	—
249	C₂×#100	C₂²×C₃₁	1	2	yes	—
250	#405	C₂²×C₃₁	1	1	yes	characteristic
251	#405	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
253	#390	C₂²×C₃₁	1	2	yes	—
254	#343	C₂²×C₃₁	1	2	yes	—
255	#339	C₂²×C₃₁	1	1	yes	characteristic
256	#339	C₂²×C₃₁	1	2	yes	—
257	#339	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₃₁	—	2	1	no	—
259	C₂×C₃₁	—	2	1	no	—
260	C₂×C₃₁	—	8	4	no	—
261	C₂×C₃₁	#2452	1	1	yes	characteristic
262	C₂×C₃₁	#3290	1	1	yes	characteristic
263	C₂×C₃₁	#3292	1	1	yes	characteristic
264	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
265	C₂×D₈	—	2	1	no	—
266	C₂×D₈	—	2	1	no	—
267	C₂×D₈	—	2	2	no	—
268	C₂×D₈	Q₈×C₃₁	1	4	yes	complemented
269	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
270	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
271	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
272	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₄⋊Q₈	—	2	2	no	—
275	C₄⋊Q₈	—	2	2	no	—
276	#125	—	2	2	no	—
277	#125	—	2	2	no	—
278	C₂²⋊Q₈	—	4	2	no	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	—	2	2	no	—
282	C₄×D₄	—	2	2	no	—
283	#117	C₂³×C₃₁	1	1	yes	characteristic
284	#117	C₂³×C₃₁	1	2	yes	—
285	#117	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
287	C₂×C₂²⋊C₄	—	4	2	no	—
288	C₂×C₂²⋊C₄	—	4	2	no	—
289	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
290	#108	—	2	2	no	—
291	#108	—	2	2	no	—
292	#107	—	2	1	no	—
293	#107	—	2	1	no	—
294	#107	—	2	2	no	—
295	#107	—	2	2	no	—
296	#107	D₄×C₃₁	1	1	yes	characteristic
297	#107	D₄×C₃₁	1	1	yes	characteristic
298	#107	D₄×C₃₁	1	1	yes	characteristic
299	#107	D₄×C₃₁	1	1	yes	characteristic
300	#106	—	2	2	no	—
301	#106	—	2	2	no	—
302	#106	D₄×C₃₁	1	2	yes	—
303	#106	D₄×C₃₁	1	1	yes	characteristic
304	#106	D₄×C₃₁	1	1	yes	characteristic
305	C₄⋊C₈	—	2	2	no	—
306	C₄⋊C₈	—	2	2	no	—
307	#102	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	2	2	no	—
311	#102	—	2	2	no	—
312	#102	—	2	2	no	—
313	#100	—	2	4	no	—
314	#97	—	2	1	no	—
315	#97	—	2	1	no	—
316	C₄×C₈	—	2	1	no	—
317	C₄×C₈	—	2	1	no	—
318	C₄×C₈	—	2	2	no	—
319	C₃₁	#26501	1	1	yes	characteristic, direct product factor, Sylow
320	C₂⁴	—	2	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	4	2	no	—
331	C₂²×C₄	—	4	2	no	—
332	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
333	D₈	—	2	4	no	—
334	D₈	—	2	4	no	—
335	M₁₆	—	2	1	no	—
336	M₁₆	—	2	2	no	—
337	M₁₆	—	2	2	no	—
338	M₁₆	—	2	1	no	—
339	M₁₆	—	2	1	no	—
340	M₁₆	#41×C₃₁	1	2	yes	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	2	no	—
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	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
351	C₂×C₈	#41×C₃₁	1	2	yes	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
360	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
361	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂²⋊C₄	—	2	1	no	—
364	C₂²⋊C₄	—	2	2	no	—
365	C₂²⋊C₄	—	2	1	no	—
366	C₂²⋊C₄	—	4	2	no	—
367	C₂²⋊C₄	—	4	2	no	—
368	C₂²⋊C₄	—	8	4	no	—
369	C₄²	—	2	1	no	—
370	C₄²	—	2	1	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	1	no	—
373	C₄²	—	2	1	no	—
374	C₂³	—	2	4	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	8	2	no	—
378	C₂³	—	8	2	no	—
379	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
380	Q₈	—	16	2	no	—
381	D₄	—	2	4	no	—
382	D₄	—	2	4	no	—
383	D₄	—	4	2	no	—
384	D₄	—	4	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	8	2	no	—
396	C₂×C₄	—	16	2	no	—
397	C₂×C₄	—	16	2	no	—
398	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
403	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
404	C₈	—	2	1	no	—
405	C₈	—	2	1	no	—
406	C₈	—	2	2	no	—
407	C₈	—	4	2	no	—
408	C₈	—	4	1	no	—
409	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
410	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
411	C₂²	—	2	1	no	—
412	C₂²	—	2	1	no	—
413	C₂²	—	4	4	no	—
414	C₂²	—	8	4	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	16	2	no	—
418	C₂²	C₃₁×#542	1	1	yes	characteristic
419	C₂²	D₄²×C₃₁	1	1	yes	characteristic
420	C₂²	C₃₁×#554	1	1	yes	characteristic
421	C₄	—	4	2	no	—
422	C₄	—	4	1	no	—
423	C₄	—	4	2	no	—
424	C₄	—	4	1	no	—
425	C₄	—	4	1	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	1	no	—
428	C₄	—	8	2	no	—
429	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
430	C₄	C₃₁×#532	1	1	yes	characteristic
431	C₄	C₃₁×#548	1	1	yes	characteristic
432	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
433	C₂	—	2	1	no	—
434	C₂	—	2	1	no	—
435	C₂	—	8	4	no	—
436	C₂	C₃₁×#2452	1	1	yes	characteristic
437	C₂	C₃₁×#3290	1	1	yes	characteristic
438	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×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
1. C₃₁×#3164	C₂	semidirect
2. C₃₁×#3081	C₂	semidirect
4. C₃₁×#2063	C₂	semidirect
5. C₃₁×#2027	C₂	semidirect
8. C₃₁×#1952	C₂	semidirect
9. C₃₁×#1945	C₂	semidirect
10. C₃₁×#1856	C₂	semidirect
11. C₃₁×#1772	C₂	semidirect
49. C₂×D₈×C₃₁	Q₈	semidirect
154. #26501	C₃₁	direct (N × H)
192. #3164	C₂×C₃₁	semidirect
193. #3081	C₂×C₃₁	semidirect
195. #2063	C₂×C₃₁	semidirect
196. #2027	C₂×C₃₁	semidirect
199. #1952	C₂×C₃₁	semidirect
200. #1945	C₂×C₃₁	semidirect
201. #1856	C₂×C₃₁	semidirect
202. #1772	C₂×C₃₁	semidirect
268. C₂×D₈	Q₈×C₃₁	semidirect
319. C₃₁	#26501	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×#3164	2
C₃₁×#3081	2
C₃₁×#2948	2
C₃₁×#2063	2
C₃₁×#2027	2
C₃₁×#1978	2
C₃₁×#1970	2
C₃₁×#1952	2
C₃₁×#1945	2
C₃₁×#1856	2
C₃₁×#1772	2
#26501	31

Subgroups — C31×#26501 #21483260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26501 #21483260