Subgroups — C₃₁×#25051 #21481810

Order 7936.

736 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3099	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3060	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2952	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2079	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2039	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1974	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1951	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1859	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1815	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
14	C₃₁×#537	C₂²	1	2	yes	—
15	C₃₁×#520	C₂²	1	2	yes	—
16	C₃₁×#485	—	2	2	no	—
17	C₃₁×#483	—	2	2	no	—
18	C₃₁×#482	—	2	2	no	—
19	C₃₁×#481	—	2	2	no	—
20	C₃₁×#470	—	2	1	no	—
21	C₃₁×#469	—	2	1	no	—
22	C₃₁×#465	—	2	2	no	—
23	C₃₁×#442	—	2	1	no	—
24	C₃₁×#441	—	2	1	no	—
25	C₄×SD₁₆×C₃₁	—	2	2	no	—
26	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
27	C₃₁×#428	C₂²	1	2	yes	—
28	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#106	C₂²	1	2	yes	—
30	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#104	C₂²	1	2	yes	—
32	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
33	C₃₁×#417	C₂²	1	2	yes	—
34	C₃₁×#417	C₂²	1	2	yes	—
35	C₃₁×#417	C₂²	1	2	yes	—
36	C₃₁×#416	C₂²	1	1	yes	characteristic
37	C₃₁×#408	C₂²	1	2	yes	—
38	C₃₁×#405	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#384	C₂²	1	2	yes	—
41	C₃₁×#343	C₂²	1	2	yes	—
42	C₃₁×#343	C₂²	1	2	yes	—
43	C₃₁×#341	C₂²	1	1	yes	characteristic
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	C₂³	1	2	yes	—
46	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#135	—	2	2	no	—
48	C₃₁×#135	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
50	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
51	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
52	C₃₁×#131	—	2	2	no	—
53	C₃₁×#131	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	—	2	2	no	—
63	C₃₁×#121	—	2	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₃₁×#121	—	4	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	—	4	2	no	—
69	C₃₁×#117	—	2	2	no	—
70	C₃₁×#117	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#108	—	2	2	no	—
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×#106	—	2	1	no	—
79	C₃₁×#106	—	2	1	no	—
80	C₃₁×#106	—	2	1	no	—
81	C₃₁×#106	—	2	1	no	—
82	C₃₁×#106	—	2	2	no	—
83	C₃₁×#106	—	2	2	no	—
84	C₃₁×#106	—	2	2	no	—
85	C₃₁×#104	—	2	4	no	—
86	C₃₁×#104	—	2	2	no	—
87	C₃₁×#103	—	2	2	no	—
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	D₄	1	2	yes	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	D₄	1	2	yes	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	—	2	2	no	—
97	C₃₁×#97	—	2	1	no	—
98	C₄×C₈×C₃₁	—	2	1	no	—
99	C₄×C₈×C₃₁	—	2	2	no	—
100	#41×C₃₁	—	2	2	no	—
101	#41×C₃₁	—	2	1	no	—
102	#41×C₃₁	—	2	1	no	—
103	#41×C₃₁	—	2	1	no	—
104	#41×C₃₁	—	2	2	no	—
105	#41×C₃₁	—	2	2	no	—
106	#41×C₃₁	—	2	2	no	—
107	#41×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	4	2	no	—
113	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	4	2	no	—
121	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
122	Q₁₆×C₃₁	—	2	2	no	—
123	SD₁₆×C₃₁	—	2	4	no	—
124	D₈×C₃₁	—	2	2	no	—
125	M₁₆×C₃₁	—	2	2	no	—
126	M₁₆×C₃₁	—	2	1	no	—
127	M₁₆×C₃₁	—	2	2	no	—
128	M₁₆×C₃₁	—	2	1	no	—
129	M₁₆×C₃₁	—	2	1	no	—
130	M₁₆×C₃₁	—	2	1	no	—
131	M₁₆×C₃₁	—	4	2	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	4	2	no	—
138	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₃₁	#41	1	2	yes	—
142	C₂×C₈×C₃₁	#41	1	2	yes	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	4	2	no	—
148	C₄⋊C₄×C₃₁	—	4	2	no	—
149	C₄⋊C₄×C₃₁	—	8	4	no	—
150	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₃₁	—	2	1	no	—
152	C₂²⋊C₄×C₃₁	—	4	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	2	no	—
154	C₂²⋊C₄×C₃₁	—	4	2	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	4	2	no	—
158	C₄²×C₃₁	#41	1	2	yes	—
159	#25051	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
163	Q₈×C₃₁	—	2	2	no	—
164	Q₈×C₃₁	—	2	1	no	—
165	Q₈×C₃₁	—	2	1	no	—
166	Q₈×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	2	2	no	—
168	D₄×C₃₁	—	2	1	no	—
169	D₄×C₃₁	—	2	1	no	—
170	D₄×C₃₁	—	2	2	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	1	no	—
173	D₄×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	4	1	no	—
175	D₄×C₃₁	—	16	2	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	8	2	no	—
188	C₂×C₄×C₃₁	—	8	2	no	—
189	C₂×C₄×C₃₁	—	16	2	no	—
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
196	C₈×C₃₁	—	2	2	no	—
197	C₈×C₃₁	—	4	1	no	—
198	C₈×C₃₁	—	4	2	no	—
199	C₈×C₃₁	—	4	1	no	—
200	C₈×C₃₁	#123	1	2	yes	—
201	C₈×C₃₁	#123	1	2	yes	—
202	#3099	C₂×C₃₁	1	2	yes	complemented
203	#3060	C₂×C₃₁	1	1	yes	characteristic
204	#2952	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#2079	C₂×C₃₁	1	2	yes	complemented
206	#2039	C₂×C₃₁	1	2	yes	complemented
207	#1974	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1951	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1859	C₂×C₃₁	1	2	yes	complemented
210	#1844	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1815	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1764	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	4	1	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	#542	1	1	yes	characteristic
220	C₂²×C₃₁	D₄²	1	1	yes	characteristic
221	C₂²×C₃₁	#553	1	1	yes	characteristic
222	C₄×C₃₁	—	2	4	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	1	no	—
225	C₄×C₃₁	—	4	2	no	—
226	C₄×C₃₁	—	8	2	no	—
227	C₄×C₃₁	—	8	2	no	—
228	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
229	C₄×C₃₁	#532	1	1	yes	characteristic
230	C₄×C₃₁	#547	1	1	yes	characteristic
231	C₄×C₃₁	#547	1	1	yes	characteristic
232	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
234	#537	C₂²×C₃₁	1	2	yes	—
235	#520	C₂²×C₃₁	1	2	yes	—
236	#485	—	2	2	no	—
237	#483	—	2	2	no	—
238	#482	—	2	2	no	—
239	#481	—	2	2	no	—
240	#470	—	2	1	no	—
241	#469	—	2	1	no	—
242	#465	—	2	2	no	—
243	#442	—	2	1	no	—
244	#441	—	2	1	no	—
245	C₄×SD₁₆	—	2	2	no	—
246	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
247	#428	C₂²×C₃₁	1	2	yes	—
248	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#106	C₂²×C₃₁	1	2	yes	—
250	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×#104	C₂²×C₃₁	1	2	yes	—
252	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
253	#417	C₂²×C₃₁	1	2	yes	—
254	#417	C₂²×C₃₁	1	2	yes	—
255	#417	C₂²×C₃₁	1	2	yes	—
256	#416	C₂²×C₃₁	1	1	yes	characteristic
257	#408	C₂²×C₃₁	1	2	yes	—
258	#405	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	#384	C₂²×C₃₁	1	2	yes	—
261	#343	C₂²×C₃₁	1	2	yes	—
262	#343	C₂²×C₃₁	1	2	yes	—
263	#341	C₂²×C₃₁	1	1	yes	characteristic
264	#339	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₃₁	—	2	1	no	—
266	C₂×C₃₁	—	2	1	no	—
267	C₂×C₃₁	—	8	2	no	—
268	C₂×C₃₁	—	8	1	no	—
269	C₂×C₃₁	#2422	1	1	yes	characteristic
270	C₂×C₃₁	#3291	1	2	yes	—
271	C₂×#41	C₂³×C₃₁	1	2	yes	—
272	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
273	#135	—	2	2	no	—
274	#135	—	2	2	no	—
275	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
276	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
277	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
278	#131	—	2	2	no	—
279	#131	—	2	2	no	—
280	C₂×M₁₆	—	2	1	no	—
281	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
284	C₂²×C₈	—	2	1	no	—
285	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₂²⋊Q₈	—	2	2	no	—
288	C₂²⋊Q₈	—	2	2	no	—
289	#121	—	2	2	no	—
290	#121	—	2	2	no	—
291	#121	—	4	2	no	—
292	C₄×Q₈	—	2	2	no	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	—	4	2	no	—
295	#117	—	2	2	no	—
296	#117	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	4	2	no	—
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
301	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
302	#108	—	2	2	no	—
303	#107	—	2	2	no	—
304	#106	—	2	1	no	—
305	#106	—	2	1	no	—
306	#106	—	2	1	no	—
307	#106	—	2	1	no	—
308	#106	—	2	2	no	—
309	#106	—	2	2	no	—
310	#106	—	2	2	no	—
311	#104	—	2	4	no	—
312	#104	—	2	2	no	—
313	#103	—	2	2	no	—
314	#103	—	2	2	no	—
315	#103	—	2	2	no	—
316	#103	D₄×C₃₁	1	2	yes	—
317	#102	—	2	2	no	—
318	#102	—	2	2	no	—
319	#102	—	2	2	no	—
320	#102	D₄×C₃₁	1	2	yes	—
321	C₂²⋊C₈	—	2	2	no	—
322	C₂²⋊C₈	—	2	2	no	—
323	#97	—	2	1	no	—
324	C₄×C₈	—	2	1	no	—
325	C₄×C₈	—	2	2	no	—
326	C₃₁	#25051	1	1	yes	characteristic, direct product factor, Sylow
327	#41	—	2	2	no	—
328	#41	—	2	1	no	—
329	#41	—	2	1	no	—
330	#41	—	2	1	no	—
331	#41	—	2	2	no	—
332	#41	—	2	2	no	—
333	#41	—	2	2	no	—
334	#41	—	2	1	no	—
335	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
349	Q₁₆	—	2	2	no	—
350	SD₁₆	—	2	4	no	—
351	D₈	—	2	2	no	—
352	M₁₆	—	2	2	no	—
353	M₁₆	—	2	1	no	—
354	M₁₆	—	2	2	no	—
355	M₁₆	—	2	1	no	—
356	M₁₆	—	2	1	no	—
357	M₁₆	—	2	1	no	—
358	M₁₆	—	4	2	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	4	2	no	—
365	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
366	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₂×C₈	#41×C₃₁	1	2	yes	—
369	C₂×C₈	#41×C₃₁	1	2	yes	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	—	4	2	no	—
376	C₄⋊C₄	—	8	4	no	—
377	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
378	C₂²⋊C₄	—	2	1	no	—
379	C₂²⋊C₄	—	4	2	no	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₄²	—	2	1	no	—
383	C₄²	—	2	1	no	—
384	C₄²	—	4	2	no	—
385	C₄²	#41×C₃₁	1	2	yes	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	1	no	—
388	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
389	Q₈	—	2	2	no	—
390	Q₈	—	2	1	no	—
391	Q₈	—	2	1	no	—
392	Q₈	—	2	2	no	—
393	D₄	—	2	2	no	—
394	D₄	—	2	1	no	—
395	D₄	—	2	1	no	—
396	D₄	—	2	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	1	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	1	no	—
401	D₄	—	16	2	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	8	2	no	—
414	C₂×C₄	—	8	2	no	—
415	C₂×C₄	—	16	2	no	—
416	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
418	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
421	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
422	C₈	—	2	2	no	—
423	C₈	—	4	1	no	—
424	C₈	—	4	2	no	—
425	C₈	—	4	1	no	—
426	C₈	C₃₁×#123	1	2	yes	—
427	C₈	C₃₁×#123	1	2	yes	—
428	C₂²	—	2	2	no	—
429	C₂²	—	4	1	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	C₃₁×#542	1	1	yes	characteristic
435	C₂²	D₄²×C₃₁	1	1	yes	characteristic
436	C₂²	C₃₁×#553	1	1	yes	characteristic
437	C₄	—	2	4	no	—
438	C₄	—	4	2	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	2	no	—
441	C₄	—	8	2	no	—
442	C₄	—	8	2	no	—
443	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
444	C₄	C₃₁×#532	1	1	yes	characteristic
445	C₄	C₃₁×#547	1	1	yes	characteristic
446	C₄	C₃₁×#547	1	1	yes	characteristic
447	C₂	—	2	1	no	—
448	C₂	—	2	1	no	—
449	C₂	—	8	2	no	—
450	C₂	—	8	1	no	—
451	C₂	C₃₁×#2422	1	1	yes	characteristic
452	C₂	C₃₁×#3291	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3060	C₂
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×#3099	C₂	semidirect
3. C₃₁×#2952	C₂	semidirect
4. C₃₁×#2079	C₂	semidirect
5. C₃₁×#2039	C₂	semidirect
6. C₃₁×#1974	C₂	semidirect
7. C₃₁×#1951	C₂	semidirect
8. C₃₁×#1859	C₂	semidirect
9. C₃₁×#1844	C₂	semidirect
10. C₃₁×#1815	C₂	semidirect
11. C₃₁×#1764	C₂	semidirect
159. #25051	C₃₁	direct (N × H)
202. #3099	C₂×C₃₁	semidirect
204. #2952	C₂×C₃₁	semidirect
205. #2079	C₂×C₃₁	semidirect
206. #2039	C₂×C₃₁	semidirect
207. #1974	C₂×C₃₁	semidirect
208. #1951	C₂×C₃₁	semidirect
209. #1859	C₂×C₃₁	semidirect
210. #1844	C₂×C₃₁	semidirect
211. #1815	C₂×C₃₁	semidirect
212. #1764	C₂×C₃₁	semidirect
326. C₃₁	#25051	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×#3099	2
C₃₁×#3060	2
C₃₁×#2952	2
C₃₁×#2079	2
C₃₁×#2039	2
C₃₁×#1974	2
C₃₁×#1951	2
C₃₁×#1859	2
C₃₁×#1844	2
C₃₁×#1815	2
C₃₁×#1764	2
#25051	31

Subgroups — C31×#25051 #21481810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25051 #21481810