Subgroups — C₃₁×#26273 #21483032

Order 7936.

748 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3156	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3097	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2952	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2052	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2038	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1975	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1882	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#536	C₂²	1	2	yes	—
16	C₃₁×#520	C₂²	1	2	yes	—
17	C₃₁×#497	—	2	2	no	—
18	C₃₁×#495	—	2	1	no	—
19	C₃₁×#494	—	2	1	no	—
20	C₃₁×#485	—	2	2	no	—
21	C₃₁×#485	—	2	2	no	—
22	C₃₁×#484	—	2	2	no	—
23	C₃₁×#483	—	2	2	no	—
24	C₃₁×#442	—	2	1	no	—
25	C₃₁×#441	—	2	1	no	—
26	C₄×SD₁₆×C₃₁	—	2	2	no	—
27	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₃₁×#428	C₂²	1	2	yes	—
30	C₃₁×#427	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#104	C₂²	1	2	yes	—
33	C₃₁×#419	C₂²	1	2	yes	—
34	C₃₁×#417	C₂²	1	2	yes	—
35	C₂×C₃₁×#103	C₂²	1	2	yes	—
36	C₂×C₃₁×#101	C₂²	1	2	yes	—
37	C₃₁×#408	C₂²	1	2	yes	—
38	C₃₁×#405	C₂²	1	1	yes	characteristic
39	C₃₁×#405	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
41	C₃₁×#391	C₂²	1	2	yes	—
42	C₃₁×#343	C₂²	1	2	yes	—
43	C₃₁×#339	C₂²	1	1	yes	characteristic
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	C₂³	1	2	yes	—
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₃₁×#137	—	2	4	no	—
48	C₃₁×#137	—	4	2	no	—
49	C₃₁×#135	—	2	2	no	—
50	C₃₁×#135	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	—	2	2	no	—
52	C₂×Q₁₆×C₃₁	—	2	1	no	—
53	C₂×Q₁₆×C₃₁	—	2	1	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×Q₁₆×C₃₁	—	2	2	no	—
56	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
59	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	—	2	2	no	—
66	C₃₁×#124	—	2	2	no	—
67	C₃₁×#123	—	4	2	no	—
68	C₃₁×C₂²⋊Q₈	—	2	2	no	—
69	C₃₁×C₂²⋊Q₈	—	2	2	no	—
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	—	4	2	no	—
73	C₃₁×#117	—	2	2	no	—
74	C₃₁×#117	C₂³	1	1	yes	characteristic
75	C₃₁×#117	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#108	—	2	2	no	—
80	C₃₁×#108	—	2	2	no	—
81	C₃₁×#107	—	2	2	no	—
82	C₃₁×#107	D₄	1	2	yes	—
83	C₃₁×#106	—	2	1	no	—
84	C₃₁×#106	—	2	1	no	—
85	C₃₁×#106	—	2	2	no	—
86	C₃₁×#106	D₄	1	2	yes	—
87	C₃₁×#104	—	4	2	no	—
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#103	—	2	2	no	—
92	C₃₁×#103	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#101	—	4	2	no	—
95	C₃₁×C₂²⋊C₈	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	—	2	2	no	—
97	C₃₁×#97	—	2	1	no	—
98	C₃₁×#97	—	2	1	no	—
99	C₃₁×#97	—	2	2	no	—
100	C₄×C₈×C₃₁	—	2	1	no	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₃₁×#95	—	4	2	no	—
103	#41×C₃₁	—	2	2	no	—
104	#41×C₃₁	—	2	2	no	—
105	#41×C₃₁	—	2	2	no	—
106	#41×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	2	no	—
112	C₂×Q₈×C₃₁	—	2	2	no	—
113	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
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	Q₁₆×C₃₁	—	4	4	no	—
124	Q₁₆×C₃₁	—	4	2	no	—
125	Q₁₆×C₃₁	—	4	2	no	—
126	SD₁₆×C₃₁	—	2	4	no	—
127	SD₁₆×C₃₁	—	4	4	no	—
128	D₈×C₃₁	—	2	2	no	—
129	M₁₆×C₃₁	—	2	1	no	—
130	M₁₆×C₃₁	—	2	1	no	—
131	M₁₆×C₃₁	—	2	2	no	—
132	M₁₆×C₃₁	—	2	1	no	—
133	M₁₆×C₃₁	—	2	2	no	—
134	M₁₆×C₃₁	#41	1	2	yes	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₃₁	#41	1	2	yes	—
145	C₂×C₈×C₃₁	#41	1	2	yes	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	4	2	no	—
149	C₄⋊C₄×C₃₁	—	4	2	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂²⋊C₄×C₃₁	—	2	1	no	—
154	C₂²⋊C₄×C₃₁	—	2	1	no	—
155	C₂²⋊C₄×C₃₁	—	4	2	no	—
156	C₂²⋊C₄×C₃₁	—	4	2	no	—
157	C₂²⋊C₄×C₃₁	—	4	2	no	—
158	C₄²×C₃₁	—	2	1	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	2	1	no	—
162	C₄²×C₃₁	—	4	2	no	—
163	#26273	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
166	Q₈×C₃₁	—	2	2	no	—
167	Q₈×C₃₁	—	2	2	no	—
168	Q₈×C₃₁	—	4	2	no	—
169	Q₈×C₃₁	—	4	2	no	—
170	Q₈×C₃₁	—	4	2	no	—
171	Q₈×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	2	2	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	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	—	16	2	no	—
189	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂×#41	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	H(5,𝔽₂)	1	1	yes	characteristic
195	C₈×C₃₁	—	2	1	no	—
196	C₈×C₃₁	—	2	1	no	—
197	C₈×C₃₁	—	2	2	no	—
198	C₈×C₃₁	—	4	1	no	—
199	C₈×C₃₁	—	4	2	no	—
200	C₈×C₃₁	#123	1	2	yes	—
201	C₈×C₃₁	#123	1	2	yes	—
202	#3156	C₂×C₃₁	1	1	yes	characteristic
203	#3097	C₂×C₃₁	1	2	yes	complemented
204	#2952	C₂×C₃₁	1	1	yes	characteristic
205	#2052	C₂×C₃₁	1	2	yes	complemented
206	#2038	C₂×C₃₁	1	2	yes	complemented
207	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1975	C₂×C₃₁	1	1	yes	characteristic
209	#1971	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1951	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1882	C₂×C₃₁	1	2	yes	complemented
212	#1773	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	#540	1	1	yes	characteristic
218	C₂²×C₃₁	D₄²	1	1	yes	characteristic
219	C₂²×C₃₁	#559	1	1	yes	characteristic
220	C₄×C₃₁	—	4	2	no	—
221	C₄×C₃₁	—	4	2	no	—
222	C₄×C₃₁	—	4	2	no	—
223	C₄×C₃₁	—	8	1	no	—
224	C₄×C₃₁	—	8	2	no	—
225	C₄×C₃₁	—	8	2	no	—
226	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
227	C₄×C₃₁	#532	1	1	yes	characteristic
228	C₄×C₃₁	#546	1	1	yes	characteristic
229	C₄×C₃₁	#550	1	1	yes	characteristic
230	C₂×#137	C₂²×C₃₁	1	2	yes	—
231	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
232	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
233	#536	C₂²×C₃₁	1	2	yes	—
234	#520	C₂²×C₃₁	1	2	yes	—
235	#497	—	2	2	no	—
236	#495	—	2	1	no	—
237	#494	—	2	1	no	—
238	#485	—	2	2	no	—
239	#485	—	2	2	no	—
240	#484	—	2	2	no	—
241	#483	—	2	2	no	—
242	#442	—	2	1	no	—
243	#441	—	2	1	no	—
244	C₄×SD₁₆	—	2	2	no	—
245	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
247	#428	C₂²×C₃₁	1	2	yes	—
248	#427	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#104	C₂²×C₃₁	1	2	yes	—
251	#419	C₂²×C₃₁	1	2	yes	—
252	#417	C₂²×C₃₁	1	2	yes	—
253	C₂×#103	C₂²×C₃₁	1	2	yes	—
254	C₂×#101	C₂²×C₃₁	1	2	yes	—
255	#408	C₂²×C₃₁	1	2	yes	—
256	#405	C₂²×C₃₁	1	1	yes	characteristic
257	#405	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
259	#391	C₂²×C₃₁	1	2	yes	—
260	#343	C₂²×C₃₁	1	2	yes	—
261	#339	C₂²×C₃₁	1	1	yes	characteristic
262	#339	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₃₁	—	2	1	no	—
264	C₂×C₃₁	—	2	1	no	—
265	C₂×C₃₁	—	8	2	no	—
266	C₂×C₃₁	#2445	1	1	yes	characteristic
267	C₂×C₃₁	#3289	1	2	yes	—
268	C₂×#41	C₂³×C₃₁	1	2	yes	—
269	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
270	#137	—	2	4	no	—
271	#137	—	4	2	no	—
272	#135	—	2	2	no	—
273	#135	—	2	2	no	—
274	C₂×Q₁₆	—	2	2	no	—
275	C₂×Q₁₆	—	2	1	no	—
276	C₂×Q₁₆	—	2	1	no	—
277	C₂×Q₁₆	—	2	2	no	—
278	C₂×Q₁₆	—	2	2	no	—
279	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
280	C₂×SD₁₆	—	2	2	no	—
281	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
282	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
283	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
284	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
285	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₄⋊Q₈	—	2	2	no	—
289	#124	—	2	2	no	—
290	#123	—	4	2	no	—
291	C₂²⋊Q₈	—	2	2	no	—
292	C₂²⋊Q₈	—	2	2	no	—
293	C₄×Q₈	—	2	2	no	—
294	C₄×D₄	—	2	2	no	—
295	C₄×D₄	—	4	2	no	—
296	#117	—	2	2	no	—
297	#117	C₂³×C₃₁	1	1	yes	characteristic
298	#117	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₄⋊C₄	—	2	2	no	—
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
301	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
302	#108	—	2	2	no	—
303	#108	—	2	2	no	—
304	#107	—	2	2	no	—
305	#107	D₄×C₃₁	1	2	yes	—
306	#106	—	2	1	no	—
307	#106	—	2	1	no	—
308	#106	—	2	2	no	—
309	#106	D₄×C₃₁	1	2	yes	—
310	#104	—	4	2	no	—
311	#103	—	2	2	no	—
312	#103	—	2	2	no	—
313	#103	—	2	2	no	—
314	#103	—	2	2	no	—
315	#103	—	2	2	no	—
316	#102	—	2	2	no	—
317	#101	—	4	2	no	—
318	C₂²⋊C₈	—	2	2	no	—
319	C₂²⋊C₈	—	2	2	no	—
320	#97	—	2	1	no	—
321	#97	—	2	1	no	—
322	#97	—	2	2	no	—
323	C₄×C₈	—	2	1	no	—
324	C₄×C₈	—	2	1	no	—
325	#95	—	4	2	no	—
326	C₃₁	#26273	1	1	yes	characteristic, direct product factor, Sylow
327	#41	—	2	2	no	—
328	#41	—	2	2	no	—
329	#41	—	2	2	no	—
330	#41	—	2	2	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
346	Q₁₆	—	2	2	no	—
347	Q₁₆	—	4	4	no	—
348	Q₁₆	—	4	2	no	—
349	Q₁₆	—	4	2	no	—
350	SD₁₆	—	2	4	no	—
351	SD₁₆	—	4	4	no	—
352	D₈	—	2	2	no	—
353	M₁₆	—	2	1	no	—
354	M₁₆	—	2	1	no	—
355	M₁₆	—	2	2	no	—
356	M₁₆	—	2	1	no	—
357	M₁₆	—	2	2	no	—
358	M₁₆	#41×C₃₁	1	2	yes	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
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	—	4	2	no	—
373	C₄⋊C₄	—	4	2	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
376	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
377	C₂²⋊C₄	—	2	1	no	—
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	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	4	2	no	—
387	C₂³	—	2	2	no	—
388	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
389	Q₈	—	2	2	no	—
390	Q₈	—	2	2	no	—
391	Q₈	—	4	2	no	—
392	Q₈	—	4	2	no	—
393	Q₈	—	4	2	no	—
394	Q₈	—	4	2	no	—
395	D₄	—	2	2	no	—
396	D₄	—	2	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	8	2	no	—
411	C₂×C₄	—	16	2	no	—
412	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
418	C₈	—	2	1	no	—
419	C₈	—	2	1	no	—
420	C₈	—	2	2	no	—
421	C₈	—	4	1	no	—
422	C₈	—	4	2	no	—
423	C₈	C₃₁×#123	1	2	yes	—
424	C₈	C₃₁×#123	1	2	yes	—
425	C₂²	—	2	2	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	C₃₁×#540	1	1	yes	characteristic
430	C₂²	D₄²×C₃₁	1	1	yes	characteristic
431	C₂²	C₃₁×#559	1	1	yes	characteristic
432	C₄	—	4	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	2	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	2	no	—
437	C₄	—	8	2	no	—
438	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
439	C₄	C₃₁×#532	1	1	yes	characteristic
440	C₄	C₃₁×#546	1	1	yes	characteristic
441	C₄	C₃₁×#550	1	1	yes	characteristic
442	C₂	—	2	1	no	—
443	C₂	—	2	1	no	—
444	C₂	—	8	2	no	—
445	C₂	C₃₁×#2445	1	1	yes	characteristic
446	C₂	C₃₁×#3289	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#135	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
2. C₃₁×#3097	C₂	semidirect
4. C₃₁×#2052	C₂	semidirect
5. C₃₁×#2038	C₂	semidirect
6. C₃₁×#1979	C₂	semidirect
8. C₃₁×#1971	C₂	semidirect
9. C₃₁×#1951	C₂	semidirect
10. C₃₁×#1882	C₂	semidirect
11. C₃₁×#1773	C₂	semidirect
163. #26273	C₃₁	direct (N × H)
203. #3097	C₂×C₃₁	semidirect
205. #2052	C₂×C₃₁	semidirect
206. #2038	C₂×C₃₁	semidirect
207. #1979	C₂×C₃₁	semidirect
209. #1971	C₂×C₃₁	semidirect
210. #1951	C₂×C₃₁	semidirect
211. #1882	C₂×C₃₁	semidirect
212. #1773	C₂×C₃₁	semidirect
326. C₃₁	#26273	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×#3156	2
C₃₁×#3097	2
C₃₁×#2952	2
C₃₁×#2052	2
C₃₁×#2038	2
C₃₁×#1979	2
C₃₁×#1975	2
C₃₁×#1971	2
C₃₁×#1951	2
C₃₁×#1882	2
C₃₁×#1773	2
#26273	31

Subgroups — C31×#26273 #21483032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26273 #21483032