Subgroups — C₃₁×#19198 #21475957

Order 7936.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1701	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1681	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#131	—	2	1	no	—
11	C₃₁×#532	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₃₁×#493	—	2	1	no	—
15	C₃₁×#492	—	2	1	no	—
16	C₃₁×#487	—	2	1	no	—
17	C₃₁×#486	—	2	1	no	—
18	C₃₁×#486	—	2	1	no	—
19	C₃₁×#486	—	2	1	no	—
20	C₃₁×#486	C₂²	1	2	yes	—
21	C₃₁×#486	C₂²	1	2	yes	—
22	C₃₁×#461	C₂²	1	4	yes	complemented
23	C₃₁×#459	C₂²	1	4	yes	—
24	C₃₁×#448	—	2	2	no	—
25	C₃₁×#447	—	2	2	no	—
26	C₃₁×#436	—	2	1	no	—
27	C₃₁×#436	C₂²	1	2	yes	complemented
28	C₈×D₄×C₃₁	—	2	1	no	—
29	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#423	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#102	—	2	1	no	—
34	C₂×C₃₁×#102	—	2	1	no	—
35	C₃₁×#408	—	2	1	no	—
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	2	yes	complemented
38	C₃₁×#334	C₂²	1	2	yes	complemented
39	C₃₁×#332	C₂²	1	4	yes	complemented
40	C₃₁×#331	C₂²	1	4	yes	—
41	C₂×#41×C₃₁	—	2	1	no	—
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂×SD₁₆×C₃₁	—	2	4	no	—
47	C₂×D₈×C₃₁	—	2	4	no	—
48	C₃₁×#131	—	4	4	no	—
49	C₂×M₁₆×C₃₁	—	2	1	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	—	2	2	no	—
52	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₃₁×C₄⋊Q₈	—	2	1	no	—
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₃₁×#127	—	2	1	no	—
60	C₃₁×#127	—	2	1	no	—
61	C₃₁×#127	—	2	1	no	—
62	C₃₁×#127	D₄	1	1	yes	characteristic
63	C₃₁×#127	C₂³	1	2	yes	—
64	C₃₁×#124	—	2	1	no	—
65	C₃₁×#121	—	2	1	no	—
66	C₃₁×#121	D₄	1	2	yes	—
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	—	2	1	no	—
73	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
74	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
75	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
76	C₃₁×#103	—	2	4	no	—
77	C₃₁×#102	—	2	2	no	—
78	C₃₁×#102	—	2	2	no	—
79	C₃₁×#102	—	2	2	no	—
80	C₃₁×#102	—	2	2	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×C₂²⋊C₈	—	2	1	no	—
83	C₃₁×C₂²⋊C₈	—	2	1	no	—
84	C₃₁×C₂²⋊C₈	—	2	2	no	—
85	C₃₁×C₂²⋊C₈	—	2	2	no	—
86	C₄×C₈×C₃₁	—	2	1	no	—
87	C₄×C₈×C₃₁	—	2	1	no	—
88	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
89	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
90	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₂⁴×C₃₁	—	4	1	no	—
93	#41×C₃₁	—	4	2	no	—
94	#41×C₃₁	—	4	2	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	4	1	no	—
98	C₂×Q₈×C₃₁	—	8	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	2	1	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	—	4	1	no	—
106	C₂×D₄×C₃₁	—	4	1	no	—
107	C₂×D₄×C₃₁	—	4	1	no	—
108	C₂×D₄×C₃₁	—	4	2	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	8	2	no	—
111	C₂×D₄×C₃₁	—	8	2	no	—
112	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	4	1	no	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
118	SD₁₆×C₃₁	—	8	4	no	—
119	D₈×C₃₁	—	8	4	no	—
120	M₁₆×C₃₁	—	4	4	no	—
121	M₁₆×C₃₁	—	4	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	4	4	no	—
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×D₄	1	1	yes	characteristic
131	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	4	1	no	—
136	C₄⋊C₄×C₃₁	—	4	1	no	—
137	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₃₁	#41	1	1	yes	characteristic
143	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
144	#19198	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	8	2	no	—
148	C₂³×C₃₁	—	8	2	no	—
149	C₂³×C₃₁	—	8	2	no	—
150	C₂³×C₃₁	#121	1	1	yes	characteristic
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	—	4	4	no	—
153	Q₈×C₃₁	—	8	2	no	—
154	D₄×C₃₁	—	2	4	no	—
155	D₄×C₃₁	—	4	4	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	4	4	no	—
159	D₄×C₃₁	—	4	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	8	2	no	—
162	D₄×C₃₁	—	8	2	no	—
163	D₄×C₃₁	—	8	2	no	—
164	D₄×C₃₁	—	8	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	8	2	no	—
172	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
173	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
174	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
175	C₂×C₄×C₃₁	#135	1	2	yes	—
176	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
180	C₈×C₃₁	—	2	1	no	—
181	C₈×C₃₁	—	2	1	no	—
182	C₈×C₃₁	—	2	1	no	—
183	C₈×C₃₁	—	2	1	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	—	4	2	no	—
186	C₈×C₃₁	—	4	2	no	—
187	#3206	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#3205	C₂×C₃₁	1	1	yes	characteristic
189	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1701	C₂×C₃₁	1	4	yes	complemented
192	#1681	C₂×C₃₁	1	4	yes	complemented
193	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
196	C₂²×C₃₁	—	2	1	no	—
197	C₂²×C₃₁	—	2	1	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	4	2	no	—
200	C₂²×C₃₁	—	4	2	no	—
201	C₂²×C₃₁	—	8	2	no	—
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	—	8	2	no	—
204	C₂²×C₃₁	—	16	4	no	—
205	C₂²×C₃₁	#463	1	2	yes	—
206	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	2	1	no	—
209	C₄×C₃₁	—	4	2	no	—
210	C₄×C₃₁	—	8	2	no	—
211	C₄×C₃₁	#463	1	2	yes	—
212	C₄×C₃₁	#471	1	2	yes	—
213	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
214	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₄×C₃₁	#576	1	2	yes	—
216	C₂×#131	—	2	1	no	—
217	#532	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
220	#493	—	2	1	no	—
221	#492	—	2	1	no	—
222	#487	—	2	1	no	—
223	#486	—	2	1	no	—
224	#486	—	2	1	no	—
225	#486	—	2	1	no	—
226	#486	C₂²×C₃₁	1	2	yes	—
227	#486	C₂²×C₃₁	1	2	yes	—
228	#461	C₂²×C₃₁	1	4	yes	complemented
229	#459	C₂²×C₃₁	1	4	yes	—
230	#448	—	2	2	no	—
231	#447	—	2	2	no	—
232	#436	—	2	1	no	—
233	#436	C₂²×C₃₁	1	2	yes	complemented
234	C₈×D₄	—	2	1	no	—
235	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
236	#431	C₂²×C₃₁	1	1	yes	characteristic
237	#431	C₂²×C₃₁	1	1	yes	characteristic
238	#423	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#102	—	2	1	no	—
240	C₂×#102	—	2	1	no	—
241	#408	—	2	1	no	—
242	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
243	#335	C₂²×C₃₁	1	2	yes	complemented
244	#334	C₂²×C₃₁	1	2	yes	complemented
245	#332	C₂²×C₃₁	1	4	yes	complemented
246	#331	C₂²×C₃₁	1	4	yes	—
247	C₂×C₃₁	—	2	2	no	—
248	C₂×C₃₁	—	8	2	no	—
249	C₂×C₃₁	—	16	2	no	—
250	C₂×C₃₁	C₂×#463	1	1	yes	characteristic
251	C₂×C₃₁	#3058	1	1	yes	characteristic
252	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
253	C₂×#41	—	2	1	no	—
254	C₂²×Q₈	—	2	1	no	—
255	C₂²×D₄	—	2	1	no	—
256	C₂²×D₄	—	2	1	no	—
257	C₂²×D₄	—	2	1	no	—
258	C₂×SD₁₆	—	2	4	no	—
259	C₂×D₈	—	2	4	no	—
260	#131	—	4	4	no	—
261	C₂×M₁₆	—	2	1	no	—
262	C₂×M₁₆	—	2	2	no	—
263	C₂²×C₈	—	2	2	no	—
264	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
265	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
266	C₄⋊Q₈	—	2	1	no	—
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	—	2	1	no	—
269	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
270	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
271	#127	—	2	1	no	—
272	#127	—	2	1	no	—
273	#127	—	2	1	no	—
274	#127	D₄×C₃₁	1	1	yes	characteristic
275	#127	C₂³×C₃₁	1	2	yes	—
276	#124	—	2	1	no	—
277	#121	—	2	1	no	—
278	#121	D₄×C₃₁	1	2	yes	—
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	C₂³×C₃₁	1	2	yes	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
286	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
287	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
288	#103	—	2	4	no	—
289	#102	—	2	2	no	—
290	#102	—	2	2	no	—
291	#102	—	2	2	no	—
292	#102	—	2	2	no	—
293	#102	—	2	4	no	—
294	C₂²⋊C₈	—	2	1	no	—
295	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₄×C₈	—	2	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
301	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
302	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₃₁	#19198	1	1	yes	characteristic, direct product factor, Sylow
305	C₂⁴	—	4	1	no	—
306	#41	—	4	2	no	—
307	#41	—	4	2	no	—
308	C₂×Q₈	—	2	1	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	4	1	no	—
311	C₂×Q₈	—	8	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	—	2	1	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂×D₄	—	4	1	no	—
321	C₂×D₄	—	4	2	no	—
322	C₂×D₄	—	4	2	no	—
323	C₂×D₄	—	8	2	no	—
324	C₂×D₄	—	8	2	no	—
325	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	4	1	no	—
328	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
331	SD₁₆	—	8	4	no	—
332	D₈	—	8	4	no	—
333	M₁₆	—	4	4	no	—
334	M₁₆	—	4	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	4	4	no	—
341	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	1	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₄²	#41×C₃₁	1	1	yes	characteristic
356	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
357	C₂³	—	2	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	8	2	no	—
360	C₂³	—	8	2	no	—
361	C₂³	—	8	2	no	—
362	C₂³	C₃₁×#121	1	1	yes	characteristic
363	Q₈	—	4	2	no	—
364	Q₈	—	4	4	no	—
365	Q₈	—	8	2	no	—
366	D₄	—	2	4	no	—
367	D₄	—	4	4	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	4	no	—
373	D₄	—	8	2	no	—
374	D₄	—	8	2	no	—
375	D₄	—	8	2	no	—
376	D₄	—	8	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
385	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
386	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
387	C₂×C₄	C₃₁×#135	1	2	yes	—
388	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
389	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
392	C₈	—	2	1	no	—
393	C₈	—	2	1	no	—
394	C₈	—	2	1	no	—
395	C₈	—	2	1	no	—
396	C₈	—	4	2	no	—
397	C₈	—	4	2	no	—
398	C₈	—	4	2	no	—
399	C₂²	—	2	1	no	—
400	C₂²	—	2	1	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	8	2	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	16	4	no	—
408	C₂²	C₃₁×#463	1	2	yes	—
409	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
410	C₄	—	2	1	no	—
411	C₄	—	2	1	no	—
412	C₄	—	4	2	no	—
413	C₄	—	8	2	no	—
414	C₄	C₃₁×#463	1	2	yes	—
415	C₄	C₃₁×#471	1	2	yes	—
416	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
417	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
418	C₄	C₃₁×#576	1	2	yes	—
419	C₂	—	2	2	no	—
420	C₂	—	8	2	no	—
421	C₂	—	16	2	no	—
422	C₂	C₂×C₃₁×#463	1	1	yes	characteristic
423	C₂	C₃₁×#3058	1	1	yes	characteristic
424	C₂	C₂×C₃₁×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3205	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3206	C₂	semidirect
3. C₂×C₃₁×#486	C₂	semidirect
4. C₃₁×#2931	C₂	semidirect
5. C₃₁×#1701	C₂	semidirect
6. C₃₁×#1681	C₂	semidirect
7. C₃₁×#1564	C₂	semidirect
8. C₃₁×#1549	C₂	semidirect
9. C₃₁×#1545	C₂	semidirect
22. C₃₁×#461	C₂²	semidirect
27. C₃₁×#436	C₂²	semidirect
29. C₈×D₄×C₃₁	C₂²	semidirect
37. C₃₁×#335	C₂²	semidirect
38. C₃₁×#334	C₂²	semidirect
39. C₃₁×#332	C₂²	semidirect
144. #19198	C₃₁	direct (N × H)
187. #3206	C₂×C₃₁	semidirect
189. C₂×#486	C₂×C₃₁	semidirect
190. #2931	C₂×C₃₁	semidirect
191. #1701	C₂×C₃₁	semidirect
192. #1681	C₂×C₃₁	semidirect
193. #1564	C₂×C₃₁	semidirect
194. #1549	C₂×C₃₁	semidirect
195. #1545	C₂×C₃₁	semidirect
228. #461	C₂²×C₃₁	semidirect
233. #436	C₂²×C₃₁	semidirect
235. C₈×D₄	C₂²×C₃₁	semidirect
243. #335	C₂²×C₃₁	semidirect
244. #334	C₂²×C₃₁	semidirect
245. #332	C₂²×C₃₁	semidirect
304. C₃₁	#19198	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×#3206	2
C₃₁×#3205	2
C₂×C₃₁×#486	2
C₃₁×#2931	2
C₃₁×#1701	2
C₃₁×#1681	2
C₃₁×#1564	2
C₃₁×#1549	2
C₃₁×#1545	2
#19198	31

Subgroups — C31×#19198 #21475957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19198 #21475957