Subgroups — C₃₁×#33785 #21490544

Order 7936.

594 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3168	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2251	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2250	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2245	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2244	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2211	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2210	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2189	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2186	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₂×C₃₁×#135	C₂²	1	2	yes	—
14	C₃₁×#532	—	2	1	no	—
15	C₃₁×#529	—	2	1	no	—
16	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
17	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
19	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
20	C₃₁×#499	C₂²	1	1	yes	characteristic
21	C₃₁×#495	—	2	1	no	—
22	C₃₁×#495	C₂²	1	2	yes	—
23	C₃₁×#495	C₂²	1	1	yes	characteristic
24	C₃₁×#494	C₂²	1	1	yes	characteristic
25	C₃₁×#493	C₂²	1	1	yes	characteristic
26	C₃₁×#492	—	2	1	no	—
27	C₃₁×#488	—	2	1	no	—
28	C₃₁×#467	C₂²	1	1	yes	characteristic
29	C₃₁×#466	C₂²	1	1	yes	characteristic
30	C₃₁×#465	—	2	1	no	—
31	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×SD₁₆×C₃₁	—	2	1	no	—
33	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
36	C₃₁×#358	C₂²	1	2	yes	—
37	C₃₁×#358	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×#357	C₂²	1	2	yes	—
40	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	C₃₁×#346	C₂²	1	1	yes	characteristic
42	C₃₁×#346	C₂²	1	1	yes	characteristic
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₃₁×#135	—	4	2	no	—
45	C₃₁×#135	—	4	2	no	—
46	C₃₁×#135	—	4	4	no	—
47	C₂×Q₁₆×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×SD₁₆×C₃₁	—	2	1	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×D₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×#127	—	2	1	no	—
60	C₃₁×#126	—	4	1	no	—
61	C₃₁×#125	—	2	1	no	—
62	C₃₁×#124	—	2	1	no	—
63	C₃₁×#124	—	2	2	no	—
64	C₃₁×#123	—	4	1	no	—
65	C₃₁×C₂²⋊Q₈	—	2	1	no	—
66	C₃₁×#121	—	2	1	no	—
67	C₃₁×#121	—	4	1	no	—
68	C₃₁×#121	—	4	1	no	—
69	C₄×Q₈×C₃₁	—	2	1	no	—
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	—	4	1	no	—
72	C₄×D₄×C₃₁	—	4	1	no	—
73	C₄×D₄×C₃₁	—	4	1	no	—
74	C₃₁×#117	—	4	1	no	—
75	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₃₁×Q₃₂	—	4	2	no	—
77	C₃₁×SD₃₂	—	4	2	no	—
78	C₃₁×SD₃₂	—	4	2	no	—
79	C₃₁×D₁₆	—	4	2	no	—
80	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
81	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
82	C₃₁×#107	C₂³	1	1	yes	characteristic
83	C₃₁×#107	C₂³	1	1	yes	characteristic
84	C₃₁×#106	—	2	1	no	—
85	C₃₁×#103	—	2	1	no	—
86	C₃₁×#103	—	2	1	no	—
87	C₃₁×#102	—	2	1	no	—
88	C₃₁×#102	—	2	1	no	—
89	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
90	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
91	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	#41×C₃₁	—	4	4	no	—
94	#41×C₃₁	—	4	4	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	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	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	—	4	2	no	—
105	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₃₁	—	2	2	no	—
109	Q₁₆×C₃₁	—	2	2	no	—
110	Q₁₆×C₃₁	—	2	2	no	—
111	Q₁₆×C₃₁	—	4	2	no	—
112	SD₁₆×C₃₁	—	4	4	no	—
113	SD₁₆×C₃₁	—	4	4	no	—
114	SD₁₆×C₃₁	—	4	2	no	—
115	D₈×C₃₁	—	2	2	no	—
116	D₈×C₃₁	—	2	2	no	—
117	D₈×C₃₁	—	2	2	no	—
118	D₈×C₃₁	—	4	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	4	1	no	—
128	C₄⋊C₄×C₃₁	—	4	1	no	—
129	C₄⋊C₄×C₃₁	—	4	1	no	—
130	C₄⋊C₄×C₃₁	—	4	1	no	—
131	C₄⋊C₄×C₃₁	—	4	1	no	—
132	C₄⋊C₄×C₃₁	—	4	1	no	—
133	C₂²⋊C₄×C₃₁	—	4	1	no	—
134	C₂²⋊C₄×C₃₁	—	4	1	no	—
135	C₂²⋊C₄×C₃₁	—	4	1	no	—
136	C₂²⋊C₄×C₃₁	—	4	1	no	—
137	C₂²⋊C₄×C₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	1	no	—
140	C₄²×C₃₁	—	4	1	no	—
141	C₄²×C₃₁	D₈	1	1	yes	characteristic
142	C₄²×C₃₁	D₈	1	1	yes	characteristic
143	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₁₆×C₃₁	—	2	2	no	—
146	C₁₆×C₃₁	—	2	2	no	—
147	#33785	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	—	4	1	no	—
153	Q₈×C₃₁	—	4	2	no	—
154	Q₈×C₃₁	—	4	1	no	—
155	D₄×C₃₁	—	4	2	no	—
156	D₄×C₃₁	—	4	1	no	—
157	D₄×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	4	1	no	—
159	D₄×C₃₁	—	8	2	no	—
160	D₄×C₃₁	—	8	2	no	—
161	D₄×C₃₁	—	8	2	no	—
162	D₄×C₃₁	—	8	4	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	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	—	4	2	no	—
171	C₂×C₄×C₃₁	—	8	2	no	—
172	C₂×C₄×C₃₁	—	8	4	no	—
173	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₈×C₃₁	—	2	1	no	—
180	C₈×C₃₁	—	2	1	no	—
181	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
182	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
183	#3168	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#2251	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#2250	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#2245	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#2244	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#2211	C₂×C₃₁	1	2	yes	complemented
190	#2210	C₂×C₃₁	1	2	yes	complemented
191	#2189	C₂×C₃₁	1	2	yes	complemented
192	#2186	C₂×C₃₁	1	2	yes	complemented
193	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	—	2	1	no	—
196	C₂²×C₃₁	—	2	1	no	—
197	C₂²×C₃₁	—	8	1	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	1	no	—
201	C₂²×C₃₁	—	8	2	no	—
202	C₂²×C₃₁	—	8	1	no	—
203	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	2	1	no	—
206	C₄×C₃₁	—	2	1	no	—
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	2	1	no	—
209	C₄×C₃₁	—	8	1	no	—
210	C₄×C₃₁	—	8	1	no	—
211	C₄×C₃₁	—	8	2	no	—
212	C₄×C₃₁	—	8	1	no	—
213	C₄×C₃₁	#535	1	1	yes	characteristic
214	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₂×#135	—	2	1	no	—
216	C₂×#135	C₂²×C₃₁	1	2	yes	—
217	#532	—	2	1	no	—
218	#529	—	2	1	no	—
219	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
220	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
221	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
222	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
223	#499	C₂²×C₃₁	1	1	yes	characteristic
224	#495	—	2	1	no	—
225	#495	C₂²×C₃₁	1	2	yes	—
226	#495	C₂²×C₃₁	1	1	yes	characteristic
227	#494	C₂²×C₃₁	1	1	yes	characteristic
228	#493	C₂²×C₃₁	1	1	yes	characteristic
229	#492	—	2	1	no	—
230	#488	—	2	1	no	—
231	#467	C₂²×C₃₁	1	1	yes	characteristic
232	#466	C₂²×C₃₁	1	1	yes	characteristic
233	#465	—	2	1	no	—
234	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
235	C₄×SD₁₆	—	2	1	no	—
236	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
239	#358	C₂²×C₃₁	1	2	yes	—
240	#358	C₂²×C₃₁	1	2	yes	—
241	#357	C₂²×C₃₁	1	2	yes	—
242	#357	C₂²×C₃₁	1	2	yes	—
243	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
244	#346	C₂²×C₃₁	1	1	yes	characteristic
245	#346	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×C₃₁	—	4	1	no	—
247	C₂×C₃₁	—	16	1	no	—
248	C₂×C₃₁	—	16	2	no	—
249	C₂×C₃₁	#3205	1	1	yes	characteristic
250	C₂×C₃₁	#3414	1	1	yes	characteristic
251	C₂×C₃₁	#3414	1	1	yes	characteristic
252	C₂×#41	—	2	2	no	—
253	#135	—	4	2	no	—
254	#135	—	4	2	no	—
255	#135	—	4	4	no	—
256	C₂×Q₁₆	—	2	1	no	—
257	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
258	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
259	C₂×SD₁₆	—	2	2	no	—
260	C₂×SD₁₆	—	2	2	no	—
261	C₂×SD₁₆	—	2	1	no	—
262	C₂×D₈	—	2	1	no	—
263	C₂×D₈	C₂³×C₃₁	1	2	yes	—
264	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
267	C₄⋊Q₈	—	2	1	no	—
268	#127	—	2	1	no	—
269	#126	—	4	1	no	—
270	#125	—	2	1	no	—
271	#124	—	2	1	no	—
272	#124	—	2	2	no	—
273	#123	—	4	1	no	—
274	C₂²⋊Q₈	—	2	1	no	—
275	#121	—	2	1	no	—
276	#121	—	4	1	no	—
277	#121	—	4	1	no	—
278	C₄×Q₈	—	2	1	no	—
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	—	4	1	no	—
281	C₄×D₄	—	4	1	no	—
282	C₄×D₄	—	4	1	no	—
283	#117	—	4	1	no	—
284	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
285	Q₃₂	—	4	2	no	—
286	SD₃₂	—	4	2	no	—
287	SD₃₂	—	4	2	no	—
288	D₁₆	—	4	2	no	—
289	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
290	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
291	#107	C₂³×C₃₁	1	1	yes	characteristic
292	#107	C₂³×C₃₁	1	1	yes	characteristic
293	#106	—	2	1	no	—
294	#103	—	2	1	no	—
295	#103	—	2	1	no	—
296	#102	—	2	1	no	—
297	#102	—	2	1	no	—
298	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
299	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
300	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
301	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₃₁	#33785	1	1	yes	characteristic, direct product factor, Sylow
303	#41	—	4	4	no	—
304	#41	—	4	4	no	—
305	C₂×Q₈	—	2	1	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×D₄	—	2	1	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	4	1	no	—
310	C₂×D₄	—	4	2	no	—
311	C₂×D₄	—	4	1	no	—
312	C₂×D₄	—	4	1	no	—
313	C₂²×C₄	—	4	1	no	—
314	C₂²×C₄	—	4	2	no	—
315	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
318	Q₁₆	—	2	2	no	—
319	Q₁₆	—	2	2	no	—
320	Q₁₆	—	2	2	no	—
321	Q₁₆	—	4	2	no	—
322	SD₁₆	—	4	4	no	—
323	SD₁₆	—	4	4	no	—
324	SD₁₆	—	4	2	no	—
325	D₈	—	2	2	no	—
326	D₈	—	2	2	no	—
327	D₈	—	2	2	no	—
328	D₈	—	4	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	4	1	no	—
338	C₄⋊C₄	—	4	1	no	—
339	C₄⋊C₄	—	4	1	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₂²⋊C₄	—	4	1	no	—
344	C₂²⋊C₄	—	4	1	no	—
345	C₂²⋊C₄	—	4	1	no	—
346	C₂²⋊C₄	—	4	1	no	—
347	C₂²⋊C₄	—	4	2	no	—
348	C₂²⋊C₄	—	4	2	no	—
349	C₂²⋊C₄	—	4	1	no	—
350	C₄²	—	4	1	no	—
351	C₄²	D₈×C₃₁	1	1	yes	characteristic
352	C₄²	D₈×C₃₁	1	1	yes	characteristic
353	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₁₆	—	2	2	no	—
356	C₁₆	—	2	2	no	—
357	C₂³	—	4	1	no	—
358	C₂³	—	4	2	no	—
359	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
360	Q₈	—	4	2	no	—
361	Q₈	—	4	1	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	1	no	—
364	D₄	—	4	2	no	—
365	D₄	—	4	1	no	—
366	D₄	—	4	2	no	—
367	D₄	—	4	1	no	—
368	D₄	—	8	2	no	—
369	D₄	—	8	2	no	—
370	D₄	—	8	2	no	—
371	D₄	—	8	4	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	8	2	no	—
381	C₂×C₄	—	8	4	no	—
382	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
383	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
384	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
385	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	2	1	no	—
390	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
391	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	8	1	no	—
396	C₂²	—	8	2	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	1	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	1	no	—
401	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	2	1	no	—
407	C₄	—	8	1	no	—
408	C₄	—	8	1	no	—
409	C₄	—	8	2	no	—
410	C₄	—	8	1	no	—
411	C₄	C₃₁×#535	1	1	yes	characteristic
412	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
413	C₂	—	4	1	no	—
414	C₂	—	16	1	no	—
415	C₂	—	16	2	no	—
416	C₂	C₃₁×#3205	1	1	yes	characteristic
417	C₂	C₃₁×#3414	1	1	yes	characteristic
418	C₂	C₃₁×#3414	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 ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3168	C₂	semidirect
2. C₃₁×#3147	C₂	semidirect
3. C₃₁×#2251	C₂	semidirect
4. C₃₁×#2250	C₂	semidirect
5. C₃₁×#2245	C₂	semidirect
6. C₃₁×#2244	C₂	semidirect
7. C₃₁×#2211	C₂	semidirect
8. C₃₁×#2210	C₂	semidirect
9. C₃₁×#2189	C₂	semidirect
10. C₃₁×#2186	C₂	semidirect
11. C₃₁×#2163	C₂	semidirect
147. #33785	C₃₁	direct (N × H)
183. #3168	C₂×C₃₁	semidirect
184. #3147	C₂×C₃₁	semidirect
185. #2251	C₂×C₃₁	semidirect
186. #2250	C₂×C₃₁	semidirect
187. #2245	C₂×C₃₁	semidirect
188. #2244	C₂×C₃₁	semidirect
189. #2211	C₂×C₃₁	semidirect
190. #2210	C₂×C₃₁	semidirect
191. #2189	C₂×C₃₁	semidirect
192. #2186	C₂×C₃₁	semidirect
193. #2163	C₂×C₃₁	semidirect
302. C₃₁	#33785	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×#3168	2
C₃₁×#3147	2
C₃₁×#2251	2
C₃₁×#2250	2
C₃₁×#2245	2
C₃₁×#2244	2
C₃₁×#2211	2
C₃₁×#2210	2
C₃₁×#2189	2
C₃₁×#2186	2
C₃₁×#2163	2
#33785	31

Subgroups — C31×#33785 #21490544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33785 #21490544