Subgroups — C₃₁×#21139 #21477898

Order 7936.

780 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2943	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2923	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1929	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1922	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1876	C₂	1	2	yes	maximal
7	C₃₁×#1866	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1778	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₃₁×#104	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×#104	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1745	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#442	C₄	1	2	yes	complemented
17	C₃₁×#441	C₄	1	2	yes	complemented
18	C₃₁×#440	C₄	1	2	yes	complemented
19	C₃₁×#440	C₄	1	2	yes	complemented
20	C₃₁×#433	—	2	1	no	—
21	C₃₁×#433	—	2	1	no	—
22	C₃₁×#432	—	2	2	no	—
23	C₂×C₃₁×#108	C₂²	1	2	yes	—
24	C₃₁×#427	C₂²	1	1	yes	characteristic
25	C₃₁×#424	C₂²	1	2	yes	—
26	C₂×C₃₁×#104	C₂²	1	2	yes	—
27	C₂×C₃₁×#104	C₂²	1	2	yes	—
28	C₃₁×#416	C₂²	1	1	yes	characteristic
29	C₃₁×#416	C₂²	1	1	yes	characteristic
30	C₃₁×#408	C₂²	1	2	yes	—
31	C₃₁×#405	—	2	2	no	—
32	C₃₁×#405	C₂²	1	2	yes	—
33	C₄×M₁₆×C₃₁	—	2	1	no	—
34	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#97	—	2	1	no	—
37	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
39	C₄³×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#341	C₂²	1	2	yes	—
41	C₃₁×#339	C₂²	1	2	yes	—
42	C₃₁×#339	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#135	—	2	4	no	—
48	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×SD₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₃₁×#131	—	2	4	no	—
53	C₃₁×#131	—	2	4	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
56	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
61	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
62	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
66	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
67	C₃₁×#117	—	2	1	no	—
68	C₃₁×#117	—	2	1	no	—
69	C₃₁×#117	C₂³	1	1	yes	characteristic
70	C₃₁×#117	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#108	—	2	4	no	—
77	C₃₁×#107	C₂×C₄	1	2	yes	—
78	C₃₁×#106	C₂×C₄	1	2	yes	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×#104	—	2	4	no	—
82	C₃₁×#104	—	2	4	no	—
83	C₃₁×#103	C₂×C₄	1	2	yes	—
84	C₃₁×#103	C₂×C₄	1	2	yes	—
85	C₃₁×#102	C₂×C₄	1	2	yes	—
86	C₃₁×#102	C₂×C₄	1	2	yes	—
87	C₃₁×C₂²⋊C₈	—	2	2	no	—
88	C₃₁×C₂²⋊C₈	—	2	2	no	—
89	C₃₁×#97	—	2	2	no	—
90	C₃₁×#97	—	2	1	no	—
91	C₃₁×#97	—	2	1	no	—
92	C₃₁×#97	C₂×C₄	1	2	yes	—
93	C₃₁×#97	C₂×C₄	1	2	yes	—
94	C₃₁×#97	D₄	1	1	yes	characteristic
95	C₃₁×#97	D₄	1	1	yes	characteristic
96	C₃₁×#97	D₄	1	2	yes	—
97	C₄×C₈×C₃₁	—	2	2	no	—
98	C₄×C₈×C₃₁	—	2	1	no	—
99	C₄×C₈×C₃₁	—	2	1	no	—
100	#41×C₃₁	—	2	2	no	—
101	#41×C₃₁	—	2	2	no	—
102	#41×C₃₁	—	2	2	no	—
103	#41×C₃₁	#41	1	4	yes	—
104	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×D₄×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	#41	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	Q₁₆×C₃₁	—	2	2	no	—
118	SD₁₆×C₃₁	—	2	2	no	—
119	SD₁₆×C₃₁	—	2	2	no	—
120	D₈×C₃₁	—	2	2	no	—
121	M₁₆×C₃₁	—	2	8	no	—
122	M₁₆×C₃₁	—	2	2	no	—
123	M₁₆×C₃₁	—	2	2	no	—
124	M₁₆×C₃₁	—	2	2	no	—
125	M₁₆×C₃₁	#41	1	4	yes	—
126	C₂×C₈×C₃₁	—	2	8	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
141	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	2	1	no	—
144	C₂²⋊C₄×C₃₁	—	2	2	no	—
145	C₂²⋊C₄×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	2	2	no	—
152	C₄²×C₃₁	—	4	4	no	—
153	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
156	#21139	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	2	1	no	—
158	C₂³×C₃₁	—	2	1	no	—
159	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
160	Q₈×C₃₁	—	2	1	no	—
161	Q₈×C₃₁	—	2	1	no	—
162	Q₈×C₃₁	—	2	1	no	—
163	Q₈×C₃₁	#117	1	2	yes	—
164	D₄×C₃₁	—	2	4	no	—
165	D₄×C₃₁	—	2	1	no	—
166	D₄×C₃₁	—	2	1	no	—
167	D₄×C₃₁	—	2	1	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	#117	1	2	yes	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	4	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	8	2	no	—
181	C₂×C₄×C₃₁	—	8	2	no	—
182	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
187	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
188	C₈×C₃₁	—	2	1	no	—
189	C₈×C₃₁	—	2	1	no	—
190	C₈×C₃₁	—	2	2	no	—
191	C₈×C₃₁	—	2	2	no	—
192	C₈×C₃₁	—	2	4	no	—
193	C₈×C₃₁	—	2	2	no	—
194	C₈×C₃₁	—	4	2	no	—
195	#2943	C₂×C₃₁	1	1	yes	characteristic
196	#2923	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#2875	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1929	C₂×C₃₁	1	2	yes	complemented
199	#1922	C₂×C₃₁	1	2	yes	complemented
200	#1876	C₂×C₃₁	1	2	yes	—
201	#1866	C₂×C₃₁	1	2	yes	complemented
202	#1778	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₄×#104	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₄×#104	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1745	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₂²×C₃₁	—	2	2	no	—
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	4	2	no	—
209	C₂²×C₃₁	—	4	2	no	—
210	C₂²×C₃₁	—	4	1	no	—
211	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂²×C₃₁	#529	1	2	yes	—
213	C₄×C₃₁	—	2	2	no	—
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	2	2	no	—
216	C₄×C₃₁	—	4	4	no	—
217	C₄×C₃₁	—	4	2	no	—
218	C₄×C₃₁	—	4	2	no	—
219	C₄×C₃₁	—	4	1	no	—
220	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
221	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
222	C₄×C₃₁	#525	1	2	yes	—
223	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#131	C₂²×C₃₁	1	2	yes	—
225	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
226	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
227	#442	C₄×C₃₁	1	2	yes	complemented
228	#441	C₄×C₃₁	1	2	yes	complemented
229	#440	C₄×C₃₁	1	2	yes	complemented
230	#440	C₄×C₃₁	1	2	yes	complemented
231	#433	—	2	1	no	—
232	#433	—	2	1	no	—
233	#432	—	2	2	no	—
234	C₂×#108	C₂²×C₃₁	1	2	yes	—
235	#427	C₂²×C₃₁	1	1	yes	characteristic
236	#424	C₂²×C₃₁	1	2	yes	—
237	C₂×#104	C₂²×C₃₁	1	2	yes	—
238	C₂×#104	C₂²×C₃₁	1	2	yes	—
239	#416	C₂²×C₃₁	1	1	yes	characteristic
240	#416	C₂²×C₃₁	1	1	yes	characteristic
241	#408	C₂²×C₃₁	1	2	yes	—
242	#405	—	2	2	no	—
243	#405	C₂²×C₃₁	1	2	yes	—
244	C₄×M₁₆	—	2	1	no	—
245	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
246	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#97	—	2	1	no	—
248	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
250	C₄³	C₂²×C₃₁	1	1	yes	characteristic
251	#341	C₂²×C₃₁	1	2	yes	—
252	#339	C₂²×C₃₁	1	2	yes	—
253	#339	C₂²×C₃₁	1	2	yes	—
254	#336	C₂²×C₃₁	1	2	yes	—
255	#336	C₂²×C₃₁	1	2	yes	—
256	C₂×C₃₁	—	2	2	no	—
257	C₂×C₃₁	—	4	2	no	—
258	C₂×C₃₁	—	8	1	no	—
259	C₂×C₃₁	#2290	1	1	yes	characteristic
260	C₂×C₃₁	#2956	1	2	yes	—
261	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
262	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
263	#135	—	2	4	no	—
264	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
267	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
268	#131	—	2	4	no	—
269	#131	—	2	4	no	—
270	C₂×M₁₆	—	2	2	no	—
271	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
272	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
273	C₂²×C₈	—	2	2	no	—
274	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
277	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
278	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
282	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
283	#117	—	2	1	no	—
284	#117	—	2	1	no	—
285	#117	C₂³×C₃₁	1	1	yes	characteristic
286	#117	C₂³×C₃₁	1	1	yes	characteristic
287	C₂×C₄²	—	2	1	no	—
288	C₂×C₄²	—	2	1	no	—
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
291	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
292	#108	—	2	4	no	—
293	#107	C₂×C₄×C₃₁	1	2	yes	—
294	#106	C₂×C₄×C₃₁	1	2	yes	—
295	C₄⋊C₈	—	2	2	no	—
296	C₄⋊C₈	—	2	2	no	—
297	#104	—	2	4	no	—
298	#104	—	2	4	no	—
299	#103	C₂×C₄×C₃₁	1	2	yes	—
300	#103	C₂×C₄×C₃₁	1	2	yes	—
301	#102	C₂×C₄×C₃₁	1	2	yes	—
302	#102	C₂×C₄×C₃₁	1	2	yes	—
303	C₂²⋊C₈	—	2	2	no	—
304	C₂²⋊C₈	—	2	2	no	—
305	#97	—	2	2	no	—
306	#97	—	2	1	no	—
307	#97	—	2	1	no	—
308	#97	C₂×C₄×C₃₁	1	2	yes	—
309	#97	C₂×C₄×C₃₁	1	2	yes	—
310	#97	D₄×C₃₁	1	1	yes	characteristic
311	#97	D₄×C₃₁	1	1	yes	characteristic
312	#97	D₄×C₃₁	1	2	yes	—
313	C₄×C₈	—	2	2	no	—
314	C₄×C₈	—	2	1	no	—
315	C₄×C₈	—	2	1	no	—
316	C₃₁	#21139	1	1	yes	characteristic, direct product factor, Sylow
317	#41	—	2	2	no	—
318	#41	—	2	2	no	—
319	#41	—	2	2	no	—
320	#41	#41×C₃₁	1	4	yes	—
321	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
322	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
326	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	#41×C₃₁	1	2	yes	—
333	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
334	Q₁₆	—	2	2	no	—
335	SD₁₆	—	2	2	no	—
336	SD₁₆	—	2	2	no	—
337	D₈	—	2	2	no	—
338	M₁₆	—	2	8	no	—
339	M₁₆	—	2	2	no	—
340	M₁₆	—	2	2	no	—
341	M₁₆	—	2	2	no	—
342	M₁₆	#41×C₃₁	1	4	yes	—
343	C₂×C₈	—	2	8	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
351	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
352	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
353	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
354	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
358	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
359	C₂²⋊C₄	—	2	1	no	—
360	C₂²⋊C₄	—	2	1	no	—
361	C₂²⋊C₄	—	2	2	no	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₄²	—	2	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	2	1	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	4	4	no	—
370	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
371	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂³	—	2	1	no	—
374	C₂³	—	2	1	no	—
375	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
376	Q₈	—	2	1	no	—
377	Q₈	—	2	1	no	—
378	Q₈	—	2	1	no	—
379	Q₈	C₃₁×#117	1	2	yes	—
380	D₄	—	2	4	no	—
381	D₄	—	2	1	no	—
382	D₄	—	2	1	no	—
383	D₄	—	2	1	no	—
384	D₄	—	4	2	no	—
385	D₄	C₃₁×#117	1	2	yes	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	8	2	no	—
397	C₂×C₄	—	8	2	no	—
398	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
399	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
400	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
403	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
404	C₈	—	2	1	no	—
405	C₈	—	2	1	no	—
406	C₈	—	2	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	2	4	no	—
409	C₈	—	2	2	no	—
410	C₈	—	4	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	1	no	—
416	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
417	C₂²	C₃₁×#529	1	2	yes	—
418	C₄	—	2	2	no	—
419	C₄	—	2	2	no	—
420	C₄	—	2	2	no	—
421	C₄	—	4	4	no	—
422	C₄	—	4	2	no	—
423	C₄	—	4	2	no	—
424	C₄	—	4	1	no	—
425	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
426	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
427	C₄	C₃₁×#525	1	2	yes	—
428	C₂	—	2	2	no	—
429	C₂	—	4	2	no	—
430	C₂	—	8	1	no	—
431	C₂	C₃₁×#2290	1	1	yes	characteristic
432	C₂	C₃₁×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2923	C₂	semidirect
3. C₃₁×#2875	C₂	semidirect
4. C₃₁×#1929	C₂	semidirect
5. C₃₁×#1922	C₂	semidirect
7. C₃₁×#1866	C₂	semidirect
8. C₃₁×#1778	C₂	semidirect
9. C₄×C₃₁×#104	C₂	semidirect
10. C₄×C₃₁×#104	C₂	semidirect
11. C₃₁×#1745	C₂	semidirect
16. C₃₁×#442	C₄	semidirect
17. C₃₁×#441	C₄	semidirect
18. C₃₁×#440	C₄	semidirect
19. C₃₁×#440	C₄	semidirect
156. #21139	C₃₁	direct (N × H)
196. #2923	C₂×C₃₁	semidirect
197. #2875	C₂×C₃₁	semidirect
198. #1929	C₂×C₃₁	semidirect
199. #1922	C₂×C₃₁	semidirect
201. #1866	C₂×C₃₁	semidirect
202. #1778	C₂×C₃₁	semidirect
203. C₄×#104	C₂×C₃₁	semidirect
204. C₄×#104	C₂×C₃₁	semidirect
205. #1745	C₂×C₃₁	semidirect
227. #442	C₄×C₃₁	semidirect
228. #441	C₄×C₃₁	semidirect
229. #440	C₄×C₃₁	semidirect
230. #440	C₄×C₃₁	semidirect
316. C₃₁	#21139	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×#2943	2
C₃₁×#2923	2
C₃₁×#2875	2
C₃₁×#1929	2
C₃₁×#1922	2
C₃₁×#1876	2
C₃₁×#1866	2
C₃₁×#1778	2
C₄×C₃₁×#104	2
C₃₁×#1745	2
#21139	31

Subgroups — C31×#21139 #21477898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21139 #21477898