Subgroups — C₃₁×#21138 #21477897

Order 7936.

772 conjugacy classes of subgroups.

The subgroup lattice has 430 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×#2922	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#131	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	C₃₁×#1759	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1759	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1746	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×#431	—	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	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₃₁	—	2	1	no	—
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#376	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	C₂³	1	2	yes	—
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#108	—	2	4	no	—
76	C₃₁×#107	C₂×C₄	1	2	yes	—
77	C₃₁×#106	C₂×C₄	1	2	yes	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×#104	—	2	4	no	—
81	C₃₁×#104	—	2	4	no	—
82	C₃₁×#103	C₂×C₄	1	2	yes	—
83	C₃₁×#103	C₂×C₄	1	2	yes	—
84	C₃₁×#102	C₂×C₄	1	2	yes	—
85	C₃₁×#102	C₂×C₄	1	2	yes	—
86	C₃₁×C₂²⋊C₈	—	2	2	no	—
87	C₃₁×C₂²⋊C₈	—	2	2	no	—
88	C₃₁×#97	—	2	2	no	—
89	C₃₁×#97	—	2	1	no	—
90	C₃₁×#97	—	2	1	no	—
91	C₃₁×#97	C₂×C₄	1	2	yes	—
92	C₃₁×#97	C₂×C₄	1	2	yes	—
93	C₄×C₈×C₃₁	—	2	2	no	—
94	C₄×C₈×C₃₁	—	2	1	no	—
95	C₄×C₈×C₃₁	—	2	1	no	—
96	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
97	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
98	C₄×C₈×C₃₁	D₄	1	2	yes	—
99	C₃₁×#95	—	2	2	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	2	no	—
113	C₂²×C₄×C₃₁	—	2	1	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	C₂²×C₄	1	2	yes	—
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	#21138	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	—	2	1	no	—
158	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
159	Q₈×C₃₁	—	2	1	no	—
160	Q₈×C₃₁	—	2	1	no	—
161	Q₈×C₃₁	—	2	1	no	—
162	Q₈×C₃₁	#117	1	2	yes	—
163	D₄×C₃₁	—	2	4	no	—
164	D₄×C₃₁	—	2	1	no	—
165	D₄×C₃₁	—	2	1	no	—
166	D₄×C₃₁	—	2	1	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	#117	1	2	yes	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	4	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	4	no	—
176	C₂×C₄×C₃₁	—	4	4	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	4	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
181	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
185	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
186	C₈×C₃₁	—	2	1	no	—
187	C₈×C₃₁	—	2	1	no	—
188	C₈×C₃₁	—	2	2	no	—
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	2	4	no	—
191	C₈×C₃₁	—	2	2	no	—
192	C₈×C₃₁	—	4	2	no	—
193	#2943	C₂×C₃₁	1	1	yes	characteristic
194	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
195	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1929	C₂×C₃₁	1	2	yes	complemented
197	#1922	C₂×C₃₁	1	2	yes	complemented
198	#1876	C₂×C₃₁	1	2	yes	—
199	#1866	C₂×C₃₁	1	2	yes	complemented
200	#1778	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1759	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1759	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂²×C₃₁	#529	1	2	yes	—
211	C₄×C₃₁	—	2	2	no	—
212	C₄×C₃₁	—	2	2	no	—
213	C₄×C₃₁	—	2	2	no	—
214	C₄×C₃₁	—	4	2	no	—
215	C₄×C₃₁	—	4	2	no	—
216	C₄×C₃₁	—	4	1	no	—
217	C₄×C₃₁	—	8	2	no	—
218	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
219	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
220	C₄×C₃₁	#525	1	2	yes	—
221	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×#131	C₂²×C₃₁	1	2	yes	—
223	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
224	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
225	#442	C₄×C₃₁	1	2	yes	complemented
226	#441	C₄×C₃₁	1	2	yes	complemented
227	#440	C₄×C₃₁	1	2	yes	complemented
228	#440	C₄×C₃₁	1	2	yes	complemented
229	#433	—	2	1	no	—
230	#433	—	2	1	no	—
231	#431	—	2	2	no	—
232	C₂×#108	C₂²×C₃₁	1	2	yes	—
233	#427	C₂²×C₃₁	1	1	yes	characteristic
234	#424	C₂²×C₃₁	1	2	yes	—
235	C₂×#104	C₂²×C₃₁	1	2	yes	—
236	C₂×#104	C₂²×C₃₁	1	2	yes	—
237	#416	C₂²×C₃₁	1	1	yes	characteristic
238	#416	C₂²×C₃₁	1	1	yes	characteristic
239	#408	C₂²×C₃₁	1	2	yes	—
240	#405	—	2	2	no	—
241	#405	C₂²×C₃₁	1	2	yes	—
242	C₄×M₁₆	—	2	1	no	—
243	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
244	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×C₄×C₈	—	2	1	no	—
247	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
248	#376	C₂²×C₃₁	1	1	yes	characteristic
249	#341	C₂²×C₃₁	1	2	yes	—
250	#339	C₂²×C₃₁	1	2	yes	—
251	#339	C₂²×C₃₁	1	2	yes	—
252	#336	C₂²×C₃₁	1	2	yes	—
253	#336	C₂²×C₃₁	1	2	yes	—
254	C₂×C₃₁	—	2	2	no	—
255	C₂×C₃₁	—	4	2	no	—
256	C₂×C₃₁	—	8	1	no	—
257	C₂×C₃₁	#2290	1	1	yes	characteristic
258	C₂×C₃₁	#2955	1	2	yes	—
259	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
260	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
261	#135	—	2	4	no	—
262	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
263	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
264	C₂×SD₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
266	#131	—	2	4	no	—
267	#131	—	2	4	no	—
268	C₂×M₁₆	—	2	2	no	—
269	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
270	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
271	C₂²×C₈	—	2	2	no	—
272	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
275	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
276	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
277	C₄×D₄	—	2	2	no	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
280	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
281	#117	—	2	1	no	—
282	#117	—	2	1	no	—
283	#117	C₂³×C₃₁	1	1	yes	characteristic
284	#117	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	—	2	1	no	—
287	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
288	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
289	#108	—	2	4	no	—
290	#107	C₂×C₄×C₃₁	1	2	yes	—
291	#106	C₂×C₄×C₃₁	1	2	yes	—
292	C₄⋊C₈	—	2	2	no	—
293	C₄⋊C₈	—	2	2	no	—
294	#104	—	2	4	no	—
295	#104	—	2	4	no	—
296	#103	C₂×C₄×C₃₁	1	2	yes	—
297	#103	C₂×C₄×C₃₁	1	2	yes	—
298	#102	C₂×C₄×C₃₁	1	2	yes	—
299	#102	C₂×C₄×C₃₁	1	2	yes	—
300	C₂²⋊C₈	—	2	2	no	—
301	C₂²⋊C₈	—	2	2	no	—
302	#97	—	2	2	no	—
303	#97	—	2	1	no	—
304	#97	—	2	1	no	—
305	#97	C₂×C₄×C₃₁	1	2	yes	—
306	#97	C₂×C₄×C₃₁	1	2	yes	—
307	C₄×C₈	—	2	2	no	—
308	C₄×C₈	—	2	1	no	—
309	C₄×C₈	—	2	1	no	—
310	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
311	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
312	C₄×C₈	D₄×C₃₁	1	2	yes	—
313	#95	—	2	2	no	—
314	C₃₁	#21138	1	1	yes	characteristic, direct product factor, Sylow
315	#41	—	2	2	no	—
316	#41	—	2	2	no	—
317	#41	—	2	2	no	—
318	#41	#41×C₃₁	1	4	yes	—
319	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
320	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
324	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	#41×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
332	Q₁₆	—	2	2	no	—
333	SD₁₆	—	2	2	no	—
334	SD₁₆	—	2	2	no	—
335	D₈	—	2	2	no	—
336	M₁₆	—	2	8	no	—
337	M₁₆	—	2	2	no	—
338	M₁₆	—	2	2	no	—
339	M₁₆	—	2	2	no	—
340	M₁₆	#41×C₃₁	1	4	yes	—
341	C₂×C₈	—	2	8	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	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	C₂²×C₄×C₃₁	1	2	yes	—
349	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
350	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
351	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
352	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
356	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
357	C₂²⋊C₄	—	2	1	no	—
358	C₂²⋊C₄	—	2	1	no	—
359	C₂²⋊C₄	—	2	2	no	—
360	C₂²⋊C₄	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
368	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂³	—	2	1	no	—
371	C₂³	—	2	1	no	—
372	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
373	Q₈	—	2	1	no	—
374	Q₈	—	2	1	no	—
375	Q₈	—	2	1	no	—
376	Q₈	C₃₁×#117	1	2	yes	—
377	D₄	—	2	4	no	—
378	D₄	—	2	1	no	—
379	D₄	—	2	1	no	—
380	D₄	—	2	1	no	—
381	D₄	—	4	2	no	—
382	D₄	C₃₁×#117	1	2	yes	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	4	no	—
390	C₂×C₄	—	4	4	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	4	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
395	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
396	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
398	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
399	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
400	C₈	—	2	1	no	—
401	C₈	—	2	1	no	—
402	C₈	—	2	2	no	—
403	C₈	—	2	2	no	—
404	C₈	—	2	4	no	—
405	C₈	—	2	2	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	1	no	—
412	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
413	C₂²	C₃₁×#529	1	2	yes	—
414	C₄	—	2	2	no	—
415	C₄	—	2	2	no	—
416	C₄	—	2	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	2	no	—
419	C₄	—	4	1	no	—
420	C₄	—	8	2	no	—
421	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
422	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
423	C₄	C₃₁×#525	1	2	yes	—
424	C₂	—	2	2	no	—
425	C₂	—	4	2	no	—
426	C₂	—	8	1	no	—
427	C₂	C₃₁×#2290	1	1	yes	characteristic
428	C₂	C₃₁×#2955	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×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2922	C₂	semidirect
3. C₄×C₃₁×#131	C₂	semidirect
4. C₃₁×#1929	C₂	semidirect
5. C₃₁×#1922	C₂	semidirect
7. C₃₁×#1866	C₂	semidirect
8. C₃₁×#1778	C₂	semidirect
9. C₃₁×#1759	C₂	semidirect
10. C₃₁×#1759	C₂	semidirect
11. C₃₁×#1746	C₂	semidirect
16. C₃₁×#442	C₄	semidirect
17. C₃₁×#441	C₄	semidirect
18. C₃₁×#440	C₄	semidirect
19. C₃₁×#440	C₄	semidirect
155. #21138	C₃₁	direct (N × H)
194. #2922	C₂×C₃₁	semidirect
195. C₄×#131	C₂×C₃₁	semidirect
196. #1929	C₂×C₃₁	semidirect
197. #1922	C₂×C₃₁	semidirect
199. #1866	C₂×C₃₁	semidirect
200. #1778	C₂×C₃₁	semidirect
201. #1759	C₂×C₃₁	semidirect
202. #1759	C₂×C₃₁	semidirect
203. #1746	C₂×C₃₁	semidirect
225. #442	C₄×C₃₁	semidirect
226. #441	C₄×C₃₁	semidirect
227. #440	C₄×C₃₁	semidirect
228. #440	C₄×C₃₁	semidirect
314. C₃₁	#21138	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×#2922	2
C₄×C₃₁×#131	2
C₃₁×#1929	2
C₃₁×#1922	2
C₃₁×#1876	2
C₃₁×#1866	2
C₃₁×#1778	2
C₃₁×#1759	2
C₃₁×#1746	2
#21138	31

Subgroups — C31×#21138 #21477897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21138 #21477897