Subgroups — C₃₁×#22608 #21479367

Order 7936.

686 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×#2924	C₂	1	2	yes	maximal
2	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1946	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1922	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1922	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1826	C₂	1	2	yes	maximal
8	C₃₁×#1811	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1753	C₂	1	1	yes	maximal, characteristic
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	2	yes	—
14	C₃₁×#433	—	2	2	no	—
15	C₃₁×#433	—	2	2	no	—
16	C₃₁×#432	—	2	2	no	—
17	C₃₁×#431	—	2	2	no	—
18	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
20	C₃₁×#428	C₄	1	2	yes	complemented
21	C₃₁×#428	C₄	1	2	yes	complemented
22	C₃₁×#428	C₄	1	2	yes	complemented
23	C₃₁×#428	C₄	1	2	yes	complemented
24	C₃₁×#427	C₂²	1	2	yes	—
25	C₃₁×#427	C₂²	1	1	yes	characteristic
26	C₃₁×#427	C₂²	1	1	yes	characteristic
27	C₃₁×#424	C₂²	1	2	yes	—
28	C₃₁×#424	C₂²	1	2	yes	—
29	C₃₁×#407	C₂²	1	2	yes	—
30	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₃₁×#405	C₂²	1	2	yes	—
32	C₃₁×#405	C₂²	1	2	yes	—
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#341	C₂²	1	2	yes	—
37	C₃₁×#339	C₂²	1	2	yes	—
38	C₃₁×#339	C₂²	1	2	yes	—
39	C₃₁×#339	C₂²	1	2	yes	—
40	C₃₁×#339	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
49	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#117	—	2	2	no	—
55	C₃₁×#117	—	2	2	no	—
56	C₃₁×#117	C₂×C₄	1	2	yes	—
57	C₃₁×#117	C₂×C₄	1	2	yes	—
58	C₃₁×#117	C₂³	1	2	yes	—
59	C₃₁×#117	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#108	—	2	2	no	—
68	C₃₁×#108	—	2	2	no	—
69	C₃₁×#107	C₂×C₄	1	2	yes	—
70	C₃₁×#107	C₂×C₄	1	2	yes	—
71	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
72	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
73	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
74	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
75	C₃₁×#106	C₂×C₄	1	2	yes	—
76	C₃₁×#106	C₂×C₄	1	2	yes	—
77	C₃₁×#106	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×C₄⋊C₈	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×C₂²⋊C₈	—	2	2	no	—
84	C₃₁×C₂²⋊C₈	—	2	2	no	—
85	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
86	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
87	C₃₁×#97	—	2	2	no	—
88	C₃₁×#97	—	2	2	no	—
89	C₃₁×#97	—	2	2	no	—
90	C₄×C₈×C₃₁	—	2	2	no	—
91	C₄×C₈×C₃₁	—	2	2	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₂⁴×C₃₁	#41	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	M₁₆×C₃₁	—	2	4	no	—
108	M₁₆×C₃₁	—	2	4	no	—
109	M₁₆×C₃₁	—	2	2	no	—
110	M₁₆×C₃₁	—	2	2	no	—
111	M₁₆×C₃₁	—	2	2	no	—
112	M₁₆×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	#41	1	2	yes	—
128	C₂×C₈×C₃₁	#41	1	2	yes	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
132	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
133	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
134	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
135	C₂²⋊C₄×C₃₁	—	2	2	no	—
136	C₂²⋊C₄×C₃₁	—	2	2	no	—
137	C₂²⋊C₄×C₃₁	—	2	2	no	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	#22608	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	2	1	no	—
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	4	1	no	—
152	C₂³×C₃₁	—	4	1	no	—
153	C₂³×C₃₁	C₄×D₄	1	2	yes	—
154	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	8	2	no	—
173	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
174	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
179	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
180	C₈×C₃₁	—	2	2	no	—
181	C₈×C₃₁	—	2	1	no	—
182	C₈×C₃₁	—	2	1	no	—
183	C₈×C₃₁	—	2	2	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	—	4	2	no	—
186	C₈×C₃₁	#117	1	2	yes	—
187	C₈×C₃₁	#117	1	2	yes	—
188	#2924	C₂×C₃₁	1	2	yes	—
189	#2911	C₂×C₃₁	1	1	yes	characteristic
190	#1946	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1922	C₂×C₃₁	1	2	yes	complemented
193	#1922	C₂×C₃₁	1	2	yes	complemented
194	#1826	C₂×C₃₁	1	2	yes	—
195	#1811	C₂×C₃₁	1	1	yes	characteristic
196	#1773	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1753	C₂×C₃₁	1	1	yes	characteristic
199	C₂²×C₃₁	—	2	1	no	—
200	C₂²×C₃₁	—	2	1	no	—
201	C₂²×C₃₁	—	2	2	no	—
202	C₂²×C₃₁	—	2	2	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	8	1	no	—
205	C₂²×C₃₁	—	8	1	no	—
206	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
207	C₂²×C₃₁	#552	1	1	yes	characteristic
208	C₂²×C₃₁	#552	1	1	yes	characteristic
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	4	2	no	—
212	C₄×C₃₁	—	4	2	no	—
213	C₄×C₃₁	—	4	2	no	—
214	C₄×C₃₁	—	4	2	no	—
215	C₄×C₃₁	—	8	1	no	—
216	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
217	C₄×C₃₁	#519	1	1	yes	characteristic
218	C₄×C₃₁	#547	1	1	yes	characteristic
219	C₄×C₃₁	#547	1	1	yes	characteristic
220	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×#117	C₂²×C₃₁	1	2	yes	—
222	#433	—	2	2	no	—
223	#433	—	2	2	no	—
224	#432	—	2	2	no	—
225	#431	—	2	2	no	—
226	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
228	#428	C₄×C₃₁	1	2	yes	complemented
229	#428	C₄×C₃₁	1	2	yes	complemented
230	#428	C₄×C₃₁	1	2	yes	complemented
231	#428	C₄×C₃₁	1	2	yes	complemented
232	#427	C₂²×C₃₁	1	2	yes	—
233	#427	C₂²×C₃₁	1	1	yes	characteristic
234	#427	C₂²×C₃₁	1	1	yes	characteristic
235	#424	C₂²×C₃₁	1	2	yes	—
236	#424	C₂²×C₃₁	1	2	yes	—
237	#407	C₂²×C₃₁	1	2	yes	—
238	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
239	#405	C₂²×C₃₁	1	2	yes	—
240	#405	C₂²×C₃₁	1	2	yes	—
241	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
244	#341	C₂²×C₃₁	1	2	yes	—
245	#339	C₂²×C₃₁	1	2	yes	—
246	#339	C₂²×C₃₁	1	2	yes	—
247	#339	C₂²×C₃₁	1	2	yes	—
248	#339	C₂²×C₃₁	1	2	yes	—
249	#336	C₂²×C₃₁	1	1	yes	characteristic
250	#336	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₃₁	—	2	1	no	—
252	C₂×C₃₁	—	2	1	no	—
253	C₂×C₃₁	—	4	2	no	—
254	C₂×C₃₁	#2344	1	1	yes	characteristic
255	C₂×C₃₁	#2955	1	1	yes	characteristic
256	C₂×C₃₁	#2956	1	1	yes	characteristic
257	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
261	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
262	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
263	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
264	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	#117	—	2	2	no	—
269	#117	—	2	2	no	—
270	#117	C₂×C₄×C₃₁	1	2	yes	—
271	#117	C₂×C₄×C₃₁	1	2	yes	—
272	#117	C₂³×C₃₁	1	2	yes	—
273	#117	C₂³×C₃₁	1	2	yes	—
274	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
275	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	—	2	1	no	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
281	#108	—	2	2	no	—
282	#108	—	2	2	no	—
283	#107	C₂×C₄×C₃₁	1	2	yes	—
284	#107	C₂×C₄×C₃₁	1	2	yes	—
285	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
286	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
287	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
288	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
289	#106	C₂×C₄×C₃₁	1	2	yes	—
290	#106	C₂×C₄×C₃₁	1	2	yes	—
291	#106	C₂×C₄×C₃₁	1	2	yes	—
292	#106	C₂×C₄×C₃₁	1	2	yes	—
293	C₄⋊C₈	—	2	2	no	—
294	C₄⋊C₈	—	2	2	no	—
295	C₄⋊C₈	—	2	2	no	—
296	C₄⋊C₈	—	2	2	no	—
297	C₂²⋊C₈	—	2	2	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
300	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
301	#97	—	2	2	no	—
302	#97	—	2	2	no	—
303	#97	—	2	2	no	—
304	C₄×C₈	—	2	2	no	—
305	C₄×C₈	—	2	2	no	—
306	C₄×C₈	—	2	2	no	—
307	#95	—	2	1	no	—
308	C₃₁	#22608	1	1	yes	characteristic, direct product factor, Sylow
309	C₂⁴	#41×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
320	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
322	M₁₆	—	2	4	no	—
323	M₁₆	—	2	4	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	2	2	no	—
326	M₁₆	—	2	2	no	—
327	M₁₆	—	2	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
338	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₈	#41×C₃₁	1	2	yes	—
343	C₂×C₈	#41×C₃₁	1	2	yes	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
347	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
348	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
349	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₂²⋊C₄	—	2	2	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	2	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	1	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	2	1	no	—
363	C₂³	—	2	1	no	—
364	C₂³	—	2	1	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	1	no	—
367	C₂³	C₄×D₄×C₃₁	1	2	yes	—
368	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	4	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	8	2	no	—
387	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
388	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
390	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
393	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
394	C₈	—	2	2	no	—
395	C₈	—	2	1	no	—
396	C₈	—	2	1	no	—
397	C₈	—	2	2	no	—
398	C₈	—	4	2	no	—
399	C₈	—	4	2	no	—
400	C₈	C₃₁×#117	1	2	yes	—
401	C₈	C₃₁×#117	1	2	yes	—
402	C₂²	—	2	1	no	—
403	C₂²	—	2	1	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	8	1	no	—
408	C₂²	—	8	1	no	—
409	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
410	C₂²	C₃₁×#552	1	1	yes	characteristic
411	C₂²	C₃₁×#552	1	1	yes	characteristic
412	C₄	—	2	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	8	1	no	—
419	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
420	C₄	C₃₁×#519	1	1	yes	characteristic
421	C₄	C₃₁×#547	1	1	yes	characteristic
422	C₄	C₃₁×#547	1	1	yes	characteristic
423	C₂	—	2	1	no	—
424	C₂	—	2	1	no	—
425	C₂	—	4	2	no	—
426	C₂	C₃₁×#2344	1	1	yes	characteristic
427	C₂	C₃₁×#2955	1	1	yes	characteristic
428	C₂	C₃₁×#2956	1	1	yes	characteristic

	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	C₂⁴×C₃₁	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×C₂²⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1946	C₂	semidirect
4. C₃₁×#1945	C₂	semidirect
5. C₃₁×#1922	C₂	semidirect
6. C₃₁×#1922	C₂	semidirect
9. C₃₁×#1773	C₂	semidirect
10. C₃₁×#1772	C₂	semidirect
20. C₃₁×#428	C₄	semidirect
21. C₃₁×#428	C₄	semidirect
22. C₃₁×#428	C₄	semidirect
23. C₃₁×#428	C₄	semidirect
148. #22608	C₃₁	direct (N × H)
190. #1946	C₂×C₃₁	semidirect
191. #1945	C₂×C₃₁	semidirect
192. #1922	C₂×C₃₁	semidirect
193. #1922	C₂×C₃₁	semidirect
196. #1773	C₂×C₃₁	semidirect
197. #1772	C₂×C₃₁	semidirect
228. #428	C₄×C₃₁	semidirect
229. #428	C₄×C₃₁	semidirect
230. #428	C₄×C₃₁	semidirect
231. #428	C₄×C₃₁	semidirect
308. C₃₁	#22608	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×#2924	2
C₃₁×#2911	2
C₃₁×#1946	2
C₃₁×#1945	2
C₃₁×#1922	2
C₃₁×#1826	2
C₃₁×#1811	2
C₃₁×#1773	2
C₃₁×#1772	2
C₃₁×#1753	2
#22608	31

Subgroups — C31×#22608 #21479367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22608 #21479367