Subgroups — C₃₁×#17876 #21474635

Order 7936.

650 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2983	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#137	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1587	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1585	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1584	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1582	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1476	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₄	1	1	yes	characteristic
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#446	—	2	1	no	—
16	C₃₁×#446	—	2	1	no	—
17	C₃₁×#446	C₂²	1	2	yes	—
18	C₃₁×#446	C₂²	1	2	yes	—
19	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#441	C₄	1	2	yes	—
21	C₃₁×#440	C₄	1	2	yes	—
22	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
23	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
24	C₈×D₄×C₃₁	—	2	2	no	—
25	C₈×D₄×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#432	—	2	2	no	—
27	C₃₁×#431	—	2	1	no	—
28	C₃₁×#431	C₂²	1	1	yes	characteristic
29	C₃₁×#431	C₂²	1	1	yes	characteristic
30	C₃₁×#428	C₄	1	1	yes	characteristic
31	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
32	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₃₁×#419	C₄	1	1	yes	characteristic
34	C₃₁×#417	C₄	1	1	yes	characteristic
35	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	2	yes	—
38	C₃₁×#334	C₂²	1	2	yes	—
39	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
40	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₃₁×#322	C₂²	1	2	yes	—
46	C₃₁×#322	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₃₁×#137	—	2	4	no	—
50	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₄×Q₈×C₃₁	—	2	1	no	—
58	C₄×Q₈×C₃₁	—	2	1	no	—
59	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
60	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
62	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₃₁	—	2	1	no	—
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#117	—	2	1	no	—
68	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#107	C₂×C₄	1	2	yes	—
75	C₃₁×#106	C₂×C₄	1	2	yes	—
76	C₃₁×C₄⋊C₈	—	2	2	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
81	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
82	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
83	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₃₁×#103	C₂×C₄	1	2	yes	—
86	C₃₁×#103	C₂×C₄	1	2	yes	—
87	C₃₁×#103	C₂×C₄	1	2	yes	—
88	C₃₁×#102	C₂×C₄	1	2	yes	—
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	2	no	—
91	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
92	C₃₁×#97	—	2	2	no	—
93	C₃₁×#97	—	2	2	no	—
94	C₃₁×#97	C₂×C₄	1	2	yes	—
95	C₄×C₈×C₃₁	—	2	2	no	—
96	C₄×C₈×C₃₁	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	1	no	—
98	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	2	yes	—
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	#41×C₃₁	—	2	2	no	—
103	#41×C₃₁	M₁₆	1	4	yes	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
119	Q₁₆×C₃₁	—	2	4	no	—
120	SD₁₆×C₃₁	—	2	4	no	—
121	M₁₆×C₃₁	—	2	2	no	—
122	M₁₆×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	4	2	no	—
129	C₂×C₈×C₃₁	—	8	2	no	—
130	C₂×C₈×C₃₁	—	8	1	no	—
131	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
141	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
142	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₃₁	—	2	1	no	—
144	C₂²⋊C₄×C₃₁	—	2	1	no	—
145	C₂²⋊C₄×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	#41	1	1	yes	characteristic
154	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
155	#17876	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
158	Q₈×C₃₁	—	2	1	no	—
159	Q₈×C₃₁	—	2	2	no	—
160	Q₈×C₃₁	—	2	2	no	—
161	Q₈×C₃₁	—	4	2	no	—
162	Q₈×C₃₁	C₂×M₁₆	1	2	yes	—
163	D₄×C₃₁	—	2	4	no	—
164	D₄×C₃₁	—	2	1	no	—
165	D₄×C₃₁	C₂×M₁₆	1	2	yes	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	8	1	no	—
179	C₂×C₄×C₃₁	—	8	1	no	—
180	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
184	C₂×C₄×C₃₁	#131	1	2	yes	—
185	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
188	C₈×C₃₁	—	2	2	no	—
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	2	2	no	—
191	C₈×C₃₁	—	2	2	no	—
192	C₈×C₃₁	—	2	2	no	—
193	C₈×C₃₁	—	4	1	no	—
194	C₈×C₃₁	—	4	1	no	—
195	C₈×C₃₁	—	4	2	no	—
196	C₈×C₃₁	—	4	1	no	—
197	#2988	C₂×C₃₁	1	1	yes	characteristic
198	#2983	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₄×#137	C₂×C₃₁	1	1	yes	characteristic
200	#1587	C₂×C₃₁	1	2	yes	complemented
201	#1585	C₂×C₃₁	1	2	yes	complemented
202	#1584	C₂×C₃₁	1	2	yes	complemented
203	#1582	C₂×C₃₁	1	2	yes	complemented
204	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1476	C₂×C₃₁	1	1	yes	characteristic
206	C₂×Q₈⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
208	C₂²×C₃₁	—	2	1	no	—
209	C₂²×C₃₁	—	2	1	no	—
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	4	1	no	—
212	C₂²×C₃₁	—	4	1	no	—
213	C₂²×C₃₁	#435	1	2	yes	—
214	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
215	C₄×C₃₁	—	2	2	no	—
216	C₄×C₃₁	—	2	1	no	—
217	C₄×C₃₁	—	2	1	no	—
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	4	2	no	—
220	C₄×C₃₁	—	4	1	no	—
221	C₄×C₃₁	—	4	1	no	—
222	C₄×C₃₁	—	4	2	no	—
223	C₄×C₃₁	#435	1	2	yes	—
224	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
225	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
226	C₄×C₃₁	#577	1	1	yes	characteristic
227	C₄×C₃₁	#578	1	1	yes	characteristic
228	C₂×#137	C₄×C₃₁	1	1	yes	characteristic
229	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
231	#446	—	2	1	no	—
232	#446	—	2	1	no	—
233	#446	C₂²×C₃₁	1	2	yes	—
234	#446	C₂²×C₃₁	1	2	yes	—
235	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
236	#441	C₄×C₃₁	1	2	yes	—
237	#440	C₄×C₃₁	1	2	yes	—
238	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
239	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
240	C₈×D₄	—	2	2	no	—
241	C₈×D₄	C₂²×C₃₁	1	2	yes	—
242	#432	—	2	2	no	—
243	#431	—	2	1	no	—
244	#431	C₂²×C₃₁	1	1	yes	characteristic
245	#431	C₂²×C₃₁	1	1	yes	characteristic
246	#428	C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄⋊C₈	—	2	1	no	—
248	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
249	#419	C₄×C₃₁	1	1	yes	characteristic
250	#417	C₄×C₃₁	1	1	yes	characteristic
251	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
253	#335	C₂²×C₃₁	1	2	yes	—
254	#334	C₂²×C₃₁	1	2	yes	—
255	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
256	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
257	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
258	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
259	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
260	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
261	#322	C₂²×C₃₁	1	2	yes	—
262	#322	C₂²×C₃₁	1	2	yes	—
263	C₂×C₃₁	—	2	2	no	—
264	C₂×C₃₁	—	4	2	no	—
265	C₂×C₃₁	C₂×#435	1	1	yes	characteristic
266	C₂×C₃₁	#2947	1	1	yes	characteristic
267	C₂×C₃₁	#2956	1	1	yes	characteristic
268	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
269	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
270	#137	—	2	4	no	—
271	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
272	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
273	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
274	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₈	—	2	1	no	—
276	C₂²×C₈	—	2	1	no	—
277	C₂²×C₈	—	2	2	no	—
278	C₄×Q₈	—	2	1	no	—
279	C₄×Q₈	—	2	1	no	—
280	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
281	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
282	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
283	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
284	C₄×D₄	—	2	1	no	—
285	C₄×D₄	—	2	1	no	—
286	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
287	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
288	#117	—	2	1	no	—
289	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
292	C₂×C₄²	—	2	1	no	—
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
295	#107	C₂×C₄×C₃₁	1	2	yes	—
296	#106	C₂×C₄×C₃₁	1	2	yes	—
297	C₄⋊C₈	—	2	2	no	—
298	C₄⋊C₈	—	2	2	no	—
299	C₄⋊C₈	—	2	2	no	—
300	C₄⋊C₈	—	2	1	no	—
301	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
302	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
303	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
306	#103	C₂×C₄×C₃₁	1	2	yes	—
307	#103	C₂×C₄×C₃₁	1	2	yes	—
308	#103	C₂×C₄×C₃₁	1	2	yes	—
309	#102	C₂×C₄×C₃₁	1	2	yes	—
310	C₂²⋊C₈	—	2	1	no	—
311	C₂²⋊C₈	—	2	2	no	—
312	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
313	#97	—	2	2	no	—
314	#97	—	2	2	no	—
315	#97	C₂×C₄×C₃₁	1	2	yes	—
316	C₄×C₈	—	2	2	no	—
317	C₄×C₈	—	2	1	no	—
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₄×C₈	C₂³×C₃₁	1	2	yes	—
322	C₄×C₈	C₂³×C₃₁	1	2	yes	—
323	C₃₁	#17876	1	1	yes	characteristic, direct product factor, Sylow
324	#41	—	2	2	no	—
325	#41	M₁₆×C₃₁	1	4	yes	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
330	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
331	C₂×D₄	—	2	1	no	—
332	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
339	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
341	Q₁₆	—	2	4	no	—
342	SD₁₆	—	2	4	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	2	2	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	4	2	no	—
351	C₂×C₈	—	8	2	no	—
352	C₂×C₈	—	8	1	no	—
353	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
354	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
355	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
356	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
363	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
364	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
365	C₂²⋊C₄	—	2	1	no	—
366	C₂²⋊C₄	—	2	1	no	—
367	C₂²⋊C₄	—	2	1	no	—
368	C₄²	—	2	1	no	—
369	C₄²	—	2	1	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	1	no	—
372	C₄²	—	2	2	no	—
373	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
374	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₄²	#41×C₃₁	1	1	yes	characteristic
376	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
377	C₂³	—	2	1	no	—
378	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
379	Q₈	—	2	1	no	—
380	Q₈	—	2	2	no	—
381	Q₈	—	2	2	no	—
382	Q₈	—	4	2	no	—
383	Q₈	C₂×M₁₆×C₃₁	1	2	yes	—
384	D₄	—	2	4	no	—
385	D₄	—	2	1	no	—
386	D₄	C₂×M₁₆×C₃₁	1	2	yes	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	8	1	no	—
400	C₂×C₄	—	8	1	no	—
401	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
402	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
403	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
405	C₂×C₄	C₃₁×#131	1	2	yes	—
406	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	—	2	2	no	—
411	C₈	—	2	2	no	—
412	C₈	—	2	2	no	—
413	C₈	—	2	2	no	—
414	C₈	—	4	1	no	—
415	C₈	—	4	1	no	—
416	C₈	—	4	2	no	—
417	C₈	—	4	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	4	1	no	—
423	C₂²	C₃₁×#435	1	2	yes	—
424	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
425	C₄	—	2	2	no	—
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	1	no	—
431	C₄	—	4	1	no	—
432	C₄	—	4	2	no	—
433	C₄	C₃₁×#435	1	2	yes	—
434	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
435	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
436	C₄	C₃₁×#577	1	1	yes	characteristic
437	C₄	C₃₁×#578	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	4	2	no	—
440	C₂	C₂×C₃₁×#435	1	1	yes	characteristic
441	C₂	C₃₁×#2947	1	1	yes	characteristic
442	C₂	C₃₁×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2983	C₂	semidirect
4. C₃₁×#1587	C₂	semidirect
5. C₃₁×#1585	C₂	semidirect
6. C₃₁×#1584	C₂	semidirect
7. C₃₁×#1582	C₂	semidirect
8. C₃₁×#1565	C₂	semidirect
10. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
11. C₃₁×#1451	C₂	semidirect
155. #17876	C₃₁	direct (N × H)
198. #2983	C₂×C₃₁	semidirect
200. #1587	C₂×C₃₁	semidirect
201. #1585	C₂×C₃₁	semidirect
202. #1584	C₂×C₃₁	semidirect
203. #1582	C₂×C₃₁	semidirect
204. #1565	C₂×C₃₁	semidirect
206. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
207. #1451	C₂×C₃₁	semidirect
323. C₃₁	#17876	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×#2988	2
C₃₁×#2983	2
C₄×C₃₁×#137	2
C₃₁×#1587	2
C₃₁×#1585	2
C₃₁×#1584	2
C₃₁×#1582	2
C₃₁×#1565	2
C₃₁×#1476	2
C₂×C₃₁×Q₈⋊C₈	2
C₃₁×#1451	2
#17876	31

Subgroups — C31×#17876 #21474635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17876 #21474635