Subgroups — C₃₁×#20504 #21477263

Order 7936.

1130 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#440	C₂	1	2	yes	maximal
2	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2276	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2275	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1776	C₂	1	2	yes	maximal, complemented
6	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1763	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1762	C₂	1	1	yes	maximal, characteristic
9	C₄×C₃₁×#103	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#102	C₂	1	2	yes	maximal
11	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#440	C₄	1	16	yes	complemented
18	C₄×SD₁₆×C₃₁	C₄	1	8	yes	complemented
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#97	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
37	C₃₁×#376	—	2	1	no	—
38	C₃₁×#376	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#95	—	2	1	no	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂³×C₄×C₃₁	—	2	2	no	—
47	C₂×SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
48	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₃₁	—	2	4	no	—
53	C₄×Q₈×C₃₁	—	2	2	no	—
54	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
55	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
56	C₄×D₄×C₃₁	—	2	4	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	—	2	1	no	—
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₃₁×#107	C₂×C₄	1	8	yes	complemented
77	C₃₁×#106	C₂×C₄	1	4	yes	complemented
78	C₃₁×#103	C₂×C₄	1	4	yes	complemented
79	C₃₁×#103	C₂×C₄	1	8	yes	complemented
80	C₃₁×#102	C₂×C₄	1	4	yes	—
81	C₃₁×#102	C₂×C₄	1	8	yes	—
82	C₃₁×#97	—	2	2	no	—
83	C₃₁×#97	C₂×C₄	1	8	yes	complemented
84	C₄×C₈×C₃₁	—	2	4	no	—
85	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	D₄	1	1	yes	characteristic
91	C₃₁×#95	D₄	1	1	yes	characteristic
92	C₃₁×#95	D₄	1	2	yes	—
93	C₂⁴×C₃₁	—	2	1	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	8	no	—
109	C₂²×C₄×C₃₁	—	8	2	no	—
110	C₂²×C₄×C₃₁	—	8	2	no	—
111	C₂²×C₄×C₃₁	—	8	2	no	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	SD₁₆×C₃₁	C₄²	1	16	yes	complemented
118	C₂×C₈×C₃₁	—	2	2	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	4	yes	—
123	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
125	C₄⋊C₄×C₃₁	—	2	4	no	—
126	C₄⋊C₄×C₃₁	—	2	8	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
132	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
133	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
134	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
135	C₂²⋊C₄×C₃₁	—	2	8	no	—
136	C₂²⋊C₄×C₃₁	—	2	4	no	—
137	C₄²×C₃₁	—	2	8	no	—
138	C₄²×C₃₁	—	2	4	no	—
139	C₄²×C₃₁	—	8	1	no	—
140	C₄²×C₃₁	—	8	1	no	—
141	C₄²×C₃₁	—	8	2	no	—
142	C₄²×C₃₁	—	8	2	no	—
143	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
144	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
145	C₄²×C₃₁	#41	1	4	yes	—
146	C₄²×C₃₁	#41	1	2	yes	—
147	#20504	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₃₁	—	2	4	no	—
149	C₂³×C₃₁	—	2	2	no	—
150	C₂³×C₃₁	—	4	1	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	—	4	1	no	—
153	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₃₁	—	2	2	no	—
155	Q₈×C₃₁	—	2	4	no	—
156	Q₈×C₃₁	C₂×C₄²	1	4	yes	—
157	D₄×C₃₁	—	2	2	no	—
158	D₄×C₃₁	—	2	4	no	—
159	D₄×C₃₁	C₂×C₄²	1	4	yes	—
160	C₂×C₄×C₃₁	—	2	4	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	1	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	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	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	1	no	—
180	C₂×C₄×C₃₁	—	8	8	no	—
181	C₂×C₄×C₃₁	—	8	4	no	—
182	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
183	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
185	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
186	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
187	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
189	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
190	C₈×C₃₁	—	2	2	no	—
191	C₈×C₃₁	—	2	4	no	—
192	C₈×C₃₁	C₂×C₄²	1	4	yes	—
193	C₂×#440	C₂×C₃₁	1	2	yes	—
194	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic
195	#2276	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#2275	C₂×C₃₁	1	1	yes	characteristic
197	#1776	C₂×C₃₁	1	2	yes	complemented
198	C₄×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1763	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1762	C₂×C₃₁	1	1	yes	characteristic
201	C₄×#103	C₂×C₃₁	1	2	yes	complemented
202	C₄×#102	C₂×C₃₁	1	2	yes	—
203	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₂²×C₃₁	—	2	4	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	—	4	4	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	2	no	—
209	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
210	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂²×C₃₁	#576	1	1	yes	characteristic
212	C₂²×C₃₁	#577	1	2	yes	—
213	C₂²×C₃₁	#578	1	1	yes	characteristic
214	C₄×C₃₁	—	2	4	no	—
215	C₄×C₃₁	—	2	4	no	—
216	C₄×C₃₁	—	2	2	no	—
217	C₄×C₃₁	—	2	2	no	—
218	C₄×C₃₁	—	2	4	no	—
219	C₄×C₃₁	—	4	4	no	—
220	C₄×C₃₁	—	4	4	no	—
221	C₄×C₃₁	—	4	2	no	—
222	C₄×C₃₁	—	4	2	no	—
223	C₄×C₃₁	C₂²×C₄²	1	1	yes	characteristic
224	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
225	C₄×C₃₁	C₄×#41	1	2	yes	—
226	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
229	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
231	#440	C₄×C₃₁	1	16	yes	complemented
232	C₄×SD₁₆	C₄×C₃₁	1	8	yes	complemented
233	C₂×#107	C₂²×C₃₁	1	2	yes	—
234	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#103	C₂²×C₃₁	1	2	yes	—
237	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×#102	C₂²×C₃₁	1	2	yes	—
239	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#97	C₂²×C₃₁	1	2	yes	—
241	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
242	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₄×C₄⋊C₄	—	2	1	no	—
244	C₄×C₄⋊C₄	—	2	2	no	—
245	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
246	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
247	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
248	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
249	C₄×C₂²⋊C₄	—	2	1	no	—
250	C₄×C₂²⋊C₄	—	2	2	no	—
251	#376	—	2	1	no	—
252	#376	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×#95	—	2	1	no	—
254	#340	C₂²×C₃₁	1	2	yes	—
255	#340	C₂²×C₃₁	1	2	yes	—
256	#340	C₂²×C₃₁	1	2	yes	—
257	C₂×C₃₁	—	4	4	no	—
258	C₂×C₃₁	C₄²×D₄	1	1	yes	characteristic
259	C₂×C₃₁	#2946	1	2	yes	—
260	C₂×C₃₁	#2947	1	2	yes	—
261	C₂×C₃₁	#2949	1	1	yes	characteristic
262	C₂×C₃₁	#2951	1	1	yes	characteristic
263	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
265	C₂³×C₄	—	2	1	no	—
266	C₂³×C₄	—	2	2	no	—
267	C₂×SD₁₆	C₂×C₄×C₃₁	1	4	yes	—
268	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	—
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
271	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
272	C₄×Q₈	—	2	4	no	—
273	C₄×Q₈	—	2	2	no	—
274	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
275	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
276	C₄×D₄	—	2	4	no	—
277	C₄×D₄	—	2	2	no	—
278	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
279	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	1	no	—
288	C₂×C₄²	—	2	1	no	—
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
296	#107	C₂×C₄×C₃₁	1	8	yes	complemented
297	#106	C₂×C₄×C₃₁	1	4	yes	complemented
298	#103	C₂×C₄×C₃₁	1	4	yes	complemented
299	#103	C₂×C₄×C₃₁	1	8	yes	complemented
300	#102	C₂×C₄×C₃₁	1	4	yes	—
301	#102	C₂×C₄×C₃₁	1	8	yes	—
302	#97	—	2	2	no	—
303	#97	C₂×C₄×C₃₁	1	8	yes	complemented
304	C₄×C₈	—	2	4	no	—
305	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
306	#95	—	2	1	no	—
307	#95	—	2	2	no	—
308	#95	—	2	1	no	—
309	#95	—	2	2	no	—
310	#95	D₄×C₃₁	1	1	yes	characteristic
311	#95	D₄×C₃₁	1	1	yes	characteristic
312	#95	D₄×C₃₁	1	2	yes	—
313	C₃₁	#20504	1	1	yes	characteristic, direct product factor, Sylow
314	C₂⁴	—	2	1	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
318	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
322	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	8	no	—
330	C₂²×C₄	—	8	2	no	—
331	C₂²×C₄	—	8	2	no	—
332	C₂²×C₄	—	8	2	no	—
333	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
338	SD₁₆	C₄²×C₃₁	1	16	yes	complemented
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
343	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
344	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
345	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	8	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
353	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
354	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
355	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
356	C₂²⋊C₄	—	2	8	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₄²	—	2	8	no	—
359	C₄²	—	2	4	no	—
360	C₄²	—	8	1	no	—
361	C₄²	—	8	1	no	—
362	C₄²	—	8	2	no	—
363	C₄²	—	8	2	no	—
364	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
365	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
366	C₄²	#41×C₃₁	1	4	yes	—
367	C₄²	#41×C₃₁	1	2	yes	—
368	C₂³	—	2	4	no	—
369	C₂³	—	2	2	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	2	no	—
372	C₂³	—	4	1	no	—
373	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
374	Q₈	—	2	2	no	—
375	Q₈	—	2	4	no	—
376	Q₈	C₂×C₄²×C₃₁	1	4	yes	—
377	D₄	—	2	2	no	—
378	D₄	—	2	4	no	—
379	D₄	C₂×C₄²×C₃₁	1	4	yes	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	8	8	no	—
401	C₂×C₄	—	8	4	no	—
402	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
403	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
404	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
405	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
406	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
407	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
409	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
410	C₈	—	2	2	no	—
411	C₈	—	2	4	no	—
412	C₈	C₂×C₄²×C₃₁	1	4	yes	—
413	C₂²	—	2	4	no	—
414	C₂²	—	4	4	no	—
415	C₂²	—	4	4	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	2	no	—
418	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
419	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
420	C₂²	C₃₁×#576	1	1	yes	characteristic
421	C₂²	C₃₁×#577	1	2	yes	—
422	C₂²	C₃₁×#578	1	1	yes	characteristic
423	C₄	—	2	4	no	—
424	C₄	—	2	4	no	—
425	C₄	—	2	2	no	—
426	C₄	—	2	2	no	—
427	C₄	—	2	4	no	—
428	C₄	—	4	4	no	—
429	C₄	—	4	4	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	2	no	—
432	C₄	C₂²×C₄²×C₃₁	1	1	yes	characteristic
433	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
434	C₄	C₄×#41×C₃₁	1	2	yes	—
435	C₂	—	4	4	no	—
436	C₂	C₄²×D₄×C₃₁	1	1	yes	characteristic
437	C₂	C₃₁×#2946	1	2	yes	—
438	C₂	C₃₁×#2947	1	2	yes	—
439	C₂	C₃₁×#2949	1	1	yes	characteristic
440	C₂	C₃₁×#2951	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2276	C₂	semidirect
5. C₃₁×#1776	C₂	semidirect
6. C₄×C₃₁×#107	C₂	semidirect
7. C₃₁×#1763	C₂	semidirect
9. C₄×C₃₁×#103	C₂	semidirect
11. C₃₁×#1726	C₂	semidirect
17. C₃₁×#440	C₄	semidirect
18. C₄×SD₁₆×C₃₁	C₄	semidirect
54. C₄×Q₈×C₃₁	C₂×C₄	semidirect
55. C₄×Q₈×C₃₁	C₂×C₄	semidirect
76. C₃₁×#107	C₂×C₄	semidirect
77. C₃₁×#106	C₂×C₄	semidirect
78. C₃₁×#103	C₂×C₄	semidirect
79. C₃₁×#103	C₂×C₄	semidirect
83. C₃₁×#97	C₂×C₄	semidirect
85. C₄×C₈×C₃₁	C₂×C₄	semidirect
117. SD₁₆×C₃₁	C₄²	semidirect
147. #20504	C₃₁	direct (N × H)
195. #2276	C₂×C₃₁	semidirect
197. #1776	C₂×C₃₁	semidirect
198. C₄×#107	C₂×C₃₁	semidirect
199. #1763	C₂×C₃₁	semidirect
201. C₄×#103	C₂×C₃₁	semidirect
203. #1726	C₂×C₃₁	semidirect
231. #440	C₄×C₃₁	semidirect
232. C₄×SD₁₆	C₄×C₃₁	semidirect
274. C₄×Q₈	C₂×C₄×C₃₁	semidirect
275. C₄×Q₈	C₂×C₄×C₃₁	semidirect
296. #107	C₂×C₄×C₃₁	semidirect
297. #106	C₂×C₄×C₃₁	semidirect
298. #103	C₂×C₄×C₃₁	semidirect
299. #103	C₂×C₄×C₃₁	semidirect
303. #97	C₂×C₄×C₃₁	semidirect
305. C₄×C₈	C₂×C₄×C₃₁	semidirect
313. C₃₁	#20504	direct (N × H)
338. SD₁₆	C₄²×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#440	2
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×#2276	2
C₃₁×#2275	2
C₃₁×#1776	2
C₄×C₃₁×#107	2
C₃₁×#1763	2
C₃₁×#1762	2
C₄×C₃₁×#103	2
C₄×C₃₁×#102	2
C₃₁×#1726	2
#20504	31

Subgroups — C31×#20504 #21477263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20504 #21477263