Subgroups — C₃₁×#24742 #21481501

Order 7936.

1104 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#460	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2541	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2351	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2041	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1961	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1880	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	2	yes	—
16	C₂×C₃₁×#121	—	2	2	no	—
17	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#460	—	2	8	no	—
21	C₃₁×#460	—	2	4	no	—
22	C₃₁×#440	—	2	4	no	—
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
33	C₃₁×#392	—	2	1	no	—
34	C₃₁×#392	—	2	1	no	—
35	C₃₁×#390	C₂²	1	2	yes	—
36	C₃₁×#389	—	2	2	no	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#382	—	2	1	no	—
39	C₃₁×#382	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	4	no	—
55	C₂×SD₁₆×C₃₁	—	2	4	no	—
56	C₂×SD₁₆×C₃₁	—	2	4	no	—
57	C₂×SD₁₆×C₃₁	D₄	1	4	yes	complemented
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#127	—	2	4	no	—
62	C₃₁×#127	—	2	2	no	—
63	C₃₁×#127	—	2	2	no	—
64	C₃₁×#127	—	4	2	no	—
65	C₃₁×#121	—	4	8	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	—	2	4	no	—
68	C₄×Q₈×C₃₁	—	4	2	no	—
69	C₄×Q₈×C₃₁	D₄	1	4	yes	complemented
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	—	2	2	no	—
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#107	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	4	no	—
88	C₃₁×C₄⋊C₈	—	2	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#102	—	2	4	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	4	2	no	—
95	C₃₁×#97	—	2	2	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₃₁×#95	—	2	2	no	—
98	C₃₁×#95	—	2	1	no	—
99	C₃₁×#95	D₄	1	1	yes	characteristic
100	C₃₁×#95	D₄	1	1	yes	characteristic
101	C₂⁴×C₃₁	—	2	1	no	—
102	C₂⁴×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×Q₈×C₃₁	#41	1	2	yes	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	—	4	4	no	—
121	C₂×D₄×C₃₁	—	4	4	no	—
122	C₂×D₄×C₃₁	—	4	4	no	—
123	C₂×D₄×C₃₁	—	4	4	no	—
124	C₂×D₄×C₃₁	—	4	2	no	—
125	C₂×D₄×C₃₁	—	4	2	no	—
126	C₂×D₄×C₃₁	—	4	4	no	—
127	C₂×D₄×C₃₁	—	8	2	no	—
128	C₂×D₄×C₃₁	—	8	2	no	—
129	C₂×D₄×C₃₁	—	8	2	no	—
130	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	4	4	no	—
136	C₂²×C₄×C₃₁	—	4	2	no	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
142	SD₁₆×C₃₁	—	2	8	no	—
143	SD₁₆×C₃₁	—	4	8	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	4	4	no	—
161	C₄⋊C₄×C₃₁	—	4	4	no	—
162	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂²⋊C₄×C₃₁	—	4	4	no	—
164	C₂²⋊C₄×C₃₁	—	4	2	no	—
165	C₂²⋊C₄×C₃₁	—	4	4	no	—
166	C₄²×C₃₁	—	2	4	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₄²×C₃₁	—	4	4	no	—
170	C₄²×C₃₁	C₂×D₄	1	2	yes	—
171	C₄²×C₃₁	#41	1	2	yes	—
172	#24742	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	2	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	2	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	4	2	no	—
185	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	Q₈×C₃₁	—	2	2	no	—
187	Q₈×C₃₁	—	2	2	no	—
188	Q₈×C₃₁	—	2	2	no	—
189	Q₈×C₃₁	#121	1	4	yes	complemented
190	D₄×C₃₁	—	2	2	no	—
191	D₄×C₃₁	—	2	2	no	—
192	D₄×C₃₁	—	4	2	no	—
193	D₄×C₃₁	—	4	1	no	—
194	D₄×C₃₁	—	4	2	no	—
195	D₄×C₃₁	—	4	2	no	—
196	D₄×C₃₁	—	4	1	no	—
197	D₄×C₃₁	—	4	2	no	—
198	D₄×C₃₁	—	8	8	no	—
199	D₄×C₃₁	—	8	4	no	—
200	D₄×C₃₁	—	8	4	no	—
201	D₄×C₃₁	—	8	8	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	8	4	no	—
220	C₂×C₄×C₃₁	#121	1	2	yes	—
221	C₂×C₄×C₃₁	#121	1	2	yes	—
222	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
227	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
228	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
229	C₈×C₃₁	—	2	2	no	—
230	C₈×C₃₁	—	4	2	no	—
231	C₈×C₃₁	—	4	1	no	—
232	C₂×#460	C₂×C₃₁	1	2	yes	complemented
233	C₂×#460	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂×#440	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#2541	C₂×C₃₁	1	1	yes	characteristic
236	#2351	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#2067	C₂×C₃₁	1	2	yes	complemented
238	#2041	C₂×C₃₁	1	2	yes	complemented
239	#1961	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1880	C₂×C₃₁	1	2	yes	complemented
242	#1804	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	4	4	no	—
245	C₂²×C₃₁	—	8	2	no	—
246	C₂²×C₃₁	—	8	1	no	—
247	C₂²×C₃₁	—	8	1	no	—
248	C₂²×C₃₁	—	8	2	no	—
249	C₂²×C₃₁	—	8	2	no	—
250	C₂²×C₃₁	—	8	2	no	—
251	C₂²×C₃₁	—	8	2	no	—
252	C₂²×C₃₁	—	8	2	no	—
253	C₂²×C₃₁	—	8	2	no	—
254	C₂²×C₃₁	—	8	2	no	—
255	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
256	C₂²×C₃₁	D₄²	1	1	yes	characteristic
257	C₂²×C₃₁	#548	1	1	yes	characteristic
258	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
259	C₂²×C₃₁	#577	1	2	yes	—
260	C₄×C₃₁	—	2	4	no	—
261	C₄×C₃₁	—	2	4	no	—
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	4	2	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	8	2	no	—
267	C₄×C₃₁	—	8	1	no	—
268	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
269	C₄×C₃₁	#537	1	1	yes	characteristic
270	C₄×C₃₁	#547	1	1	yes	characteristic
271	C₄×C₃₁	#553	1	1	yes	characteristic
272	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
273	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#127	C₂²×C₃₁	1	2	yes	—
276	C₂×#121	—	2	2	no	—
277	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
278	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
280	#460	—	2	8	no	—
281	#460	—	2	4	no	—
282	#440	—	2	4	no	—
283	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
286	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×#102	C₂²×C₃₁	1	2	yes	—
288	C₂×#102	C₂²×C₃₁	1	2	yes	—
289	C₂×#102	C₂²×C₃₁	1	2	yes	—
290	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×#102	C₂²×C₃₁	1	2	yes	—
292	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
293	#392	—	2	1	no	—
294	#392	—	2	1	no	—
295	#390	C₂²×C₃₁	1	2	yes	—
296	#389	—	2	2	no	—
297	#389	C₂²×C₃₁	1	2	yes	—
298	#382	—	2	1	no	—
299	#382	C₂²×C₃₁	1	1	yes	characteristic
300	C₄×C₄⋊C₄	—	2	1	no	—
301	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
302	#340	C₂²×C₃₁	1	2	yes	—
303	#336	C₂²×C₃₁	1	2	yes	—
304	C₂×C₃₁	—	8	2	no	—
305	C₂×C₃₁	—	16	2	no	—
306	C₂×C₃₁	#2416	1	1	yes	characteristic
307	C₂×C₃₁	#3038	1	1	yes	characteristic
308	C₂×C₃₁	#3043	1	1	yes	characteristic
309	C₂×C₃₁	#3265	1	2	yes	—
310	C₂×C₃₁	#3368	1	2	yes	—
311	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×D₄	—	2	1	no	—
313	C₂²×D₄	—	2	2	no	—
314	C₂²×D₄	—	2	2	no	—
315	C₂²×D₄	—	2	1	no	—
316	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
317	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
318	C₂³×C₄	—	2	1	no	—
319	C₂×SD₁₆	—	2	2	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	4	no	—
322	C₂×SD₁₆	—	2	4	no	—
323	C₂×SD₁₆	—	2	4	no	—
324	C₂×SD₁₆	D₄×C₃₁	1	4	yes	complemented
325	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
326	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
327	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
328	#127	—	2	4	no	—
329	#127	—	2	2	no	—
330	#127	—	2	2	no	—
331	#127	—	4	2	no	—
332	#121	—	4	8	no	—
333	C₄×Q₈	—	2	2	no	—
334	C₄×Q₈	—	2	4	no	—
335	C₄×Q₈	—	4	2	no	—
336	C₄×Q₈	D₄×C₃₁	1	4	yes	complemented
337	C₄×D₄	—	2	2	no	—
338	C₄×D₄	—	2	2	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	2	no	—
341	C₂×C₄⋊C₄	—	2	2	no	—
342	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	—	2	2	no	—
351	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
352	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
353	#107	—	2	2	no	—
354	C₄⋊C₈	—	2	4	no	—
355	C₄⋊C₈	—	2	2	no	—
356	#103	—	2	2	no	—
357	#102	—	2	4	no	—
358	#102	—	2	4	no	—
359	#102	—	2	2	no	—
360	#102	—	2	4	no	—
361	#102	—	4	2	no	—
362	#97	—	2	2	no	—
363	#95	—	2	1	no	—
364	#95	—	2	2	no	—
365	#95	—	2	1	no	—
366	#95	D₄×C₃₁	1	1	yes	characteristic
367	#95	D₄×C₃₁	1	1	yes	characteristic
368	C₃₁	#24742	1	1	yes	characteristic, direct product factor, Sylow
369	C₂⁴	—	2	1	no	—
370	C₂⁴	—	2	2	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
375	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
376	C₂×Q₈	#41×C₃₁	1	2	yes	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	4	4	no	—
389	C₂×D₄	—	4	4	no	—
390	C₂×D₄	—	4	4	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	—	4	2	no	—
394	C₂×D₄	—	4	4	no	—
395	C₂×D₄	—	8	2	no	—
396	C₂×D₄	—	8	2	no	—
397	C₂×D₄	—	8	2	no	—
398	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	4	4	no	—
404	C₂²×C₄	—	4	2	no	—
405	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
410	SD₁₆	—	2	8	no	—
411	SD₁₆	—	4	8	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	4	4	no	—
429	C₄⋊C₄	—	4	4	no	—
430	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₂²⋊C₄	—	4	2	no	—
433	C₂²⋊C₄	—	4	4	no	—
434	C₄²	—	2	4	no	—
435	C₄²	—	2	2	no	—
436	C₄²	—	2	2	no	—
437	C₄²	—	4	4	no	—
438	C₄²	C₂×D₄×C₃₁	1	2	yes	—
439	C₄²	#41×C₃₁	1	2	yes	—
440	C₂³	—	2	2	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	4	1	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	2	no	—
452	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	Q₈	—	2	2	no	—
455	Q₈	—	2	2	no	—
456	Q₈	C₃₁×#121	1	4	yes	complemented
457	D₄	—	2	2	no	—
458	D₄	—	2	2	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	1	no	—
464	D₄	—	4	2	no	—
465	D₄	—	8	8	no	—
466	D₄	—	8	4	no	—
467	D₄	—	8	4	no	—
468	D₄	—	8	8	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	8	4	no	—
487	C₂×C₄	C₃₁×#121	1	2	yes	—
488	C₂×C₄	C₃₁×#121	1	2	yes	—
489	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
490	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
495	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
496	C₈	—	2	2	no	—
497	C₈	—	4	2	no	—
498	C₈	—	4	1	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	8	2	no	—
502	C₂²	—	8	1	no	—
503	C₂²	—	8	1	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	2	no	—
509	C₂²	—	8	2	no	—
510	C₂²	—	8	2	no	—
511	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
512	C₂²	D₄²×C₃₁	1	1	yes	characteristic
513	C₂²	C₃₁×#548	1	1	yes	characteristic
514	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
515	C₂²	C₃₁×#577	1	2	yes	—
516	C₄	—	2	4	no	—
517	C₄	—	2	4	no	—
518	C₄	—	2	2	no	—
519	C₄	—	2	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	2	no	—
522	C₄	—	8	2	no	—
523	C₄	—	8	1	no	—
524	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
525	C₄	C₃₁×#537	1	1	yes	characteristic
526	C₄	C₃₁×#547	1	1	yes	characteristic
527	C₄	C₃₁×#553	1	1	yes	characteristic
528	C₂	—	8	2	no	—
529	C₂	—	16	2	no	—
530	C₂	C₃₁×#2416	1	1	yes	characteristic
531	C₂	C₃₁×#3038	1	1	yes	characteristic
532	C₂	C₃₁×#3043	1	1	yes	characteristic
533	C₂	C₃₁×#3265	1	2	yes	—
534	C₂	C₃₁×#3368	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2541	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#460	C₂	semidirect
2. C₂×C₃₁×#460	C₂	semidirect
3. C₂×C₃₁×#440	C₂	semidirect
5. C₃₁×#2351	C₂	semidirect
6. C₃₁×#2067	C₂	semidirect
7. C₃₁×#2041	C₂	semidirect
8. C₃₁×#1961	C₂	semidirect
9. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1880	C₂	semidirect
11. C₃₁×#1804	C₂	semidirect
57. C₂×SD₁₆×C₃₁	D₄	semidirect
69. C₄×Q₈×C₃₁	D₄	semidirect
172. #24742	C₃₁	direct (N × H)
189. Q₈×C₃₁	#121	semidirect
232. C₂×#460	C₂×C₃₁	semidirect
233. C₂×#460	C₂×C₃₁	semidirect
234. C₂×#440	C₂×C₃₁	semidirect
236. #2351	C₂×C₃₁	semidirect
237. #2067	C₂×C₃₁	semidirect
238. #2041	C₂×C₃₁	semidirect
239. #1961	C₂×C₃₁	semidirect
240. #1940	C₂×C₃₁	semidirect
241. #1880	C₂×C₃₁	semidirect
242. #1804	C₂×C₃₁	semidirect
324. C₂×SD₁₆	D₄×C₃₁	semidirect
336. C₄×Q₈	D₄×C₃₁	semidirect
368. C₃₁	#24742	direct (N × H)
456. Q₈	C₃₁×#121	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#460	2
C₂×C₃₁×#440	2
C₃₁×#2541	2
C₃₁×#2351	2
C₃₁×#2067	2
C₃₁×#2041	2
C₃₁×#1961	2
C₃₁×#1940	2
C₃₁×#1880	2
C₃₁×#1804	2
#24742	31

Subgroups — C31×#24742 #21481501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24742 #21481501