Subgroups — C₃₁×#21258 #21478017

Order 7936.

1706 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#417	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1983	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1982	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1981	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#532	—	2	4	no	—
11	C₃₁×#532	C₄	1	8	yes	complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#124	C₂²	1	2	yes	—
15	C₂×C₃₁×#121	—	2	1	no	—
16	C₂×C₃₁×#121	—	2	1	no	—
17	C₂×C₄×D₄×C₃₁	—	2	1	no	—
18	C₂×C₄×D₄×C₃₁	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₄²×C₃₁	—	2	1	no	—
21	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#423	—	2	4	no	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
25	C₃₁×#417	—	2	16	no	—
26	C₂×C₃₁×#103	C₂²	1	8	yes	—
27	C₂×C₃₁×#102	C₂²	1	8	yes	—
28	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₃₁×#95	—	2	1	no	—
33	C₂×#41×C₃₁	—	2	4	no	—
34	C₂×#41×C₃₁	—	2	8	no	—
35	C₂×#41×C₃₁	—	2	8	no	—
36	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
37	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×D₄×C₃₁	—	2	2	no	—
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	—	2	1	no	—
41	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	—	2	2	no	—
43	C₂³×C₄×C₃₁	—	2	2	no	—
44	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₃₁	—	2	8	no	—
48	C₂×M₁₆×C₃₁	—	2	4	no	—
49	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
50	C₃₁×C₄⋊Q₈	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
52	C₃₁×#127	—	2	2	no	—
53	C₃₁×#127	C₂×C₄	1	4	yes	complemented
54	C₃₁×#124	—	2	4	no	—
55	C₃₁×#124	C₂×C₄	1	8	yes	complemented
56	C₃₁×#121	—	2	8	no	—
57	C₃₁×#121	—	2	8	no	—
58	C₄×D₄×C₃₁	—	2	8	no	—
59	C₄×D₄×C₃₁	—	2	8	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	complemented
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	4	4	no	—
71	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
72	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	D₄	1	4	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊C₈	—	2	4	no	—
78	C₃₁×C₄⋊C₈	—	2	4	no	—
79	C₃₁×#103	—	2	16	no	—
80	C₃₁×#102	—	2	16	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₂⁴×C₃₁	—	2	2	no	—
84	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
85	#41×C₃₁	—	2	32	no	—
86	#41×C₃₁	—	4	8	no	—
87	#41×C₃₁	—	4	8	no	—
88	C₂×Q₈×C₃₁	—	2	2	no	—
89	C₂×Q₈×C₃₁	—	2	4	no	—
90	C₂×Q₈×C₃₁	—	2	4	no	—
91	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
92	C₂×D₄×C₃₁	—	2	8	no	—
93	C₂×D₄×C₃₁	—	2	4	no	—
94	C₂×D₄×C₃₁	—	2	4	no	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	4	4	no	—
99	C₂×D₄×C₃₁	—	4	8	no	—
100	C₂×D₄×C₃₁	—	4	2	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	8	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	8	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	C₂²×C₄×C₃₁	—	4	4	no	—
111	C₂²×C₄×C₃₁	—	4	4	no	—
112	C₂²×C₄×C₃₁	—	4	8	no	—
113	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
114	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
115	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₃₁	#41	1	4	yes	—
124	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	M₁₆×C₃₁	—	4	8	no	—
126	C₂×C₈×C₃₁	—	2	8	no	—
127	C₂×C₈×C₃₁	—	2	4	no	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₄⋊C₄×C₃₁	—	2	8	no	—
131	C₄⋊C₄×C₃₁	—	2	4	no	—
132	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
133	C₂²⋊C₄×C₃₁	—	2	8	no	—
134	C₂²⋊C₄×C₃₁	—	2	8	no	—
135	C₄²×C₃₁	—	4	4	no	—
136	C₄²×C₃₁	—	4	4	no	—
137	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
138	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
139	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
140	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
141	C₄²×C₃₁	C₂×D₄	1	2	yes	—
142	C₄²×C₃₁	C₂×D₄	1	2	yes	—
143	#21258	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	1	no	—
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	2	4	no	—
148	C₂³×C₃₁	—	4	4	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	4	no	—
152	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂³×C₃₁	#121	1	4	yes	—
154	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	Q₈×C₃₁	—	2	8	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	2	8	no	—
159	D₄×C₃₁	—	4	8	no	—
160	D₄×C₃₁	—	4	16	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	4	8	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	8	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	2	8	no	—
171	C₂×C₄×C₃₁	—	4	4	no	—
172	C₂×C₄×C₃₁	—	4	16	no	—
173	C₂×C₄×C₃₁	—	4	4	no	—
174	C₂×C₄×C₃₁	—	4	4	no	—
175	C₂×C₄×C₃₁	—	4	4	no	—
176	C₂×C₄×C₃₁	—	4	16	no	—
177	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
183	C₂×C₄×C₃₁	#121	1	4	yes	—
184	C₂×C₄×C₃₁	#124	1	4	yes	—
185	C₂×C₄×C₃₁	#127	1	4	yes	—
186	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
187	C₂×C₄×C₃₁	#137	1	2	yes	—
188	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
192	C₈×C₃₁	—	4	4	no	—
193	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
194	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
195	C₂×#417	C₂×C₃₁	1	4	yes	complemented
196	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1983	C₂×C₃₁	1	4	yes	complemented
198	#1982	C₂×C₃₁	1	2	yes	complemented
199	#1981	C₂×C₃₁	1	2	yes	complemented
200	C₂²×C₃₁	—	2	2	no	—
201	C₂²×C₃₁	—	2	2	no	—
202	C₂²×C₃₁	—	2	4	no	—
203	C₂²×C₃₁	—	2	4	no	—
204	C₂²×C₃₁	—	4	4	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	—	4	4	no	—
207	C₂²×C₃₁	—	8	4	no	—
208	C₂²×C₃₁	—	8	4	no	—
209	C₂²×C₃₁	#390	1	4	yes	—
210	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
211	C₂²×C₃₁	C₂×#121	1	2	yes	—
212	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
214	C₂²×C₃₁	#575	1	2	yes	—
215	C₄×C₃₁	—	2	2	no	—
216	C₄×C₃₁	—	2	2	no	—
217	C₄×C₃₁	—	2	2	no	—
218	C₄×C₃₁	—	4	8	no	—
219	C₄×C₃₁	—	4	4	no	—
220	C₄×C₃₁	#390	1	4	yes	—
221	C₄×C₃₁	#417	1	4	yes	complemented
222	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
223	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
224	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
225	C₂²×#41	C₂²×C₃₁	1	2	yes	—
226	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
227	#532	—	2	4	no	—
228	#532	C₄×C₃₁	1	8	yes	complemented
229	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×#124	C₂²×C₃₁	1	2	yes	—
232	C₂×#121	—	2	1	no	—
233	C₂×#121	—	2	1	no	—
234	C₂×C₄×D₄	—	2	1	no	—
235	C₂×C₄×D₄	—	2	1	no	—
236	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
237	C₂²×C₄²	—	2	1	no	—
238	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
239	#423	—	2	4	no	—
240	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
241	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
242	#417	—	2	16	no	—
243	C₂×#103	C₂²×C₃₁	1	8	yes	—
244	C₂×#102	C₂²×C₃₁	1	8	yes	—
245	C₄×C₄⋊C₄	—	2	1	no	—
246	C₄×C₄⋊C₄	—	2	1	no	—
247	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
248	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
249	C₂×#95	—	2	1	no	—
250	C₂×C₃₁	—	2	4	no	—
251	C₂×C₃₁	—	8	4	no	—
252	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
253	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
254	C₂×C₃₁	#2896	1	1	yes	characteristic
255	C₂×C₃₁	#3036	1	2	yes	—
256	C₂×C₃₁	#3037	1	2	yes	—
257	C₂×#41	—	2	4	no	—
258	C₂×#41	—	2	8	no	—
259	C₂×#41	—	2	8	no	—
260	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
261	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
262	C₂²×D₄	—	2	2	no	—
263	C₂²×D₄	—	2	1	no	—
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
266	C₂³×C₄	—	2	2	no	—
267	C₂³×C₄	—	2	2	no	—
268	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
269	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
270	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×M₁₆	—	2	8	no	—
272	C₂×M₁₆	—	2	4	no	—
273	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
274	C₄⋊Q₈	—	2	2	no	—
275	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
276	#127	—	2	2	no	—
277	#127	C₂×C₄×C₃₁	1	4	yes	complemented
278	#124	—	2	4	no	—
279	#124	C₂×C₄×C₃₁	1	8	yes	complemented
280	#121	—	2	8	no	—
281	#121	—	2	8	no	—
282	C₄×D₄	—	2	8	no	—
283	C₄×D₄	—	2	8	no	—
284	C₂×C₄⋊C₄	—	2	2	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	4	no	—
288	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	complemented
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	—	4	4	no	—
295	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
296	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
298	C₂×C₄²	D₄×C₃₁	1	4	yes	—
299	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
301	C₄⋊C₈	—	2	4	no	—
302	C₄⋊C₈	—	2	4	no	—
303	#103	—	2	16	no	—
304	#102	—	2	16	no	—
305	#95	—	2	2	no	—
306	#95	—	2	2	no	—
307	C₃₁	#21258	1	1	yes	characteristic, direct product factor, Sylow
308	C₂⁴	—	2	2	no	—
309	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
310	#41	—	2	32	no	—
311	#41	—	4	8	no	—
312	#41	—	4	8	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	4	no	—
315	C₂×Q₈	—	2	4	no	—
316	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
317	C₂×D₄	—	2	8	no	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	2	4	no	—
320	C₂×D₄	—	2	4	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	4	no	—
323	C₂×D₄	—	4	4	no	—
324	C₂×D₄	—	4	8	no	—
325	C₂×D₄	—	4	2	no	—
326	C₂×D₄	—	4	2	no	—
327	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	8	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	—	4	4	no	—
336	C₂²×C₄	—	4	4	no	—
337	C₂²×C₄	—	4	8	no	—
338	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
339	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
340	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
348	C₂²×C₄	#41×C₃₁	1	4	yes	—
349	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
350	M₁₆	—	4	8	no	—
351	C₂×C₈	—	2	8	no	—
352	C₂×C₈	—	2	4	no	—
353	C₄⋊C₄	—	2	4	no	—
354	C₄⋊C₄	—	2	4	no	—
355	C₄⋊C₄	—	2	8	no	—
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
358	C₂²⋊C₄	—	2	8	no	—
359	C₂²⋊C₄	—	2	8	no	—
360	C₄²	—	4	4	no	—
361	C₄²	—	4	4	no	—
362	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
363	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
364	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
365	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
366	C₄²	C₂×D₄×C₃₁	1	2	yes	—
367	C₄²	C₂×D₄×C₃₁	1	2	yes	—
368	C₂³	—	2	1	no	—
369	C₂³	—	2	1	no	—
370	C₂³	—	2	2	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	4	4	no	—
373	C₂³	—	4	2	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	4	no	—
376	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
377	C₂³	C₃₁×#121	1	4	yes	—
378	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
379	Q₈	—	2	8	no	—
380	Q₈	—	4	2	no	—
381	Q₈	—	4	2	no	—
382	D₄	—	2	8	no	—
383	D₄	—	4	8	no	—
384	D₄	—	4	16	no	—
385	D₄	—	4	8	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	8	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	8	no	—
395	C₂×C₄	—	4	4	no	—
396	C₂×C₄	—	4	16	no	—
397	C₂×C₄	—	4	4	no	—
398	C₂×C₄	—	4	4	no	—
399	C₂×C₄	—	4	4	no	—
400	C₂×C₄	—	4	16	no	—
401	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
402	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
403	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
404	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
405	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
406	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
407	C₂×C₄	C₃₁×#121	1	4	yes	—
408	C₂×C₄	C₃₁×#124	1	4	yes	—
409	C₂×C₄	C₃₁×#127	1	4	yes	—
410	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
411	C₂×C₄	C₃₁×#137	1	2	yes	—
412	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
414	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
416	C₈	—	4	4	no	—
417	C₂²	—	2	2	no	—
418	C₂²	—	2	2	no	—
419	C₂²	—	2	4	no	—
420	C₂²	—	2	4	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	4	4	no	—
423	C₂²	—	4	4	no	—
424	C₂²	—	8	4	no	—
425	C₂²	—	8	4	no	—
426	C₂²	C₃₁×#390	1	4	yes	—
427	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
428	C₂²	C₂×C₃₁×#121	1	2	yes	—
429	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
430	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
431	C₂²	C₃₁×#575	1	2	yes	—
432	C₄	—	2	2	no	—
433	C₄	—	2	2	no	—
434	C₄	—	2	2	no	—
435	C₄	—	4	8	no	—
436	C₄	—	4	4	no	—
437	C₄	C₃₁×#390	1	4	yes	—
438	C₄	C₃₁×#417	1	4	yes	complemented
439	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
440	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
441	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
442	C₂	—	2	4	no	—
443	C₂	—	8	4	no	—
444	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
445	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
446	C₂	C₃₁×#2896	1	1	yes	characteristic
447	C₂	C₃₁×#3036	1	2	yes	—
448	C₂	C₃₁×#3037	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#423	C₂	semidirect
3. C₂×C₃₁×#417	C₂	semidirect
4. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#1983	C₂	semidirect
6. C₃₁×#1982	C₂	semidirect
7. C₃₁×#1981	C₂	semidirect
11. C₃₁×#532	C₄	semidirect
51. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
53. C₃₁×#127	C₂×C₄	semidirect
55. C₃₁×#124	C₂×C₄	semidirect
64. C₂×C₄⋊C₄×C₃₁	D₄	semidirect
143. #21258	C₃₁	direct (N × H)
194. C₂×#423	C₂×C₃₁	semidirect
195. C₂×#417	C₂×C₃₁	semidirect
196. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
197. #1983	C₂×C₃₁	semidirect
198. #1982	C₂×C₃₁	semidirect
199. #1981	C₂×C₃₁	semidirect
221. C₄×C₃₁	#417	semidirect
228. #532	C₄×C₃₁	semidirect
275. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
277. #127	C₂×C₄×C₃₁	semidirect
279. #124	C₂×C₄×C₃₁	semidirect
288. C₂×C₄⋊C₄	D₄×C₃₁	semidirect
307. C₃₁	#21258	direct (N × H)
438. C₄	C₃₁×#417	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#532	2
C₂×C₃₁×#423	2
C₂×C₃₁×#417	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1983	2
C₃₁×#1982	2
C₃₁×#1981	2
#21258	31

Subgroups — C31×#21258 #21478017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21258 #21478017