Subgroups — C₃₁×#22184 #21478943

Order 7936.

1062 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#121	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1893	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#121	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	—	2	2	no	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#466	C₄	1	8	yes	complemented
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₃₁×#390	C₂²	1	2	yes	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#388	—	2	2	no	—
27	C₃₁×#386	—	2	2	no	—
28	C₃₁×#380	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂³×C₄×C₃₁	D₄	1	2	yes	—
40	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×D₈×C₃₁	—	4	1	no	—
42	C₂×D₈×C₃₁	—	4	1	no	—
43	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
44	C₂²×C₈×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	—	4	1	no	—
46	C₂²×C₈×C₃₁	—	4	1	no	—
47	C₂²×C₈×C₃₁	—	4	1	no	—
48	C₂²×C₈×C₃₁	—	4	1	no	—
49	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#121	—	2	4	no	—
54	C₃₁×#121	C₂×C₄	1	8	yes	complemented
55	C₄×D₄×C₃₁	—	4	2	no	—
56	C₄×D₄×C₃₁	—	4	2	no	—
57	C₄×D₄×C₃₁	—	4	8	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	2	no	—
68	C₂×C₄²×C₃₁	—	4	1	no	—
69	C₂×C₄²×C₃₁	—	4	1	no	—
70	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
73	C₃₁×#107	C₂×C₄	1	4	yes	complemented
74	C₃₁×#106	—	4	1	no	—
75	C₃₁×#106	—	4	1	no	—
76	C₃₁×#102	—	4	2	no	—
77	C₃₁×#102	—	4	2	no	—
78	C₃₁×#102	C₂×C₄	1	8	yes	complemented
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₂⁴×C₃₁	—	2	2	no	—
83	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×D₄×C₃₁	—	2	8	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	4	4	no	—
91	C₂×D₄×C₃₁	—	4	4	no	—
92	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	4	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	4	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	4	2	no	—
103	C₂²×C₄×C₃₁	—	4	4	no	—
104	C₂²×C₄×C₃₁	—	4	4	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	#41	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	D₈×C₃₁	—	4	4	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	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	4	4	no	—
121	C₂×C₈×C₃₁	—	4	4	no	—
122	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	4	4	no	—
130	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
131	C₂²⋊C₄×C₃₁	—	2	8	no	—
132	C₂²⋊C₄×C₃₁	—	4	4	no	—
133	C₂²⋊C₄×C₃₁	—	4	4	no	—
134	C₂²⋊C₄×C₃₁	—	4	4	no	—
135	C₂²⋊C₄×C₃₁	—	4	4	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	4	1	no	—
139	C₄²×C₃₁	—	4	1	no	—
140	C₄²×C₃₁	—	4	4	no	—
141	#22184	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	4	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	4	2	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	C₄×D₄	1	2	yes	—
153	C₂³×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
154	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	D₄×C₃₁	—	2	4	no	—
156	D₄×C₃₁	—	2	4	no	—
157	D₄×C₃₁	—	4	16	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	1	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	4	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	4	8	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	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	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	8	8	no	—
182	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₃₁	#135	1	4	yes	—
184	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
190	C₈×C₃₁	—	2	2	no	—
191	C₈×C₃₁	—	2	2	no	—
192	C₈×C₃₁	—	4	2	no	—
193	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
194	C₄×#121	C₂×C₃₁	1	2	yes	complemented
195	#1926	C₂×C₃₁	1	2	yes	complemented
196	#1921	C₂×C₃₁	1	2	yes	complemented
197	#1893	C₂×C₃₁	1	2	yes	complemented
198	#1879	C₂×C₃₁	1	2	yes	complemented
199	#1864	C₂×C₃₁	1	2	yes	complemented
200	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂×#336	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂²×C₃₁	—	2	2	no	—
203	C₂²×C₃₁	—	2	4	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	2	4	no	—
206	C₂²×C₃₁	—	4	4	no	—
207	C₂²×C₃₁	—	4	4	no	—
208	C₂²×C₃₁	—	4	4	no	—
209	C₂²×C₃₁	—	8	2	no	—
210	C₂²×C₃₁	—	8	2	no	—
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	#442	1	4	yes	—
214	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₃₁	#534	1	1	yes	characteristic
216	C₂²×C₃₁	#537	1	1	yes	characteristic
217	C₂²×C₃₁	C₂×#135	1	2	yes	—
218	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	2	4	no	—
221	C₄×C₃₁	—	4	2	no	—
222	C₄×C₃₁	—	4	4	no	—
223	C₄×C₃₁	—	8	2	no	—
224	C₄×C₃₁	—	8	2	no	—
225	C₄×C₃₁	#518	1	1	yes	characteristic
226	C₄×C₃₁	#520	1	1	yes	characteristic
227	C₄×C₃₁	#538	1	1	yes	characteristic
228	C₄×C₃₁	#539	1	1	yes	characteristic
229	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
231	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×#121	C₂²×C₃₁	1	2	yes	—
233	C₂×C₄×D₄	—	2	2	no	—
234	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
235	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
236	#466	C₄×C₃₁	1	8	yes	complemented
237	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#102	C₂²×C₃₁	1	2	yes	—
241	C₂×#102	C₂²×C₃₁	1	2	yes	—
242	C₂×#102	C₂²×C₃₁	1	2	yes	—
243	#390	C₂²×C₃₁	1	2	yes	—
244	#389	C₂²×C₃₁	1	2	yes	—
245	#388	—	2	2	no	—
246	#386	—	2	2	no	—
247	#380	C₂²×C₃₁	1	2	yes	—
248	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
249	C₄×C₂²⋊C₄	—	2	2	no	—
250	#340	C₂²×C₃₁	1	2	yes	—
251	#340	C₂²×C₃₁	1	2	yes	—
252	#340	C₂²×C₃₁	1	2	yes	—
253	#336	C₂²×C₃₁	1	2	yes	—
254	#336	C₂²×C₃₁	1	2	yes	—
255	C₂×C₃₁	—	2	4	no	—
256	C₂×C₃₁	—	8	4	no	—
257	C₂×C₃₁	#2325	1	1	yes	characteristic
258	C₂×C₃₁	C₄×#135	1	1	yes	characteristic
259	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
260	C₂×C₃₁	#3188	1	2	yes	—
261	C₂×C₃₁	#3247	1	2	yes	—
262	C₂²×D₄	—	2	2	no	—
263	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
264	C₂³×C₄	—	2	2	no	—
265	C₂³×C₄	D₄×C₃₁	1	2	yes	—
266	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×D₈	—	4	1	no	—
268	C₂×D₈	—	4	1	no	—
269	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
270	C₂²×C₈	—	2	2	no	—
271	C₂²×C₈	—	4	1	no	—
272	C₂²×C₈	—	4	1	no	—
273	C₂²×C₈	—	4	1	no	—
274	C₂²×C₈	—	4	1	no	—
275	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
276	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
277	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
279	#121	—	2	4	no	—
280	#121	C₂×C₄×C₃₁	1	8	yes	complemented
281	C₄×D₄	—	4	2	no	—
282	C₄×D₄	—	4	2	no	—
283	C₄×D₄	—	4	8	no	—
284	C₂×C₄⋊C₄	—	2	2	no	—
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₄²	—	2	2	no	—
294	C₂×C₄²	—	4	1	no	—
295	C₂×C₄²	—	4	1	no	—
296	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
298	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
299	#107	C₂×C₄×C₃₁	1	4	yes	complemented
300	#106	—	4	1	no	—
301	#106	—	4	1	no	—
302	#102	—	4	2	no	—
303	#102	—	4	2	no	—
304	#102	C₂×C₄×C₃₁	1	8	yes	complemented
305	#95	—	2	2	no	—
306	#95	—	2	2	no	—
307	#95	—	2	2	no	—
308	C₃₁	#22184	1	1	yes	characteristic, direct product factor, Sylow
309	C₂⁴	—	2	2	no	—
310	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂×D₄	—	2	8	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	4	4	no	—
318	C₂×D₄	—	4	4	no	—
319	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	4	2	no	—
330	C₂²×C₄	—	4	4	no	—
331	C₂²×C₄	—	4	4	no	—
332	C₂²×C₄	—	4	4	no	—
333	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
334	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₂²×C₄	#41×C₃₁	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
339	D₈	—	4	4	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	4	4	no	—
348	C₂×C₈	—	4	4	no	—
349	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	4	4	no	—
357	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
358	C₂²⋊C₄	—	2	8	no	—
359	C₂²⋊C₄	—	4	4	no	—
360	C₂²⋊C₄	—	4	4	no	—
361	C₂²⋊C₄	—	4	4	no	—
362	C₂²⋊C₄	—	4	4	no	—
363	C₄²	—	2	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	4	1	no	—
366	C₄²	—	4	1	no	—
367	C₄²	—	4	4	no	—
368	C₂³	—	2	1	no	—
369	C₂³	—	2	1	no	—
370	C₂³	—	2	2	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	2	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	C₄×D₄×C₃₁	1	2	yes	—
379	C₂³	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
380	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
381	D₄	—	2	4	no	—
382	D₄	—	2	4	no	—
383	D₄	—	4	16	no	—
384	D₄	—	4	2	no	—
385	D₄	—	4	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	4	8	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	8	8	no	—
408	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
409	C₂×C₄	C₃₁×#135	1	4	yes	—
410	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
411	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
413	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
414	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
415	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	4	2	no	—
419	C₂²	—	2	2	no	—
420	C₂²	—	2	4	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	4	4	no	—
424	C₂²	—	4	4	no	—
425	C₂²	—	4	4	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	8	2	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	8	2	no	—
430	C₂²	C₃₁×#442	1	4	yes	—
431	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
432	C₂²	C₃₁×#534	1	1	yes	characteristic
433	C₂²	C₃₁×#537	1	1	yes	characteristic
434	C₂²	C₂×C₃₁×#135	1	2	yes	—
435	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
436	C₄	—	2	2	no	—
437	C₄	—	2	4	no	—
438	C₄	—	4	2	no	—
439	C₄	—	4	4	no	—
440	C₄	—	8	2	no	—
441	C₄	—	8	2	no	—
442	C₄	C₃₁×#518	1	1	yes	characteristic
443	C₄	C₃₁×#520	1	1	yes	characteristic
444	C₄	C₃₁×#538	1	1	yes	characteristic
445	C₄	C₃₁×#539	1	1	yes	characteristic
446	C₂	—	2	4	no	—
447	C₂	—	8	4	no	—
448	C₂	C₃₁×#2325	1	1	yes	characteristic
449	C₂	C₄×C₃₁×#135	1	1	yes	characteristic
450	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
451	C₂	C₃₁×#3188	1	2	yes	—
452	C₂	C₃₁×#3247	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×#466	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. C₄×C₃₁×#121	C₂	semidirect
3. C₃₁×#1926	C₂	semidirect
4. C₃₁×#1921	C₂	semidirect
5. C₃₁×#1893	C₂	semidirect
6. C₃₁×#1879	C₂	semidirect
7. C₃₁×#1864	C₂	semidirect
8. C₃₁×#1807	C₂	semidirect
9. C₂×C₃₁×#336	C₂	semidirect
17. C₃₁×#466	C₄	semidirect
43. C₂×D₈×C₃₁	C₂×C₄	semidirect
49. C₂²×C₈×C₃₁	C₂×C₄	semidirect
54. C₃₁×#121	C₂×C₄	semidirect
73. C₃₁×#107	C₂×C₄	semidirect
78. C₃₁×#102	C₂×C₄	semidirect
141. #22184	C₃₁	direct (N × H)
194. C₄×#121	C₂×C₃₁	semidirect
195. #1926	C₂×C₃₁	semidirect
196. #1921	C₂×C₃₁	semidirect
197. #1893	C₂×C₃₁	semidirect
198. #1879	C₂×C₃₁	semidirect
199. #1864	C₂×C₃₁	semidirect
200. #1807	C₂×C₃₁	semidirect
201. C₂×#336	C₂×C₃₁	semidirect
236. #466	C₄×C₃₁	semidirect
269. C₂×D₈	C₂×C₄×C₃₁	semidirect
275. C₂²×C₈	C₂×C₄×C₃₁	semidirect
280. #121	C₂×C₄×C₃₁	semidirect
299. #107	C₂×C₄×C₃₁	semidirect
304. #102	C₂×C₄×C₃₁	semidirect
308. C₃₁	#22184	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#466	2
C₄×C₃₁×#121	2
C₃₁×#1926	2
C₃₁×#1921	2
C₃₁×#1893	2
C₃₁×#1879	2
C₃₁×#1864	2
C₃₁×#1807	2
C₂×C₃₁×#336	2
#22184	31

Subgroups — C31×#22184 #21478943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22184 #21478943