Subgroups — C₃₁×#26021 #21482780

Order 7936.

734 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	C₃₁×#2552	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2065	C₂	1	2	yes	maximal
4	C₃₁×#2025	C₂	1	2	yes	maximal
5	C₃₁×#1989	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1919	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#125	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	—	2	1	no	—
15	C₂×C₃₁×#106	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#398	—	2	2	no	—
23	C₃₁×#390	C₂²	1	2	yes	—
24	C₃₁×#389	—	2	1	no	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#388	—	2	2	no	—
27	C₃₁×#384	—	2	1	no	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#384	C₂²	1	1	yes	characteristic
30	C₃₁×#380	—	2	1	no	—
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	—	2	1	no	—
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₃₁×#125	—	2	4	no	—
47	C₃₁×#125	—	4	2	no	—
48	C₄×D₄×C₃₁	—	2	2	no	—
49	C₄×D₄×C₃₁	Q₈	1	4	yes	—
50	C₃₁×#117	—	4	4	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
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	1	no	—
70	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#106	—	2	4	no	—
73	C₃₁×C₄⋊C₈	—	2	2	no	—
74	C₃₁×C₄⋊C₈	—	4	2	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₃₁×#102	—	4	2	no	—
78	C₄×C₈×C₃₁	—	2	2	no	—
79	C₃₁×#95	—	2	1	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	D₄	1	2	yes	—
83	C₂⁴×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	2	1	no	—
85	C₂×D₄×C₃₁	—	2	1	no	—
86	C₂×D₄×C₃₁	—	2	1	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	—	2	1	no	—
89	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₃₁	—	2	1	no	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	2	4	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	4	2	no	—
97	C₂²×C₄×C₃₁	—	4	2	no	—
98	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₂×C₈×C₃₁	—	2	1	no	—
104	C₂×C₈×C₃₁	—	2	1	no	—
105	C₂×C₈×C₃₁	—	2	1	no	—
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	1	no	—
115	C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₄⋊C₄×C₃₁	—	2	1	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	4	4	no	—
122	C₄⋊C₄×C₃₁	—	4	4	no	—
123	C₄⋊C₄×C₃₁	—	4	2	no	—
124	C₄⋊C₄×C₃₁	—	4	2	no	—
125	C₄⋊C₄×C₃₁	—	8	1	no	—
126	C₄⋊C₄×C₃₁	—	8	1	no	—
127	C₄⋊C₄×C₃₁	—	8	2	no	—
128	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
129	C₄⋊C₄×C₃₁	#41	1	2	yes	—
130	C₄⋊C₄×C₃₁	#41	1	2	yes	—
131	C₂²⋊C₄×C₃₁	—	2	4	no	—
132	C₂²⋊C₄×C₃₁	—	4	4	no	—
133	C₂²⋊C₄×C₃₁	—	8	2	no	—
134	C₄²×C₃₁	—	2	2	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	4	4	no	—
137	C₄²×C₃₁	—	4	2	no	—
138	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
139	C₄²×C₃₁	#41	1	2	yes	—
140	#26021	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	4	1	no	—
143	C₂³×C₃₁	—	4	1	no	—
144	C₂³×C₃₁	—	4	1	no	—
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	—	4	1	no	—
147	C₂³×C₃₁	—	4	1	no	—
148	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	D₄×C₃₁	—	2	2	no	—
150	D₄×C₃₁	—	2	2	no	—
151	D₄×C₃₁	—	4	1	no	—
152	D₄×C₃₁	—	4	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	8	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	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	C₂²⋊Q₈	1	2	yes	—
178	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
179	C₂×C₄×C₃₁	#135	1	2	yes	—
180	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
181	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
187	C₈×C₃₁	—	4	1	no	—
188	C₈×C₃₁	—	4	2	no	—
189	C₈×C₃₁	—	4	1	no	—
190	#2552	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#2354	C₂×C₃₁	1	1	yes	characteristic
192	#2065	C₂×C₃₁	1	2	yes	—
193	#2025	C₂×C₃₁	1	2	yes	—
194	#1989	C₂×C₃₁	1	2	yes	complemented
195	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1919	C₂×C₃₁	1	2	yes	complemented
198	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1802	C₂×C₃₁	1	1	yes	characteristic
200	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
201	C₂²×C₃₁	—	4	2	no	—
202	C₂²×C₃₁	—	4	2	no	—
203	C₂²×C₃₁	—	4	2	no	—
204	C₂²×C₃₁	—	8	1	no	—
205	C₂²×C₃₁	—	8	1	no	—
206	C₂²×C₃₁	—	8	1	no	—
207	C₂²×C₃₁	—	8	1	no	—
208	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
209	C₂²×C₃₁	#546	1	1	yes	characteristic
210	C₂²×C₃₁	#548	1	1	yes	characteristic
211	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
212	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₂²×C₃₁	#577	1	1	yes	characteristic
214	C₂²×C₃₁	#577	1	1	yes	characteristic
215	C₄×C₃₁	—	2	2	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	—	4	2	no	—
218	C₄×C₃₁	—	4	2	no	—
219	C₄×C₃₁	—	4	2	no	—
220	C₄×C₃₁	—	4	2	no	—
221	C₄×C₃₁	—	8	2	no	—
222	C₄×C₃₁	—	8	2	no	—
223	C₄×C₃₁	#478	1	4	yes	—
224	C₄×C₃₁	#533	1	1	yes	characteristic
225	C₄×C₃₁	#544	1	1	yes	characteristic
226	C₄×C₃₁	#555	1	1	yes	characteristic
227	C₄×C₃₁	#556	1	1	yes	characteristic
228	C₂×#125	C₂²×C₃₁	1	2	yes	—
229	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#117	—	2	1	no	—
231	C₂×#106	C₂²×C₃₁	1	2	yes	—
232	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
234	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#102	C₂²×C₃₁	1	2	yes	—
237	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
238	#398	—	2	2	no	—
239	#390	C₂²×C₃₁	1	2	yes	—
240	#389	—	2	1	no	—
241	#389	C₂²×C₃₁	1	2	yes	—
242	#388	—	2	2	no	—
243	#384	—	2	1	no	—
244	#384	C₂²×C₃₁	1	1	yes	characteristic
245	#384	C₂²×C₃₁	1	1	yes	characteristic
246	#380	—	2	1	no	—
247	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
248	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
249	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
250	#340	C₂²×C₃₁	1	2	yes	—
251	#340	C₂²×C₃₁	1	2	yes	—
252	#340	C₂²×C₃₁	1	2	yes	—
253	#340	C₂²×C₃₁	1	1	yes	characteristic
254	#340	C₂²×C₃₁	1	2	yes	—
255	#340	C₂²×C₃₁	1	1	yes	characteristic
256	#336	C₂²×C₃₁	1	2	yes	—
257	C₂×C₃₁	—	8	2	no	—
258	C₂×C₃₁	#2439	1	1	yes	characteristic
259	C₂×C₃₁	C₂×#478	1	1	yes	characteristic
260	C₂×C₃₁	#3084	1	1	yes	characteristic
261	C₂×C₃₁	#3303	1	1	yes	characteristic
262	C₂×C₃₁	#3332	1	1	yes	characteristic
263	C₂×C₃₁	#3360	1	1	yes	characteristic
264	C₂×C₃₁	#3368	1	1	yes	characteristic
265	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
266	C₂³×C₄	—	2	1	no	—
267	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	#125	—	2	4	no	—
271	#125	—	4	2	no	—
272	C₄×D₄	—	2	2	no	—
273	C₄×D₄	Q₈×C₃₁	1	4	yes	—
274	#117	—	4	4	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
295	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
296	#106	—	2	4	no	—
297	C₄⋊C₈	—	2	2	no	—
298	C₄⋊C₈	—	4	2	no	—
299	#102	—	2	2	no	—
300	#102	—	2	2	no	—
301	#102	—	4	2	no	—
302	C₄×C₈	—	2	2	no	—
303	#95	—	2	2	no	—
304	#95	—	2	1	no	—
305	#95	—	2	2	no	—
306	#95	D₄×C₃₁	1	2	yes	—
307	C₃₁	#26021	1	1	yes	characteristic, direct product factor, Sylow
308	C₂⁴	—	2	1	no	—
309	C₂×D₄	—	2	1	no	—
310	C₂×D₄	—	2	1	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	4	2	no	—
322	C₂²×C₄	—	4	2	no	—
323	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
324	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
325	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	4	4	no	—
347	C₄⋊C₄	—	4	4	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	—	4	2	no	—
350	C₄⋊C₄	—	8	1	no	—
351	C₄⋊C₄	—	8	1	no	—
352	C₄⋊C₄	—	8	2	no	—
353	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
354	C₄⋊C₄	#41×C₃₁	1	2	yes	—
355	C₄⋊C₄	#41×C₃₁	1	2	yes	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	4	4	no	—
358	C₂²⋊C₄	—	8	2	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	4	4	no	—
362	C₄²	—	4	2	no	—
363	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
364	C₄²	#41×C₃₁	1	2	yes	—
365	C₂³	—	2	2	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	1	no	—
368	C₂³	—	4	1	no	—
369	C₂³	—	4	1	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
373	D₄	—	2	2	no	—
374	D₄	—	2	2	no	—
375	D₄	—	4	1	no	—
376	D₄	—	4	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	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	—	4	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	8	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	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
402	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
403	C₂×C₄	C₃₁×#135	1	2	yes	—
404	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
405	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
409	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
410	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
411	C₈	—	4	1	no	—
412	C₈	—	4	2	no	—
413	C₈	—	4	1	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	8	1	no	—
418	C₂²	—	8	1	no	—
419	C₂²	—	8	1	no	—
420	C₂²	—	8	1	no	—
421	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
422	C₂²	C₃₁×#546	1	1	yes	characteristic
423	C₂²	C₃₁×#548	1	1	yes	characteristic
424	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
425	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
426	C₂²	C₃₁×#577	1	1	yes	characteristic
427	C₂²	C₃₁×#577	1	1	yes	characteristic
428	C₄	—	2	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	8	2	no	—
435	C₄	—	8	2	no	—
436	C₄	C₃₁×#478	1	4	yes	—
437	C₄	C₃₁×#533	1	1	yes	characteristic
438	C₄	C₃₁×#544	1	1	yes	characteristic
439	C₄	C₃₁×#555	1	1	yes	characteristic
440	C₄	C₃₁×#556	1	1	yes	characteristic
441	C₂	—	8	2	no	—
442	C₂	C₃₁×#2439	1	1	yes	characteristic
443	C₂	C₂×C₃₁×#478	1	1	yes	characteristic
444	C₂	C₃₁×#3084	1	1	yes	characteristic
445	C₂	C₃₁×#3303	1	1	yes	characteristic
446	C₂	C₃₁×#3332	1	1	yes	characteristic
447	C₂	C₃₁×#3360	1	1	yes	characteristic
448	C₂	C₃₁×#3368	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
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. C₃₁×#2552	C₂	semidirect
5. C₃₁×#1989	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
7. C₃₁×#1939	C₂	semidirect
8. C₃₁×#1919	C₂	semidirect
9. C₃₁×#1834	C₂	semidirect
140. #26021	C₃₁	direct (N × H)
190. #2552	C₂×C₃₁	semidirect
194. #1989	C₂×C₃₁	semidirect
195. #1948	C₂×C₃₁	semidirect
196. #1939	C₂×C₃₁	semidirect
197. #1919	C₂×C₃₁	semidirect
198. #1834	C₂×C₃₁	semidirect
307. C₃₁	#26021	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×#2552	2
C₃₁×#2354	2
C₃₁×#2065	2
C₃₁×#2025	2
C₃₁×#1989	2
C₃₁×#1948	2
C₃₁×#1939	2
C₃₁×#1919	2
C₃₁×#1834	2
C₃₁×#1802	2
C₄×C₃₁×#102	2
#26021	31

Subgroups — C31×#26021 #21482780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26021 #21482780