Subgroups — C₃₁×#60798 #21517557

Order 7936.

2822 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3217	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3208	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3205	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3151	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3138	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3038	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3003	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₃₁	—	2	2	no	—
12	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
14	C₃₁×#533	C₂²	1	2	yes	complemented
15	C₃₁×#533	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#532	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#532	C₂²	1	2	yes	complemented
18	C₂×C₃₁×#127	—	2	2	no	—
19	C₂×C₃₁×#127	—	2	2	no	—
20	C₂×C₃₁×#127	—	2	2	no	—
21	C₂×C₃₁×#127	C₂²	1	2	yes	—
22	C₂×C₃₁×#127	C₂²	1	4	yes	—
23	C₂×C₃₁×#127	C₂²	1	4	yes	—
24	C₂×C₃₁×#127	C₂²	1	2	yes	—
25	C₃₁×#529	—	2	2	no	—
26	C₃₁×#529	—	2	2	no	—
27	C₃₁×#529	C₂²	1	4	yes	complemented
28	C₃₁×#525	—	2	2	no	—
29	C₃₁×#525	C₂²	1	2	yes	—
30	C₄×#41×C₃₁	—	2	2	no	—
31	C₄×#41×C₃₁	C₂²	1	2	yes	complemented
32	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#496	C₂²	1	8	yes	complemented
34	C₃₁×#493	C₂²	1	4	yes	complemented
35	C₃₁×#492	C₂²	1	4	yes	complemented
36	C₃₁×#490	C₂²	1	8	yes	complemented
37	C₃₁×#486	C₂²	1	8	yes	complemented
38	C₃₁×#460	C₂²	1	16	yes	complemented
39	C₃₁×#459	C₂²	1	16	yes	complemented
40	C₃₁×#450	C₂²	1	16	yes	complemented
41	C₃₁×#447	C₂²	1	16	yes	complemented
42	C₃₁×#432	C₂²	1	4	yes	complemented
43	C₃₁×#431	C₂²	1	4	yes	complemented
44	C₃₁×#423	C₂²	1	2	yes	complemented
45	C₃₁×#418	C₂²	1	8	yes	complemented
46	C₃₁×#407	C₂²	1	2	yes	—
47	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
48	C₂⁵×C₃₁	—	2	2	no	—
49	C₂×#41×C₃₁	—	2	2	no	—
50	C₂×#41×C₃₁	C₂³	1	2	yes	—
51	C₂²×D₄×C₃₁	—	2	4	no	—
52	C₂²×D₄×C₃₁	—	2	2	no	—
53	C₂²×D₄×C₃₁	—	2	4	no	—
54	C₂²×D₄×C₃₁	—	2	4	no	—
55	C₂²×D₄×C₃₁	—	2	4	no	—
56	C₂²×D₄×C₃₁	—	2	4	no	—
57	C₂²×D₄×C₃₁	—	2	4	no	—
58	C₂²×D₄×C₃₁	—	2	4	no	—
59	C₂²×D₄×C₃₁	—	4	8	no	—
60	C₂²×D₄×C₃₁	—	4	16	no	—
61	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
62	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
63	C₂³×C₄×C₃₁	D₄	1	2	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×AGL(1,ℤ/8)	—	4	16	no	—
66	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	complemented
67	C₂×D₈×C₃₁	C₂³	1	16	yes	complemented
68	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic, complemented
70	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	complemented
71	C₃₁×#127	—	2	2	no	—
72	C₃₁×#127	—	2	4	no	—
73	C₃₁×#127	—	2	2	no	—
74	C₃₁×#127	—	2	4	no	—
75	C₃₁×#127	—	2	4	no	—
76	C₃₁×#127	—	2	4	no	—
77	C₃₁×#127	—	2	4	no	—
78	C₃₁×#127	C₂³	1	8	yes	—
79	C₃₁×#127	C₂³	1	4	yes	—
80	C₃₁×#127	C₂³	1	4	yes	—
81	C₃₁×#127	C₂³	1	1	yes	characteristic, complemented
82	C₃₁×#126	—	2	4	no	—
83	C₃₁×#125	—	2	2	no	—
84	C₃₁×#125	C₂³	1	2	yes	complemented
85	C₃₁×#124	—	2	2	no	—
86	C₃₁×#124	C₂³	1	2	yes	complemented
87	C₃₁×#123	—	2	4	no	—
88	C₃₁×#123	—	2	2	no	—
89	C₃₁×C₂²⋊Q₈	—	2	2	no	—
90	C₃₁×C₂²⋊Q₈	—	2	2	no	—
91	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
92	C₃₁×#121	—	2	2	no	—
93	C₃₁×#121	—	2	2	no	—
94	C₃₁×#121	C₂³	1	4	yes	—
95	C₃₁×#120	—	2	2	no	—
96	C₄×Q₈×C₃₁	—	2	2	no	—
97	C₄×Q₈×C₃₁	—	2	2	no	—
98	C₄×Q₈×C₃₁	C₂³	1	4	yes	complemented
99	C₄×D₄×C₃₁	—	2	4	no	—
100	C₄×D₄×C₃₁	—	2	4	no	—
101	C₄×D₄×C₃₁	—	2	4	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	2	no	—
104	C₄×D₄×C₃₁	C₂³	1	4	yes	complemented
105	C₃₁×#117	—	2	2	no	—
106	C₃₁×#117	—	2	4	no	—
107	C₃₁×#117	—	2	2	no	—
108	C₃₁×#117	C₂³	1	2	yes	—
109	C₂×C₄²×C₃₁	—	2	2	no	—
110	C₂×C₄²×C₃₁	D₄	1	2	yes	—
111	C₂×C₄²×C₃₁	D₄	1	2	yes	—
112	C₂×C₄²×C₃₁	D₄	1	2	yes	—
113	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
117	C₃₁×C₄⋊C₈	C₂³	1	8	yes	complemented
118	C₃₁×#102	C₂³	1	32	yes	complemented
119	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
120	C₃₁×#97	C₂³	1	4	yes	complemented
121	C₄×C₈×C₃₁	C₂³	1	4	yes	complemented
122	C₂⁴×C₃₁	—	2	4	no	—
123	C₂⁴×C₃₁	—	2	8	no	—
124	C₂⁴×C₃₁	—	4	8	no	—
125	C₂⁴×C₃₁	—	4	16	no	—
126	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
127	#41×C₃₁	—	2	8	no	—
128	#41×C₃₁	—	4	8	no	—
129	C₂×Q₈×C₃₁	—	2	2	no	—
130	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
131	C₂×D₄×C₃₁	—	2	2	no	—
132	C₂×D₄×C₃₁	—	2	4	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	8	no	—
135	C₂×D₄×C₃₁	—	2	4	no	—
136	C₂×D₄×C₃₁	—	2	8	no	—
137	C₂×D₄×C₃₁	—	2	8	no	—
138	C₂×D₄×C₃₁	—	2	8	no	—
139	C₂×D₄×C₃₁	—	2	8	no	—
140	C₂×D₄×C₃₁	—	2	16	no	—
141	C₂×D₄×C₃₁	—	4	16	no	—
142	C₂×D₄×C₃₁	—	4	8	no	—
143	C₂×D₄×C₃₁	—	4	16	no	—
144	C₂×D₄×C₃₁	—	4	16	no	—
145	C₂×D₄×C₃₁	—	4	16	no	—
146	C₂×D₄×C₃₁	—	4	16	no	—
147	C₂×D₄×C₃₁	—	4	16	no	—
148	C₂×D₄×C₃₁	—	4	16	no	—
149	C₂×D₄×C₃₁	—	4	16	no	—
150	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
151	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₃₁	—	2	8	no	—
153	C₂²×C₄×C₃₁	—	2	4	no	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
166	SD₁₆×C₃₁	—	4	16	no	—
167	D₈×C₃₁	—	4	16	no	—
168	M₁₆×C₃₁	—	2	8	no	—
169	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	4	no	—
174	C₄⋊C₄×C₃₁	—	2	4	no	—
175	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
176	C₂²⋊C₄×C₃₁	—	2	4	no	—
177	C₂²⋊C₄×C₃₁	—	2	2	no	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₄²×C₃₁	—	2	4	no	—
182	C₄²×C₃₁	—	2	2	no	—
183	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₃₁	C₂×D₄	1	2	yes	—
185	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂×D₄	1	2	yes	—
187	C₄²×C₃₁	C₂×D₄	1	2	yes	—
188	C₄²×C₃₁	C₂×D₄	1	2	yes	—
189	C₄²×C₃₁	C₂×D₄	1	2	yes	—
190	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
191	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
192	C₄²×C₃₁	C₂⁴	1	2	yes	—
193	#60798	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	2	4	no	—
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	—	2	8	no	—
198	C₂³×C₃₁	—	4	16	no	—
199	C₂³×C₃₁	—	4	16	no	—
200	C₂³×C₃₁	—	4	16	no	—
201	C₂³×C₃₁	—	4	8	no	—
202	C₂³×C₃₁	—	4	16	no	—
203	C₂³×C₃₁	—	8	32	no	—
204	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
206	Q₈×C₃₁	—	2	4	no	—
207	Q₈×C₃₁	—	4	4	no	—
208	D₄×C₃₁	—	2	4	no	—
209	D₄×C₃₁	—	2	16	no	—
210	D₄×C₃₁	—	4	16	no	—
211	D₄×C₃₁	—	4	16	no	—
212	D₄×C₃₁	—	4	16	no	—
213	D₄×C₃₁	—	4	4	no	—
214	D₄×C₃₁	—	4	8	no	—
215	D₄×C₃₁	—	4	8	no	—
216	D₄×C₃₁	—	4	16	no	—
217	D₄×C₃₁	—	4	16	no	—
218	D₄×C₃₁	—	4	4	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	4	4	no	—
230	C₂×C₄×C₃₁	—	4	8	no	—
231	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	8	yes	complemented
232	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
239	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
240	C₈×C₃₁	—	2	8	no	—
241	#3444	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂²×#127	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#3217	C₂×C₃₁	1	8	yes	complemented
244	#3208	C₂×C₃₁	1	4	yes	complemented
245	#3205	C₂×C₃₁	1	4	yes	complemented
246	#3151	C₂×C₃₁	1	2	yes	complemented
247	#3138	C₂×C₃₁	1	2	yes	complemented
248	#3038	C₂×C₃₁	1	4	yes	complemented
249	#3003	C₂×C₃₁	1	4	yes	complemented
250	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×C₃₁	—	2	8	no	—
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	8	no	—
257	C₂²×C₃₁	—	4	4	no	—
258	C₂²×C₃₁	—	4	16	no	—
259	C₂²×C₃₁	—	8	16	no	—
260	C₂²×C₃₁	—	8	32	no	—
261	C₂²×C₃₁	#535	1	2	yes	—
262	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
263	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
264	C₄×C₃₁	—	2	2	no	—
265	C₄×C₃₁	—	2	2	no	—
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	4	2	no	—
268	C₄×C₃₁	—	4	4	no	—
269	C₄×C₃₁	#535	1	2	yes	—
270	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
271	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
272	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
273	C₂³×D₄	—	2	2	no	—
274	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
275	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	8	yes	complemented
276	#533	C₂²×C₃₁	1	2	yes	complemented
277	#533	C₂²×C₃₁	1	1	yes	characteristic, complemented
278	#532	C₂²×C₃₁	1	1	yes	characteristic, complemented
279	#532	C₂²×C₃₁	1	2	yes	complemented
280	C₂×#127	—	2	2	no	—
281	C₂×#127	—	2	2	no	—
282	C₂×#127	—	2	2	no	—
283	C₂×#127	C₂²×C₃₁	1	2	yes	—
284	C₂×#127	C₂²×C₃₁	1	4	yes	—
285	C₂×#127	C₂²×C₃₁	1	4	yes	—
286	C₂×#127	C₂²×C₃₁	1	2	yes	—
287	#529	—	2	2	no	—
288	#529	—	2	2	no	—
289	#529	C₂²×C₃₁	1	4	yes	complemented
290	#525	—	2	2	no	—
291	#525	C₂²×C₃₁	1	2	yes	—
292	C₄×#41	—	2	2	no	—
293	C₄×#41	C₂²×C₃₁	1	2	yes	complemented
294	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
295	#496	C₂²×C₃₁	1	8	yes	complemented
296	#493	C₂²×C₃₁	1	4	yes	complemented
297	#492	C₂²×C₃₁	1	4	yes	complemented
298	#490	C₂²×C₃₁	1	8	yes	complemented
299	#486	C₂²×C₃₁	1	8	yes	complemented
300	#460	C₂²×C₃₁	1	16	yes	complemented
301	#459	C₂²×C₃₁	1	16	yes	complemented
302	#450	C₂²×C₃₁	1	16	yes	complemented
303	#447	C₂²×C₃₁	1	16	yes	complemented
304	#432	C₂²×C₃₁	1	4	yes	complemented
305	#431	C₂²×C₃₁	1	4	yes	complemented
306	#423	C₂²×C₃₁	1	2	yes	complemented
307	#418	C₂²×C₃₁	1	8	yes	complemented
308	#407	C₂²×C₃₁	1	2	yes	—
309	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
310	C₂×C₃₁	—	2	2	no	—
311	C₂×C₃₁	—	2	4	no	—
312	C₂×C₃₁	—	8	2	no	—
313	C₂×C₃₁	—	8	8	no	—
314	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
315	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	2	yes	—
316	C₂⁵	—	2	2	no	—
317	C₂×#41	—	2	2	no	—
318	C₂×#41	C₂³×C₃₁	1	2	yes	—
319	C₂²×D₄	—	2	4	no	—
320	C₂²×D₄	—	2	2	no	—
321	C₂²×D₄	—	2	4	no	—
322	C₂²×D₄	—	2	4	no	—
323	C₂²×D₄	—	2	4	no	—
324	C₂²×D₄	—	2	4	no	—
325	C₂²×D₄	—	2	4	no	—
326	C₂²×D₄	—	2	4	no	—
327	C₂²×D₄	—	4	8	no	—
328	C₂²×D₄	—	4	16	no	—
329	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
330	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
331	C₂³×C₄	D₄×C₃₁	1	2	yes	—
332	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
333	AGL(1,ℤ/8)	—	4	16	no	—
334	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	complemented
335	C₂×D₈	C₂³×C₃₁	1	16	yes	complemented
336	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
337	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic, complemented
338	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	complemented
339	#127	—	2	2	no	—
340	#127	—	2	4	no	—
341	#127	—	2	2	no	—
342	#127	—	2	4	no	—
343	#127	—	2	4	no	—
344	#127	—	2	4	no	—
345	#127	—	2	4	no	—
346	#127	C₂³×C₃₁	1	8	yes	—
347	#127	C₂³×C₃₁	1	4	yes	—
348	#127	C₂³×C₃₁	1	4	yes	—
349	#127	C₂³×C₃₁	1	1	yes	characteristic, complemented
350	#126	—	2	4	no	—
351	#125	—	2	2	no	—
352	#125	C₂³×C₃₁	1	2	yes	complemented
353	#124	—	2	2	no	—
354	#124	C₂³×C₃₁	1	2	yes	complemented
355	#123	—	2	4	no	—
356	#123	—	2	2	no	—
357	C₂²⋊Q₈	—	2	2	no	—
358	C₂²⋊Q₈	—	2	2	no	—
359	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
360	#121	—	2	2	no	—
361	#121	—	2	2	no	—
362	#121	C₂³×C₃₁	1	4	yes	—
363	#120	—	2	2	no	—
364	C₄×Q₈	—	2	2	no	—
365	C₄×Q₈	—	2	2	no	—
366	C₄×Q₈	C₂³×C₃₁	1	4	yes	complemented
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	4	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	C₂³×C₃₁	1	4	yes	complemented
373	#117	—	2	2	no	—
374	#117	—	2	4	no	—
375	#117	—	2	2	no	—
376	#117	C₂³×C₃₁	1	2	yes	—
377	C₂×C₄²	—	2	2	no	—
378	C₂×C₄²	D₄×C₃₁	1	2	yes	—
379	C₂×C₄²	D₄×C₃₁	1	2	yes	—
380	C₂×C₄²	D₄×C₃₁	1	2	yes	—
381	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
382	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
383	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
384	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
385	C₄⋊C₈	C₂³×C₃₁	1	8	yes	complemented
386	#102	C₂³×C₃₁	1	32	yes	complemented
387	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
388	#97	C₂³×C₃₁	1	4	yes	complemented
389	C₄×C₈	C₂³×C₃₁	1	4	yes	complemented
390	C₃₁	#60798	1	1	yes	characteristic, direct product factor, Sylow
391	C₂⁴	—	2	4	no	—
392	C₂⁴	—	2	8	no	—
393	C₂⁴	—	4	8	no	—
394	C₂⁴	—	4	16	no	—
395	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
396	#41	—	2	8	no	—
397	#41	—	4	8	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	8	no	—
406	C₂×D₄	—	2	8	no	—
407	C₂×D₄	—	2	8	no	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	2	16	no	—
410	C₂×D₄	—	4	16	no	—
411	C₂×D₄	—	4	8	no	—
412	C₂×D₄	—	4	16	no	—
413	C₂×D₄	—	4	16	no	—
414	C₂×D₄	—	4	16	no	—
415	C₂×D₄	—	4	16	no	—
416	C₂×D₄	—	4	16	no	—
417	C₂×D₄	—	4	16	no	—
418	C₂×D₄	—	4	16	no	—
419	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
420	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
421	C₂²×C₄	—	2	8	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
427	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
428	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
430	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
431	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
432	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
433	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
435	SD₁₆	—	4	16	no	—
436	D₈	—	4	16	no	—
437	M₁₆	—	2	8	no	—
438	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₄²	—	2	4	no	—
451	C₄²	—	2	2	no	—
452	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
453	C₄²	C₂×D₄×C₃₁	1	2	yes	—
454	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₄²	C₂×D₄×C₃₁	1	2	yes	—
456	C₄²	C₂×D₄×C₃₁	1	2	yes	—
457	C₄²	C₂×D₄×C₃₁	1	2	yes	—
458	C₄²	C₂×D₄×C₃₁	1	2	yes	—
459	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
460	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
461	C₄²	C₂⁴×C₃₁	1	2	yes	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	4	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	8	no	—
466	C₂³	—	4	16	no	—
467	C₂³	—	4	16	no	—
468	C₂³	—	4	16	no	—
469	C₂³	—	4	8	no	—
470	C₂³	—	4	16	no	—
471	C₂³	—	8	32	no	—
472	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
473	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
474	Q₈	—	2	4	no	—
475	Q₈	—	4	4	no	—
476	D₄	—	2	4	no	—
477	D₄	—	2	16	no	—
478	D₄	—	4	16	no	—
479	D₄	—	4	16	no	—
480	D₄	—	4	16	no	—
481	D₄	—	4	4	no	—
482	D₄	—	4	8	no	—
483	D₄	—	4	8	no	—
484	D₄	—	4	16	no	—
485	D₄	—	4	16	no	—
486	D₄	—	4	4	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	8	no	—
499	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	8	yes	complemented
500	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
501	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
503	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
506	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
507	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
508	C₈	—	2	8	no	—
509	C₂²	—	2	8	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	4	8	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	16	no	—
517	C₂²	—	8	16	no	—
518	C₂²	—	8	32	no	—
519	C₂²	C₃₁×#535	1	2	yes	—
520	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
521	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	4	no	—
527	C₄	C₃₁×#535	1	2	yes	—
528	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
529	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
530	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
531	C₂	—	2	2	no	—
532	C₂	—	2	4	no	—
533	C₂	—	8	8	no	—
534	C₂	—	8	2	no	—
535	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
536	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3444	C₂	semidirect
2. C₂²×C₃₁×#127	C₂	semidirect
3. C₃₁×#3217	C₂	semidirect
4. C₃₁×#3208	C₂	semidirect
5. C₃₁×#3205	C₂	semidirect
6. C₃₁×#3151	C₂	semidirect
7. C₃₁×#3138	C₂	semidirect
8. C₃₁×#3038	C₂	semidirect
9. C₃₁×#3003	C₂	semidirect
10. C₃₁×#2920	C₂	semidirect
13. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
14. C₃₁×#533	C₂²	semidirect
15. C₃₁×#533	C₂²	semidirect
16. C₃₁×#532	C₂²	semidirect
17. C₃₁×#532	C₂²	semidirect
27. C₃₁×#529	C₂²	semidirect
31. C₄×#41×C₃₁	C₂²	semidirect
33. C₃₁×#496	C₂²	semidirect
34. C₃₁×#493	C₂²	semidirect
35. C₃₁×#492	C₂²	semidirect
36. C₃₁×#490	C₂²	semidirect
37. C₃₁×#486	C₂²	semidirect
38. C₃₁×#460	C₂²	semidirect
39. C₃₁×#459	C₂²	semidirect
40. C₃₁×#450	C₂²	semidirect
41. C₃₁×#447	C₂²	semidirect
42. C₃₁×#432	C₂²	semidirect
43. C₃₁×#431	C₂²	semidirect
44. C₃₁×#423	C₂²	semidirect
45. C₃₁×#418	C₂²	semidirect
47. C₄×M₁₆×C₃₁	C₂²	semidirect
66. C₂×SD₁₆×C₃₁	C₂³	semidirect
67. C₂×D₈×C₃₁	C₂³	semidirect
69. C₃₁×C₄⋊Q₈	C₂³	semidirect
70. C₃₁×C₄⋊Q₈	C₂³	semidirect
81. C₃₁×#127	C₂³	semidirect
84. C₃₁×#125	C₂³	semidirect
86. C₃₁×#124	C₂³	semidirect
98. C₄×Q₈×C₃₁	C₂³	semidirect
104. C₄×D₄×C₃₁	C₂³	semidirect
117. C₃₁×C₄⋊C₈	C₂³	semidirect
118. C₃₁×#102	C₂³	semidirect
120. C₃₁×#97	C₂³	semidirect
121. C₄×C₈×C₃₁	C₂³	semidirect
193. #60798	C₃₁	direct (N × H)
231. C₂×C₄×C₃₁	AGL(1,ℤ/8)	semidirect
241. #3444	C₂×C₃₁	semidirect
242. C₂²×#127	C₂×C₃₁	semidirect
243. #3217	C₂×C₃₁	semidirect
244. #3208	C₂×C₃₁	semidirect
245. #3205	C₂×C₃₁	semidirect
246. #3151	C₂×C₃₁	semidirect
247. #3138	C₂×C₃₁	semidirect
248. #3038	C₂×C₃₁	semidirect
249. #3003	C₂×C₃₁	semidirect
250. #2920	C₂×C₃₁	semidirect
275. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
276. #533	C₂²×C₃₁	semidirect
277. #533	C₂²×C₃₁	semidirect
278. #532	C₂²×C₃₁	semidirect
279. #532	C₂²×C₃₁	semidirect
289. #529	C₂²×C₃₁	semidirect
293. C₄×#41	C₂²×C₃₁	semidirect
295. #496	C₂²×C₃₁	semidirect
296. #493	C₂²×C₃₁	semidirect
297. #492	C₂²×C₃₁	semidirect
298. #490	C₂²×C₃₁	semidirect
299. #486	C₂²×C₃₁	semidirect
300. #460	C₂²×C₃₁	semidirect
301. #459	C₂²×C₃₁	semidirect
302. #450	C₂²×C₃₁	semidirect
303. #447	C₂²×C₃₁	semidirect
304. #432	C₂²×C₃₁	semidirect
305. #431	C₂²×C₃₁	semidirect
306. #423	C₂²×C₃₁	semidirect
307. #418	C₂²×C₃₁	semidirect
309. C₄×M₁₆	C₂²×C₃₁	semidirect
334. C₂×SD₁₆	C₂³×C₃₁	semidirect
335. C₂×D₈	C₂³×C₃₁	semidirect
337. C₄⋊Q₈	C₂³×C₃₁	semidirect
338. C₄⋊Q₈	C₂³×C₃₁	semidirect
349. #127	C₂³×C₃₁	semidirect
352. #125	C₂³×C₃₁	semidirect
354. #124	C₂³×C₃₁	semidirect
366. C₄×Q₈	C₂³×C₃₁	semidirect
372. C₄×D₄	C₂³×C₃₁	semidirect
385. C₄⋊C₈	C₂³×C₃₁	semidirect
386. #102	C₂³×C₃₁	semidirect
388. #97	C₂³×C₃₁	semidirect
389. C₄×C₈	C₂³×C₃₁	semidirect
390. C₃₁	#60798	direct (N × H)
499. C₂×C₄	C₃₁×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3444	2
C₂²×C₃₁×#127	2
C₃₁×#3217	2
C₃₁×#3208	2
C₃₁×#3205	2
C₃₁×#3151	2
C₃₁×#3138	2
C₃₁×#3038	2
C₃₁×#3003	2
C₃₁×#2920	2
#60798	31

Subgroups — C31×#60798 #21517557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60798 #21517557