Subgroups — C₃₁×#60920 #21517679

Order 7936.

1886 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#540	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3260	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3259	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3257	C₂	1	8	yes	maximal, complemented
6	C₃₁×#3080	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3077	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3055	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3038	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3036	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#542	C₂²	1	4	yes	complemented
15	C₃₁×#541	C₂²	1	2	yes	complemented
16	C₃₁×#540	—	2	4	no	—
17	C₃₁×#540	C₂²	1	4	yes	—
18	C₃₁×#540	C₂²	1	4	yes	—
19	C₃₁×#540	C₂²	1	2	yes	complemented
20	C₃₁×#539	—	2	2	no	—
21	C₃₁×#538	—	2	2	no	—
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
25	C₃₁×#529	—	2	4	no	—
26	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
27	C₃₁×#525	C₂²	1	2	yes	—
28	C₂×C₃₁×#123	—	2	2	no	—
29	C₂×C₃₁×#121	—	2	2	no	—
30	C₂×C₃₁×#121	C₂²	1	2	yes	—
31	C₄×#41×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₃₁×#479	C₂²	1	8	yes	complemented
35	C₃₁×#478	C₂²	1	8	yes	complemented
36	C₃₁×#478	C₂²	1	8	yes	complemented
37	C₃₁×#476	C₂²	1	4	yes	complemented
38	C₃₁×#474	C₂²	1	4	yes	complemented
39	C₃₁×#469	C₂²	1	8	yes	complemented
40	C₃₁×#468	C₂²	1	8	yes	complemented
41	C₃₁×#466	C₂²	1	8	yes	complemented
42	C₃₁×#465	C₂²	1	8	yes	complemented
43	C₃₁×#463	C₂²	1	8	yes	complemented
44	C₃₁×#461	C₂²	1	4	yes	complemented
45	C₃₁×#460	C₂²	1	8	yes	complemented
46	C₃₁×#459	C₂²	1	8	yes	complemented
47	C₃₁×#459	C₂²	1	4	yes	complemented
48	C₃₁×#428	C₂²	1	4	yes	—
49	C₃₁×#424	C₂²	1	4	yes	complemented
50	C₃₁×#424	C₂²	1	4	yes	complemented
51	C₃₁×#423	C₂²	1	2	yes	—
52	C₃₁×#423	C₂²	1	2	yes	—
53	C₃₁×#418	C₂²	1	4	yes	—
54	C₃₁×#417	C₂²	1	4	yes	—
55	C₂×#41×C₃₁	C₂³	1	2	yes	—
56	C₂²×D₄×C₃₁	—	2	2	no	—
57	C₂²×D₄×C₃₁	—	2	2	no	—
58	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₃₁	—	2	2	no	—
60	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
62	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
63	C₂×D₈×C₃₁	C₂³	1	8	yes	—
64	C₂×M₁₆×C₃₁	—	4	4	no	—
65	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
66	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
67	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
68	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
70	C₃₁×#127	—	2	2	no	—
71	C₃₁×#127	C₂³	1	1	yes	characteristic
72	C₃₁×#127	C₂³	1	4	yes	—
73	C₃₁×#126	—	2	4	no	—
74	C₃₁×#126	—	2	4	no	—
75	C₃₁×#125	—	2	4	no	—
76	C₃₁×#125	—	2	2	no	—
77	C₃₁×#125	C₂³	1	2	yes	—
78	C₃₁×#125	C₂³	1	4	yes	—
79	C₃₁×#124	—	2	4	no	—
80	C₃₁×#124	C₂³	1	2	yes	—
81	C₃₁×#123	—	2	2	no	—
82	C₃₁×#123	—	2	8	no	—
83	C₃₁×#123	—	2	8	no	—
84	C₃₁×#123	—	2	4	no	—
85	C₃₁×#123	—	2	4	no	—
86	C₃₁×C₂²⋊Q₈	—	2	2	no	—
87	C₃₁×C₂²⋊Q₈	—	2	4	no	—
88	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
89	C₃₁×#121	—	2	4	no	—
90	C₃₁×#121	—	2	2	no	—
91	C₃₁×#121	—	2	8	no	—
92	C₃₁×#121	—	2	8	no	—
93	C₃₁×#121	—	2	4	no	—
94	C₃₁×#121	C₂³	1	8	yes	—
95	C₃₁×#121	C₂³	1	4	yes	—
96	C₃₁×#120	—	2	2	no	—
97	C₄×Q₈×C₃₁	—	2	2	no	—
98	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
99	C₄×D₄×C₃₁	—	2	4	no	—
100	C₄×D₄×C₃₁	—	2	4	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	2	no	—
104	C₄×D₄×C₃₁	C₂³	1	4	yes	—
105	C₄×D₄×C₃₁	C₂³	1	4	yes	—
106	C₄×D₄×C₃₁	C₂³	1	4	yes	—
107	C₃₁×#117	—	2	2	no	—
108	C₃₁×#117	D₄	1	4	yes	—
109	C₃₁×#117	C₂³	1	2	yes	—
110	C₃₁×#117	C₂³	1	2	yes	—
111	C₃₁×#117	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
118	C₂×C₄²×C₃₁	—	2	2	no	—
119	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₃₁×#107	C₂³	1	8	yes	—
122	C₃₁×#106	C₂³	1	8	yes	—
123	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
124	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
125	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
126	C₃₁×#103	C₂³	1	8	yes	—
127	C₃₁×#102	C₂³	1	8	yes	—
128	C₃₁×#102	C₂³	1	8	yes	—
129	C₃₁×#102	C₂³	1	8	yes	—
130	C₂⁴×C₃₁	—	2	2	no	—
131	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
132	#41×C₃₁	—	2	4	no	—
133	#41×C₃₁	—	2	4	no	—
134	C₂×Q₈×C₃₁	—	2	2	no	—
135	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	8	no	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	8	no	—
142	C₂×D₄×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	2	4	no	—
144	C₂×D₄×C₃₁	—	8	4	no	—
145	C₂×D₄×C₃₁	—	8	4	no	—
146	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	2	4	no	—
154	C₂²×C₄×C₃₁	—	2	4	no	—
155	C₂²×C₄×C₃₁	—	8	4	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	SD₁₆×C₃₁	—	4	8	no	—
163	D₈×C₃₁	—	4	8	no	—
164	M₁₆×C₃₁	—	2	8	no	—
165	M₁₆×C₃₁	—	4	4	no	—
166	C₂×C₈×C₃₁	—	4	4	no	—
167	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
168	C₄⋊C₄×C₃₁	—	2	8	no	—
169	C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₄⋊C₄×C₃₁	—	2	4	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	2	no	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
177	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
178	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
179	C₂²⋊C₄×C₃₁	—	2	2	no	—
180	C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₂²⋊C₄×C₃₁	—	2	4	no	—
182	C₂²⋊C₄×C₃₁	—	2	4	no	—
183	C₂²⋊C₄×C₃₁	—	2	8	no	—
184	C₂²⋊C₄×C₃₁	—	2	4	no	—
185	C₂²⋊C₄×C₃₁	—	2	4	no	—
186	C₂²⋊C₄×C₃₁	—	2	4	no	—
187	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
188	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
189	C₄²×C₃₁	—	2	2	no	—
190	C₄²×C₃₁	—	2	4	no	—
191	C₄²×C₃₁	C₂×D₄	1	2	yes	—
192	C₄²×C₃₁	C₂×D₄	1	2	yes	—
193	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
194	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₄²×C₃₁	C₂⁴	1	2	yes	—
196	#60920	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	4	no	—
200	C₂³×C₃₁	—	4	2	no	—
201	C₂³×C₃₁	—	4	4	no	—
202	C₂³×C₃₁	—	4	4	no	—
203	C₂³×C₃₁	—	4	4	no	—
204	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
205	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
206	Q₈×C₃₁	—	2	2	no	—
207	Q₈×C₃₁	—	2	2	no	—
208	Q₈×C₃₁	—	8	2	no	—
209	D₄×C₃₁	—	2	2	no	—
210	D₄×C₃₁	—	2	4	no	—
211	D₄×C₃₁	—	2	4	no	—
212	D₄×C₃₁	—	2	2	no	—
213	D₄×C₃₁	—	4	4	no	—
214	D₄×C₃₁	—	4	4	no	—
215	D₄×C₃₁	—	8	2	no	—
216	D₄×C₃₁	—	8	8	no	—
217	D₄×C₃₁	—	8	4	no	—
218	D₄×C₃₁	—	8	8	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	—	4	4	no	—
229	C₂×C₄×C₃₁	—	4	4	no	—
230	C₂×C₄×C₃₁	—	8	4	no	—
231	C₂×C₄×C₃₁	—	8	4	no	—
232	C₂×C₄×C₃₁	—	8	4	no	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
239	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
240	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
241	C₈×C₃₁	—	2	8	no	—
242	#3455	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂×#540	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#3260	C₂×C₃₁	1	4	yes	complemented
245	#3259	C₂×C₃₁	1	4	yes	complemented
246	#3257	C₂×C₃₁	1	8	yes	complemented
247	#3080	C₂×C₃₁	1	2	yes	complemented
248	#3077	C₂×C₃₁	1	2	yes	complemented
249	#3055	C₂×C₃₁	1	4	yes	complemented
250	#3038	C₂×C₃₁	1	2	yes	complemented
251	#3036	C₂×C₃₁	1	2	yes	complemented
252	#2905	C₂×C₃₁	1	1	yes	characteristic, complemented
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	2	4	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	4	4	no	—
257	C₂²×C₃₁	—	4	4	no	—
258	C₂²×C₃₁	—	4	4	no	—
259	C₂²×C₃₁	—	4	2	no	—
260	C₂²×C₃₁	—	4	4	no	—
261	C₂²×C₃₁	—	8	2	no	—
262	C₂²×C₃₁	—	8	8	no	—
263	C₂²×C₃₁	#537	1	2	yes	—
264	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
265	C₄×C₃₁	—	2	2	no	—
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	2	4	no	—
268	C₄×C₃₁	—	2	2	no	—
269	C₄×C₃₁	—	4	4	no	—
270	C₄×C₃₁	—	4	4	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	#537	1	2	yes	—
273	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
274	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
275	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
276	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
277	#542	C₂²×C₃₁	1	4	yes	complemented
278	#541	C₂²×C₃₁	1	2	yes	complemented
279	#540	—	2	4	no	—
280	#540	C₂²×C₃₁	1	4	yes	—
281	#540	C₂²×C₃₁	1	4	yes	—
282	#540	C₂²×C₃₁	1	2	yes	complemented
283	#539	—	2	2	no	—
284	#538	—	2	2	no	—
285	#533	C₂²×C₃₁	1	1	yes	characteristic
286	#532	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
288	#529	—	2	4	no	—
289	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
290	#525	C₂²×C₃₁	1	2	yes	—
291	C₂×#123	—	2	2	no	—
292	C₂×#121	—	2	2	no	—
293	C₂×#121	C₂²×C₃₁	1	2	yes	—
294	C₄×#41	C₂²×C₃₁	1	2	yes	—
295	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
296	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
297	#479	C₂²×C₃₁	1	8	yes	complemented
298	#478	C₂²×C₃₁	1	8	yes	complemented
299	#478	C₂²×C₃₁	1	8	yes	complemented
300	#476	C₂²×C₃₁	1	4	yes	complemented
301	#474	C₂²×C₃₁	1	4	yes	complemented
302	#469	C₂²×C₃₁	1	8	yes	complemented
303	#468	C₂²×C₃₁	1	8	yes	complemented
304	#466	C₂²×C₃₁	1	8	yes	complemented
305	#465	C₂²×C₃₁	1	8	yes	complemented
306	#463	C₂²×C₃₁	1	8	yes	complemented
307	#461	C₂²×C₃₁	1	4	yes	complemented
308	#460	C₂²×C₃₁	1	8	yes	complemented
309	#459	C₂²×C₃₁	1	8	yes	complemented
310	#459	C₂²×C₃₁	1	4	yes	complemented
311	#428	C₂²×C₃₁	1	4	yes	—
312	#424	C₂²×C₃₁	1	4	yes	complemented
313	#424	C₂²×C₃₁	1	4	yes	complemented
314	#423	C₂²×C₃₁	1	2	yes	—
315	#423	C₂²×C₃₁	1	2	yes	—
316	#418	C₂²×C₃₁	1	4	yes	—
317	#417	C₂²×C₃₁	1	4	yes	—
318	C₂×C₃₁	—	2	2	no	—
319	C₂×C₃₁	—	4	2	no	—
320	C₂×C₃₁	—	8	2	no	—
321	C₂×C₃₁	—	8	4	no	—
322	C₂×C₃₁	C₂×#537	1	1	yes	characteristic
323	C₂×C₃₁	#3585	1	2	yes	—
324	C₂×#41	C₂³×C₃₁	1	2	yes	—
325	C₂²×D₄	—	2	2	no	—
326	C₂²×D₄	—	2	2	no	—
327	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
328	C₂³×C₄	—	2	2	no	—
329	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
330	AGL(1,ℤ/8)	—	4	8	no	—
331	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
332	C₂×D₈	C₂³×C₃₁	1	8	yes	—
333	C₂×M₁₆	—	4	4	no	—
334	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
335	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
336	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
337	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
338	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
339	#127	—	2	2	no	—
340	#127	C₂³×C₃₁	1	4	yes	—
341	#127	C₂³×C₃₁	1	1	yes	characteristic
342	#126	—	2	4	no	—
343	#126	—	2	4	no	—
344	#125	—	2	4	no	—
345	#125	—	2	2	no	—
346	#125	C₂³×C₃₁	1	4	yes	—
347	#125	C₂³×C₃₁	1	2	yes	—
348	#124	—	2	4	no	—
349	#124	C₂³×C₃₁	1	2	yes	—
350	#123	—	2	2	no	—
351	#123	—	2	8	no	—
352	#123	—	2	8	no	—
353	#123	—	2	4	no	—
354	#123	—	2	4	no	—
355	C₂²⋊Q₈	—	2	2	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
358	#121	—	2	4	no	—
359	#121	—	2	2	no	—
360	#121	—	2	8	no	—
361	#121	—	2	8	no	—
362	#121	—	2	4	no	—
363	#121	C₂³×C₃₁	1	8	yes	—
364	#121	C₂³×C₃₁	1	4	yes	—
365	#120	—	2	2	no	—
366	C₄×Q₈	—	2	2	no	—
367	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
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	—	2	2	no	—
373	C₄×D₄	C₂³×C₃₁	1	4	yes	—
374	C₄×D₄	C₂³×C₃₁	1	4	yes	—
375	C₄×D₄	C₂³×C₃₁	1	4	yes	—
376	#117	—	2	2	no	—
377	#117	D₄×C₃₁	1	4	yes	—
378	#117	C₂³×C₃₁	1	2	yes	—
379	#117	C₂³×C₃₁	1	2	yes	—
380	#117	C₂³×C₃₁	1	2	yes	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
387	C₂×C₄²	—	2	2	no	—
388	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
390	#107	C₂³×C₃₁	1	8	yes	—
391	#106	C₂³×C₃₁	1	8	yes	—
392	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
393	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
394	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
395	#103	C₂³×C₃₁	1	8	yes	—
396	#102	C₂³×C₃₁	1	8	yes	—
397	#102	C₂³×C₃₁	1	8	yes	—
398	#102	C₂³×C₃₁	1	8	yes	—
399	C₃₁	#60920	1	1	yes	characteristic, direct product factor, Sylow
400	C₂⁴	—	2	2	no	—
401	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
402	#41	—	2	4	no	—
403	#41	—	2	4	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	8	4	no	—
415	C₂×D₄	—	8	4	no	—
416	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
417	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	8	4	no	—
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	C₂⁴×C₃₁	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
432	SD₁₆	—	4	8	no	—
433	D₈	—	4	8	no	—
434	M₁₆	—	2	8	no	—
435	M₁₆	—	4	4	no	—
436	C₂×C₈	—	4	4	no	—
437	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
446	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
447	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
448	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	4	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
458	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
459	C₄²	—	2	2	no	—
460	C₄²	—	2	4	no	—
461	C₄²	C₂×D₄×C₃₁	1	2	yes	—
462	C₄²	C₂×D₄×C₃₁	1	2	yes	—
463	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
464	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
465	C₄²	C₂⁴×C₃₁	1	2	yes	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	4	2	no	—
470	C₂³	—	4	4	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	4	4	no	—
473	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
474	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
475	Q₈	—	2	2	no	—
476	Q₈	—	2	2	no	—
477	Q₈	—	8	2	no	—
478	D₄	—	2	2	no	—
479	D₄	—	2	4	no	—
480	D₄	—	2	4	no	—
481	D₄	—	2	2	no	—
482	D₄	—	4	4	no	—
483	D₄	—	4	4	no	—
484	D₄	—	8	2	no	—
485	D₄	—	8	8	no	—
486	D₄	—	8	4	no	—
487	D₄	—	8	8	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	8	4	no	—
500	C₂×C₄	—	8	4	no	—
501	C₂×C₄	—	8	4	no	—
502	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
503	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
504	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
506	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
507	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
508	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
509	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
510	C₈	—	2	8	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	8	2	no	—
520	C₂²	—	8	8	no	—
521	C₂²	C₃₁×#537	1	2	yes	—
522	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	4	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	4	no	—
529	C₄	—	4	4	no	—
530	C₄	C₃₁×#537	1	2	yes	—
531	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
532	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
533	C₂	—	2	2	no	—
534	C₂	—	4	2	no	—
535	C₂	—	8	4	no	—
536	C₂	—	8	2	no	—
537	C₂	C₂×C₃₁×#537	1	1	yes	characteristic
538	C₂	C₃₁×#3585	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 ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3455	C₂	semidirect
2. C₂×C₃₁×#540	C₂	semidirect
3. C₃₁×#3260	C₂	semidirect
4. C₃₁×#3259	C₂	semidirect
5. C₃₁×#3257	C₂	semidirect
6. C₃₁×#3080	C₂	semidirect
7. C₃₁×#3077	C₂	semidirect
8. C₃₁×#3055	C₂	semidirect
9. C₃₁×#3038	C₂	semidirect
10. C₃₁×#3036	C₂	semidirect
11. C₃₁×#2905	C₂	semidirect
14. C₃₁×#542	C₂²	semidirect
15. C₃₁×#541	C₂²	semidirect
19. C₃₁×#540	C₂²	semidirect
34. C₃₁×#479	C₂²	semidirect
35. C₃₁×#478	C₂²	semidirect
36. C₃₁×#478	C₂²	semidirect
37. C₃₁×#476	C₂²	semidirect
38. C₃₁×#474	C₂²	semidirect
39. C₃₁×#469	C₂²	semidirect
40. C₃₁×#468	C₂²	semidirect
41. C₃₁×#466	C₂²	semidirect
42. C₃₁×#465	C₂²	semidirect
43. C₃₁×#463	C₂²	semidirect
44. C₃₁×#461	C₂²	semidirect
45. C₃₁×#460	C₂²	semidirect
46. C₃₁×#459	C₂²	semidirect
47. C₃₁×#459	C₂²	semidirect
49. C₃₁×#424	C₂²	semidirect
50. C₃₁×#424	C₂²	semidirect
196. #60920	C₃₁	direct (N × H)
242. #3455	C₂×C₃₁	semidirect
243. C₂×#540	C₂×C₃₁	semidirect
244. #3260	C₂×C₃₁	semidirect
245. #3259	C₂×C₃₁	semidirect
246. #3257	C₂×C₃₁	semidirect
247. #3080	C₂×C₃₁	semidirect
248. #3077	C₂×C₃₁	semidirect
249. #3055	C₂×C₃₁	semidirect
250. #3038	C₂×C₃₁	semidirect
251. #3036	C₂×C₃₁	semidirect
252. #2905	C₂×C₃₁	semidirect
277. #542	C₂²×C₃₁	semidirect
278. #541	C₂²×C₃₁	semidirect
282. #540	C₂²×C₃₁	semidirect
297. #479	C₂²×C₃₁	semidirect
298. #478	C₂²×C₃₁	semidirect
299. #478	C₂²×C₃₁	semidirect
300. #476	C₂²×C₃₁	semidirect
301. #474	C₂²×C₃₁	semidirect
302. #469	C₂²×C₃₁	semidirect
303. #468	C₂²×C₃₁	semidirect
304. #466	C₂²×C₃₁	semidirect
305. #465	C₂²×C₃₁	semidirect
306. #463	C₂²×C₃₁	semidirect
307. #461	C₂²×C₃₁	semidirect
308. #460	C₂²×C₃₁	semidirect
309. #459	C₂²×C₃₁	semidirect
310. #459	C₂²×C₃₁	semidirect
312. #424	C₂²×C₃₁	semidirect
313. #424	C₂²×C₃₁	semidirect
399. C₃₁	#60920	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×#3455	2
C₂×C₃₁×#540	2
C₃₁×#3260	2
C₃₁×#3259	2
C₃₁×#3257	2
C₃₁×#3080	2
C₃₁×#3077	2
C₃₁×#3055	2
C₃₁×#3038	2
C₃₁×#3036	2
C₃₁×#2905	2
#60920	31

Subgroups — C31×#60920 #21517679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60920 #21517679