Subgroups — C₃₁×#24297 #21481056

Order 7936.

1334 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3067	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3058	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3019	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2955	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2048	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2008	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1976	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1814	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1795	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	—
13	C₃₁×#576	—	2	2	no	—
14	C₃₁×#576	—	2	2	no	—
15	C₃₁×#576	—	2	2	no	—
16	C₃₁×#576	—	2	2	no	—
17	C₃₁×#575	C₂²	1	1	yes	characteristic
18	C₃₁×#575	C₂²	1	2	yes	complemented
19	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
20	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
21	C₂×C₃₁×#135	—	2	1	no	—
22	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
23	C₂²×D₈×C₃₁	—	2	1	no	—
24	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#535	C₂²	1	2	yes	complemented
28	C₃₁×#473	—	2	1	no	—
29	C₃₁×#472	—	2	1	no	—
30	C₃₁×#471	—	2	2	no	—
31	C₃₁×#469	—	2	2	no	—
32	C₃₁×#466	—	2	2	no	—
33	C₃₁×#455	—	2	4	no	—
34	C₃₁×#453	—	2	4	no	—
35	C₃₁×#444	—	2	2	no	—
36	C₃₁×#443	—	2	2	no	—
37	C₃₁×#430	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#427	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#421	C₂²	1	1	yes	characteristic
41	C₃₁×#421	C₂²	1	2	yes	—
42	C₃₁×#421	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#421	C₂²	1	2	yes	complemented
44	C₃₁×#418	C₂²	1	2	yes	complemented
45	C₃₁×#418	C₂²	1	2	yes	complemented
46	C₃₁×#413	C₂²	1	2	yes	complemented
47	C₂×C₃₁×#100	C₂²	1	2	yes	complemented
48	C₃₁×#410	C₂²	1	2	yes	complemented
49	C₃₁×#405	C₂²	1	1	yes	characteristic, complemented
50	C₃₁×#344	C₂²	1	2	yes	complemented
51	C₃₁×H(5,𝔽₂)	—	2	4	no	—
52	C₃₁×H(5,𝔽₂)	—	2	4	no	—
53	C₃₁×H(5,𝔽₂)	—	2	4	no	—
54	C₃₁×H(5,𝔽₂)	—	4	2	no	—
55	C₂×#41×C₃₁	—	2	2	no	—
56	C₂×#41×C₃₁	—	2	2	no	—
57	C₂×#41×C₃₁	C₂³	1	2	yes	—
58	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₃₁	—	2	2	no	—
61	C₂²×D₄×C₃₁	—	2	2	no	—
62	C₂²×D₄×C₃₁	—	4	2	no	—
63	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
64	C₃₁×#137	—	2	1	no	—
65	C₃₁×#137	—	2	2	no	—
66	C₃₁×#137	—	2	2	no	—
67	C₃₁×#137	D₄	1	2	yes	—
68	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
69	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
70	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
71	C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
72	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
73	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
74	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
75	C₃₁×AGL(1,ℤ/8)	D₄	1	2	yes	—
76	C₃₁×#135	—	2	4	no	—
77	C₃₁×#135	—	2	4	no	—
78	C₃₁×#135	—	2	2	no	—
79	C₃₁×#135	—	2	1	no	—
80	C₃₁×#135	—	2	1	no	—
81	C₃₁×#135	—	2	1	no	—
82	C₃₁×#135	—	2	1	no	—
83	C₃₁×#135	—	4	2	no	—
84	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic, complemented
85	C₂×SD₁₆×C₃₁	—	2	2	no	—
86	C₂×SD₁₆×C₃₁	D₄	1	2	yes	complemented
87	C₂×D₈×C₃₁	—	2	4	no	—
88	C₂×D₈×C₃₁	—	2	4	no	—
89	C₂×D₈×C₃₁	—	2	2	no	—
90	C₂×D₈×C₃₁	—	2	1	no	—
91	C₂×D₈×C₃₁	—	2	1	no	—
92	C₂×D₈×C₃₁	—	2	1	no	—
93	C₂×D₈×C₃₁	—	2	1	no	—
94	C₂×D₈×C₃₁	D₄	1	2	yes	—
95	C₂×D₈×C₃₁	D₄	1	2	yes	—
96	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic, complemented
97	C₃₁×#131	—	2	1	no	—
98	C₃₁×#131	—	2	1	no	—
99	C₃₁×#131	—	2	1	no	—
100	C₃₁×#131	—	2	1	no	—
101	C₃₁×#131	—	2	1	no	—
102	C₃₁×#131	—	2	1	no	—
103	C₃₁×#131	D₄	1	2	yes	complemented
104	C₃₁×#131	D₄	1	2	yes	complemented
105	C₂×M₁₆×C₃₁	—	2	1	no	—
106	C₂×M₁₆×C₃₁	—	2	1	no	—
107	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
108	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
111	C₂²×C₈×C₃₁	—	2	1	no	—
112	C₂²×C₈×C₃₁	—	2	1	no	—
113	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
114	C₃₁×#127	—	2	2	no	—
115	C₃₁×#124	—	2	2	no	—
116	C₃₁×#121	—	2	4	no	—
117	C₃₁×#120	—	4	2	no	—
118	C₃₁×#117	C₂³	1	1	yes	characteristic
119	C₃₁×#117	C₂³	1	2	yes	—
120	C₃₁×#108	—	2	2	no	—
121	C₃₁×#108	—	2	1	no	—
122	C₃₁×#108	—	2	1	no	—
123	C₃₁×#107	—	2	1	no	—
124	C₃₁×#106	—	2	1	no	—
125	C₃₁×#104	—	2	2	no	—
126	C₃₁×#104	—	2	4	no	—
127	C₃₁×#104	—	2	2	no	—
128	C₃₁×#104	—	2	4	no	—
129	C₃₁×#102	—	2	4	no	—
130	C₃₁×#102	—	4	2	no	—
131	C₃₁×#101	—	2	2	no	—
132	C₃₁×#100	—	2	2	no	—
133	C₃₁×#100	—	2	4	no	—
134	C₃₁×#99	—	4	2	no	—
135	C₃₁×#97	—	2	1	no	—
136	C₄×C₈×C₃₁	—	2	1	no	—
137	C₂⁴×C₃₁	—	2	2	no	—
138	C₂⁴×C₃₁	—	4	2	no	—
139	#41×C₃₁	—	2	1	no	—
140	#41×C₃₁	—	2	1	no	—
141	#41×C₃₁	—	2	8	no	—
142	#41×C₃₁	—	2	1	no	—
143	#41×C₃₁	—	2	2	no	—
144	#41×C₃₁	—	2	1	no	—
145	#41×C₃₁	—	2	2	no	—
146	#41×C₃₁	—	2	2	no	—
147	#41×C₃₁	—	2	1	no	—
148	#41×C₃₁	—	2	1	no	—
149	#41×C₃₁	—	2	2	no	—
150	#41×C₃₁	—	4	4	no	—
151	#41×C₃₁	—	4	4	no	—
152	#41×C₃₁	—	4	4	no	—
153	#41×C₃₁	C₂×D₄	1	2	yes	—
154	#41×C₃₁	C₂×D₄	1	2	yes	—
155	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
156	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
158	C₂×D₄×C₃₁	—	2	1	no	—
159	C₂×D₄×C₃₁	—	2	8	no	—
160	C₂×D₄×C₃₁	—	2	2	no	—
161	C₂×D₄×C₃₁	—	2	1	no	—
162	C₂×D₄×C₃₁	—	2	2	no	—
163	C₂×D₄×C₃₁	—	2	2	no	—
164	C₂×D₄×C₃₁	—	2	2	no	—
165	C₂×D₄×C₃₁	—	2	2	no	—
166	C₂×D₄×C₃₁	—	2	2	no	—
167	C₂×D₄×C₃₁	—	4	4	no	—
168	C₂×D₄×C₃₁	—	4	2	no	—
169	C₂×D₄×C₃₁	—	4	2	no	—
170	C₂×D₄×C₃₁	—	4	4	no	—
171	C₂×D₄×C₃₁	—	4	8	no	—
172	C₂×D₄×C₃₁	—	4	8	no	—
173	C₂×D₄×C₃₁	—	4	4	no	—
174	C₂×D₄×C₃₁	—	4	4	no	—
175	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
177	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	—	2	1	no	—
180	C₂²×C₄×C₃₁	—	2	1	no	—
181	C₂²×C₄×C₃₁	—	2	2	no	—
182	C₂²×C₄×C₃₁	—	4	2	no	—
183	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
184	Q₁₆×C₃₁	—	2	2	no	—
185	Q₁₆×C₃₁	—	2	1	no	—
186	Q₁₆×C₃₁	—	2	1	no	—
187	Q₁₆×C₃₁	—	4	2	no	—
188	SD₁₆×C₃₁	—	2	8	no	—
189	SD₁₆×C₃₁	—	2	2	no	—
190	SD₁₆×C₃₁	—	2	2	no	—
191	SD₁₆×C₃₁	—	2	2	no	—
192	SD₁₆×C₃₁	—	2	2	no	—
193	SD₁₆×C₃₁	—	4	2	no	—
194	SD₁₆×C₃₁	—	4	2	no	—
195	D₈×C₃₁	—	2	8	no	—
196	D₈×C₃₁	—	2	2	no	—
197	D₈×C₃₁	—	2	2	no	—
198	D₈×C₃₁	—	2	2	no	—
199	D₈×C₃₁	—	2	1	no	—
200	D₈×C₃₁	—	2	1	no	—
201	D₈×C₃₁	—	2	2	no	—
202	D₈×C₃₁	—	2	2	no	—
203	D₈×C₃₁	—	4	2	no	—
204	M₁₆×C₃₁	—	2	1	no	—
205	M₁₆×C₃₁	—	2	1	no	—
206	M₁₆×C₃₁	—	2	2	no	—
207	M₁₆×C₃₁	—	2	2	no	—
208	M₁₆×C₃₁	—	2	1	no	—
209	M₁₆×C₃₁	—	2	2	no	—
210	M₁₆×C₃₁	—	2	2	no	—
211	M₁₆×C₃₁	—	2	1	no	—
212	M₁₆×C₃₁	—	4	1	no	—
213	M₁₆×C₃₁	—	4	1	no	—
214	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
215	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
216	C₂×C₈×C₃₁	—	2	1	no	—
217	C₂×C₈×C₃₁	—	2	1	no	—
218	C₂×C₈×C₃₁	—	2	2	no	—
219	C₂×C₈×C₃₁	—	2	2	no	—
220	C₂×C₈×C₃₁	—	2	1	no	—
221	C₂×C₈×C₃₁	—	2	2	no	—
222	C₂×C₈×C₃₁	—	4	1	no	—
223	C₂×C₈×C₃₁	—	4	1	no	—
224	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
225	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
226	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
227	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
228	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
229	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
230	C₄⋊C₄×C₃₁	—	2	2	no	—
231	C₄⋊C₄×C₃₁	—	2	1	no	—
232	C₂²⋊C₄×C₃₁	—	2	1	no	—
233	C₂²⋊C₄×C₃₁	—	2	2	no	—
234	C₂²⋊C₄×C₃₁	—	4	4	no	—
235	C₄²×C₃₁	—	2	1	no	—
236	C₄²×C₃₁	—	2	2	no	—
237	#24297	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂³×C₃₁	—	2	1	no	—
239	C₂³×C₃₁	—	2	1	no	—
240	C₂³×C₃₁	—	2	4	no	—
241	C₂³×C₃₁	—	2	2	no	—
242	C₂³×C₃₁	—	4	8	no	—
243	C₂³×C₃₁	—	4	2	no	—
244	C₂³×C₃₁	—	4	2	no	—
245	C₂³×C₃₁	—	16	2	no	—
246	C₂³×C₃₁	—	16	4	no	—
247	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
248	Q₈×C₃₁	—	2	1	no	—
249	Q₈×C₃₁	—	2	2	no	—
250	Q₈×C₃₁	—	2	1	no	—
251	Q₈×C₃₁	—	2	1	no	—
252	Q₈×C₃₁	—	2	2	no	—
253	Q₈×C₃₁	#121	1	2	yes	—
254	D₄×C₃₁	—	2	2	no	—
255	D₄×C₃₁	—	2	2	no	—
256	D₄×C₃₁	—	2	2	no	—
257	D₄×C₃₁	—	2	4	no	—
258	D₄×C₃₁	—	2	1	no	—
259	D₄×C₃₁	—	2	1	no	—
260	D₄×C₃₁	—	2	2	no	—
261	D₄×C₃₁	—	2	1	no	—
262	D₄×C₃₁	—	2	4	no	—
263	D₄×C₃₁	—	4	8	no	—
264	D₄×C₃₁	—	4	8	no	—
265	D₄×C₃₁	—	4	2	no	—
266	D₄×C₃₁	—	4	1	no	—
267	D₄×C₃₁	—	4	2	no	—
268	D₄×C₃₁	—	4	1	no	—
269	D₄×C₃₁	—	4	2	no	—
270	D₄×C₃₁	—	4	2	no	—
271	D₄×C₃₁	—	16	4	no	—
272	D₄×C₃₁	#121	1	2	yes	—
273	C₂×C₄×C₃₁	—	2	1	no	—
274	C₂×C₄×C₃₁	—	2	2	no	—
275	C₂×C₄×C₃₁	—	2	2	no	—
276	C₂×C₄×C₃₁	—	2	1	no	—
277	C₂×C₄×C₃₁	—	2	2	no	—
278	C₂×C₄×C₃₁	—	4	8	no	—
279	C₂×C₄×C₃₁	—	4	1	no	—
280	C₂×C₄×C₃₁	—	4	2	no	—
281	C₂×C₄×C₃₁	—	4	2	no	—
282	C₂×C₄×C₃₁	—	4	1	no	—
283	C₂×C₄×C₃₁	—	4	1	no	—
284	C₂×C₄×C₃₁	—	4	2	no	—
285	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
290	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
291	C₈×C₃₁	—	2	2	no	—
292	C₈×C₃₁	—	2	2	no	—
293	C₈×C₃₁	—	4	1	no	—
294	C₈×C₃₁	—	4	2	no	—
295	C₈×C₃₁	#120	1	1	yes	characteristic
296	C₈×C₃₁	#120	1	1	yes	characteristic
297	C₈×C₃₁	#120	1	1	yes	characteristic
298	C₈×C₃₁	#120	1	1	yes	characteristic
299	C₂×#576	C₂×C₃₁	1	1	yes	characteristic, complemented
300	#3067	C₂×C₃₁	1	1	yes	characteristic, complemented
301	#3058	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#3019	C₂×C₃₁	1	2	yes	complemented
303	#2955	C₂×C₃₁	1	1	yes	characteristic, complemented
304	#2048	C₂×C₃₁	1	2	yes	complemented
305	#2008	C₂×C₃₁	1	2	yes	complemented
306	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
307	#1976	C₂×C₃₁	1	1	yes	characteristic, complemented
308	#1814	C₂×C₃₁	1	1	yes	characteristic, complemented
309	#1795	C₂×C₃₁	1	2	yes	complemented
310	C₂²×C₃₁	—	2	2	no	—
311	C₂²×C₃₁	—	4	1	no	—
312	C₂²×C₃₁	—	4	2	no	—
313	C₂²×C₃₁	—	4	2	no	—
314	C₂²×C₃₁	—	4	4	no	—
315	C₂²×C₃₁	—	4	2	no	—
316	C₂²×C₃₁	—	4	1	no	—
317	C₂²×C₃₁	—	8	4	no	—
318	C₂²×C₃₁	—	8	2	no	—
319	C₂²×C₃₁	—	8	4	no	—
320	C₂²×C₃₁	—	16	2	no	—
321	C₂²×C₃₁	—	16	8	no	—
322	C₂²×C₃₁	#535	1	1	yes	characteristic
323	C₂²×C₃₁	D₄²	1	1	yes	characteristic
324	C₂²×C₃₁	#546	1	1	yes	characteristic
325	C₄×C₃₁	—	2	2	no	—
326	C₄×C₃₁	—	4	2	no	—
327	C₄×C₃₁	—	4	1	no	—
328	C₄×C₃₁	—	8	1	no	—
329	C₄×C₃₁	—	8	2	no	—
330	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
331	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
332	C₄×C₃₁	D₄²	1	1	yes	characteristic
333	C₄×C₃₁	#546	1	1	yes	characteristic
334	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	—
335	#576	—	2	2	no	—
336	#576	—	2	2	no	—
337	#576	—	2	2	no	—
338	#576	—	2	2	no	—
339	#575	C₂²×C₃₁	1	1	yes	characteristic
340	#575	C₂²×C₃₁	1	2	yes	complemented
341	C₂×AGL(1,ℤ/8)	—	2	2	no	—
342	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
343	C₂×#135	—	2	1	no	—
344	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
345	C₂²×D₈	—	2	1	no	—
346	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
347	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
348	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic, complemented
349	#535	C₂²×C₃₁	1	2	yes	complemented
350	#473	—	2	1	no	—
351	#472	—	2	1	no	—
352	#471	—	2	2	no	—
353	#469	—	2	2	no	—
354	#466	—	2	2	no	—
355	#455	—	2	4	no	—
356	#453	—	2	4	no	—
357	#444	—	2	2	no	—
358	#443	—	2	2	no	—
359	#430	C₂²×C₃₁	1	1	yes	characteristic, complemented
360	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic, complemented
361	#427	C₂²×C₃₁	1	1	yes	characteristic, complemented
362	#421	C₂²×C₃₁	1	1	yes	characteristic
363	#421	C₂²×C₃₁	1	2	yes	—
364	#421	C₂²×C₃₁	1	2	yes	complemented
365	#421	C₂²×C₃₁	1	1	yes	characteristic, complemented
366	#418	C₂²×C₃₁	1	2	yes	complemented
367	#418	C₂²×C₃₁	1	2	yes	complemented
368	#413	C₂²×C₃₁	1	2	yes	complemented
369	C₂×#100	C₂²×C₃₁	1	2	yes	complemented
370	#410	C₂²×C₃₁	1	2	yes	complemented
371	#405	C₂²×C₃₁	1	1	yes	characteristic, complemented
372	#344	C₂²×C₃₁	1	2	yes	complemented
373	C₂×C₃₁	—	2	1	no	—
374	C₂×C₃₁	—	2	1	no	—
375	C₂×C₃₁	—	2	1	no	—
376	C₂×C₃₁	—	4	2	no	—
377	C₂×C₃₁	—	8	1	no	—
378	C₂×C₃₁	—	8	4	no	—
379	C₂×C₃₁	—	8	2	no	—
380	C₂×C₃₁	#2408	1	1	yes	characteristic
381	H(5,𝔽₂)	—	2	4	no	—
382	H(5,𝔽₂)	—	2	4	no	—
383	H(5,𝔽₂)	—	2	4	no	—
384	H(5,𝔽₂)	—	4	2	no	—
385	C₂×#41	—	2	2	no	—
386	C₂×#41	—	2	2	no	—
387	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
388	C₂×#41	C₂³×C₃₁	1	2	yes	—
389	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
390	C₂²×D₄	—	2	2	no	—
391	C₂²×D₄	—	2	2	no	—
392	C₂²×D₄	—	4	2	no	—
393	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
394	#137	—	2	1	no	—
395	#137	—	2	2	no	—
396	#137	—	2	2	no	—
397	#137	D₄×C₃₁	1	2	yes	—
398	AGL(1,ℤ/8)	—	2	8	no	—
399	AGL(1,ℤ/8)	—	2	8	no	—
400	AGL(1,ℤ/8)	—	2	4	no	—
401	AGL(1,ℤ/8)	—	2	1	no	—
402	AGL(1,ℤ/8)	—	2	2	no	—
403	AGL(1,ℤ/8)	—	2	2	no	—
404	AGL(1,ℤ/8)	—	2	4	no	—
405	AGL(1,ℤ/8)	D₄×C₃₁	1	2	yes	—
406	#135	—	2	4	no	—
407	#135	—	2	4	no	—
408	#135	—	2	2	no	—
409	#135	—	2	1	no	—
410	#135	—	2	1	no	—
411	#135	—	2	1	no	—
412	#135	—	2	1	no	—
413	#135	—	4	2	no	—
414	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic, complemented
415	C₂×SD₁₆	—	2	2	no	—
416	C₂×SD₁₆	D₄×C₃₁	1	2	yes	complemented
417	C₂×D₈	—	2	4	no	—
418	C₂×D₈	—	2	4	no	—
419	C₂×D₈	—	2	2	no	—
420	C₂×D₈	—	2	1	no	—
421	C₂×D₈	—	2	1	no	—
422	C₂×D₈	—	2	1	no	—
423	C₂×D₈	—	2	1	no	—
424	C₂×D₈	D₄×C₃₁	1	2	yes	—
425	C₂×D₈	D₄×C₃₁	1	2	yes	—
426	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic, complemented
427	#131	—	2	1	no	—
428	#131	—	2	1	no	—
429	#131	—	2	1	no	—
430	#131	—	2	1	no	—
431	#131	—	2	1	no	—
432	#131	—	2	1	no	—
433	#131	D₄×C₃₁	1	2	yes	complemented
434	#131	D₄×C₃₁	1	2	yes	complemented
435	C₂×M₁₆	—	2	1	no	—
436	C₂×M₁₆	—	2	1	no	—
437	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
439	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
440	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
441	C₂²×C₈	—	2	1	no	—
442	C₂²×C₈	—	2	1	no	—
443	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
444	#127	—	2	2	no	—
445	#124	—	2	2	no	—
446	#121	—	2	4	no	—
447	#120	—	4	2	no	—
448	#117	C₂³×C₃₁	1	2	yes	—
449	#117	C₂³×C₃₁	1	1	yes	characteristic
450	#108	—	2	2	no	—
451	#108	—	2	1	no	—
452	#108	—	2	1	no	—
453	#107	—	2	1	no	—
454	#106	—	2	1	no	—
455	#104	—	2	2	no	—
456	#104	—	2	4	no	—
457	#104	—	2	2	no	—
458	#104	—	2	4	no	—
459	#102	—	2	4	no	—
460	#102	—	4	2	no	—
461	#101	—	2	2	no	—
462	#100	—	2	2	no	—
463	#100	—	2	4	no	—
464	#99	—	4	2	no	—
465	#97	—	2	1	no	—
466	C₄×C₈	—	2	1	no	—
467	C₃₁	#24297	1	1	yes	characteristic, direct product factor, Sylow
468	C₂⁴	—	2	2	no	—
469	C₂⁴	—	4	2	no	—
470	#41	—	2	1	no	—
471	#41	—	2	1	no	—
472	#41	—	2	8	no	—
473	#41	—	2	1	no	—
474	#41	—	2	2	no	—
475	#41	—	2	1	no	—
476	#41	—	2	2	no	—
477	#41	—	2	2	no	—
478	#41	—	2	1	no	—
479	#41	—	2	1	no	—
480	#41	—	2	2	no	—
481	#41	—	4	4	no	—
482	#41	—	4	4	no	—
483	#41	—	4	4	no	—
484	#41	C₂×D₄×C₃₁	1	2	yes	—
485	#41	C₂×D₄×C₃₁	1	2	yes	—
486	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
487	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	8	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	2	no	—
494	C₂×D₄	—	2	2	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	2	no	—
498	C₂×D₄	—	4	4	no	—
499	C₂×D₄	—	4	2	no	—
500	C₂×D₄	—	4	2	no	—
501	C₂×D₄	—	4	4	no	—
502	C₂×D₄	—	4	8	no	—
503	C₂×D₄	—	4	8	no	—
504	C₂×D₄	—	4	4	no	—
505	C₂×D₄	—	4	4	no	—
506	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
507	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
508	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
509	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	4	2	no	—
514	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
515	Q₁₆	—	2	2	no	—
516	Q₁₆	—	2	1	no	—
517	Q₁₆	—	2	1	no	—
518	Q₁₆	—	4	2	no	—
519	SD₁₆	—	2	8	no	—
520	SD₁₆	—	2	2	no	—
521	SD₁₆	—	2	2	no	—
522	SD₁₆	—	2	2	no	—
523	SD₁₆	—	2	2	no	—
524	SD₁₆	—	4	2	no	—
525	SD₁₆	—	4	2	no	—
526	D₈	—	2	8	no	—
527	D₈	—	2	2	no	—
528	D₈	—	2	2	no	—
529	D₈	—	2	2	no	—
530	D₈	—	2	1	no	—
531	D₈	—	2	1	no	—
532	D₈	—	2	2	no	—
533	D₈	—	2	2	no	—
534	D₈	—	4	2	no	—
535	M₁₆	—	2	1	no	—
536	M₁₆	—	2	1	no	—
537	M₁₆	—	2	2	no	—
538	M₁₆	—	2	2	no	—
539	M₁₆	—	2	1	no	—
540	M₁₆	—	2	2	no	—
541	M₁₆	—	2	2	no	—
542	M₁₆	—	2	1	no	—
543	M₁₆	—	4	1	no	—
544	M₁₆	—	4	1	no	—
545	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
546	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	2	no	—
550	C₂×C₈	—	2	2	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	2	no	—
553	C₂×C₈	—	4	1	no	—
554	C₂×C₈	—	4	1	no	—
555	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
556	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
557	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
558	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
559	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
560	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	1	no	—
563	C₂²⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	2	no	—
565	C₂²⋊C₄	—	4	4	no	—
566	C₄²	—	2	1	no	—
567	C₄²	—	2	2	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	4	no	—
571	C₂³	—	2	2	no	—
572	C₂³	—	4	8	no	—
573	C₂³	—	4	2	no	—
574	C₂³	—	4	2	no	—
575	C₂³	—	16	2	no	—
576	C₂³	—	16	4	no	—
577	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
578	Q₈	—	2	1	no	—
579	Q₈	—	2	2	no	—
580	Q₈	—	2	1	no	—
581	Q₈	—	2	1	no	—
582	Q₈	—	2	2	no	—
583	Q₈	C₃₁×#121	1	2	yes	—
584	D₄	—	2	2	no	—
585	D₄	—	2	2	no	—
586	D₄	—	2	2	no	—
587	D₄	—	2	4	no	—
588	D₄	—	2	1	no	—
589	D₄	—	2	1	no	—
590	D₄	—	2	2	no	—
591	D₄	—	2	1	no	—
592	D₄	—	2	4	no	—
593	D₄	—	4	8	no	—
594	D₄	—	4	8	no	—
595	D₄	—	4	2	no	—
596	D₄	—	4	1	no	—
597	D₄	—	4	2	no	—
598	D₄	—	4	1	no	—
599	D₄	—	4	2	no	—
600	D₄	—	4	2	no	—
601	D₄	—	16	4	no	—
602	D₄	C₃₁×#121	1	2	yes	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	4	8	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	2	no	—
615	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
616	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
617	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
618	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
619	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
620	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
621	C₈	—	2	2	no	—
622	C₈	—	2	2	no	—
623	C₈	—	4	1	no	—
624	C₈	—	4	2	no	—
625	C₈	C₃₁×#120	1	1	yes	characteristic
626	C₈	C₃₁×#120	1	1	yes	characteristic
627	C₈	C₃₁×#120	1	1	yes	characteristic
628	C₈	C₃₁×#120	1	1	yes	characteristic
629	C₂²	—	2	2	no	—
630	C₂²	—	4	1	no	—
631	C₂²	—	4	2	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	1	no	—
636	C₂²	—	8	4	no	—
637	C₂²	—	8	2	no	—
638	C₂²	—	8	4	no	—
639	C₂²	—	16	2	no	—
640	C₂²	—	16	8	no	—
641	C₂²	C₃₁×#535	1	1	yes	characteristic
642	C₂²	D₄²×C₃₁	1	1	yes	characteristic
643	C₂²	C₃₁×#546	1	1	yes	characteristic
644	C₄	—	2	2	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	1	no	—
647	C₄	—	8	2	no	—
648	C₄	—	8	1	no	—
649	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
650	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
651	C₄	D₄²×C₃₁	1	1	yes	characteristic
652	C₄	C₃₁×#546	1	1	yes	characteristic
653	C₂	—	2	1	no	—
654	C₂	—	2	1	no	—
655	C₂	—	2	1	no	—
656	C₂	—	4	2	no	—
657	C₂	—	8	1	no	—
658	C₂	—	8	4	no	—
659	C₂	—	8	2	no	—
660	C₂	C₃₁×#2408	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#2408
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#576	C₂	semidirect
2. C₃₁×#3067	C₂	semidirect
3. C₃₁×#3058	C₂	semidirect
4. C₃₁×#3019	C₂	semidirect
5. C₃₁×#2955	C₂	semidirect
6. C₃₁×#2048	C₂	semidirect
7. C₃₁×#2008	C₂	semidirect
8. C₃₁×#1978	C₂	semidirect
9. C₃₁×#1976	C₂	semidirect
10. C₃₁×#1814	C₂	semidirect
11. C₃₁×#1795	C₂	semidirect
18. C₃₁×#575	C₂²	semidirect
26. C₂×C₃₁×#131	C₂²	semidirect
27. C₃₁×#535	C₂²	semidirect
37. C₃₁×#430	C₂²	semidirect
38. C₂×C₃₁×#108	C₂²	semidirect
39. C₃₁×#427	C₂²	semidirect
42. C₃₁×#421	C₂²	semidirect
43. C₃₁×#421	C₂²	semidirect
44. C₃₁×#418	C₂²	semidirect
45. C₃₁×#418	C₂²	semidirect
46. C₃₁×#413	C₂²	semidirect
47. C₂×C₃₁×#100	C₂²	semidirect
48. C₃₁×#410	C₂²	semidirect
49. C₃₁×#405	C₂²	semidirect
50. C₃₁×#344	C₂²	semidirect
84. C₂×Q₁₆×C₃₁	D₄	semidirect
86. C₂×SD₁₆×C₃₁	D₄	semidirect
96. C₂×D₈×C₃₁	D₄	semidirect
103. C₃₁×#131	D₄	semidirect
104. C₃₁×#131	D₄	semidirect
237. #24297	C₃₁	direct (N × H)
299. C₂×#576	C₂×C₃₁	semidirect
300. #3067	C₂×C₃₁	semidirect
301. #3058	C₂×C₃₁	semidirect
302. #3019	C₂×C₃₁	semidirect
303. #2955	C₂×C₃₁	semidirect
304. #2048	C₂×C₃₁	semidirect
305. #2008	C₂×C₃₁	semidirect
306. #1978	C₂×C₃₁	semidirect
307. #1976	C₂×C₃₁	semidirect
308. #1814	C₂×C₃₁	semidirect
309. #1795	C₂×C₃₁	semidirect
340. #575	C₂²×C₃₁	semidirect
348. C₂×#131	C₂²×C₃₁	semidirect
349. #535	C₂²×C₃₁	semidirect
359. #430	C₂²×C₃₁	semidirect
360. C₂×#108	C₂²×C₃₁	semidirect
361. #427	C₂²×C₃₁	semidirect
364. #421	C₂²×C₃₁	semidirect
365. #421	C₂²×C₃₁	semidirect
366. #418	C₂²×C₃₁	semidirect
367. #418	C₂²×C₃₁	semidirect
368. #413	C₂²×C₃₁	semidirect
369. C₂×#100	C₂²×C₃₁	semidirect
370. #410	C₂²×C₃₁	semidirect
371. #405	C₂²×C₃₁	semidirect
372. #344	C₂²×C₃₁	semidirect
414. C₂×Q₁₆	D₄×C₃₁	semidirect
416. C₂×SD₁₆	D₄×C₃₁	semidirect
426. C₂×D₈	D₄×C₃₁	semidirect
433. #131	D₄×C₃₁	semidirect
434. #131	D₄×C₃₁	semidirect
467. C₃₁	#24297	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×#576	2
C₃₁×#3067	2
C₃₁×#3058	2
C₃₁×#3019	2
C₃₁×#2955	2
C₃₁×#2048	2
C₃₁×#2008	2
C₃₁×#1978	2
C₃₁×#1976	2
C₃₁×#1814	2
C₃₁×#1795	2
#24297	31

Subgroups — C31×#24297 #21481056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24297 #21481056