Subgroups — C₃₁×#24803 #21481562

Order 7936.

910 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3055	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2063	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2051	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1900	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1816	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#525	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#121	C₂²	1	2	yes	—
19	C₃₁×#472	—	2	4	no	—
20	C₃₁×#469	—	2	2	no	—
21	C₃₁×#468	—	2	2	no	—
22	C₃₁×#466	—	2	1	no	—
23	C₃₁×#466	—	2	1	no	—
24	C₃₁×#466	—	2	1	no	—
25	C₃₁×#466	—	2	1	no	—
26	C₃₁×#466	—	2	2	no	—
27	C₃₁×#465	—	2	2	no	—
28	C₃₁×#435	—	2	1	no	—
29	C₃₁×#435	—	2	1	no	—
30	C₈×D₄×C₃₁	—	2	1	no	—
31	C₈×D₄×C₃₁	—	2	1	no	—
32	C₂×C₃₁×#108	—	2	4	no	—
33	C₂×C₃₁×#108	C₂²	1	2	yes	—
34	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
36	C₃₁×#428	C₂²	1	2	yes	—
37	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
38	C₃₁×#424	C₂²	1	1	yes	characteristic
39	C₃₁×#418	C₂²	1	2	yes	—
40	C₃₁×#417	C₂²	1	2	yes	—
41	C₂×C₃₁×#102	C₂²	1	2	yes	—
42	C₂×C₃₁×#100	C₂²	1	2	yes	—
43	C₂×C₃₁×#100	C₂²	1	2	yes	—
44	C₃₁×#408	C₂²	1	1	yes	characteristic
45	C₃₁×#407	C₂²	1	1	yes	characteristic
46	C₃₁×#405	C₂²	1	1	yes	characteristic
47	C₃₁×#343	C₂²	1	2	yes	—
48	C₃₁×#343	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₃₁	—	2	2	no	—
51	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
54	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂×D₈×C₃₁	—	2	1	no	—
58	C₂×D₈×C₃₁	—	2	1	no	—
59	C₂×D₈×C₃₁	—	2	2	no	—
60	C₂×D₈×C₃₁	—	2	2	no	—
61	C₃₁×#131	—	2	2	no	—
62	C₃₁×#131	—	2	2	no	—
63	C₂×M₁₆×C₃₁	—	2	2	no	—
64	C₂×M₁₆×C₃₁	—	2	1	no	—
65	C₂×M₁₆×C₃₁	—	2	4	no	—
66	C₂×M₁₆×C₃₁	—	4	2	no	—
67	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂²×C₈×C₃₁	—	2	2	no	—
72	C₂²×C₈×C₃₁	—	2	2	no	—
73	C₂²×C₈×C₃₁	—	2	2	no	—
74	C₂²×C₈×C₃₁	—	2	1	no	—
75	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#123	—	2	1	no	—
82	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
83	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
84	C₃₁×#121	—	2	2	no	—
85	C₃₁×#121	—	2	2	no	—
86	C₃₁×#121	—	2	2	no	—
87	C₃₁×#121	—	2	2	no	—
88	C₃₁×#121	D₄	1	1	yes	characteristic
89	C₃₁×#121	D₄	1	1	yes	characteristic
90	C₃₁×#120	—	2	1	no	—
91	C₄×D₄×C₃₁	—	2	1	no	—
92	C₄×D₄×C₃₁	—	2	1	no	—
93	C₃₁×#117	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
95	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₃₁×#108	—	2	4	no	—
97	C₃₁×#108	—	2	2	no	—
98	C₃₁×#108	D₄	1	4	yes	complemented
99	C₃₁×#107	—	2	1	no	—
100	C₃₁×#107	—	2	1	no	—
101	C₃₁×#107	—	2	2	no	—
102	C₃₁×#106	—	2	2	no	—
103	C₃₁×C₄⋊C₈	—	2	1	no	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×#103	—	2	2	no	—
106	C₃₁×#102	—	2	2	no	—
107	C₃₁×#102	—	2	2	no	—
108	C₃₁×#102	—	2	2	no	—
109	C₃₁×#102	—	2	2	no	—
110	C₃₁×#102	—	2	2	no	—
111	C₃₁×#100	—	2	4	no	—
112	C₃₁×#100	—	4	2	no	—
113	C₃₁×C₂²⋊C₈	—	2	1	no	—
114	C₃₁×C₂²⋊C₈	—	2	1	no	—
115	C₃₁×C₂²⋊C₈	—	2	1	no	—
116	C₃₁×C₂²⋊C₈	—	2	1	no	—
117	C₃₁×#97	—	2	1	no	—
118	C₄×C₈×C₃₁	—	2	1	no	—
119	C₂⁴×C₃₁	—	2	2	no	—
120	C₂⁴×C₃₁	#41	1	1	yes	characteristic
121	#41×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	1	no	—
123	#41×C₃₁	—	2	1	no	—
124	#41×C₃₁	—	2	1	no	—
125	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×D₄×C₃₁	—	2	1	no	—
127	C₂×D₄×C₃₁	—	2	1	no	—
128	C₂×D₄×C₃₁	—	2	1	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	2	2	no	—
131	C₂×D₄×C₃₁	—	2	2	no	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	2	no	—
135	C₂×D₄×C₃₁	—	4	2	no	—
136	C₂×D₄×C₃₁	—	4	2	no	—
137	C₂×D₄×C₃₁	—	8	4	no	—
138	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	2	no	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
151	SD₁₆×C₃₁	—	4	4	no	—
152	D₈×C₃₁	—	4	4	no	—
153	D₈×C₃₁	—	4	2	no	—
154	D₈×C₃₁	—	4	2	no	—
155	M₁₆×C₃₁	—	2	2	no	—
156	M₁₆×C₃₁	—	2	1	no	—
157	M₁₆×C₃₁	—	2	2	no	—
158	M₁₆×C₃₁	—	2	1	no	—
159	M₁₆×C₃₁	—	2	1	no	—
160	M₁₆×C₃₁	—	2	2	no	—
161	M₁₆×C₃₁	—	2	1	no	—
162	M₁₆×C₃₁	—	4	4	no	—
163	M₁₆×C₃₁	—	4	2	no	—
164	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	—	2	1	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	2	no	—
169	C₂×C₈×C₃₁	—	2	2	no	—
170	C₂×C₈×C₃₁	—	2	2	no	—
171	C₂×C₈×C₃₁	—	2	2	no	—
172	C₂×C₈×C₃₁	—	2	1	no	—
173	C₂×C₈×C₃₁	—	2	1	no	—
174	C₂×C₈×C₃₁	—	2	2	no	—
175	C₂×C₈×C₃₁	—	4	4	no	—
176	C₂×C₈×C₃₁	—	4	2	no	—
177	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₃₁	#41	1	2	yes	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₂²⋊C₄×C₃₁	—	2	1	no	—
190	C₂²⋊C₄×C₃₁	—	2	1	no	—
191	C₂²⋊C₄×C₃₁	—	2	1	no	—
192	C₂²⋊C₄×C₃₁	—	2	1	no	—
193	C₂²⋊C₄×C₃₁	—	2	1	no	—
194	C₂²⋊C₄×C₃₁	—	4	2	no	—
195	C₂²⋊C₄×C₃₁	—	4	2	no	—
196	C₄²×C₃₁	—	2	1	no	—
197	#24803	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₃₁	—	2	1	no	—
199	C₂³×C₃₁	—	2	1	no	—
200	C₂³×C₃₁	—	2	2	no	—
201	C₂³×C₃₁	—	2	1	no	—
202	C₂³×C₃₁	—	4	1	no	—
203	C₂³×C₃₁	—	4	2	no	—
204	C₂³×C₃₁	—	4	2	no	—
205	C₂³×C₃₁	—	4	2	no	—
206	C₂³×C₃₁	—	8	2	no	—
207	C₂³×C₃₁	—	8	2	no	—
208	C₂³×C₃₁	#121	1	1	yes	characteristic
209	C₂³×C₃₁	#121	1	1	yes	characteristic
210	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
211	Q₈×C₃₁	—	2	1	no	—
212	Q₈×C₃₁	—	2	1	no	—
213	D₄×C₃₁	—	2	1	no	—
214	D₄×C₃₁	—	2	1	no	—
215	D₄×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	4	1	no	—
217	D₄×C₃₁	—	4	1	no	—
218	D₄×C₃₁	—	4	2	no	—
219	D₄×C₃₁	—	4	2	no	—
220	D₄×C₃₁	—	4	2	no	—
221	D₄×C₃₁	—	4	2	no	—
222	D₄×C₃₁	—	4	2	no	—
223	D₄×C₃₁	—	8	4	no	—
224	D₄×C₃₁	—	8	4	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	1	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	2	1	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	2	1	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	1	no	—
240	C₂×C₄×C₃₁	—	4	2	no	—
241	C₂×C₄×C₃₁	—	8	2	no	—
242	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
243	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
249	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
250	C₈×C₃₁	—	2	1	no	—
251	C₈×C₃₁	—	2	1	no	—
252	C₈×C₃₁	—	2	4	no	—
253	C₈×C₃₁	—	4	1	no	—
254	C₈×C₃₁	—	4	1	no	—
255	C₈×C₃₁	—	4	2	no	—
256	C₈×C₃₁	#121	1	2	yes	—
257	C₈×C₃₁	#121	1	2	yes	—
258	C₂×#472	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#3055	C₂×C₃₁	1	2	yes	complemented
260	C₂×#466	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#2929	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#2063	C₂×C₃₁	1	2	yes	complemented
264	#2051	C₂×C₃₁	1	2	yes	complemented
265	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#1900	C₂×C₃₁	1	2	yes	complemented
268	#1816	C₂×C₃₁	1	1	yes	characteristic, complemented
269	C₂²×C₃₁	—	2	1	no	—
270	C₂²×C₃₁	—	2	1	no	—
271	C₂²×C₃₁	—	2	1	no	—
272	C₂²×C₃₁	—	2	1	no	—
273	C₂²×C₃₁	—	2	1	no	—
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	2	1	no	—
276	C₂²×C₃₁	—	4	2	no	—
277	C₂²×C₃₁	—	4	2	no	—
278	C₂²×C₃₁	—	4	1	no	—
279	C₂²×C₃₁	—	4	1	no	—
280	C₂²×C₃₁	—	4	1	no	—
281	C₂²×C₃₁	—	8	2	no	—
282	C₂²×C₃₁	—	8	2	no	—
283	C₂²×C₃₁	—	8	2	no	—
284	C₂²×C₃₁	—	16	2	no	—
285	C₂²×C₃₁	—	16	2	no	—
286	C₂²×C₃₁	#472	1	2	yes	complemented
287	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
288	C₂²×C₃₁	#548	1	1	yes	characteristic
289	C₂²×C₃₁	#553	1	1	yes	characteristic
290	C₄×C₃₁	—	2	1	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	4	1	no	—
293	C₄×C₃₁	—	4	1	no	—
294	C₄×C₃₁	—	4	1	no	—
295	C₄×C₃₁	—	8	2	no	—
296	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
297	C₄×C₃₁	#537	1	1	yes	characteristic
298	C₄×C₃₁	D₄²	1	1	yes	characteristic
299	C₄×C₃₁	#547	1	1	yes	characteristic
300	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
301	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
302	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
303	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
304	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
305	#525	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×#121	C₂²×C₃₁	1	2	yes	—
307	#472	—	2	4	no	—
308	#469	—	2	2	no	—
309	#468	—	2	2	no	—
310	#466	—	2	1	no	—
311	#466	—	2	1	no	—
312	#466	—	2	1	no	—
313	#466	—	2	1	no	—
314	#466	—	2	2	no	—
315	#465	—	2	2	no	—
316	#435	—	2	1	no	—
317	#435	—	2	1	no	—
318	C₈×D₄	—	2	1	no	—
319	C₈×D₄	—	2	1	no	—
320	C₂×#108	—	2	4	no	—
321	C₂×#108	C₂²×C₃₁	1	2	yes	—
322	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
324	#428	C₂²×C₃₁	1	2	yes	—
325	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
326	#424	C₂²×C₃₁	1	1	yes	characteristic
327	#418	C₂²×C₃₁	1	2	yes	—
328	#417	C₂²×C₃₁	1	2	yes	—
329	C₂×#102	C₂²×C₃₁	1	2	yes	—
330	C₂×#100	C₂²×C₃₁	1	2	yes	—
331	C₂×#100	C₂²×C₃₁	1	2	yes	—
332	#408	C₂²×C₃₁	1	1	yes	characteristic
333	#407	C₂²×C₃₁	1	1	yes	characteristic
334	#405	C₂²×C₃₁	1	1	yes	characteristic
335	#343	C₂²×C₃₁	1	2	yes	—
336	#343	C₂²×C₃₁	1	2	yes	—
337	C₂×C₃₁	—	2	1	no	—
338	C₂×C₃₁	—	2	1	no	—
339	C₂×C₃₁	—	2	2	no	—
340	C₂×C₃₁	—	4	1	no	—
341	C₂×C₃₁	—	8	1	no	—
342	C₂×C₃₁	—	16	2	no	—
343	C₂×C₃₁	#2416	1	1	yes	characteristic
344	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
345	C₂×C₃₁	#3067	1	1	yes	characteristic
346	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
347	C₂²×D₄	—	2	2	no	—
348	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
349	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
350	AGL(1,ℤ/8)	—	2	4	no	—
351	AGL(1,ℤ/8)	—	4	2	no	—
352	C₂×SD₁₆	—	2	2	no	—
353	C₂×D₈	—	2	2	no	—
354	C₂×D₈	—	2	1	no	—
355	C₂×D₈	—	2	1	no	—
356	C₂×D₈	—	2	2	no	—
357	C₂×D₈	—	2	2	no	—
358	#131	—	2	2	no	—
359	#131	—	2	2	no	—
360	C₂×M₁₆	—	2	2	no	—
361	C₂×M₁₆	—	2	1	no	—
362	C₂×M₁₆	—	2	4	no	—
363	C₂×M₁₆	—	4	2	no	—
364	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
365	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
366	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
367	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
368	C₂²×C₈	—	2	2	no	—
369	C₂²×C₈	—	2	2	no	—
370	C₂²×C₈	—	2	2	no	—
371	C₂²×C₈	—	2	1	no	—
372	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
375	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
378	#123	—	2	1	no	—
379	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
380	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
381	#121	—	2	2	no	—
382	#121	—	2	2	no	—
383	#121	—	2	2	no	—
384	#121	—	2	2	no	—
385	#121	D₄×C₃₁	1	1	yes	characteristic
386	#121	D₄×C₃₁	1	1	yes	characteristic
387	#120	—	2	1	no	—
388	C₄×D₄	—	2	1	no	—
389	C₄×D₄	—	2	1	no	—
390	#117	C₂³×C₃₁	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
392	C₂×C₂²⋊C₄	—	2	2	no	—
393	#108	—	2	4	no	—
394	#108	—	2	2	no	—
395	#108	D₄×C₃₁	1	4	yes	complemented
396	#107	—	2	1	no	—
397	#107	—	2	1	no	—
398	#107	—	2	2	no	—
399	#106	—	2	2	no	—
400	C₄⋊C₈	—	2	1	no	—
401	C₄⋊C₈	—	2	1	no	—
402	#103	—	2	2	no	—
403	#102	—	2	2	no	—
404	#102	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	#100	—	2	4	no	—
409	#100	—	4	2	no	—
410	C₂²⋊C₈	—	2	1	no	—
411	C₂²⋊C₈	—	2	1	no	—
412	C₂²⋊C₈	—	2	1	no	—
413	C₂²⋊C₈	—	2	1	no	—
414	#97	—	2	1	no	—
415	C₄×C₈	—	2	1	no	—
416	C₃₁	#24803	1	1	yes	characteristic, direct product factor, Sylow
417	C₂⁴	—	2	2	no	—
418	C₂⁴	#41×C₃₁	1	1	yes	characteristic
419	#41	—	2	1	no	—
420	#41	—	2	1	no	—
421	#41	—	2	1	no	—
422	#41	—	2	1	no	—
423	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	4	2	no	—
434	C₂×D₄	—	4	2	no	—
435	C₂×D₄	—	8	4	no	—
436	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
444	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
449	SD₁₆	—	4	4	no	—
450	D₈	—	4	4	no	—
451	D₈	—	4	2	no	—
452	D₈	—	4	2	no	—
453	M₁₆	—	2	2	no	—
454	M₁₆	—	2	1	no	—
455	M₁₆	—	2	2	no	—
456	M₁₆	—	2	1	no	—
457	M₁₆	—	2	1	no	—
458	M₁₆	—	2	2	no	—
459	M₁₆	—	2	1	no	—
460	M₁₆	—	4	4	no	—
461	M₁₆	—	4	2	no	—
462	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	4	4	no	—
474	C₂×C₈	—	4	2	no	—
475	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
480	C₂×C₈	#41×C₃₁	1	2	yes	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	2	1	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₂²⋊C₄	—	4	2	no	—
494	C₄²	—	2	1	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	4	1	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	4	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	8	2	no	—
504	C₂³	—	8	2	no	—
505	C₂³	C₃₁×#121	1	1	yes	characteristic
506	C₂³	C₃₁×#121	1	1	yes	characteristic
507	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
508	Q₈	—	2	1	no	—
509	Q₈	—	2	1	no	—
510	D₄	—	2	1	no	—
511	D₄	—	2	1	no	—
512	D₄	—	4	2	no	—
513	D₄	—	4	2	no	—
514	D₄	—	4	1	no	—
515	D₄	—	4	1	no	—
516	D₄	—	4	2	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	8	4	no	—
521	D₄	—	8	4	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
540	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
543	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
544	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
545	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
546	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
547	C₈	—	2	1	no	—
548	C₈	—	2	1	no	—
549	C₈	—	2	4	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	1	no	—
552	C₈	—	4	2	no	—
553	C₈	C₃₁×#121	1	2	yes	—
554	C₈	C₃₁×#121	1	2	yes	—
555	C₂²	—	2	1	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	2	1	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	1	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	—	8	2	no	—
568	C₂²	—	8	2	no	—
569	C₂²	—	8	2	no	—
570	C₂²	—	16	2	no	—
571	C₂²	—	16	2	no	—
572	C₂²	C₃₁×#472	1	2	yes	complemented
573	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
574	C₂²	C₃₁×#548	1	1	yes	characteristic
575	C₂²	C₃₁×#553	1	1	yes	characteristic
576	C₄	—	2	1	no	—
577	C₄	—	2	1	no	—
578	C₄	—	4	1	no	—
579	C₄	—	4	1	no	—
580	C₄	—	4	1	no	—
581	C₄	—	8	2	no	—
582	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
583	C₄	C₃₁×#537	1	1	yes	characteristic
584	C₄	D₄²×C₃₁	1	1	yes	characteristic
585	C₄	C₃₁×#547	1	1	yes	characteristic
586	C₂	—	2	1	no	—
587	C₂	—	2	1	no	—
588	C₂	—	2	2	no	—
589	C₂	—	4	1	no	—
590	C₂	—	8	1	no	—
591	C₂	—	16	2	no	—
592	C₂	C₃₁×#2416	1	1	yes	characteristic
593	C₂	C₂×C₃₁×#472	1	1	yes	characteristic
594	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#472	C₂	semidirect
2. C₃₁×#3055	C₂	semidirect
3. C₂×C₃₁×#466	C₂	semidirect
4. C₃₁×#2929	C₂	semidirect
5. C₂²×C₃₁×#108	C₂	semidirect
6. C₃₁×#2063	C₂	semidirect
7. C₃₁×#2051	C₂	semidirect
8. C₃₁×#1978	C₂	semidirect
9. C₃₁×#1952	C₂	semidirect
10. C₃₁×#1900	C₂	semidirect
11. C₃₁×#1816	C₂	semidirect
98. C₃₁×#108	D₄	semidirect
197. #24803	C₃₁	direct (N × H)
258. C₂×#472	C₂×C₃₁	semidirect
259. #3055	C₂×C₃₁	semidirect
260. C₂×#466	C₂×C₃₁	semidirect
261. #2929	C₂×C₃₁	semidirect
262. C₂²×#108	C₂×C₃₁	semidirect
263. #2063	C₂×C₃₁	semidirect
264. #2051	C₂×C₃₁	semidirect
265. #1978	C₂×C₃₁	semidirect
266. #1952	C₂×C₃₁	semidirect
267. #1900	C₂×C₃₁	semidirect
268. #1816	C₂×C₃₁	semidirect
286. C₂²×C₃₁	#472	semidirect
395. #108	D₄×C₃₁	semidirect
416. C₃₁	#24803	direct (N × H)
572. C₂²	C₃₁×#472	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#472	2
C₃₁×#3055	2
C₂×C₃₁×#466	2
C₃₁×#2929	2
C₂²×C₃₁×#108	2
C₃₁×#2063	2
C₃₁×#2051	2
C₃₁×#1978	2
C₃₁×#1952	2
C₃₁×#1900	2
C₃₁×#1816	2
#24803	31

Subgroups — C31×#24803 #21481562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24803 #21481562