Subgroups — C₃₁×#24857 #21481616

Order 7936.

720 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#137	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2089	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2064	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1908	C₂	1	2	yes	maximal
10	C₃₁×#1816	C₂	1	2	yes	maximal
11	C₃₁×#1801	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#477	—	2	1	no	—
18	C₃₁×#477	—	2	1	no	—
19	C₃₁×#477	—	2	1	no	—
20	C₃₁×#477	—	2	2	no	—
21	C₃₁×#476	—	2	1	no	—
22	C₃₁×#475	—	2	1	no	—
23	C₃₁×#474	—	2	1	no	—
24	C₃₁×#441	—	2	1	no	—
25	C₃₁×#440	—	2	1	no	—
26	C₄×Q₁₆×C₃₁	—	2	1	no	—
27	C₄×SD₁₆×C₃₁	—	2	1	no	—
28	C₂×C₃₁×#108	C₂²	1	2	yes	—
29	C₃₁×#428	C₂²	1	2	yes	—
30	C₃₁×#428	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#107	C₂²	1	2	yes	—
32	C₃₁×#424	—	2	2	no	—
33	C₃₁×#424	C₂²	1	2	yes	—
34	C₃₁×#423	—	2	1	no	—
35	C₃₁×#423	C₂²	1	1	yes	characteristic
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃₁×#419	C₂²	1	1	yes	characteristic
40	C₃₁×#417	C₂²	1	2	yes	—
41	C₃₁×#417	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#103	C₂²	1	2	yes	—
43	C₂×C₃₁×#101	C₂²	1	2	yes	—
44	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#343	C₂²	1	2	yes	—
46	C₃₁×#343	C₂²	1	2	yes	—
47	C₃₁×#343	C₂²	1	2	yes	—
48	C₃₁×#341	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	—	2	1	no	—
51	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#137	—	2	2	no	—
53	C₃₁×#137	Q₈	1	4	yes	complemented
54	C₂×Q₁₆×C₃₁	—	2	1	no	—
55	C₂×SD₁₆×C₃₁	—	2	1	no	—
56	C₃₁×#131	—	2	4	no	—
57	C₃₁×#131	—	4	2	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
65	C₃₁×C₄⋊Q₈	—	2	1	no	—
66	C₃₁×C₄⋊Q₈	—	2	1	no	—
67	C₃₁×C₄⋊Q₈	—	2	1	no	—
68	C₃₁×C₄⋊Q₈	—	2	1	no	—
69	C₄×Q₈×C₃₁	—	2	1	no	—
70	C₄×Q₈×C₃₁	—	2	1	no	—
71	C₄×Q₈×C₃₁	—	2	1	no	—
72	C₄×Q₈×C₃₁	—	2	1	no	—
73	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
78	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
79	C₃₁×#117	—	2	1	no	—
80	C₃₁×#117	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	—	2	1	no	—
87	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
88	C₃₁×#108	—	4	2	no	—
89	C₃₁×#107	—	2	2	no	—
90	C₃₁×#107	—	2	2	no	—
91	C₃₁×#107	—	2	1	no	—
92	C₃₁×#107	—	2	2	no	—
93	C₃₁×#106	—	2	1	no	—
94	C₃₁×#106	—	2	2	no	—
95	C₃₁×C₄⋊C₈	—	2	2	no	—
96	C₃₁×C₄⋊C₈	—	2	2	no	—
97	C₃₁×C₄⋊C₈	—	2	1	no	—
98	C₃₁×C₄⋊C₈	—	2	1	no	—
99	C₃₁×C₄⋊C₈	—	2	1	no	—
100	C₃₁×C₄⋊C₈	—	2	1	no	—
101	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
102	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
103	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
104	C₃₁×#103	—	2	1	no	—
105	C₃₁×#103	—	2	2	no	—
106	C₃₁×#103	—	2	2	no	—
107	C₃₁×#103	—	2	2	no	—
108	C₃₁×#103	—	2	1	no	—
109	C₃₁×#103	—	2	1	no	—
110	C₃₁×#102	—	2	1	no	—
111	C₃₁×#102	—	2	2	no	—
112	C₃₁×#101	—	4	2	no	—
113	C₃₁×#97	—	2	1	no	—
114	C₄×C₈×C₃₁	—	2	1	no	—
115	#41×C₃₁	—	2	1	no	—
116	#41×C₃₁	—	2	1	no	—
117	#41×C₃₁	C₂×Q₈	1	2	yes	—
118	#41×C₃₁	#41	1	2	yes	—
119	C₂×Q₈×C₃₁	—	2	1	no	—
120	C₂×Q₈×C₃₁	—	2	1	no	—
121	C₂×Q₈×C₃₁	—	2	1	no	—
122	C₂×Q₈×C₃₁	—	2	1	no	—
123	C₂×Q₈×C₃₁	—	2	1	no	—
124	C₂×Q₈×C₃₁	—	4	1	no	—
125	C₂×Q₈×C₃₁	—	4	1	no	—
126	C₂×Q₈×C₃₁	—	8	2	no	—
127	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
129	C₂×D₄×C₃₁	—	2	1	no	—
130	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	2	2	no	—
137	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₃₁	—	2	4	no	—
141	SD₁₆×C₃₁	—	2	4	no	—
142	M₁₆×C₃₁	—	2	1	no	—
143	M₁₆×C₃₁	—	2	2	no	—
144	M₁₆×C₃₁	—	2	2	no	—
145	M₁₆×C₃₁	—	2	2	no	—
146	M₁₆×C₃₁	—	2	1	no	—
147	M₁₆×C₃₁	—	2	1	no	—
148	M₁₆×C₃₁	—	2	2	no	—
149	M₁₆×C₃₁	—	4	4	no	—
150	M₁₆×C₃₁	C₂×Q₈	1	2	yes	—
151	C₂×C₈×C₃₁	—	2	1	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	1	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	4	4	no	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	4	1	no	—
167	C₄⋊C₄×C₃₁	—	4	1	no	—
168	C₄⋊C₄×C₃₁	—	4	1	no	—
169	C₄⋊C₄×C₃₁	—	4	1	no	—
170	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²⋊C₄×C₃₁	—	2	1	no	—
173	C₂²⋊C₄×C₃₁	—	2	1	no	—
174	C₂²⋊C₄×C₃₁	—	2	1	no	—
175	C₄²×C₃₁	—	2	1	no	—
176	C₄²×C₃₁	—	2	1	no	—
177	C₄²×C₃₁	—	2	1	no	—
178	C₄²×C₃₁	—	4	1	no	—
179	C₄²×C₃₁	—	4	1	no	—
180	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
184	#24857	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₃₁	—	2	1	no	—
186	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
187	Q₈×C₃₁	—	2	1	no	—
188	Q₈×C₃₁	—	2	2	no	—
189	Q₈×C₃₁	—	2	2	no	—
190	Q₈×C₃₁	—	4	1	no	—
191	Q₈×C₃₁	—	4	1	no	—
192	Q₈×C₃₁	—	8	2	no	—
193	Q₈×C₃₁	—	8	2	no	—
194	Q₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
195	D₄×C₃₁	—	2	2	no	—
196	D₄×C₃₁	—	2	2	no	—
197	D₄×C₃₁	—	2	1	no	—
198	D₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	8	1	no	—
213	C₂×C₄×C₃₁	—	8	1	no	—
214	C₂×C₄×C₃₁	—	8	2	no	—
215	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
216	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
217	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
218	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₃₁	—	2	2	no	—
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	2	2	no	—
228	C₈×C₃₁	—	4	1	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₈×C₃₁	—	4	2	no	—
231	#3079	C₂×C₃₁	1	1	yes	characteristic
232	C₂×#477	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₄×#137	C₂×C₃₁	1	1	yes	characteristic
234	#2906	C₂×C₃₁	1	1	yes	characteristic
235	#2089	C₂×C₃₁	1	2	yes	complemented
236	#2064	C₂×C₃₁	1	2	yes	complemented
237	#1997	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1908	C₂×C₃₁	1	2	yes	—
240	#1816	C₂×C₃₁	1	2	yes	—
241	#1801	C₂×C₃₁	1	1	yes	characteristic, complemented
242	C₂²×C₃₁	—	2	1	no	—
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	4	1	no	—
246	C₂²×C₃₁	—	4	1	no	—
247	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
248	C₂²×C₃₁	#552	1	1	yes	characteristic
249	C₂²×C₃₁	#554	1	1	yes	characteristic
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	2	1	no	—
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	1	no	—
254	C₄×C₃₁	—	4	2	no	—
255	C₄×C₃₁	—	4	1	no	—
256	C₄×C₃₁	—	4	1	no	—
257	C₄×C₃₁	—	8	1	no	—
258	C₄×C₃₁	—	8	2	no	—
259	C₄×C₃₁	#473	1	2	yes	—
260	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
261	C₄×C₃₁	#537	1	1	yes	characteristic
262	C₄×C₃₁	#546	1	1	yes	characteristic
263	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
264	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#131	C₂²×C₃₁	1	2	yes	—
266	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
267	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
269	#477	—	2	1	no	—
270	#477	—	2	1	no	—
271	#477	—	2	1	no	—
272	#477	—	2	2	no	—
273	#476	—	2	1	no	—
274	#475	—	2	1	no	—
275	#474	—	2	1	no	—
276	#441	—	2	1	no	—
277	#440	—	2	1	no	—
278	C₄×Q₁₆	—	2	1	no	—
279	C₄×SD₁₆	—	2	1	no	—
280	C₂×#108	C₂²×C₃₁	1	2	yes	—
281	#428	C₂²×C₃₁	1	2	yes	—
282	#428	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×#107	C₂²×C₃₁	1	2	yes	—
284	#424	—	2	2	no	—
285	#424	C₂²×C₃₁	1	2	yes	—
286	#423	—	2	1	no	—
287	#423	C₂²×C₃₁	1	1	yes	characteristic
288	#423	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₈	—	2	1	no	—
290	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
291	#419	C₂²×C₃₁	1	1	yes	characteristic
292	#417	C₂²×C₃₁	1	2	yes	—
293	#417	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×#103	C₂²×C₃₁	1	2	yes	—
295	C₂×#101	C₂²×C₃₁	1	2	yes	—
296	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
297	#343	C₂²×C₃₁	1	2	yes	—
298	#343	C₂²×C₃₁	1	2	yes	—
299	#343	C₂²×C₃₁	1	2	yes	—
300	#341	C₂²×C₃₁	1	2	yes	—
301	C₂×C₃₁	—	2	1	no	—
302	C₂×C₃₁	—	2	1	no	—
303	C₂×C₃₁	—	4	2	no	—
304	C₂×C₃₁	#2417	1	1	yes	characteristic
305	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
306	C₂×C₃₁	#3068	1	1	yes	characteristic
307	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
308	C₂²×Q₈	—	2	1	no	—
309	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
310	#137	—	2	2	no	—
311	#137	Q₈×C₃₁	1	4	yes	complemented
312	C₂×Q₁₆	—	2	1	no	—
313	C₂×SD₁₆	—	2	1	no	—
314	#131	—	2	4	no	—
315	#131	—	4	2	no	—
316	C₂×M₁₆	—	2	2	no	—
317	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
318	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
319	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
320	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
321	C₂²×C₈	—	2	2	no	—
322	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
323	C₄⋊Q₈	—	2	1	no	—
324	C₄⋊Q₈	—	2	1	no	—
325	C₄⋊Q₈	—	2	1	no	—
326	C₄⋊Q₈	—	2	1	no	—
327	C₄×Q₈	—	2	1	no	—
328	C₄×Q₈	—	2	1	no	—
329	C₄×Q₈	—	2	1	no	—
330	C₄×Q₈	—	2	1	no	—
331	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
332	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
333	C₄×D₄	—	2	1	no	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
336	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
337	#117	—	2	1	no	—
338	#117	C₂³×C₃₁	1	1	yes	characteristic
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
343	C₂×C₄²	—	2	1	no	—
344	C₂×C₄²	—	2	1	no	—
345	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
346	#108	—	4	2	no	—
347	#107	—	2	2	no	—
348	#107	—	2	2	no	—
349	#107	—	2	1	no	—
350	#107	—	2	2	no	—
351	#106	—	2	1	no	—
352	#106	—	2	2	no	—
353	C₄⋊C₈	—	2	2	no	—
354	C₄⋊C₈	—	2	2	no	—
355	C₄⋊C₈	—	2	1	no	—
356	C₄⋊C₈	—	2	1	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
360	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
361	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
362	#103	—	2	1	no	—
363	#103	—	2	2	no	—
364	#103	—	2	2	no	—
365	#103	—	2	2	no	—
366	#103	—	2	1	no	—
367	#103	—	2	1	no	—
368	#102	—	2	1	no	—
369	#102	—	2	2	no	—
370	#101	—	4	2	no	—
371	#97	—	2	1	no	—
372	C₄×C₈	—	2	1	no	—
373	C₃₁	#24857	1	1	yes	characteristic, direct product factor, Sylow
374	#41	—	2	1	no	—
375	#41	—	2	1	no	—
376	#41	C₂×Q₈×C₃₁	1	2	yes	—
377	#41	#41×C₃₁	1	2	yes	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	—	4	1	no	—
384	C₂×Q₈	—	4	1	no	—
385	C₂×Q₈	—	8	2	no	—
386	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
399	Q₁₆	—	2	4	no	—
400	SD₁₆	—	2	4	no	—
401	M₁₆	—	2	1	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	2	1	no	—
406	M₁₆	—	2	2	no	—
407	M₁₆	—	2	1	no	—
408	M₁₆	—	4	4	no	—
409	M₁₆	C₂×Q₈×C₃₁	1	2	yes	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	4	4	no	—
417	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
418	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	4	1	no	—
426	C₄⋊C₄	—	4	1	no	—
427	C₄⋊C₄	—	4	1	no	—
428	C₄⋊C₄	—	4	1	no	—
429	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
430	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₂²⋊C₄	—	2	1	no	—
432	C₂²⋊C₄	—	2	1	no	—
433	C₂²⋊C₄	—	2	1	no	—
434	C₄²	—	2	1	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	1	no	—
437	C₄²	—	4	1	no	—
438	C₄²	—	4	1	no	—
439	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₂³	—	2	1	no	—
444	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
445	Q₈	—	2	1	no	—
446	Q₈	—	2	2	no	—
447	Q₈	—	2	2	no	—
448	Q₈	—	4	1	no	—
449	Q₈	—	4	1	no	—
450	Q₈	—	8	2	no	—
451	Q₈	—	8	2	no	—
452	Q₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
453	D₄	—	2	2	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	1	no	—
456	D₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	8	1	no	—
471	C₂×C₄	—	8	1	no	—
472	C₂×C₄	—	8	2	no	—
473	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
474	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
475	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
476	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
482	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
483	C₈	—	2	2	no	—
484	C₈	—	2	2	no	—
485	C₈	—	2	2	no	—
486	C₈	—	4	1	no	—
487	C₈	—	4	2	no	—
488	C₈	—	4	2	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	2	1	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	4	1	no	—
493	C₂²	—	4	1	no	—
494	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
495	C₂²	C₃₁×#552	1	1	yes	characteristic
496	C₂²	C₃₁×#554	1	1	yes	characteristic
497	C₄	—	2	2	no	—
498	C₄	—	2	1	no	—
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	1	no	—
503	C₄	—	4	1	no	—
504	C₄	—	8	2	no	—
505	C₄	—	8	1	no	—
506	C₄	C₃₁×#473	1	2	yes	—
507	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
508	C₄	C₃₁×#537	1	1	yes	characteristic
509	C₄	C₃₁×#546	1	1	yes	characteristic
510	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
511	C₂	—	2	1	no	—
512	C₂	—	2	1	no	—
513	C₂	—	4	2	no	—
514	C₂	C₃₁×#2417	1	1	yes	characteristic
515	C₂	C₂×C₃₁×#473	1	1	yes	characteristic
516	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#477	C₂	semidirect
5. C₃₁×#2089	C₂	semidirect
6. C₃₁×#2064	C₂	semidirect
7. C₃₁×#1997	C₂	semidirect
8. C₃₁×#1950	C₂	semidirect
11. C₃₁×#1801	C₂	semidirect
53. C₃₁×#137	Q₈	semidirect
184. #24857	C₃₁	direct (N × H)
232. C₂×#477	C₂×C₃₁	semidirect
235. #2089	C₂×C₃₁	semidirect
236. #2064	C₂×C₃₁	semidirect
237. #1997	C₂×C₃₁	semidirect
238. #1950	C₂×C₃₁	semidirect
241. #1801	C₂×C₃₁	semidirect
311. #137	Q₈×C₃₁	semidirect
373. C₃₁	#24857	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×#3079	2
C₂×C₃₁×#477	2
C₄×C₃₁×#137	2
C₃₁×#2906	2
C₃₁×#2089	2
C₃₁×#2064	2
C₃₁×#1997	2
C₃₁×#1950	2
C₃₁×#1908	2
C₃₁×#1816	2
C₃₁×#1801	2
#24857	31

Subgroups — C31×#24857 #21481616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24857 #21481616