Subgroups — C₃₁×#24673 #21481432

Order 7936.

806 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2051	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2019	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1977	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1931	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1801	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#460	—	2	1	no	—
19	C₃₁×#460	—	2	1	no	—
20	C₃₁×#460	—	2	1	no	—
21	C₃₁×#460	—	2	1	no	—
22	C₃₁×#460	—	2	2	no	—
23	C₃₁×#459	—	2	1	no	—
24	C₃₁×#459	—	2	1	no	—
25	C₃₁×#441	—	2	1	no	—
26	C₃₁×#440	—	2	1	no	—
27	C₄×Q₁₆×C₃₁	—	2	1	no	—
28	C₄×SD₁₆×C₃₁	—	2	1	no	—
29	C₃₁×#433	—	2	2	no	—
30	C₃₁×#432	—	2	1	no	—
31	C₃₁×#431	—	2	1	no	—
32	C₂×C₃₁×#108	C₂²	1	2	yes	—
33	C₃₁×#428	C₂²	1	1	yes	characteristic
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₃₁×#423	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×#419	C₂²	1	1	yes	characteristic
38	C₃₁×#418	C₂²	1	2	yes	—
39	C₃₁×#417	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#102	C₂²	1	2	yes	—
41	C₂×C₃₁×#101	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	2	yes	—
45	C₃₁×#405	C₂²	1	2	yes	—
46	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
54	C₃₁×#137	—	2	2	no	—
55	C₃₁×#137	—	2	2	no	—
56	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
57	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
58	C₂×Q₁₆×C₃₁	—	2	1	no	—
59	C₂×SD₁₆×C₃₁	—	2	1	no	—
60	C₂×SD₁₆×C₃₁	—	2	2	no	—
61	C₂×SD₁₆×C₃₁	—	2	4	no	—
62	C₂×SD₁₆×C₃₁	—	2	2	no	—
63	C₂×SD₁₆×C₃₁	—	2	2	no	—
64	C₂×SD₁₆×C₃₁	—	4	2	no	—
65	C₂×D₈×C₃₁	—	2	2	no	—
66	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
68	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
71	C₃₁×#127	—	2	1	no	—
72	C₃₁×#127	—	2	1	no	—
73	C₃₁×#127	—	2	1	no	—
74	C₃₁×#127	—	2	1	no	—
75	C₄×Q₈×C₃₁	—	2	1	no	—
76	C₄×Q₈×C₃₁	—	2	1	no	—
77	C₄×Q₈×C₃₁	—	2	1	no	—
78	C₄×Q₈×C₃₁	—	2	1	no	—
79	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
81	C₄×D₄×C₃₁	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
84	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
85	C₃₁×#117	—	2	1	no	—
86	C₃₁×#117	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	—	2	1	no	—
90	C₂×C₄²×C₃₁	—	2	1	no	—
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#108	—	4	2	no	—
93	C₃₁×#107	—	2	1	no	—
94	C₃₁×#106	—	2	1	no	—
95	C₃₁×C₄⋊C₈	—	2	1	no	—
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	—	2	1	no	—
98	C₃₁×C₄⋊C₈	—	2	1	no	—
99	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
100	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
101	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
102	C₃₁×#103	—	2	1	no	—
103	C₃₁×#103	—	2	1	no	—
104	C₃₁×#103	—	2	1	no	—
105	C₃₁×#102	—	2	1	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×#101	—	4	2	no	—
111	C₃₁×#100	—	2	4	no	—
112	C₃₁×#100	—	4	2	no	—
113	C₃₁×C₂²⋊C₈	—	2	2	no	—
114	C₃₁×C₂²⋊C₈	—	2	2	no	—
115	C₃₁×#97	—	2	2	no	—
116	C₃₁×#97	—	2	1	no	—
117	C₄×C₈×C₃₁	—	2	2	no	—
118	C₄×C₈×C₃₁	—	2	1	no	—
119	C₂⁴×C₃₁	—	2	2	no	—
120	#41×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	1	no	—
123	#41×C₃₁	—	2	1	no	—
124	C₂×Q₈×C₃₁	—	2	1	no	—
125	C₂×Q₈×C₃₁	—	2	1	no	—
126	C₂×Q₈×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Q₈×C₃₁	#41	1	2	yes	—
131	C₂×D₄×C₃₁	—	2	1	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	—	2	2	no	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	4	1	no	—
139	C₂×D₄×C₃₁	—	4	1	no	—
140	C₂×D₄×C₃₁	—	4	1	no	—
141	C₂×D₄×C₃₁	—	4	1	no	—
142	C₂×D₄×C₃₁	—	8	2	no	—
143	C₂×D₄×C₃₁	—	8	2	no	—
144	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	—	2	1	no	—
146	C₂²×C₄×C₃₁	—	2	1	no	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	1	no	—
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
153	Q₁₆×C₃₁	—	2	4	no	—
154	SD₁₆×C₃₁	—	2	4	no	—
155	SD₁₆×C₃₁	—	4	4	no	—
156	SD₁₆×C₃₁	—	4	4	no	—
157	SD₁₆×C₃₁	—	4	4	no	—
158	D₈×C₃₁	—	4	4	no	—
159	M₁₆×C₃₁	—	2	1	no	—
160	M₁₆×C₃₁	—	2	2	no	—
161	M₁₆×C₃₁	—	2	1	no	—
162	M₁₆×C₃₁	—	2	2	no	—
163	M₁₆×C₃₁	—	2	2	no	—
164	M₁₆×C₃₁	—	2	1	no	—
165	M₁₆×C₃₁	—	2	2	no	—
166	M₁₆×C₃₁	#41	1	2	yes	—
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	1	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
174	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	4	1	no	—
180	C₄⋊C₄×C₃₁	—	4	1	no	—
181	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₂²⋊C₄×C₃₁	—	2	1	no	—
184	C₂²⋊C₄×C₃₁	—	2	1	no	—
185	C₂²⋊C₄×C₃₁	—	2	1	no	—
186	C₄²×C₃₁	—	2	1	no	—
187	C₄²×C₃₁	—	2	1	no	—
188	C₄²×C₃₁	—	2	1	no	—
189	C₄²×C₃₁	—	4	1	no	—
190	C₄²×C₃₁	—	4	1	no	—
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	#24673	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	1	no	—
197	C₂³×C₃₁	—	4	2	no	—
198	C₂³×C₃₁	—	4	2	no	—
199	C₂³×C₃₁	—	4	2	no	—
200	C₂³×C₃₁	—	8	2	no	—
201	C₂³×C₃₁	—	8	2	no	—
202	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
203	Q₈×C₃₁	—	2	2	no	—
204	Q₈×C₃₁	—	2	1	no	—
205	Q₈×C₃₁	—	2	1	no	—
206	Q₈×C₃₁	—	2	2	no	—
207	Q₈×C₃₁	—	4	1	no	—
208	Q₈×C₃₁	—	4	1	no	—
209	D₄×C₃₁	—	2	1	no	—
210	D₄×C₃₁	—	2	1	no	—
211	D₄×C₃₁	—	4	1	no	—
212	D₄×C₃₁	—	4	1	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	4	2	no	—
215	D₄×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	8	2	no	—
218	D₄×C₃₁	—	8	2	no	—
219	D₄×C₃₁	—	8	2	no	—
220	D₄×C₃₁	—	8	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	4	2	no	—
232	C₂×C₄×C₃₁	—	4	1	no	—
233	C₂×C₄×C₃₁	—	8	1	no	—
234	C₂×C₄×C₃₁	—	8	1	no	—
235	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
236	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
237	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
238	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
245	C₈×C₃₁	—	2	2	no	—
246	C₈×C₃₁	—	2	2	no	—
247	C₈×C₃₁	—	2	2	no	—
248	C₈×C₃₁	—	4	1	no	—
249	C₈×C₃₁	—	4	2	no	—
250	C₈×C₃₁	—	4	2	no	—
251	#3038	C₂×C₃₁	1	1	yes	characteristic
252	C₂×#460	C₂×C₃₁	1	1	yes	characteristic, complemented
253	C₄×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#2051	C₂×C₃₁	1	2	yes	complemented
256	#2019	C₂×C₃₁	1	2	yes	complemented
257	#1977	C₂×C₃₁	1	2	yes	complemented
258	#1965	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1931	C₂×C₃₁	1	2	yes	complemented
261	#1801	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂²×C₃₁	—	2	1	no	—
263	C₂²×C₃₁	—	2	1	no	—
264	C₂²×C₃₁	—	4	1	no	—
265	C₂²×C₃₁	—	4	1	no	—
266	C₂²×C₃₁	—	4	1	no	—
267	C₂²×C₃₁	—	8	2	no	—
268	C₂²×C₃₁	—	8	2	no	—
269	C₂²×C₃₁	—	8	2	no	—
270	C₂²×C₃₁	—	16	2	no	—
271	C₂²×C₃₁	—	16	2	no	—
272	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
273	C₂²×C₃₁	#553	1	1	yes	characteristic
274	C₂²×C₃₁	#555	1	1	yes	characteristic
275	C₄×C₃₁	—	2	1	no	—
276	C₄×C₃₁	—	2	1	no	—
277	C₄×C₃₁	—	2	1	no	—
278	C₄×C₃₁	—	4	1	no	—
279	C₄×C₃₁	—	4	2	no	—
280	C₄×C₃₁	—	4	1	no	—
281	C₄×C₃₁	—	4	1	no	—
282	C₄×C₃₁	—	8	1	no	—
283	C₄×C₃₁	#471	1	2	yes	—
284	C₄×C₃₁	#525	1	1	yes	characteristic
285	C₄×C₃₁	#535	1	1	yes	characteristic
286	C₄×C₃₁	#546	1	1	yes	characteristic
287	C₄×C₃₁	#550	1	1	yes	characteristic
288	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
290	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
291	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
292	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
294	#460	—	2	1	no	—
295	#460	—	2	1	no	—
296	#460	—	2	1	no	—
297	#460	—	2	1	no	—
298	#460	—	2	2	no	—
299	#459	—	2	1	no	—
300	#459	—	2	1	no	—
301	#441	—	2	1	no	—
302	#440	—	2	1	no	—
303	C₄×Q₁₆	—	2	1	no	—
304	C₄×SD₁₆	—	2	1	no	—
305	#433	—	2	2	no	—
306	#432	—	2	1	no	—
307	#431	—	2	1	no	—
308	C₂×#108	C₂²×C₃₁	1	2	yes	—
309	#428	C₂²×C₃₁	1	1	yes	characteristic
310	#423	C₂²×C₃₁	1	1	yes	characteristic
311	#423	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
313	#419	C₂²×C₃₁	1	1	yes	characteristic
314	#418	C₂²×C₃₁	1	2	yes	—
315	#417	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×#102	C₂²×C₃₁	1	2	yes	—
317	C₂×#101	C₂²×C₃₁	1	2	yes	—
318	C₂×#100	C₂²×C₃₁	1	2	yes	—
319	C₂×#100	C₂²×C₃₁	1	2	yes	—
320	#408	C₂²×C₃₁	1	2	yes	—
321	#405	C₂²×C₃₁	1	2	yes	—
322	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
323	#343	C₂²×C₃₁	1	2	yes	—
324	#341	C₂²×C₃₁	1	2	yes	—
325	C₂×C₃₁	—	2	1	no	—
326	C₂×C₃₁	—	2	1	no	—
327	C₂×C₃₁	—	8	1	no	—
328	C₂×C₃₁	—	16	2	no	—
329	C₂×C₃₁	#2414	1	1	yes	characteristic
330	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
331	C₂×C₃₁	#3068	1	1	yes	characteristic
332	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₂²×D₄	—	2	1	no	—
335	C₂²×D₄	—	2	1	no	—
336	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
337	#137	—	2	2	no	—
338	#137	—	2	2	no	—
339	AGL(1,ℤ/8)	—	4	2	no	—
340	AGL(1,ℤ/8)	—	4	2	no	—
341	C₂×Q₁₆	—	2	1	no	—
342	C₂×SD₁₆	—	2	1	no	—
343	C₂×SD₁₆	—	2	2	no	—
344	C₂×SD₁₆	—	2	4	no	—
345	C₂×SD₁₆	—	2	2	no	—
346	C₂×SD₁₆	—	2	2	no	—
347	C₂×SD₁₆	—	4	2	no	—
348	C₂×D₈	—	2	2	no	—
349	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
350	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
351	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
352	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
354	#127	—	2	1	no	—
355	#127	—	2	1	no	—
356	#127	—	2	1	no	—
357	#127	—	2	1	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×Q₈	—	2	1	no	—
361	C₄×Q₈	—	2	1	no	—
362	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
363	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
367	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
368	#117	—	2	1	no	—
369	#117	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
372	C₂×C₄²	—	2	1	no	—
373	C₂×C₄²	—	2	1	no	—
374	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
375	#108	—	4	2	no	—
376	#107	—	2	1	no	—
377	#106	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	—	2	1	no	—
380	C₄⋊C₈	—	2	1	no	—
381	C₄⋊C₈	—	2	1	no	—
382	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
383	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
384	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
385	#103	—	2	1	no	—
386	#103	—	2	1	no	—
387	#103	—	2	1	no	—
388	#102	—	2	1	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	#101	—	4	2	no	—
394	#100	—	2	4	no	—
395	#100	—	4	2	no	—
396	C₂²⋊C₈	—	2	2	no	—
397	C₂²⋊C₈	—	2	2	no	—
398	#97	—	2	2	no	—
399	#97	—	2	1	no	—
400	C₄×C₈	—	2	2	no	—
401	C₄×C₈	—	2	1	no	—
402	C₃₁	#24673	1	1	yes	characteristic, direct product factor, Sylow
403	C₂⁴	—	2	2	no	—
404	#41	—	2	1	no	—
405	#41	—	2	1	no	—
406	#41	—	2	1	no	—
407	#41	—	2	1	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	1	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
414	C₂×Q₈	#41×C₃₁	1	2	yes	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	4	1	no	—
423	C₂×D₄	—	4	1	no	—
424	C₂×D₄	—	4	1	no	—
425	C₂×D₄	—	4	1	no	—
426	C₂×D₄	—	8	2	no	—
427	C₂×D₄	—	8	2	no	—
428	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
437	Q₁₆	—	2	4	no	—
438	SD₁₆	—	2	4	no	—
439	SD₁₆	—	4	4	no	—
440	SD₁₆	—	4	4	no	—
441	SD₁₆	—	4	4	no	—
442	D₈	—	4	4	no	—
443	M₁₆	—	2	1	no	—
444	M₁₆	—	2	2	no	—
445	M₁₆	—	2	1	no	—
446	M₁₆	—	2	2	no	—
447	M₁₆	—	2	2	no	—
448	M₁₆	—	2	1	no	—
449	M₁₆	—	2	2	no	—
450	M₁₆	#41×C₃₁	1	2	yes	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
458	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	4	1	no	—
464	C₄⋊C₄	—	4	1	no	—
465	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
467	C₂²⋊C₄	—	2	1	no	—
468	C₂²⋊C₄	—	2	1	no	—
469	C₂²⋊C₄	—	2	1	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	—	4	1	no	—
474	C₄²	—	4	1	no	—
475	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₂³	—	2	1	no	—
480	C₂³	—	4	2	no	—
481	C₂³	—	4	2	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	8	2	no	—
484	C₂³	—	8	2	no	—
485	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
486	Q₈	—	2	2	no	—
487	Q₈	—	2	1	no	—
488	Q₈	—	2	1	no	—
489	Q₈	—	2	2	no	—
490	Q₈	—	4	1	no	—
491	Q₈	—	4	1	no	—
492	D₄	—	2	1	no	—
493	D₄	—	2	1	no	—
494	D₄	—	4	1	no	—
495	D₄	—	4	1	no	—
496	D₄	—	4	2	no	—
497	D₄	—	4	2	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	2	no	—
500	D₄	—	8	2	no	—
501	D₄	—	8	2	no	—
502	D₄	—	8	2	no	—
503	D₄	—	8	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	8	1	no	—
517	C₂×C₄	—	8	1	no	—
518	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
519	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
520	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
521	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
524	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
526	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
527	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
528	C₈	—	2	2	no	—
529	C₈	—	2	2	no	—
530	C₈	—	2	2	no	—
531	C₈	—	4	1	no	—
532	C₈	—	4	2	no	—
533	C₈	—	4	2	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	4	1	no	—
537	C₂²	—	4	1	no	—
538	C₂²	—	4	1	no	—
539	C₂²	—	8	2	no	—
540	C₂²	—	8	2	no	—
541	C₂²	—	8	2	no	—
542	C₂²	—	16	2	no	—
543	C₂²	—	16	2	no	—
544	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
545	C₂²	C₃₁×#553	1	1	yes	characteristic
546	C₂²	C₃₁×#555	1	1	yes	characteristic
547	C₄	—	2	1	no	—
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	4	1	no	—
551	C₄	—	4	2	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	8	1	no	—
555	C₄	C₃₁×#471	1	2	yes	—
556	C₄	C₃₁×#525	1	1	yes	characteristic
557	C₄	C₃₁×#535	1	1	yes	characteristic
558	C₄	C₃₁×#546	1	1	yes	characteristic
559	C₄	C₃₁×#550	1	1	yes	characteristic
560	C₂	—	2	1	no	—
561	C₂	—	2	1	no	—
562	C₂	—	8	1	no	—
563	C₂	—	16	2	no	—
564	C₂	C₃₁×#2414	1	1	yes	characteristic
565	C₂	C₂×C₃₁×#471	1	1	yes	characteristic
566	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	C₃₁×#3038	C₂
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×#460	C₂	semidirect
3. C₄×C₃₁×#137	C₂	semidirect
4. C₃₁×#2925	C₂	semidirect
5. C₃₁×#2051	C₂	semidirect
6. C₃₁×#2019	C₂	semidirect
7. C₃₁×#1977	C₂	semidirect
8. C₃₁×#1965	C₂	semidirect
9. C₃₁×#1950	C₂	semidirect
10. C₃₁×#1931	C₂	semidirect
11. C₃₁×#1801	C₂	semidirect
195. #24673	C₃₁	direct (N × H)
252. C₂×#460	C₂×C₃₁	semidirect
253. C₄×#137	C₂×C₃₁	semidirect
254. #2925	C₂×C₃₁	semidirect
255. #2051	C₂×C₃₁	semidirect
256. #2019	C₂×C₃₁	semidirect
257. #1977	C₂×C₃₁	semidirect
258. #1965	C₂×C₃₁	semidirect
259. #1950	C₂×C₃₁	semidirect
260. #1931	C₂×C₃₁	semidirect
261. #1801	C₂×C₃₁	semidirect
402. C₃₁	#24673	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×#3038	2
C₂×C₃₁×#460	2
C₄×C₃₁×#137	2
C₃₁×#2925	2
C₃₁×#2051	2
C₃₁×#2019	2
C₃₁×#1977	2
C₃₁×#1965	2
C₃₁×#1950	2
C₃₁×#1931	2
C₃₁×#1801	2
#24673	31

Subgroups — C31×#24673 #21481432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24673 #21481432