Subgroups — C₃₁×#25441 #21482200

Order 7936.

938 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3155	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3062	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3061	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2037	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2006	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1907	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
14	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
15	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#131	C₂²	1	2	yes	—
18	C₃₁×#537	C₂²	1	2	yes	—
19	C₃₁×#535	C₂²	1	2	yes	—
20	C₃₁×#496	—	2	2	no	—
21	C₃₁×#495	—	2	1	no	—
22	C₃₁×#493	—	2	1	no	—
23	C₃₁×#472	—	2	4	no	—
24	C₃₁×#469	—	2	2	no	—
25	C₃₁×#468	—	2	2	no	—
26	C₃₁×#468	—	2	2	no	—
27	C₃₁×#467	—	2	1	no	—
28	C₃₁×#466	—	2	1	no	—
29	C₃₁×#466	—	2	2	no	—
30	C₃₁×#465	—	2	2	no	—
31	C₃₁×#443	—	2	4	no	—
32	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
33	C₃₁×#428	C₂²	1	2	yes	—
34	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#104	C₂²	1	2	yes	—
36	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
37	C₃₁×#417	C₂²	1	2	yes	—
38	C₃₁×#416	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#102	C₂²	1	2	yes	—
40	C₂×C₃₁×#100	C₂²	1	2	yes	—
41	C₃₁×#405	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#390	C₂²	1	2	yes	—
44	C₃₁×#343	C₂²	1	2	yes	—
45	C₃₁×#341	C₂²	1	1	yes	characteristic
46	C₃₁×#339	C₂²	1	1	yes	characteristic
47	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×#41×C₃₁	C₂³	1	2	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
50	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
51	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
52	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
53	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
54	C₃₁×#135	—	2	2	no	—
55	C₃₁×#135	—	2	2	no	—
56	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	—	2	2	no	—
59	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
60	C₂×D₈×C₃₁	—	2	2	no	—
61	C₂×D₈×C₃₁	—	2	2	no	—
62	C₂×D₈×C₃₁	—	2	2	no	—
63	C₂×D₈×C₃₁	—	2	1	no	—
64	C₂×D₈×C₃₁	—	2	1	no	—
65	C₂×D₈×C₃₁	—	2	2	no	—
66	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
67	C₃₁×#131	—	2	4	no	—
68	C₃₁×#131	—	2	2	no	—
69	C₃₁×#131	—	2	2	no	—
70	C₃₁×#131	—	4	2	no	—
71	C₂×M₁₆×C₃₁	—	2	1	no	—
72	C₂×M₁₆×C₃₁	—	2	2	no	—
73	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
75	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₃₁	—	2	2	no	—
77	C₂²×C₈×C₃₁	—	2	1	no	—
78	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#127	—	2	2	no	—
81	C₃₁×#124	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	—	2	2	no	—
83	C₃₁×C₂²⋊Q₈	—	2	2	no	—
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	—	4	2	no	—
89	C₃₁×#120	—	4	2	no	—
90	C₃₁×#117	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
93	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
94	C₃₁×#108	D₄	1	4	yes	complemented
95	C₃₁×#107	—	2	1	no	—
96	C₃₁×#107	—	2	1	no	—
97	C₃₁×#107	—	2	2	no	—
98	C₃₁×#106	—	2	2	no	—
99	C₃₁×#104	—	2	4	no	—
100	C₃₁×#104	—	2	2	no	—
101	C₃₁×#103	—	2	2	no	—
102	C₃₁×#103	D₄	1	1	yes	characteristic
103	C₃₁×#103	D₄	1	1	yes	characteristic
104	C₃₁×#102	—	2	2	no	—
105	C₃₁×#102	—	2	2	no	—
106	C₃₁×#102	—	2	2	no	—
107	C₃₁×#102	D₄	1	1	yes	characteristic
108	C₃₁×#102	D₄	1	1	yes	characteristic
109	C₃₁×#100	—	2	4	no	—
110	C₃₁×#97	—	2	1	no	—
111	C₄×C₈×C₃₁	—	2	2	no	—
112	C₄×C₈×C₃₁	—	2	1	no	—
113	C₂⁴×C₃₁	—	2	2	no	—
114	#41×C₃₁	—	2	2	no	—
115	#41×C₃₁	—	2	1	no	—
116	#41×C₃₁	—	2	1	no	—
117	#41×C₃₁	—	2	1	no	—
118	#41×C₃₁	—	2	2	no	—
119	#41×C₃₁	—	2	2	no	—
120	#41×C₃₁	—	2	2	no	—
121	#41×C₃₁	—	2	1	no	—
122	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
123	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	1	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	2	2	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	—	4	2	no	—
133	C₂×D₄×C₃₁	—	4	2	no	—
134	C₂×D₄×C₃₁	—	4	2	no	—
135	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
142	Q₁₆×C₃₁	—	2	2	no	—
143	SD₁₆×C₃₁	—	2	4	no	—
144	SD₁₆×C₃₁	—	4	4	no	—
145	SD₁₆×C₃₁	—	4	4	no	—
146	D₈×C₃₁	—	2	2	no	—
147	D₈×C₃₁	—	4	4	no	—
148	D₈×C₃₁	—	4	4	no	—
149	D₈×C₃₁	—	4	2	no	—
150	D₈×C₃₁	—	4	2	no	—
151	M₁₆×C₃₁	—	2	2	no	—
152	M₁₆×C₃₁	—	2	1	no	—
153	M₁₆×C₃₁	—	2	1	no	—
154	M₁₆×C₃₁	—	2	1	no	—
155	M₁₆×C₃₁	—	4	4	no	—
156	M₁₆×C₃₁	—	4	2	no	—
157	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
158	M₁₆×C₃₁	#41	1	2	yes	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	4	4	no	—
165	C₂×C₈×C₃₁	—	4	2	no	—
166	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
170	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂×C₈×C₃₁	#41	1	2	yes	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₄⋊C₄×C₃₁	—	4	2	no	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²⋊C₄×C₃₁	—	2	1	no	—
178	C₂²⋊C₄×C₃₁	—	4	2	no	—
179	C₂²⋊C₄×C₃₁	—	4	2	no	—
180	C₂²⋊C₄×C₃₁	—	4	2	no	—
181	C₂²⋊C₄×C₃₁	—	4	2	no	—
182	C₂²⋊C₄×C₃₁	—	8	4	no	—
183	C₄²×C₃₁	—	2	1	no	—
184	C₄²×C₃₁	—	2	1	no	—
185	C₄²×C₃₁	—	2	1	no	—
186	#25441	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂³×C₃₁	—	2	1	no	—
188	C₂³×C₃₁	—	2	2	no	—
189	C₂³×C₃₁	—	4	2	no	—
190	C₂³×C₃₁	—	4	2	no	—
191	C₂³×C₃₁	—	4	2	no	—
192	C₂³×C₃₁	—	8	2	no	—
193	C₂³×C₃₁	—	8	2	no	—
194	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
195	Q₈×C₃₁	—	2	2	no	—
196	Q₈×C₃₁	—	2	1	no	—
197	Q₈×C₃₁	—	2	1	no	—
198	Q₈×C₃₁	—	2	2	no	—
199	D₄×C₃₁	—	2	2	no	—
200	D₄×C₃₁	—	2	1	no	—
201	D₄×C₃₁	—	2	1	no	—
202	D₄×C₃₁	—	2	2	no	—
203	D₄×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	4	1	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	4	1	no	—
207	D₄×C₃₁	—	4	2	no	—
208	D₄×C₃₁	—	4	2	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	2	no	—
211	D₄×C₃₁	—	16	2	no	—
212	D₄×C₃₁	—	16	2	no	—
213	D₄×C₃₁	—	16	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	8	2	no	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
232	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
233	C₈×C₃₁	—	2	1	no	—
234	C₈×C₃₁	—	2	1	no	—
235	C₈×C₃₁	—	2	4	no	—
236	C₈×C₃₁	—	2	2	no	—
237	C₈×C₃₁	—	4	1	no	—
238	C₈×C₃₁	#121	1	2	yes	—
239	C₈×C₃₁	#121	1	2	yes	—
240	#3155	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂×#472	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#3062	C₂×C₃₁	1	2	yes	complemented
243	#3061	C₂×C₃₁	1	1	yes	characteristic, complemented
244	C₂×#443	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#2037	C₂×C₃₁	1	2	yes	complemented
246	#2006	C₂×C₃₁	1	2	yes	complemented
247	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1907	C₂×C₃₁	1	2	yes	complemented
250	#1764	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×C₃₁	—	2	1	no	—
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	4	1	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	1	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	1	no	—
259	C₂²×C₃₁	—	8	4	no	—
260	C₂²×C₃₁	—	8	2	no	—
261	C₂²×C₃₁	—	8	2	no	—
262	C₂²×C₃₁	—	8	2	no	—
263	C₂²×C₃₁	—	16	2	no	—
264	C₂²×C₃₁	#540	1	1	yes	characteristic
265	C₂²×C₃₁	D₄²	1	1	yes	characteristic
266	C₂²×C₃₁	#550	1	1	yes	characteristic
267	C₄×C₃₁	—	4	2	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	1	no	—
270	C₄×C₃₁	—	4	1	no	—
271	C₄×C₃₁	—	4	1	no	—
272	C₄×C₃₁	—	8	2	no	—
273	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
274	C₄×C₃₁	#532	1	1	yes	characteristic
275	C₄×C₃₁	D₄²	1	1	yes	characteristic
276	C₄×C₃₁	#546	1	1	yes	characteristic
277	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
278	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
279	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
280	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×#131	C₂²×C₃₁	1	2	yes	—
283	#537	C₂²×C₃₁	1	2	yes	—
284	#535	C₂²×C₃₁	1	2	yes	—
285	#496	—	2	2	no	—
286	#495	—	2	1	no	—
287	#493	—	2	1	no	—
288	#472	—	2	4	no	—
289	#469	—	2	2	no	—
290	#468	—	2	2	no	—
291	#468	—	2	2	no	—
292	#467	—	2	1	no	—
293	#466	—	2	1	no	—
294	#466	—	2	2	no	—
295	#465	—	2	2	no	—
296	#443	—	2	4	no	—
297	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
298	#428	C₂²×C₃₁	1	2	yes	—
299	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×#104	C₂²×C₃₁	1	2	yes	—
301	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
302	#417	C₂²×C₃₁	1	2	yes	—
303	#416	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#102	C₂²×C₃₁	1	2	yes	—
305	C₂×#100	C₂²×C₃₁	1	2	yes	—
306	#405	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
308	#390	C₂²×C₃₁	1	2	yes	—
309	#343	C₂²×C₃₁	1	2	yes	—
310	#341	C₂²×C₃₁	1	1	yes	characteristic
311	#339	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×C₃₁	—	2	1	no	—
313	C₂×C₃₁	—	2	1	no	—
314	C₂×C₃₁	—	8	2	no	—
315	C₂×C₃₁	—	8	1	no	—
316	C₂×C₃₁	—	16	2	no	—
317	C₂×C₃₁	#2428	1	1	yes	characteristic
318	C₂×C₃₁	#3290	1	1	yes	characteristic
319	C₂×C₃₁	#3292	1	1	yes	characteristic
320	C₂×#41	C₂³×C₃₁	1	2	yes	—
321	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
322	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
323	AGL(1,ℤ/8)	—	2	4	no	—
324	AGL(1,ℤ/8)	—	2	4	no	—
325	AGL(1,ℤ/8)	—	4	2	no	—
326	AGL(1,ℤ/8)	—	4	2	no	—
327	#135	—	2	2	no	—
328	#135	—	2	2	no	—
329	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
330	C₂×SD₁₆	—	2	2	no	—
331	C₂×SD₁₆	—	2	2	no	—
332	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
333	C₂×D₈	—	2	2	no	—
334	C₂×D₈	—	2	2	no	—
335	C₂×D₈	—	2	2	no	—
336	C₂×D₈	—	2	1	no	—
337	C₂×D₈	—	2	1	no	—
338	C₂×D₈	—	2	2	no	—
339	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
340	#131	—	2	4	no	—
341	#131	—	2	2	no	—
342	#131	—	2	2	no	—
343	#131	—	4	2	no	—
344	C₂×M₁₆	—	2	1	no	—
345	C₂×M₁₆	—	2	2	no	—
346	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
347	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
348	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
349	C₂²×C₈	—	2	2	no	—
350	C₂²×C₈	—	2	1	no	—
351	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
353	#127	—	2	2	no	—
354	#124	—	2	2	no	—
355	C₂²⋊Q₈	—	2	2	no	—
356	C₂²⋊Q₈	—	2	2	no	—
357	#121	—	2	2	no	—
358	#121	—	2	2	no	—
359	#121	—	2	2	no	—
360	#121	—	2	2	no	—
361	#121	—	4	2	no	—
362	#120	—	4	2	no	—
363	#117	C₂³×C₃₁	1	1	yes	characteristic
364	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
365	C₂×C₂²⋊C₄	—	4	2	no	—
366	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
367	#108	D₄×C₃₁	1	4	yes	complemented
368	#107	—	2	1	no	—
369	#107	—	2	1	no	—
370	#107	—	2	2	no	—
371	#106	—	2	2	no	—
372	#104	—	2	4	no	—
373	#104	—	2	2	no	—
374	#103	—	2	2	no	—
375	#103	D₄×C₃₁	1	1	yes	characteristic
376	#103	D₄×C₃₁	1	1	yes	characteristic
377	#102	—	2	2	no	—
378	#102	—	2	2	no	—
379	#102	—	2	2	no	—
380	#102	D₄×C₃₁	1	1	yes	characteristic
381	#102	D₄×C₃₁	1	1	yes	characteristic
382	#100	—	2	4	no	—
383	#97	—	2	1	no	—
384	C₄×C₈	—	2	2	no	—
385	C₄×C₈	—	2	1	no	—
386	C₃₁	#25441	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	—	2	2	no	—
388	#41	—	2	2	no	—
389	#41	—	2	1	no	—
390	#41	—	2	1	no	—
391	#41	—	2	1	no	—
392	#41	—	2	2	no	—
393	#41	—	2	2	no	—
394	#41	—	2	2	no	—
395	#41	—	2	1	no	—
396	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
397	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	4	2	no	—
407	C₂×D₄	—	4	2	no	—
408	C₂×D₄	—	4	2	no	—
409	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
410	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
416	Q₁₆	—	2	2	no	—
417	SD₁₆	—	2	4	no	—
418	SD₁₆	—	4	4	no	—
419	SD₁₆	—	4	4	no	—
420	D₈	—	2	2	no	—
421	D₈	—	4	4	no	—
422	D₈	—	4	4	no	—
423	D₈	—	4	2	no	—
424	D₈	—	4	2	no	—
425	M₁₆	—	2	2	no	—
426	M₁₆	—	2	1	no	—
427	M₁₆	—	2	1	no	—
428	M₁₆	—	2	1	no	—
429	M₁₆	—	4	4	no	—
430	M₁₆	—	4	2	no	—
431	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
432	M₁₆	#41×C₃₁	1	2	yes	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	4	4	no	—
439	C₂×C₈	—	4	2	no	—
440	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
444	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
445	C₂×C₈	#41×C₃₁	1	2	yes	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	4	2	no	—
449	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
450	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
451	C₂²⋊C₄	—	2	1	no	—
452	C₂²⋊C₄	—	4	2	no	—
453	C₂²⋊C₄	—	4	2	no	—
454	C₂²⋊C₄	—	4	2	no	—
455	C₂²⋊C₄	—	4	2	no	—
456	C₂²⋊C₄	—	8	4	no	—
457	C₄²	—	2	1	no	—
458	C₄²	—	2	1	no	—
459	C₄²	—	2	1	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	2	no	—
465	C₂³	—	8	2	no	—
466	C₂³	—	8	2	no	—
467	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
468	Q₈	—	2	2	no	—
469	Q₈	—	2	1	no	—
470	Q₈	—	2	1	no	—
471	Q₈	—	2	2	no	—
472	D₄	—	2	2	no	—
473	D₄	—	2	1	no	—
474	D₄	—	2	1	no	—
475	D₄	—	2	2	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	1	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	1	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	2	no	—
483	D₄	—	4	2	no	—
484	D₄	—	16	2	no	—
485	D₄	—	16	2	no	—
486	D₄	—	16	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	8	2	no	—
500	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
503	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
505	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
506	C₈	—	2	1	no	—
507	C₈	—	2	1	no	—
508	C₈	—	2	4	no	—
509	C₈	—	2	2	no	—
510	C₈	—	4	1	no	—
511	C₈	C₃₁×#121	1	2	yes	—
512	C₈	C₃₁×#121	1	2	yes	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	1	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	1	no	—
521	C₂²	—	8	4	no	—
522	C₂²	—	8	2	no	—
523	C₂²	—	8	2	no	—
524	C₂²	—	8	2	no	—
525	C₂²	—	16	2	no	—
526	C₂²	C₃₁×#540	1	1	yes	characteristic
527	C₂²	D₄²×C₃₁	1	1	yes	characteristic
528	C₂²	C₃₁×#550	1	1	yes	characteristic
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	1	no	—
532	C₄	—	4	1	no	—
533	C₄	—	4	1	no	—
534	C₄	—	8	2	no	—
535	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
536	C₄	C₃₁×#532	1	1	yes	characteristic
537	C₄	D₄²×C₃₁	1	1	yes	characteristic
538	C₄	C₃₁×#546	1	1	yes	characteristic
539	C₂	—	2	1	no	—
540	C₂	—	2	1	no	—
541	C₂	—	8	2	no	—
542	C₂	—	8	1	no	—
543	C₂	—	16	2	no	—
544	C₂	C₃₁×#2428	1	1	yes	characteristic
545	C₂	C₃₁×#3290	1	1	yes	characteristic
546	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3155	C₂	semidirect
2. C₂×C₃₁×#472	C₂	semidirect
3. C₃₁×#3062	C₂	semidirect
4. C₃₁×#3061	C₂	semidirect
5. C₂×C₃₁×#443	C₂	semidirect
6. C₃₁×#2037	C₂	semidirect
7. C₃₁×#2006	C₂	semidirect
8. C₃₁×#1970	C₂	semidirect
9. C₃₁×#1952	C₂	semidirect
10. C₃₁×#1907	C₂	semidirect
11. C₃₁×#1764	C₂	semidirect
94. C₃₁×#108	D₄	semidirect
186. #25441	C₃₁	direct (N × H)
240. #3155	C₂×C₃₁	semidirect
241. C₂×#472	C₂×C₃₁	semidirect
242. #3062	C₂×C₃₁	semidirect
243. #3061	C₂×C₃₁	semidirect
244. C₂×#443	C₂×C₃₁	semidirect
245. #2037	C₂×C₃₁	semidirect
246. #2006	C₂×C₃₁	semidirect
247. #1970	C₂×C₃₁	semidirect
248. #1952	C₂×C₃₁	semidirect
249. #1907	C₂×C₃₁	semidirect
250. #1764	C₂×C₃₁	semidirect
367. #108	D₄×C₃₁	semidirect
386. C₃₁	#25441	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×#3155	2
C₂×C₃₁×#472	2
C₃₁×#3062	2
C₃₁×#3061	2
C₂×C₃₁×#443	2
C₃₁×#2037	2
C₃₁×#2006	2
C₃₁×#1970	2
C₃₁×#1952	2
C₃₁×#1907	2
C₃₁×#1764	2
#25441	31

Subgroups — C31×#25441 #21482200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25441 #21482200