Subgroups — C₃₁×#24285 #21481044

Order 7936.

1336 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3050	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3025	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2053	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2010	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1976	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1812	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1794	C₂	1	2	yes	maximal, complemented
10	C₃₁×#585	C₂²	1	2	yes	—
11	C₃₁×#578	—	2	2	no	—
12	C₃₁×#577	—	2	4	no	—
13	C₃₁×#576	—	2	2	no	—
14	C₃₁×#575	—	2	2	no	—
15	C₃₁×#575	C₂²	1	2	yes	—
16	C₂×C₃₁×#135	—	2	2	no	—
17	C₂×C₃₁×#135	C₂²	1	2	yes	—
18	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#525	C₂²	1	1	yes	characteristic
21	C₃₁×#525	C₂²	1	4	yes	complemented
22	C₃₁×#467	—	2	2	no	—
23	C₃₁×#466	—	2	2	no	—
24	C₃₁×#465	—	2	4	no	—
25	C₃₁×#458	—	2	4	no	—
26	C₃₁×B(4,𝔽₂)	—	2	4	no	—
27	C₃₁×#435	—	2	2	no	—
28	C₈×D₄×C₃₁	—	2	2	no	—
29	C₃₁×#427	C₂²	1	2	yes	complemented
30	C₃₁×#424	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#421	C₂²	1	4	yes	complemented
32	C₃₁×#416	C₂²	1	4	yes	complemented
33	C₃₁×#410	C₂²	1	2	yes	—
34	C₃₁×#410	C₂²	1	4	yes	complemented
35	C₃₁×#408	C₂²	1	1	yes	characteristic
36	C₃₁×#407	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#405	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×H(3,ℤ/4)	C₂²	1	2	yes	complemented
39	C₃₁×2₋¹⁺⁴	—	2	4	no	—
40	C₃₁×2₋¹⁺⁴	—	2	2	no	—
41	C₃₁×H(5,𝔽₂)	—	2	4	no	—
42	C₃₁×H(5,𝔽₂)	—	2	4	no	—
43	C₃₁×H(5,𝔽₂)	—	2	2	no	—
44	C₂×#41×C₃₁	—	2	4	no	—
45	C₂×#41×C₃₁	—	4	4	no	—
46	C₂×#41×C₃₁	C₂³	1	4	yes	—
47	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×#137	—	2	8	no	—
50	C₃₁×#137	—	2	4	no	—
51	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
52	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
53	C₃₁×#135	—	2	4	no	—
54	C₃₁×#135	—	2	4	no	—
55	C₃₁×#135	—	2	2	no	—
56	C₃₁×#135	—	2	4	no	—
57	C₃₁×#135	—	2	2	no	—
58	C₃₁×#135	—	2	2	no	—
59	C₃₁×#135	—	2	2	no	—
60	C₂×Q₁₆×C₃₁	—	2	2	no	—
61	C₂×Q₁₆×C₃₁	D₄	1	2	yes	complemented
62	C₂×SD₁₆×C₃₁	—	2	4	no	—
63	C₂×SD₁₆×C₃₁	D₄	1	4	yes	complemented
64	C₂×D₈×C₃₁	—	2	2	no	—
65	C₂×D₈×C₃₁	D₄	1	2	yes	complemented
66	C₃₁×#131	—	2	2	no	—
67	C₃₁×#131	—	2	2	no	—
68	C₂×M₁₆×C₃₁	D₄	1	2	yes	complemented
69	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₃₁	—	2	1	no	—
72	C₂²×C₈×C₃₁	—	2	1	no	—
73	C₂²×C₈×C₃₁	—	2	2	no	—
74	C₂²×C₈×C₃₁	—	2	2	no	—
75	C₂²×C₈×C₃₁	D₄	1	2	yes	complemented
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#123	—	2	4	no	—
79	C₃₁×#123	D₄	1	2	yes	—
80	C₃₁×C₂²⋊Q₈	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	—	2	4	no	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	—	2	4	no	—
84	C₃₁×#120	—	2	4	no	—
85	C₃₁×#120	D₄	1	2	yes	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	4	4	no	—
88	C₃₁×#117	C₂³	1	4	yes	—
89	C₃₁×#117	C₂³	1	1	yes	characteristic
90	C₃₁×#107	—	2	2	no	—
91	C₃₁×#106	—	2	2	no	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×#104	—	4	4	no	—
94	C₃₁×#103	—	2	4	no	—
95	C₃₁×#102	—	2	4	no	—
96	C₃₁×#99	—	2	4	no	—
97	C₃₁×#99	—	2	8	no	—
98	C₃₁×C₂²⋊C₈	—	2	2	no	—
99	C₃₁×C₂²⋊C₈	—	2	2	no	—
100	C₃₁×#97	—	2	1	no	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
103	#41×C₃₁	—	2	2	no	—
104	#41×C₃₁	—	2	2	no	—
105	#41×C₃₁	—	2	8	no	—
106	#41×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	2	4	no	—
108	#41×C₃₁	—	2	4	no	—
109	#41×C₃₁	—	2	4	no	—
110	#41×C₃₁	—	4	8	no	—
111	#41×C₃₁	—	4	4	no	—
112	#41×C₃₁	—	4	4	no	—
113	#41×C₃₁	—	4	4	no	—
114	#41×C₃₁	—	4	4	no	—
115	C₂×Q₈×C₃₁	—	2	4	no	—
116	C₂×Q₈×C₃₁	—	4	4	no	—
117	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
118	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
119	C₂×D₄×C₃₁	—	2	4	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	—	4	4	no	—
123	C₂×D₄×C₃₁	—	4	8	no	—
124	C₂×D₄×C₃₁	—	4	4	no	—
125	C₂×D₄×C₃₁	—	4	4	no	—
126	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
127	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
128	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	4	4	no	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
138	Q₁₆×C₃₁	—	2	4	no	—
139	Q₁₆×C₃₁	—	2	2	no	—
140	Q₁₆×C₃₁	—	2	2	no	—
141	SD₁₆×C₃₁	—	2	8	no	—
142	SD₁₆×C₃₁	—	2	4	no	—
143	SD₁₆×C₃₁	—	2	4	no	—
144	D₈×C₃₁	—	2	4	no	—
145	D₈×C₃₁	—	2	2	no	—
146	D₈×C₃₁	—	2	2	no	—
147	M₁₆×C₃₁	—	2	2	no	—
148	M₁₆×C₃₁	—	2	4	no	—
149	M₁₆×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	4	1	no	—
156	C₂×C₈×C₃₁	—	4	1	no	—
157	C₂×C₈×C₃₁	—	4	2	no	—
158	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
162	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	4	no	—
166	C₄⋊C₄×C₃₁	—	4	4	no	—
167	C₂²⋊C₄×C₃₁	—	2	2	no	—
168	C₂²⋊C₄×C₃₁	—	2	2	no	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₂²⋊C₄×C₃₁	—	4	4	no	—
171	C₂²⋊C₄×C₃₁	—	4	4	no	—
172	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₄²×C₃₁	—	2	1	no	—
174	C₄²×C₃₁	—	2	4	no	—
175	#24285	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	4	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	4	no	—
181	C₂³×C₃₁	#121	1	2	yes	—
182	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₃₁	—	2	2	no	—
184	Q₈×C₃₁	—	2	4	no	—
185	Q₈×C₃₁	—	2	4	no	—
186	Q₈×C₃₁	—	4	4	no	—
187	D₄×C₃₁	—	2	4	no	—
188	D₄×C₃₁	—	2	4	no	—
189	D₄×C₃₁	—	2	2	no	—
190	D₄×C₃₁	—	2	4	no	—
191	D₄×C₃₁	—	4	4	no	—
192	D₄×C₃₁	—	4	4	no	—
193	D₄×C₃₁	—	4	8	no	—
194	D₄×C₃₁	—	4	4	no	—
195	D₄×C₃₁	—	4	4	no	—
196	D₄×C₃₁	—	8	8	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	4	8	no	—
204	C₂×C₄×C₃₁	—	4	4	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	4	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	4	no	—
210	C₂×C₄×C₃₁	—	4	4	no	—
211	C₂×C₄×C₃₁	#121	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
216	C₈×C₃₁	—	2	2	no	—
217	C₈×C₃₁	—	4	1	no	—
218	C₈×C₃₁	—	4	1	no	—
219	C₈×C₃₁	#120	1	1	yes	characteristic
220	C₈×C₃₁	#120	1	1	yes	characteristic
221	C₈×C₃₁	#120	1	2	yes	—
222	#3584	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#3050	C₂×C₃₁	1	2	yes	complemented
224	#3025	C₂×C₃₁	1	2	yes	complemented
225	#2929	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#2053	C₂×C₃₁	1	2	yes	complemented
227	#2010	C₂×C₃₁	1	2	yes	complemented
228	#1976	C₂×C₃₁	1	2	yes	complemented
229	#1812	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1794	C₂×C₃₁	1	2	yes	complemented
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	4	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	4	1	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	1	no	—
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	1	no	—
239	C₂²×C₃₁	—	4	4	no	—
240	C₂²×C₃₁	—	8	4	no	—
241	C₂²×C₃₁	—	16	4	no	—
242	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
243	C₂²×C₃₁	D₄²	1	2	yes	—
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	4	1	no	—
247	C₄×C₃₁	—	4	4	no	—
248	C₄×C₃₁	—	8	4	no	—
249	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
250	C₄×C₃₁	#535	1	1	yes	characteristic
251	C₄×C₃₁	#546	1	2	yes	—
252	#585	C₂²×C₃₁	1	2	yes	—
253	#578	—	2	2	no	—
254	#577	—	2	4	no	—
255	#576	—	2	2	no	—
256	#575	—	2	2	no	—
257	#575	C₂²×C₃₁	1	2	yes	—
258	C₂×#135	—	2	2	no	—
259	C₂×#135	C₂²×C₃₁	1	2	yes	—
260	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
261	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
262	#525	C₂²×C₃₁	1	1	yes	characteristic
263	#525	C₂²×C₃₁	1	4	yes	complemented
264	#467	—	2	2	no	—
265	#466	—	2	2	no	—
266	#465	—	2	4	no	—
267	#458	—	2	4	no	—
268	B(4,𝔽₂)	—	2	4	no	—
269	#435	—	2	2	no	—
270	C₈×D₄	—	2	2	no	—
271	#427	C₂²×C₃₁	1	2	yes	complemented
272	#424	C₂²×C₃₁	1	1	yes	characteristic, complemented
273	#421	C₂²×C₃₁	1	4	yes	complemented
274	#416	C₂²×C₃₁	1	4	yes	complemented
275	#410	C₂²×C₃₁	1	2	yes	—
276	#410	C₂²×C₃₁	1	4	yes	complemented
277	#408	C₂²×C₃₁	1	1	yes	characteristic
278	#407	C₂²×C₃₁	1	1	yes	characteristic, complemented
279	#405	C₂²×C₃₁	1	1	yes	characteristic, complemented
280	H(3,ℤ/4)	C₂²×C₃₁	1	2	yes	complemented
281	C₂×C₃₁	—	2	1	no	—
282	C₂×C₃₁	—	2	2	no	—
283	C₂×C₃₁	—	4	2	no	—
284	C₂×C₃₁	—	4	2	no	—
285	C₂×C₃₁	—	8	4	no	—
286	C₂×C₃₁	#2408	1	1	yes	characteristic
287	2₋¹⁺⁴	—	2	4	no	—
288	2₋¹⁺⁴	—	2	2	no	—
289	H(5,𝔽₂)	—	2	4	no	—
290	H(5,𝔽₂)	—	2	4	no	—
291	H(5,𝔽₂)	—	2	2	no	—
292	C₂×#41	—	2	4	no	—
293	C₂×#41	—	4	4	no	—
294	C₂×#41	C₂³×C₃₁	1	4	yes	—
295	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
296	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
297	#137	—	2	8	no	—
298	#137	—	2	4	no	—
299	AGL(1,ℤ/8)	—	2	8	no	—
300	AGL(1,ℤ/8)	—	2	4	no	—
301	#135	—	2	4	no	—
302	#135	—	2	4	no	—
303	#135	—	2	2	no	—
304	#135	—	2	4	no	—
305	#135	—	2	2	no	—
306	#135	—	2	2	no	—
307	#135	—	2	2	no	—
308	C₂×Q₁₆	—	2	2	no	—
309	C₂×Q₁₆	D₄×C₃₁	1	2	yes	complemented
310	C₂×SD₁₆	—	2	4	no	—
311	C₂×SD₁₆	D₄×C₃₁	1	4	yes	complemented
312	C₂×D₈	—	2	2	no	—
313	C₂×D₈	D₄×C₃₁	1	2	yes	complemented
314	#131	—	2	2	no	—
315	#131	—	2	2	no	—
316	C₂×M₁₆	D₄×C₃₁	1	2	yes	complemented
317	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
318	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
319	C₂²×C₈	—	2	1	no	—
320	C₂²×C₈	—	2	1	no	—
321	C₂²×C₈	—	2	2	no	—
322	C₂²×C₈	—	2	2	no	—
323	C₂²×C₈	D₄×C₃₁	1	2	yes	complemented
324	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
326	#123	—	2	4	no	—
327	#123	D₄×C₃₁	1	2	yes	—
328	C₂²⋊Q₈	—	2	1	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	#121	—	2	1	no	—
331	#121	—	2	4	no	—
332	#120	—	2	4	no	—
333	#120	D₄×C₃₁	1	2	yes	—
334	C₄×D₄	—	2	2	no	—
335	C₄×D₄	—	4	4	no	—
336	#117	C₂³×C₃₁	1	4	yes	—
337	#117	C₂³×C₃₁	1	1	yes	characteristic
338	#107	—	2	2	no	—
339	#106	—	2	2	no	—
340	C₄⋊C₈	—	2	2	no	—
341	#104	—	4	4	no	—
342	#103	—	2	4	no	—
343	#102	—	2	4	no	—
344	#99	—	2	4	no	—
345	#99	—	2	8	no	—
346	C₂²⋊C₈	—	2	2	no	—
347	C₂²⋊C₈	—	2	2	no	—
348	#97	—	2	1	no	—
349	C₄×C₈	—	2	1	no	—
350	C₃₁	#24285	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
352	#41	—	2	2	no	—
353	#41	—	2	2	no	—
354	#41	—	2	8	no	—
355	#41	—	2	4	no	—
356	#41	—	2	4	no	—
357	#41	—	2	4	no	—
358	#41	—	2	4	no	—
359	#41	—	4	8	no	—
360	#41	—	4	4	no	—
361	#41	—	4	4	no	—
362	#41	—	4	4	no	—
363	#41	—	4	4	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	—	4	4	no	—
366	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
367	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	—	4	8	no	—
373	C₂×D₄	—	4	4	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
376	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
377	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
386	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
387	Q₁₆	—	2	4	no	—
388	Q₁₆	—	2	2	no	—
389	Q₁₆	—	2	2	no	—
390	SD₁₆	—	2	8	no	—
391	SD₁₆	—	2	4	no	—
392	SD₁₆	—	2	4	no	—
393	D₈	—	2	4	no	—
394	D₈	—	2	2	no	—
395	D₈	—	2	2	no	—
396	M₁₆	—	2	2	no	—
397	M₁₆	—	2	4	no	—
398	M₁₆	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	4	1	no	—
405	C₂×C₈	—	4	1	no	—
406	C₂×C₈	—	4	2	no	—
407	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
408	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
409	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
410	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
411	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	4	4	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₂²⋊C₄	—	4	4	no	—
421	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	4	no	—
424	C₂³	—	2	1	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	4	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	4	no	—
429	C₂³	C₃₁×#121	1	2	yes	—
430	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
431	Q₈	—	2	2	no	—
432	Q₈	—	2	4	no	—
433	Q₈	—	2	4	no	—
434	Q₈	—	4	4	no	—
435	D₄	—	2	4	no	—
436	D₄	—	2	4	no	—
437	D₄	—	2	2	no	—
438	D₄	—	2	4	no	—
439	D₄	—	4	4	no	—
440	D₄	—	4	4	no	—
441	D₄	—	4	8	no	—
442	D₄	—	4	4	no	—
443	D₄	—	4	4	no	—
444	D₄	—	8	8	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	4	8	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	—	4	4	no	—
459	C₂×C₄	C₃₁×#121	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
462	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
463	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
464	C₈	—	2	2	no	—
465	C₈	—	4	1	no	—
466	C₈	—	4	1	no	—
467	C₈	C₃₁×#120	1	1	yes	characteristic
468	C₈	C₃₁×#120	1	1	yes	characteristic
469	C₈	C₃₁×#120	1	2	yes	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	4	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	1	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	8	4	no	—
480	C₂²	—	16	4	no	—
481	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
482	C₂²	D₄²×C₃₁	1	2	yes	—
483	C₄	—	2	2	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	4	no	—
487	C₄	—	8	4	no	—
488	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
489	C₄	C₃₁×#535	1	1	yes	characteristic
490	C₄	C₃₁×#546	1	2	yes	—
491	C₂	—	2	1	no	—
492	C₂	—	2	2	no	—
493	C₂	—	4	2	no	—
494	C₂	—	4	2	no	—
495	C₂	—	8	4	no	—
496	C₂	C₃₁×#2408	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	#2408
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3584	C₂	semidirect
2. C₃₁×#3050	C₂	semidirect
3. C₃₁×#3025	C₂	semidirect
4. C₃₁×#2929	C₂	semidirect
5. C₃₁×#2053	C₂	semidirect
6. C₃₁×#2010	C₂	semidirect
7. C₃₁×#1976	C₂	semidirect
8. C₃₁×#1812	C₂	semidirect
9. C₃₁×#1794	C₂	semidirect
19. C₂³×C₈×C₃₁	C₂²	semidirect
21. C₃₁×#525	C₂²	semidirect
29. C₃₁×#427	C₂²	semidirect
30. C₃₁×#424	C₂²	semidirect
31. C₃₁×#421	C₂²	semidirect
32. C₃₁×#416	C₂²	semidirect
34. C₃₁×#410	C₂²	semidirect
36. C₃₁×#407	C₂²	semidirect
37. C₃₁×#405	C₂²	semidirect
38. C₃₁×H(3,ℤ/4)	C₂²	semidirect
61. C₂×Q₁₆×C₃₁	D₄	semidirect
63. C₂×SD₁₆×C₃₁	D₄	semidirect
65. C₂×D₈×C₃₁	D₄	semidirect
68. C₂×M₁₆×C₃₁	D₄	semidirect
75. C₂²×C₈×C₃₁	D₄	semidirect
175. #24285	C₃₁	direct (N × H)
222. #3584	C₂×C₃₁	semidirect
223. #3050	C₂×C₃₁	semidirect
224. #3025	C₂×C₃₁	semidirect
225. #2929	C₂×C₃₁	semidirect
226. #2053	C₂×C₃₁	semidirect
227. #2010	C₂×C₃₁	semidirect
228. #1976	C₂×C₃₁	semidirect
229. #1812	C₂×C₃₁	semidirect
230. #1794	C₂×C₃₁	semidirect
261. C₂³×C₈	C₂²×C₃₁	semidirect
263. #525	C₂²×C₃₁	semidirect
271. #427	C₂²×C₃₁	semidirect
272. #424	C₂²×C₃₁	semidirect
273. #421	C₂²×C₃₁	semidirect
274. #416	C₂²×C₃₁	semidirect
276. #410	C₂²×C₃₁	semidirect
278. #407	C₂²×C₃₁	semidirect
279. #405	C₂²×C₃₁	semidirect
280. H(3,ℤ/4)	C₂²×C₃₁	semidirect
309. C₂×Q₁₆	D₄×C₃₁	semidirect
311. C₂×SD₁₆	D₄×C₃₁	semidirect
313. C₂×D₈	D₄×C₃₁	semidirect
316. C₂×M₁₆	D₄×C₃₁	semidirect
323. C₂²×C₈	D₄×C₃₁	semidirect
350. C₃₁	#24285	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×#3584	2
C₃₁×#3050	2
C₃₁×#3025	2
C₃₁×#2929	2
C₃₁×#2053	2
C₃₁×#2010	2
C₃₁×#1976	2
C₃₁×#1812	2
C₃₁×#1794	2
#24285	31

Subgroups — C31×#24285 #21481044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24285 #21481044