Subgroups — C₃₁×#22898 #21479657

Order 7936.

996 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3018	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2956	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2901	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1963	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1932	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1906	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1884	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1861	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1759	C₂	1	2	yes	maximal, complemented
10	C₃₁×#585	C₂²	1	1	yes	characteristic
11	C₃₁×#575	C₂²	1	2	yes	—
12	C₃₁×#568	C₂²	1	2	yes	—
13	C₃₁×#568	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#456	C₄	1	2	yes	complemented
17	C₃₁×#455	C₄	1	2	yes	complemented
18	C₃₁×#454	C₄	1	2	yes	complemented
19	C₃₁×#453	C₄	1	2	yes	complemented
20	C₃₁×#444	—	2	4	no	—
21	C₃₁×#443	—	2	4	no	—
22	C₃₁×#430	C₂²	1	2	yes	complemented
23	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#421	—	2	4	no	—
25	C₃₁×#421	C₂²	1	2	yes	—
26	C₃₁×#421	C₂²	1	2	yes	—
27	C₂×C₃₁×#104	—	2	4	no	—
28	C₂×C₃₁×#104	C₂²	1	2	yes	—
29	C₂×C₃₁×#104	C₂²	1	2	yes	—
30	C₂×C₃₁×#104	C₂²	1	2	yes	—
31	C₂×C₃₁×#104	C₂²	1	2	yes	complemented
32	C₃₁×#413	C₂²	1	1	yes	characteristic
33	C₃₁×#410	C₂²	1	1	yes	characteristic, complemented
34	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
35	C₃₁×#376	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#344	C₂²	1	2	yes	complemented
37	C₃₁×#339	C₂²	1	2	yes	complemented
38	C₃₁×#339	C₂²	1	2	yes	complemented
39	C₃₁×2₋¹⁺⁴	—	2	2	no	—
40	C₃₁×2₋¹⁺⁴	C₂×C₄	1	2	yes	—
41	C₃₁×H(5,𝔽₂)	—	2	2	no	—
42	C₃₁×H(5,𝔽₂)	—	4	1	no	—
43	C₃₁×H(5,𝔽₂)	C₂×C₄	1	2	yes	—
44	C₂×#41×C₃₁	—	2	4	no	—
45	C₂×#41×C₃₁	—	4	1	no	—
46	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×#41×C₃₁	C₂³	1	2	yes	—
49	C₃₁×#137	—	2	2	no	—
50	C₃₁×#137	C₂×C₄	1	4	yes	complemented
51	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
52	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	complemented
53	C₃₁×#135	—	2	4	no	—
54	C₃₁×#131	—	2	4	no	—
55	C₃₁×#131	—	2	4	no	—
56	C₃₁×#131	—	4	2	no	—
57	C₂×M₁₆×C₃₁	—	2	4	no	—
58	C₂×M₁₆×C₃₁	—	4	1	no	—
59	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
63	C₃₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
64	C₃₁×#124	C₂×C₄	1	1	yes	characteristic, complemented
65	C₃₁×#124	C₂×C₄	1	1	yes	characteristic, complemented
66	C₃₁×#121	—	4	1	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	—	2	2	no	—
69	C₄×D₄×C₃₁	—	4	1	no	—
70	C₄×D₄×C₃₁	—	4	2	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#117	C₂³	1	2	yes	—
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₃₁×#108	—	2	4	no	—
77	C₃₁×C₄⋊C₈	—	4	1	no	—
78	C₃₁×#104	—	2	4	no	—
79	C₃₁×#104	—	2	4	no	—
80	C₃₁×#104	—	2	2	no	—
81	C₃₁×#104	—	2	2	no	—
82	C₃₁×#104	—	2	2	no	—
83	C₃₁×#104	—	2	2	no	—
84	C₃₁×#104	C₂×C₄	1	4	yes	complemented
85	C₃₁×#104	C₂×C₄	1	4	yes	complemented
86	C₃₁×#104	D₄	1	4	yes	complemented
87	C₃₁×#104	D₄	1	4	yes	complemented
88	C₃₁×#101	C₂×C₄	1	2	yes	complemented
89	C₃₁×#100	C₂×C₄	1	2	yes	complemented
90	C₃₁×#99	—	4	1	no	—
91	C₃₁×#97	—	2	2	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₃₁×#95	—	4	1	no	—
94	#41×C₃₁	—	2	2	no	—
95	#41×C₃₁	—	2	2	no	—
96	#41×C₃₁	—	2	4	no	—
97	#41×C₃₁	—	2	4	no	—
98	#41×C₃₁	—	2	2	no	—
99	#41×C₃₁	—	2	2	no	—
100	#41×C₃₁	—	2	2	no	—
101	#41×C₃₁	—	2	2	no	—
102	#41×C₃₁	—	4	2	no	—
103	#41×C₃₁	—	4	2	no	—
104	#41×C₃₁	—	4	4	no	—
105	#41×C₃₁	—	4	2	no	—
106	#41×C₃₁	C₂²×C₄	1	4	yes	—
107	#41×C₃₁	C₂×D₄	1	4	yes	—
108	C₂×Q₈×C₃₁	—	2	4	no	—
109	C₂×Q₈×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×Q₈×C₃₁	#41	1	2	yes	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	4	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	4	1	no	—
118	C₂×D₄×C₃₁	—	4	2	no	—
119	C₂×D₄×C₃₁	—	4	1	no	—
120	C₂×D₄×C₃₁	—	4	4	no	—
121	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×D₄×C₃₁	#41	1	2	yes	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	—	4	2	no	—
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₃₁	—	2	4	no	—
133	SD₁₆×C₃₁	—	2	4	no	—
134	SD₁₆×C₃₁	—	2	4	no	—
135	D₈×C₃₁	—	2	4	no	—
136	M₁₆×C₃₁	—	2	2	no	—
137	M₁₆×C₃₁	—	2	2	no	—
138	M₁₆×C₃₁	—	2	2	no	—
139	M₁₆×C₃₁	—	2	2	no	—
140	M₁₆×C₃₁	—	4	2	no	—
141	M₁₆×C₃₁	—	4	4	no	—
142	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
143	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	4	2	no	—
148	C₂×C₈×C₃₁	—	4	4	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	4	1	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₂²⋊C₄×C₃₁	—	2	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	1	no	—
154	C₂²⋊C₄×C₃₁	—	4	1	no	—
155	C₂²⋊C₄×C₃₁	—	4	2	no	—
156	C₄²×C₃₁	—	2	2	no	—
157	C₄²×C₃₁	—	2	1	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	4	2	no	—
160	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
161	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₃₁	C₂×D₄	1	2	yes	—
163	C₄²×C₃₁	C₂×D₄	1	2	yes	—
164	#22898	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	2	1	no	—
167	C₂³×C₃₁	—	2	1	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
170	Q₈×C₃₁	—	2	2	no	—
171	Q₈×C₃₁	—	2	2	no	—
172	Q₈×C₃₁	—	2	2	no	—
173	Q₈×C₃₁	—	4	2	no	—
174	Q₈×C₃₁	C₄×D₄	1	4	yes	—
175	D₄×C₃₁	—	2	4	no	—
176	D₄×C₃₁	—	2	4	no	—
177	D₄×C₃₁	—	2	2	no	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	2	2	no	—
180	D₄×C₃₁	—	4	4	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	—	8	2	no	—
185	D₄×C₃₁	C₄×D₄	1	4	yes	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	4	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	4	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	8	2	no	—
203	C₂×C₄×C₃₁	—	8	1	no	—
204	C₂×C₄×C₃₁	—	8	4	no	—
205	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
207	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
208	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
209	C₈×C₃₁	—	2	4	no	—
210	C₈×C₃₁	—	2	4	no	—
211	C₈×C₃₁	—	4	2	no	—
212	#3018	C₂×C₃₁	1	1	yes	characteristic
213	#2956	C₂×C₃₁	1	2	yes	complemented
214	#2901	C₂×C₃₁	1	2	yes	complemented
215	#1963	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1932	C₂×C₃₁	1	2	yes	complemented
217	#1906	C₂×C₃₁	1	2	yes	complemented
218	#1884	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1861	C₂×C₃₁	1	2	yes	complemented
220	#1759	C₂×C₃₁	1	2	yes	complemented
221	C₂²×C₃₁	—	2	4	no	—
222	C₂²×C₃₁	—	4	1	no	—
223	C₂²×C₃₁	—	4	1	no	—
224	C₂²×C₃₁	—	4	1	no	—
225	C₂²×C₃₁	—	8	2	no	—
226	C₂²×C₃₁	—	8	1	no	—
227	C₂²×C₃₁	—	8	1	no	—
228	C₂²×C₃₁	—	16	2	no	—
229	C₂²×C₃₁	#518	1	1	yes	characteristic
230	C₂²×C₃₁	#546	1	2	yes	—
231	C₄×C₃₁	—	2	4	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	1	no	—
234	C₄×C₃₁	—	4	2	no	—
235	C₄×C₃₁	—	4	1	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	—	4	1	no	—
238	C₄×C₃₁	—	8	2	no	—
239	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
240	C₄×C₃₁	D₄²	1	1	yes	characteristic
241	C₄×C₃₁	#551	1	1	yes	characteristic
242	#585	C₂²×C₃₁	1	1	yes	characteristic
243	#575	C₂²×C₃₁	1	2	yes	—
244	#568	C₂²×C₃₁	1	2	yes	—
245	#568	C₂²×C₃₁	1	1	yes	characteristic, complemented
246	#532	C₂²×C₃₁	1	1	yes	characteristic
247	C₄×#41	C₂²×C₃₁	1	2	yes	—
248	#456	C₄×C₃₁	1	2	yes	complemented
249	#455	C₄×C₃₁	1	2	yes	complemented
250	#454	C₄×C₃₁	1	2	yes	complemented
251	#453	C₄×C₃₁	1	2	yes	complemented
252	#444	—	2	4	no	—
253	#443	—	2	4	no	—
254	#430	C₂²×C₃₁	1	2	yes	complemented
255	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
256	#421	—	2	4	no	—
257	#421	C₂²×C₃₁	1	2	yes	—
258	#421	C₂²×C₃₁	1	2	yes	—
259	C₂×#104	—	2	4	no	—
260	C₂×#104	C₂²×C₃₁	1	2	yes	—
261	C₂×#104	C₂²×C₃₁	1	2	yes	—
262	C₂×#104	C₂²×C₃₁	1	2	yes	—
263	C₂×#104	C₂²×C₃₁	1	2	yes	complemented
264	#413	C₂²×C₃₁	1	1	yes	characteristic
265	#410	C₂²×C₃₁	1	1	yes	characteristic, complemented
266	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
267	#376	C₂²×C₃₁	1	1	yes	characteristic, complemented
268	#344	C₂²×C₃₁	1	2	yes	complemented
269	#339	C₂²×C₃₁	1	2	yes	complemented
270	#339	C₂²×C₃₁	1	2	yes	complemented
271	C₂×C₃₁	—	2	2	no	—
272	C₂×C₃₁	—	2	1	no	—
273	C₂×C₃₁	—	4	4	no	—
274	C₂×C₃₁	—	8	1	no	—
275	C₂×C₃₁	—	8	1	no	—
276	C₂×C₃₁	#2358	1	1	yes	characteristic
277	2₋¹⁺⁴	—	2	2	no	—
278	2₋¹⁺⁴	C₂×C₄×C₃₁	1	2	yes	—
279	H(5,𝔽₂)	—	2	2	no	—
280	H(5,𝔽₂)	—	4	1	no	—
281	H(5,𝔽₂)	C₂×C₄×C₃₁	1	2	yes	—
282	C₂×#41	—	2	4	no	—
283	C₂×#41	—	4	1	no	—
284	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×#41	C₂³×C₃₁	1	2	yes	—
287	#137	—	2	2	no	—
288	#137	C₂×C₄×C₃₁	1	4	yes	complemented
289	AGL(1,ℤ/8)	—	2	2	no	—
290	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	4	yes	complemented
291	#135	—	2	4	no	—
292	#131	—	2	4	no	—
293	#131	—	2	4	no	—
294	#131	—	4	2	no	—
295	C₂×M₁₆	—	2	4	no	—
296	C₂×M₁₆	—	4	1	no	—
297	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
298	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
299	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
300	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
301	#127	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
302	#124	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
303	#124	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
304	#121	—	4	1	no	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×D₄	—	2	2	no	—
307	C₄×D₄	—	4	1	no	—
308	C₄×D₄	—	4	2	no	—
309	#117	—	2	2	no	—
310	#117	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄²	—	2	2	no	—
312	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
314	#108	—	2	4	no	—
315	C₄⋊C₈	—	4	1	no	—
316	#104	—	2	4	no	—
317	#104	—	2	4	no	—
318	#104	—	2	2	no	—
319	#104	—	2	2	no	—
320	#104	—	2	2	no	—
321	#104	—	2	2	no	—
322	#104	C₂×C₄×C₃₁	1	4	yes	complemented
323	#104	C₂×C₄×C₃₁	1	4	yes	complemented
324	#104	D₄×C₃₁	1	4	yes	complemented
325	#104	D₄×C₃₁	1	4	yes	complemented
326	#101	C₂×C₄×C₃₁	1	2	yes	complemented
327	#100	C₂×C₄×C₃₁	1	2	yes	complemented
328	#99	—	4	1	no	—
329	#97	—	2	2	no	—
330	C₄×C₈	—	2	2	no	—
331	#95	—	4	1	no	—
332	C₃₁	#22898	1	1	yes	characteristic, direct product factor, Sylow
333	#41	—	2	2	no	—
334	#41	—	2	2	no	—
335	#41	—	2	4	no	—
336	#41	—	2	4	no	—
337	#41	—	2	2	no	—
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	#41	—	2	2	no	—
341	#41	—	4	2	no	—
342	#41	—	4	2	no	—
343	#41	—	4	4	no	—
344	#41	—	4	2	no	—
345	#41	C₂²×C₄×C₃₁	1	4	yes	—
346	#41	C₂×D₄×C₃₁	1	4	yes	—
347	C₂×Q₈	—	2	4	no	—
348	C₂×Q₈	—	4	1	no	—
349	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
350	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
351	C₂×Q₈	#41×C₃₁	1	2	yes	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	4	1	no	—
357	C₂×D₄	—	4	2	no	—
358	C₂×D₄	—	4	1	no	—
359	C₂×D₄	—	4	4	no	—
360	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
361	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
362	C₂×D₄	#41×C₃₁	1	2	yes	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
371	Q₁₆	—	2	4	no	—
372	SD₁₆	—	2	4	no	—
373	SD₁₆	—	2	4	no	—
374	D₈	—	2	4	no	—
375	M₁₆	—	2	2	no	—
376	M₁₆	—	2	2	no	—
377	M₁₆	—	2	2	no	—
378	M₁₆	—	2	2	no	—
379	M₁₆	—	4	2	no	—
380	M₁₆	—	4	4	no	—
381	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
382	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	4	2	no	—
387	C₂×C₈	—	4	4	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	4	1	no	—
390	C₄⋊C₄	—	4	2	no	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	4	1	no	—
393	C₂²⋊C₄	—	4	1	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	4	2	no	—
399	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
400	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
401	C₄²	C₂×D₄×C₃₁	1	2	yes	—
402	C₄²	C₂×D₄×C₃₁	1	2	yes	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	1	no	—
405	C₂³	—	2	1	no	—
406	C₂³	—	4	2	no	—
407	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
408	Q₈	—	2	2	no	—
409	Q₈	—	2	2	no	—
410	Q₈	—	2	2	no	—
411	Q₈	—	4	2	no	—
412	Q₈	C₄×D₄×C₃₁	1	4	yes	—
413	D₄	—	2	4	no	—
414	D₄	—	2	4	no	—
415	D₄	—	2	2	no	—
416	D₄	—	2	2	no	—
417	D₄	—	2	2	no	—
418	D₄	—	4	4	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	2	no	—
421	D₄	—	4	2	no	—
422	D₄	—	8	2	no	—
423	D₄	C₄×D₄×C₃₁	1	4	yes	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	8	2	no	—
441	C₂×C₄	—	8	1	no	—
442	C₂×C₄	—	8	4	no	—
443	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
445	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
446	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
447	C₈	—	2	4	no	—
448	C₈	—	2	4	no	—
449	C₈	—	4	2	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	4	1	no	—
453	C₂²	—	4	1	no	—
454	C₂²	—	8	2	no	—
455	C₂²	—	8	1	no	—
456	C₂²	—	8	1	no	—
457	C₂²	—	16	2	no	—
458	C₂²	C₃₁×#518	1	1	yes	characteristic
459	C₂²	C₃₁×#546	1	2	yes	—
460	C₄	—	2	4	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	1	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	1	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	8	2	no	—
468	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
469	C₄	D₄²×C₃₁	1	1	yes	characteristic
470	C₄	C₃₁×#551	1	1	yes	characteristic
471	C₂	—	2	2	no	—
472	C₂	—	2	1	no	—
473	C₂	—	4	4	no	—
474	C₂	—	8	1	no	—
475	C₂	—	8	1	no	—
476	C₂	C₃₁×#2358	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	#2358
prime radical	C₃₁×#3018	C₂
LCS	G ▷ C₂×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
2. C₃₁×#2956	C₂	semidirect
3. C₃₁×#2901	C₂	semidirect
4. C₃₁×#1963	C₂	semidirect
5. C₃₁×#1932	C₂	semidirect
6. C₃₁×#1906	C₂	semidirect
7. C₃₁×#1884	C₂	semidirect
8. C₃₁×#1861	C₂	semidirect
9. C₃₁×#1759	C₂	semidirect
13. C₃₁×#568	C₂²	semidirect
16. C₃₁×#456	C₄	semidirect
17. C₃₁×#455	C₄	semidirect
18. C₃₁×#454	C₄	semidirect
19. C₃₁×#453	C₄	semidirect
22. C₃₁×#430	C₂²	semidirect
23. C₃₁×#423	C₂²	semidirect
31. C₂×C₃₁×#104	C₂²	semidirect
33. C₃₁×#410	C₂²	semidirect
34. C₄×M₁₆×C₃₁	C₂²	semidirect
35. C₃₁×#376	C₂²	semidirect
36. C₃₁×#344	C₂²	semidirect
37. C₃₁×#339	C₂²	semidirect
38. C₃₁×#339	C₂²	semidirect
50. C₃₁×#137	C₂×C₄	semidirect
52. C₃₁×AGL(1,ℤ/8)	C₂×C₄	semidirect
62. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
63. C₃₁×#127	C₂×C₄	semidirect
64. C₃₁×#124	C₂×C₄	semidirect
65. C₃₁×#124	C₂×C₄	semidirect
84. C₃₁×#104	C₂×C₄	semidirect
85. C₃₁×#104	C₂×C₄	semidirect
86. C₃₁×#104	D₄	semidirect
87. C₃₁×#104	D₄	semidirect
88. C₃₁×#101	C₂×C₄	semidirect
89. C₃₁×#100	C₂×C₄	semidirect
164. #22898	C₃₁	direct (N × H)
213. #2956	C₂×C₃₁	semidirect
214. #2901	C₂×C₃₁	semidirect
215. #1963	C₂×C₃₁	semidirect
216. #1932	C₂×C₃₁	semidirect
217. #1906	C₂×C₃₁	semidirect
218. #1884	C₂×C₃₁	semidirect
219. #1861	C₂×C₃₁	semidirect
220. #1759	C₂×C₃₁	semidirect
245. #568	C₂²×C₃₁	semidirect
248. #456	C₄×C₃₁	semidirect
249. #455	C₄×C₃₁	semidirect
250. #454	C₄×C₃₁	semidirect
251. #453	C₄×C₃₁	semidirect
254. #430	C₂²×C₃₁	semidirect
255. #423	C₂²×C₃₁	semidirect
263. C₂×#104	C₂²×C₃₁	semidirect
265. #410	C₂²×C₃₁	semidirect
266. C₄×M₁₆	C₂²×C₃₁	semidirect
267. #376	C₂²×C₃₁	semidirect
268. #344	C₂²×C₃₁	semidirect
269. #339	C₂²×C₃₁	semidirect
270. #339	C₂²×C₃₁	semidirect
288. #137	C₂×C₄×C₃₁	semidirect
290. AGL(1,ℤ/8)	C₂×C₄×C₃₁	semidirect
300. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
301. #127	C₂×C₄×C₃₁	semidirect
302. #124	C₂×C₄×C₃₁	semidirect
303. #124	C₂×C₄×C₃₁	semidirect
322. #104	C₂×C₄×C₃₁	semidirect
323. #104	C₂×C₄×C₃₁	semidirect
324. #104	D₄×C₃₁	semidirect
325. #104	D₄×C₃₁	semidirect
326. #101	C₂×C₄×C₃₁	semidirect
327. #100	C₂×C₄×C₃₁	semidirect
332. C₃₁	#22898	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×#3018	2
C₃₁×#2956	2
C₃₁×#2901	2
C₃₁×#1963	2
C₃₁×#1932	2
C₃₁×#1906	2
C₃₁×#1884	2
C₃₁×#1861	2
C₃₁×#1759	2
#22898	31

Subgroups — C31×#22898 #21479657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22898 #21479657