Subgroups — C₃₁×#60624 #21517383

Order 7936.

2038 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3500	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3142	C₂	1	3	yes	maximal, complemented
4	C₃₁×#3140	C₂	1	3	yes	maximal, complemented
5	C₃₁×#3137	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3136	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3131	C₂	1	3	yes	maximal, complemented
8	C₂×C₃₁×#488	C₂	1	3	yes	maximal, complemented
9	C₃₁×#2898	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2897	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	2	yes	—
15	C₃₁×#557	—	2	2	no	—
16	C₃₁×#554	—	2	6	no	—
17	C₃₁×#548	—	2	2	no	—
18	C₃₁×#546	—	2	6	no	—
19	C₃₁×#544	C₂²	1	6	yes	complemented
20	C₃₁×#536	C₂²	1	3	yes	—
21	C₃₁×#535	C₂²	1	3	yes	—
22	C₃₁×#533	C₂²	1	3	yes	complemented
23	C₃₁×#532	C₂²	1	3	yes	—
24	C₂×C₃₁×#125	C₂²	1	3	yes	complemented
25	C₂×C₃₁×#124	C₂²	1	3	yes	—
26	C₃₁×#520	C₂²	1	2	yes	—
27	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#491	C₂²	1	6	yes	complemented
29	C₃₁×#490	C₂²	1	6	yes	complemented
30	C₃₁×#489	C₂²	1	12	yes	complemented
31	C₃₁×#488	C₂²	1	3	yes	complemented
32	C₃₁×#488	C₂²	1	12	yes	complemented
33	C₃₁×#488	C₂²	1	6	yes	complemented
34	C₃₁×#488	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#488	C₂²	1	3	yes	complemented
36	C₃₁×#488	C₂²	1	3	yes	complemented
37	C₃₁×#487	C₂²	1	6	yes	complemented
38	C₃₁×#486	C₂²	1	6	yes	complemented
39	C₃₁×#419	C₂²	1	6	yes	complemented
40	C₃₁×#418	C₂²	1	6	yes	—
41	C₃₁×#417	C₂²	1	6	yes	—
42	C₃₁×#417	C₂²	1	6	yes	complemented
43	C₃₁×#416	C₂²	1	6	yes	—
44	C₃₁×#416	C₂²	1	6	yes	complemented
45	C₂×C₃₁×#103	C₂²	1	6	yes	complemented
46	C₂×C₃₁×#102	C₂²	1	6	yes	—
47	C₃₁×#405	C₂²	1	6	yes	complemented
48	C₄×M₁₆×C₃₁	C₂²	1	3	yes	complemented
49	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	complemented
50	C₃₁×2₋¹⁺⁴	—	2	8	no	—
51	C₃₁×H(5,𝔽₂)	—	2	8	no	—
52	C₂×#41×C₃₁	—	2	1	no	—
53	C₂×#41×C₃₁	—	2	3	no	—
54	C₂×#41×C₃₁	C₂³	1	3	yes	—
55	C₂×#41×C₃₁	C₂³	1	3	yes	—
56	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₃₁	—	2	1	no	—
58	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
59	C₂²×D₄×C₃₁	—	2	3	no	—
60	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
61	C₃₁×#131	—	2	8	no	—
62	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
63	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
64	C₃₁×C₄⋊Q₈	—	2	6	no	—
65	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
66	C₃₁×C₄⋊Q₈	C₂³	1	6	yes	—
67	C₃₁×#127	C₂³	1	3	yes	—
68	C₃₁×#125	—	2	6	no	—
69	C₃₁×#125	—	2	6	no	—
70	C₃₁×#125	C₂³	1	3	yes	—
71	C₃₁×#125	C₂³	1	1	yes	characteristic
72	C₃₁×#125	C₂³	1	6	yes	—
73	C₃₁×#124	—	2	6	no	—
74	C₃₁×#124	C₂³	1	3	yes	—
75	C₃₁×#124	C₂³	1	1	yes	characteristic
76	C₃₁×#124	C₂³	1	6	yes	—
77	C₃₁×#123	—	2	6	no	—
78	C₃₁×C₂²⋊Q₈	—	2	6	no	—
79	C₃₁×C₂²⋊Q₈	—	2	6	no	—
80	C₃₁×C₂²⋊Q₈	—	2	12	no	—
81	C₃₁×#121	—	2	6	no	—
82	C₃₁×#121	—	2	6	no	—
83	C₃₁×#120	—	2	6	no	—
84	C₄×Q₈×C₃₁	—	2	2	no	—
85	C₄×Q₈×C₃₁	—	2	6	no	—
86	C₄×Q₈×C₃₁	D₄	1	2	yes	—
87	C₄×D₄×C₃₁	—	2	6	no	—
88	C₄×D₄×C₃₁	—	2	6	no	—
89	C₄×D₄×C₃₁	D₄	1	6	yes	—
90	C₃₁×#117	C₂³	1	6	yes	—
91	C₃₁×#117	C₂³	1	3	yes	—
92	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	6	no	—
95	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
96	C₃₁×#103	C₂³	1	6	yes	—
97	C₃₁×#103	C₂³	1	12	yes	—
98	C₃₁×#103	C₂³	1	2	yes	—
99	C₃₁×#103	C₂³	1	6	yes	—
100	C₃₁×#103	C₂³	1	6	yes	—
101	C₃₁×#102	C₂³	1	6	yes	—
102	C₃₁×#102	C₂³	1	12	yes	—
103	C₃₁×#102	C₂³	1	6	yes	—
104	C₃₁×#102	C₂³	1	2	yes	—
105	C₃₁×#102	C₂³	1	6	yes	—
106	C₃₁×#97	C₂³	1	6	yes	—
107	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₃₁	C₂³	1	3	yes	—
109	C₄×C₈×C₃₁	C₂³	1	6	yes	—
110	C₂⁴×C₃₁	—	2	3	no	—
111	#41×C₃₁	—	2	12	no	—
112	#41×C₃₁	—	2	12	no	—
113	#41×C₃₁	—	4	4	no	—
114	#41×C₃₁	—	4	12	no	—
115	#41×C₃₁	#41	1	8	yes	—
116	C₂×Q₈×C₃₁	—	2	3	no	—
117	C₂×Q₈×C₃₁	—	2	6	no	—
118	C₂×Q₈×C₃₁	—	2	12	no	—
119	C₂×Q₈×C₃₁	—	4	4	no	—
120	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Q₈×C₃₁	C₂⁴	1	3	yes	—
122	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
123	C₂×D₄×C₃₁	—	2	3	no	—
124	C₂×D₄×C₃₁	—	2	3	no	—
125	C₂×D₄×C₃₁	—	2	3	no	—
126	C₂×D₄×C₃₁	—	2	6	no	—
127	C₂×D₄×C₃₁	—	2	12	no	—
128	C₂×D₄×C₃₁	—	4	12	no	—
129	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
130	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
131	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	—	2	3	no	—
133	C₂²×C₄×C₃₁	—	2	6	no	—
134	C₂²×C₄×C₃₁	—	2	3	no	—
135	C₂²×C₄×C₃₁	—	2	3	no	—
136	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
138	M₁₆×C₃₁	—	2	12	no	—
139	C₂×C₈×C₃₁	—	2	12	no	—
140	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
141	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
142	C₄⋊C₄×C₃₁	—	2	6	no	—
143	C₄⋊C₄×C₃₁	—	2	12	no	—
144	C₄⋊C₄×C₃₁	—	2	6	no	—
145	C₄⋊C₄×C₃₁	—	2	6	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	6	no	—
148	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
149	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
150	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
151	C₂²⋊C₄×C₃₁	—	2	6	no	—
152	C₂²⋊C₄×C₃₁	—	2	6	no	—
153	C₂²⋊C₄×C₃₁	—	2	12	no	—
154	C₂²⋊C₄×C₃₁	—	2	2	no	—
155	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
156	C₄²×C₃₁	—	2	6	no	—
157	C₄²×C₃₁	C₂×D₄	1	6	yes	—
158	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
159	C₄²×C₃₁	C₂⁴	1	3	yes	—
160	#60624	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	3	no	—
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	—	4	6	no	—
164	C₂³×C₃₁	—	8	6	no	—
165	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
166	Q₈×C₃₁	—	2	6	no	—
167	Q₈×C₃₁	—	2	2	no	—
168	Q₈×C₃₁	—	4	6	no	—
169	Q₈×C₃₁	—	8	6	no	—
170	Q₈×C₃₁	#124	1	4	yes	—
171	D₄×C₃₁	—	2	6	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	4	6	no	—
174	D₄×C₃₁	—	4	6	no	—
175	D₄×C₃₁	—	4	6	no	—
176	D₄×C₃₁	—	8	6	no	—
177	D₄×C₃₁	#124	1	12	yes	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	6	no	—
180	C₂×C₄×C₃₁	—	2	3	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	4	6	no	—
183	C₂×C₄×C₃₁	—	4	6	no	—
184	C₂×C₄×C₃₁	—	4	6	no	—
185	C₂×C₄×C₃₁	—	4	6	no	—
186	C₂×C₄×C₃₁	—	8	6	no	—
187	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
189	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
190	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
191	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
192	C₈×C₃₁	—	2	2	no	—
193	C₈×C₃₁	—	2	6	no	—
194	#3504	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#3500	C₂×C₃₁	1	1	yes	characteristic
196	#3142	C₂×C₃₁	1	3	yes	complemented
197	#3140	C₂×C₃₁	1	3	yes	complemented
198	#3137	C₂×C₃₁	1	6	yes	complemented
199	#3136	C₂×C₃₁	1	6	yes	complemented
200	#3131	C₂×C₃₁	1	3	yes	complemented
201	C₂×#488	C₂×C₃₁	1	3	yes	complemented
202	#2898	C₂×C₃₁	1	2	yes	complemented
203	#2897	C₂×C₃₁	1	2	yes	complemented
204	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₂²×C₃₁	—	2	6	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	—	4	6	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	—	4	3	no	—
210	C₂²×C₃₁	—	16	6	no	—
211	C₂²×C₃₁	C₂×#124	1	6	yes	—
212	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
213	C₄×C₃₁	—	2	2	no	—
214	C₄×C₃₁	—	2	6	no	—
215	C₄×C₃₁	—	4	6	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	—	4	1	no	—
218	C₄×C₃₁	—	4	3	no	—
219	C₄×C₃₁	C₂×#124	1	6	yes	—
220	C₄×C₃₁	C₂²×#41	1	2	yes	—
221	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#131	C₂²×C₃₁	1	2	yes	—
224	#557	—	2	2	no	—
225	#554	—	2	6	no	—
226	#548	—	2	2	no	—
227	#546	—	2	6	no	—
228	#544	C₂²×C₃₁	1	6	yes	complemented
229	#536	C₂²×C₃₁	1	3	yes	—
230	#535	C₂²×C₃₁	1	3	yes	—
231	#533	C₂²×C₃₁	1	3	yes	complemented
232	#532	C₂²×C₃₁	1	3	yes	—
233	C₂×#125	C₂²×C₃₁	1	3	yes	complemented
234	C₂×#124	C₂²×C₃₁	1	3	yes	—
235	#520	C₂²×C₃₁	1	2	yes	—
236	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
237	#491	C₂²×C₃₁	1	6	yes	complemented
238	#490	C₂²×C₃₁	1	6	yes	complemented
239	#489	C₂²×C₃₁	1	12	yes	complemented
240	#488	C₂²×C₃₁	1	3	yes	complemented
241	#488	C₂²×C₃₁	1	12	yes	complemented
242	#488	C₂²×C₃₁	1	6	yes	complemented
243	#488	C₂²×C₃₁	1	1	yes	characteristic, complemented
244	#488	C₂²×C₃₁	1	3	yes	complemented
245	#488	C₂²×C₃₁	1	3	yes	complemented
246	#487	C₂²×C₃₁	1	6	yes	complemented
247	#486	C₂²×C₃₁	1	6	yes	complemented
248	#419	C₂²×C₃₁	1	6	yes	complemented
249	#418	C₂²×C₃₁	1	6	yes	—
250	#417	C₂²×C₃₁	1	6	yes	—
251	#417	C₂²×C₃₁	1	6	yes	complemented
252	#416	C₂²×C₃₁	1	6	yes	—
253	#416	C₂²×C₃₁	1	6	yes	complemented
254	C₂×#103	C₂²×C₃₁	1	6	yes	complemented
255	C₂×#102	C₂²×C₃₁	1	6	yes	—
256	#405	C₂²×C₃₁	1	6	yes	complemented
257	C₄×M₁₆	C₂²×C₃₁	1	3	yes	complemented
258	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	complemented
259	C₂×C₃₁	—	2	6	no	—
260	C₂×C₃₁	—	8	1	no	—
261	C₂×C₃₁	—	8	3	no	—
262	C₂×C₃₁	C₂²×#124	1	1	yes	characteristic
263	C₂×C₃₁	#3584	1	2	yes	—
264	2₋¹⁺⁴	—	2	8	no	—
265	H(5,𝔽₂)	—	2	8	no	—
266	C₂×#41	—	2	1	no	—
267	C₂×#41	—	2	3	no	—
268	C₂×#41	C₂³×C₃₁	1	3	yes	—
269	C₂×#41	C₂³×C₃₁	1	3	yes	—
270	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
273	C₂²×D₄	—	2	3	no	—
274	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
275	#131	—	2	8	no	—
276	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
277	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
278	C₄⋊Q₈	—	2	6	no	—
279	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
280	C₄⋊Q₈	C₂³×C₃₁	1	6	yes	—
281	#127	C₂³×C₃₁	1	3	yes	—
282	#125	—	2	6	no	—
283	#125	—	2	6	no	—
284	#125	C₂³×C₃₁	1	3	yes	—
285	#125	C₂³×C₃₁	1	6	yes	—
286	#125	C₂³×C₃₁	1	1	yes	characteristic
287	#124	—	2	6	no	—
288	#124	C₂³×C₃₁	1	3	yes	—
289	#124	C₂³×C₃₁	1	6	yes	—
290	#124	C₂³×C₃₁	1	1	yes	characteristic
291	#123	—	2	6	no	—
292	C₂²⋊Q₈	—	2	6	no	—
293	C₂²⋊Q₈	—	2	6	no	—
294	C₂²⋊Q₈	—	2	12	no	—
295	#121	—	2	6	no	—
296	#121	—	2	6	no	—
297	#120	—	2	6	no	—
298	C₄×Q₈	—	2	2	no	—
299	C₄×Q₈	—	2	6	no	—
300	C₄×Q₈	D₄×C₃₁	1	2	yes	—
301	C₄×D₄	—	2	6	no	—
302	C₄×D₄	—	2	6	no	—
303	C₄×D₄	D₄×C₃₁	1	6	yes	—
304	#117	C₂³×C₃₁	1	6	yes	—
305	#117	C₂³×C₃₁	1	3	yes	—
306	C₂×C₄⋊C₄	—	2	6	no	—
307	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
308	C₂×C₂²⋊C₄	—	2	6	no	—
309	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
310	#103	C₂³×C₃₁	1	6	yes	—
311	#103	C₂³×C₃₁	1	12	yes	—
312	#103	C₂³×C₃₁	1	2	yes	—
313	#103	C₂³×C₃₁	1	6	yes	—
314	#103	C₂³×C₃₁	1	6	yes	—
315	#102	C₂³×C₃₁	1	6	yes	—
316	#102	C₂³×C₃₁	1	12	yes	—
317	#102	C₂³×C₃₁	1	6	yes	—
318	#102	C₂³×C₃₁	1	2	yes	—
319	#102	C₂³×C₃₁	1	6	yes	—
320	#97	C₂³×C₃₁	1	6	yes	—
321	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₄×C₈	C₂³×C₃₁	1	3	yes	—
323	C₄×C₈	C₂³×C₃₁	1	6	yes	—
324	C₃₁	#60624	1	1	yes	characteristic, direct product factor, Sylow
325	C₂⁴	—	2	3	no	—
326	#41	—	2	12	no	—
327	#41	—	2	12	no	—
328	#41	—	4	4	no	—
329	#41	—	4	12	no	—
330	#41	#41×C₃₁	1	8	yes	—
331	C₂×Q₈	—	2	3	no	—
332	C₂×Q₈	—	2	6	no	—
333	C₂×Q₈	—	2	12	no	—
334	C₂×Q₈	—	4	4	no	—
335	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂×Q₈	C₂⁴×C₃₁	1	3	yes	—
337	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
338	C₂×D₄	—	2	3	no	—
339	C₂×D₄	—	2	3	no	—
340	C₂×D₄	—	2	3	no	—
341	C₂×D₄	—	2	6	no	—
342	C₂×D₄	—	2	12	no	—
343	C₂×D₄	—	4	12	no	—
344	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
345	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
346	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	—	2	3	no	—
348	C₂²×C₄	—	2	6	no	—
349	C₂²×C₄	—	2	3	no	—
350	C₂²×C₄	—	2	3	no	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
352	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
353	M₁₆	—	2	12	no	—
354	C₂×C₈	—	2	12	no	—
355	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
356	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
357	C₄⋊C₄	—	2	6	no	—
358	C₄⋊C₄	—	2	12	no	—
359	C₄⋊C₄	—	2	6	no	—
360	C₄⋊C₄	—	2	6	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	6	no	—
363	C₄⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
364	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
365	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
366	C₂²⋊C₄	—	2	6	no	—
367	C₂²⋊C₄	—	2	6	no	—
368	C₂²⋊C₄	—	2	12	no	—
369	C₂²⋊C₄	—	2	2	no	—
370	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
371	C₄²	—	2	6	no	—
372	C₄²	C₂×D₄×C₃₁	1	6	yes	—
373	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
374	C₄²	C₂⁴×C₃₁	1	3	yes	—
375	C₂³	—	2	3	no	—
376	C₂³	—	2	1	no	—
377	C₂³	—	4	6	no	—
378	C₂³	—	8	6	no	—
379	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
380	Q₈	—	2	6	no	—
381	Q₈	—	2	2	no	—
382	Q₈	—	4	6	no	—
383	Q₈	—	8	6	no	—
384	Q₈	C₃₁×#124	1	4	yes	—
385	D₄	—	2	6	no	—
386	D₄	—	2	2	no	—
387	D₄	—	4	6	no	—
388	D₄	—	4	6	no	—
389	D₄	—	4	6	no	—
390	D₄	—	8	6	no	—
391	D₄	C₃₁×#124	1	12	yes	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	6	no	—
394	C₂×C₄	—	2	3	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	4	6	no	—
397	C₂×C₄	—	4	6	no	—
398	C₂×C₄	—	4	6	no	—
399	C₂×C₄	—	4	6	no	—
400	C₂×C₄	—	8	6	no	—
401	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
402	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
403	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
404	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
405	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	—	2	6	no	—
408	C₂²	—	2	6	no	—
409	C₂²	—	4	1	no	—
410	C₂²	—	4	3	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	6	no	—
413	C₂²	—	16	6	no	—
414	C₂²	C₂×C₃₁×#124	1	6	yes	—
415	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	2	6	no	—
418	C₄	—	4	1	no	—
419	C₄	—	4	3	no	—
420	C₄	—	4	6	no	—
421	C₄	—	4	2	no	—
422	C₄	C₂×C₃₁×#124	1	6	yes	—
423	C₄	C₂²×#41×C₃₁	1	2	yes	—
424	C₂	—	2	6	no	—
425	C₂	—	8	3	no	—
426	C₂	—	8	1	no	—
427	C₂	C₂²×C₃₁×#124	1	1	yes	characteristic
428	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3504	C₂	semidirect
3. C₃₁×#3142	C₂	semidirect
4. C₃₁×#3140	C₂	semidirect
5. C₃₁×#3137	C₂	semidirect
6. C₃₁×#3136	C₂	semidirect
7. C₃₁×#3131	C₂	semidirect
8. C₂×C₃₁×#488	C₂	semidirect
9. C₃₁×#2898	C₂	semidirect
10. C₃₁×#2897	C₂	semidirect
11. C₄×C₃₁×#131	C₂	semidirect
19. C₃₁×#544	C₂²	semidirect
22. C₃₁×#533	C₂²	semidirect
24. C₂×C₃₁×#125	C₂²	semidirect
28. C₃₁×#491	C₂²	semidirect
29. C₃₁×#490	C₂²	semidirect
30. C₃₁×#489	C₂²	semidirect
31. C₃₁×#488	C₂²	semidirect
32. C₃₁×#488	C₂²	semidirect
33. C₃₁×#488	C₂²	semidirect
34. C₃₁×#488	C₂²	semidirect
35. C₃₁×#488	C₂²	semidirect
36. C₃₁×#488	C₂²	semidirect
37. C₃₁×#487	C₂²	semidirect
38. C₃₁×#486	C₂²	semidirect
39. C₃₁×#419	C₂²	semidirect
42. C₃₁×#417	C₂²	semidirect
44. C₃₁×#416	C₂²	semidirect
45. C₂×C₃₁×#103	C₂²	semidirect
47. C₃₁×#405	C₂²	semidirect
48. C₄×M₁₆×C₃₁	C₂²	semidirect
49. C₂×C₄×C₈×C₃₁	C₂²	semidirect
160. #60624	C₃₁	direct (N × H)
194. #3504	C₂×C₃₁	semidirect
196. #3142	C₂×C₃₁	semidirect
197. #3140	C₂×C₃₁	semidirect
198. #3137	C₂×C₃₁	semidirect
199. #3136	C₂×C₃₁	semidirect
200. #3131	C₂×C₃₁	semidirect
201. C₂×#488	C₂×C₃₁	semidirect
202. #2898	C₂×C₃₁	semidirect
203. #2897	C₂×C₃₁	semidirect
204. C₄×#131	C₂×C₃₁	semidirect
228. #544	C₂²×C₃₁	semidirect
231. #533	C₂²×C₃₁	semidirect
233. C₂×#125	C₂²×C₃₁	semidirect
237. #491	C₂²×C₃₁	semidirect
238. #490	C₂²×C₃₁	semidirect
239. #489	C₂²×C₃₁	semidirect
240. #488	C₂²×C₃₁	semidirect
241. #488	C₂²×C₃₁	semidirect
242. #488	C₂²×C₃₁	semidirect
243. #488	C₂²×C₃₁	semidirect
244. #488	C₂²×C₃₁	semidirect
245. #488	C₂²×C₃₁	semidirect
246. #487	C₂²×C₃₁	semidirect
247. #486	C₂²×C₃₁	semidirect
248. #419	C₂²×C₃₁	semidirect
251. #417	C₂²×C₃₁	semidirect
253. #416	C₂²×C₃₁	semidirect
254. C₂×#103	C₂²×C₃₁	semidirect
256. #405	C₂²×C₃₁	semidirect
257. C₄×M₁₆	C₂²×C₃₁	semidirect
258. C₂×C₄×C₈	C₂²×C₃₁	semidirect
324. C₃₁	#60624	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×#3504	2
C₃₁×#3500	2
C₃₁×#3142	2
C₃₁×#3140	2
C₃₁×#3137	2
C₃₁×#3136	2
C₃₁×#3131	2
C₂×C₃₁×#488	2
C₃₁×#2898	2
C₃₁×#2897	2
C₄×C₃₁×#131	2
#60624	31

Subgroups — C31×#60624 #21517383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60624 #21517383