Subgroups — C₃₁×#60685 #21517444

Order 7936.

1878 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3455	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3179	C₂	1	8	yes	maximal, complemented
3	C₃₁×#3178	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3162	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3131	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3054	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3050	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2943	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#135	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	2	yes	—
13	C₂×C₃₁×#135	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#542	C₂²	1	8	yes	—
17	C₃₁×#541	C₂²	1	4	yes	—
18	C₃₁×#540	C₂²	1	4	yes	—
19	C₃₁×#539	—	2	4	no	—
20	C₃₁×#538	—	2	4	no	—
21	C₃₁×#533	C₂²	1	2	yes	—
22	C₃₁×#532	C₂²	1	2	yes	—
23	C₃₁×#529	—	2	8	no	—
24	C₃₁×#525	C₂²	1	4	yes	—
25	C₄×#41×C₃₁	C₂²	1	4	yes	—
26	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
27	C₃₁×#501	C₂²	1	8	yes	complemented
28	C₃₁×#497	C₂²	1	4	yes	complemented
29	C₃₁×#496	C₂²	1	4	yes	complemented
30	C₃₁×#488	C₂²	1	8	yes	complemented
31	C₃₁×#487	C₂²	1	4	yes	complemented
32	C₃₁×#486	C₂²	1	4	yes	complemented
33	C₃₁×#470	C₂²	1	4	yes	—
34	C₃₁×#469	C₂²	1	4	yes	—
35	C₃₁×#468	C₂²	1	8	yes	—
36	C₃₁×#467	C₂²	1	4	yes	—
37	C₃₁×#466	C₂²	1	4	yes	—
38	C₃₁×#465	C₂²	1	8	yes	—
39	C₃₁×#442	C₂²	1	4	yes	complemented
40	C₃₁×#441	C₂²	1	4	yes	complemented
41	C₃₁×#440	C₂²	1	8	yes	complemented
42	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
43	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
44	C₄×D₈×C₃₁	C₂²	1	4	yes	complemented
45	C₃₁×#427	C₂²	1	2	yes	—
46	C₃₁×#427	C₂²	1	2	yes	—
47	C₃₁×#416	C₂²	1	4	yes	—
48	C₃₁×#416	C₂²	1	4	yes	—
49	C₃₁×#405	C₂²	1	8	yes	—
50	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
51	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
52	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
54	C₂×#41×C₃₁	C₂³	1	4	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#135	—	2	8	no	—
57	C₃₁×#135	—	2	8	no	—
58	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
59	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
60	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
61	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
62	C₂×D₈×C₃₁	C₂³	1	2	yes	—
63	C₂×D₈×C₃₁	C₂³	1	2	yes	—
64	C₂×M₁₆×C₃₁	—	2	4	no	—
65	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
66	C₂²×C₈×C₃₁	—	2	4	no	—
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
70	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
73	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
74	C₃₁×#127	C₂³	1	2	yes	—
75	C₃₁×#126	—	2	8	no	—
76	C₃₁×#126	—	2	8	no	—
77	C₃₁×#125	—	2	8	no	—
78	C₃₁×#125	C₂³	1	4	yes	—
79	C₃₁×#124	—	2	8	no	—
80	C₃₁×#124	C₂³	1	4	yes	—
81	C₃₁×#123	—	2	4	no	—
82	C₃₁×#123	—	2	8	no	—
83	C₃₁×#123	—	2	8	no	—
84	C₃₁×C₂²⋊Q₈	—	2	4	no	—
85	C₃₁×C₂²⋊Q₈	—	2	8	no	—
86	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
87	C₃₁×#121	—	2	4	no	—
88	C₃₁×#121	—	2	8	no	—
89	C₃₁×#121	C₂³	1	8	yes	—
90	C₃₁×#120	—	2	4	no	—
91	C₄×Q₈×C₃₁	—	2	4	no	—
92	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
93	C₄×D₄×C₃₁	—	2	8	no	—
94	C₄×D₄×C₃₁	—	2	8	no	—
95	C₄×D₄×C₃₁	—	2	4	no	—
96	C₄×D₄×C₃₁	C₂³	1	8	yes	—
97	C₃₁×#117	—	2	4	no	—
98	C₃₁×#117	D₄	1	4	yes	—
99	C₃₁×#117	C₂³	1	4	yes	—
100	C₃₁×#117	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
103	C₂×C₄²×C₃₁	—	2	4	no	—
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
106	C₃₁×#107	C₂³	1	4	yes	—
107	C₃₁×#107	C₂³	1	4	yes	—
108	C₃₁×#106	C₂³	1	4	yes	—
109	C₃₁×#106	C₂³	1	4	yes	—
110	C₃₁×#103	C₂³	1	8	yes	—
111	C₃₁×#103	C₂³	1	8	yes	—
112	C₃₁×#102	C₂³	1	8	yes	—
113	C₃₁×#102	C₂³	1	8	yes	—
114	C₃₁×#97	C₂³	1	4	yes	—
115	C₃₁×#97	C₂³	1	4	yes	—
116	C₄×C₈×C₃₁	C₂³	1	4	yes	—
117	C₄×C₈×C₃₁	C₂³	1	4	yes	—
118	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
119	#41×C₃₁	—	2	8	no	—
120	#41×C₃₁	—	2	8	no	—
121	C₂×Q₈×C₃₁	—	2	4	no	—
122	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	8	no	—
125	C₂×D₄×C₃₁	—	2	4	no	—
126	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	8	no	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
137	Q₁₆×C₃₁	—	2	4	no	—
138	Q₁₆×C₃₁	—	2	4	no	—
139	SD₁₆×C₃₁	—	2	8	no	—
140	SD₁₆×C₃₁	—	2	8	no	—
141	D₈×C₃₁	—	2	4	no	—
142	D₈×C₃₁	—	2	4	no	—
143	M₁₆×C₃₁	—	2	8	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	8	no	—
147	C₂×C₈×C₃₁	#41	1	4	yes	—
148	C₂×C₈×C₃₁	#41	1	4	yes	—
149	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
150	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
151	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
152	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
153	C₄⋊C₄×C₃₁	—	2	8	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	8	no	—
156	C₄⋊C₄×C₃₁	—	2	8	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
159	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
160	C₂²⋊C₄×C₃₁	—	2	4	no	—
161	C₂²⋊C₄×C₃₁	—	2	8	no	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	8	no	—
165	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
166	C₄²×C₃₁	—	2	8	no	—
167	C₄²×C₃₁	—	2	4	no	—
168	C₄²×C₃₁	C₂×D₄	1	2	yes	—
169	C₄²×C₃₁	C₂×D₄	1	2	yes	—
170	C₄²×C₃₁	C₂⁴	1	2	yes	—
171	C₄²×C₃₁	C₂⁴	1	2	yes	—
172	#60685	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	2	4	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
177	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₃₁	—	2	4	no	—
179	Q₈×C₃₁	—	2	4	no	—
180	Q₈×C₃₁	—	8	4	no	—
181	D₄×C₃₁	—	2	4	no	—
182	D₄×C₃₁	—	2	4	no	—
183	D₄×C₃₁	—	4	8	no	—
184	D₄×C₃₁	—	8	4	no	—
185	D₄×C₃₁	—	8	8	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	4	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	4	no	—
191	C₂×C₄×C₃₁	—	2	8	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	8	no	—
195	C₂×C₄×C₃₁	—	8	8	no	—
196	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
197	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
201	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
202	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
203	C₈×C₃₁	—	2	2	no	—
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	C₂×#41	1	4	yes	—
206	C₈×C₃₁	C₂×#41	1	4	yes	—
207	#3455	C₂×C₃₁	1	2	yes	complemented
208	#3179	C₂×C₃₁	1	8	yes	complemented
209	#3178	C₂×C₃₁	1	8	yes	complemented
210	#3162	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#3150	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#3131	C₂×C₃₁	1	2	yes	complemented
213	#3054	C₂×C₃₁	1	2	yes	complemented
214	#3050	C₂×C₃₁	1	2	yes	complemented
215	#2943	C₂×C₃₁	1	2	yes	complemented
216	C₄×#135	C₂×C₃₁	1	2	yes	complemented
217	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
218	C₂²×C₃₁	—	2	2	no	—
219	C₂²×C₃₁	—	2	4	no	—
220	C₂²×C₃₁	—	2	2	no	—
221	C₂²×C₃₁	—	4	4	no	—
222	C₂²×C₃₁	—	4	8	no	—
223	C₂²×C₃₁	—	4	4	no	—
224	C₂²×C₃₁	#532	1	2	yes	—
225	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
226	C₄×C₃₁	—	2	4	no	—
227	C₄×C₃₁	—	2	4	no	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	4	8	no	—
230	C₄×C₃₁	—	4	4	no	—
231	C₄×C₃₁	#532	1	2	yes	—
232	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
233	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
234	C₂×#135	C₂²×C₃₁	1	2	yes	—
235	C₂×#135	C₂²×C₃₁	1	2	yes	—
236	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
237	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
238	#542	C₂²×C₃₁	1	8	yes	—
239	#541	C₂²×C₃₁	1	4	yes	—
240	#540	C₂²×C₃₁	1	4	yes	—
241	#539	—	2	4	no	—
242	#538	—	2	4	no	—
243	#533	C₂²×C₃₁	1	2	yes	—
244	#532	C₂²×C₃₁	1	2	yes	—
245	#529	—	2	8	no	—
246	#525	C₂²×C₃₁	1	4	yes	—
247	C₄×#41	C₂²×C₃₁	1	4	yes	—
248	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
249	#501	C₂²×C₃₁	1	8	yes	complemented
250	#497	C₂²×C₃₁	1	4	yes	complemented
251	#496	C₂²×C₃₁	1	4	yes	complemented
252	#488	C₂²×C₃₁	1	8	yes	complemented
253	#487	C₂²×C₃₁	1	4	yes	complemented
254	#486	C₂²×C₃₁	1	4	yes	complemented
255	#470	C₂²×C₃₁	1	4	yes	—
256	#469	C₂²×C₃₁	1	4	yes	—
257	#468	C₂²×C₃₁	1	8	yes	—
258	#467	C₂²×C₃₁	1	4	yes	—
259	#466	C₂²×C₃₁	1	4	yes	—
260	#465	C₂²×C₃₁	1	8	yes	—
261	#442	C₂²×C₃₁	1	4	yes	complemented
262	#441	C₂²×C₃₁	1	4	yes	complemented
263	#440	C₂²×C₃₁	1	8	yes	complemented
264	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
265	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
266	C₄×D₈	C₂²×C₃₁	1	4	yes	complemented
267	#427	C₂²×C₃₁	1	2	yes	—
268	#427	C₂²×C₃₁	1	2	yes	—
269	#416	C₂²×C₃₁	1	4	yes	—
270	#416	C₂²×C₃₁	1	4	yes	—
271	#405	C₂²×C₃₁	1	8	yes	—
272	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₃₁	—	2	2	no	—
277	C₂×C₃₁	—	4	2	no	—
278	C₂×C₃₁	—	8	4	no	—
279	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
280	C₂×C₃₁	#3584	1	2	yes	—
281	C₂×#41	C₂³×C₃₁	1	4	yes	—
282	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
283	#135	—	2	8	no	—
284	#135	—	2	8	no	—
285	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
286	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
287	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
288	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
289	C₂×D₈	C₂³×C₃₁	1	2	yes	—
290	C₂×D₈	C₂³×C₃₁	1	2	yes	—
291	C₂×M₁₆	—	2	4	no	—
292	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
293	C₂²×C₈	—	2	4	no	—
294	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
296	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
300	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
301	#127	C₂³×C₃₁	1	2	yes	—
302	#126	—	2	8	no	—
303	#126	—	2	8	no	—
304	#125	—	2	8	no	—
305	#125	C₂³×C₃₁	1	4	yes	—
306	#124	—	2	8	no	—
307	#124	C₂³×C₃₁	1	4	yes	—
308	#123	—	2	4	no	—
309	#123	—	2	8	no	—
310	#123	—	2	8	no	—
311	C₂²⋊Q₈	—	2	4	no	—
312	C₂²⋊Q₈	—	2	8	no	—
313	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
314	#121	—	2	4	no	—
315	#121	—	2	8	no	—
316	#121	C₂³×C₃₁	1	8	yes	—
317	#120	—	2	4	no	—
318	C₄×Q₈	—	2	4	no	—
319	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
320	C₄×D₄	—	2	8	no	—
321	C₄×D₄	—	2	8	no	—
322	C₄×D₄	—	2	4	no	—
323	C₄×D₄	C₂³×C₃₁	1	8	yes	—
324	#117	—	2	4	no	—
325	#117	D₄×C₃₁	1	4	yes	—
326	#117	C₂³×C₃₁	1	4	yes	—
327	#117	C₂³×C₃₁	1	4	yes	—
328	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
329	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
330	C₂×C₄²	—	2	4	no	—
331	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
333	#107	C₂³×C₃₁	1	4	yes	—
334	#107	C₂³×C₃₁	1	4	yes	—
335	#106	C₂³×C₃₁	1	4	yes	—
336	#106	C₂³×C₃₁	1	4	yes	—
337	#103	C₂³×C₃₁	1	8	yes	—
338	#103	C₂³×C₃₁	1	8	yes	—
339	#102	C₂³×C₃₁	1	8	yes	—
340	#102	C₂³×C₃₁	1	8	yes	—
341	#97	C₂³×C₃₁	1	4	yes	—
342	#97	C₂³×C₃₁	1	4	yes	—
343	C₄×C₈	C₂³×C₃₁	1	4	yes	—
344	C₄×C₈	C₂³×C₃₁	1	4	yes	—
345	C₃₁	#60685	1	1	yes	characteristic, direct product factor, Sylow
346	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
347	#41	—	2	8	no	—
348	#41	—	2	8	no	—
349	C₂×Q₈	—	2	4	no	—
350	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
351	C₂×D₄	—	2	4	no	—
352	C₂×D₄	—	2	8	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	8	no	—
359	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
361	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
362	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
363	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
364	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
365	Q₁₆	—	2	4	no	—
366	Q₁₆	—	2	4	no	—
367	SD₁₆	—	2	8	no	—
368	SD₁₆	—	2	8	no	—
369	D₈	—	2	4	no	—
370	D₈	—	2	4	no	—
371	M₁₆	—	2	8	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	8	no	—
375	C₂×C₈	#41×C₃₁	1	4	yes	—
376	C₂×C₈	#41×C₃₁	1	4	yes	—
377	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
378	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
379	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
380	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	8	no	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
387	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₂²⋊C₄	—	2	8	no	—
390	C₂²⋊C₄	—	2	8	no	—
391	C₂²⋊C₄	—	2	8	no	—
392	C₂²⋊C₄	—	2	8	no	—
393	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
394	C₄²	—	2	4	no	—
395	C₄²	—	2	8	no	—
396	C₄²	C₂×D₄×C₃₁	1	2	yes	—
397	C₄²	C₂×D₄×C₃₁	1	2	yes	—
398	C₄²	C₂⁴×C₃₁	1	2	yes	—
399	C₄²	C₂⁴×C₃₁	1	2	yes	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	4	no	—
402	C₂³	—	4	2	no	—
403	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
404	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
405	Q₈	—	2	4	no	—
406	Q₈	—	2	4	no	—
407	Q₈	—	8	4	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	4	no	—
410	D₄	—	4	8	no	—
411	D₄	—	8	4	no	—
412	D₄	—	8	8	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	8	no	—
422	C₂×C₄	—	8	8	no	—
423	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
424	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
425	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
426	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
428	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
429	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
430	C₈	—	2	2	no	—
431	C₈	—	2	2	no	—
432	C₈	C₂×#41×C₃₁	1	4	yes	—
433	C₈	C₂×#41×C₃₁	1	4	yes	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	4	no	—
437	C₂²	—	4	4	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	8	no	—
440	C₂²	C₃₁×#532	1	2	yes	—
441	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
442	C₄	—	2	2	no	—
443	C₄	—	2	4	no	—
444	C₄	—	2	4	no	—
445	C₄	—	4	4	no	—
446	C₄	—	4	8	no	—
447	C₄	C₃₁×#532	1	2	yes	—
448	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
449	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
450	C₂	—	2	2	no	—
451	C₂	—	4	2	no	—
452	C₂	—	8	4	no	—
453	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
454	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3455	C₂	semidirect
2. C₃₁×#3179	C₂	semidirect
3. C₃₁×#3178	C₂	semidirect
4. C₃₁×#3162	C₂	semidirect
5. C₃₁×#3150	C₂	semidirect
6. C₃₁×#3131	C₂	semidirect
7. C₃₁×#3054	C₂	semidirect
8. C₃₁×#3050	C₂	semidirect
9. C₃₁×#2943	C₂	semidirect
10. C₄×C₃₁×#135	C₂	semidirect
11. C₂×C₃₁×#405	C₂	semidirect
27. C₃₁×#501	C₂²	semidirect
28. C₃₁×#497	C₂²	semidirect
29. C₃₁×#496	C₂²	semidirect
30. C₃₁×#488	C₂²	semidirect
31. C₃₁×#487	C₂²	semidirect
32. C₃₁×#486	C₂²	semidirect
39. C₃₁×#442	C₂²	semidirect
40. C₃₁×#441	C₂²	semidirect
41. C₃₁×#440	C₂²	semidirect
42. C₄×Q₁₆×C₃₁	C₂²	semidirect
43. C₄×SD₁₆×C₃₁	C₂²	semidirect
44. C₄×D₈×C₃₁	C₂²	semidirect
172. #60685	C₃₁	direct (N × H)
207. #3455	C₂×C₃₁	semidirect
208. #3179	C₂×C₃₁	semidirect
209. #3178	C₂×C₃₁	semidirect
210. #3162	C₂×C₃₁	semidirect
211. #3150	C₂×C₃₁	semidirect
212. #3131	C₂×C₃₁	semidirect
213. #3054	C₂×C₃₁	semidirect
214. #3050	C₂×C₃₁	semidirect
215. #2943	C₂×C₃₁	semidirect
216. C₄×#135	C₂×C₃₁	semidirect
217. C₂×#405	C₂×C₃₁	semidirect
249. #501	C₂²×C₃₁	semidirect
250. #497	C₂²×C₃₁	semidirect
251. #496	C₂²×C₃₁	semidirect
252. #488	C₂²×C₃₁	semidirect
253. #487	C₂²×C₃₁	semidirect
254. #486	C₂²×C₃₁	semidirect
261. #442	C₂²×C₃₁	semidirect
262. #441	C₂²×C₃₁	semidirect
263. #440	C₂²×C₃₁	semidirect
264. C₄×Q₁₆	C₂²×C₃₁	semidirect
265. C₄×SD₁₆	C₂²×C₃₁	semidirect
266. C₄×D₈	C₂²×C₃₁	semidirect
345. C₃₁	#60685	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×#3455	2
C₃₁×#3179	2
C₃₁×#3178	2
C₃₁×#3162	2
C₃₁×#3150	2
C₃₁×#3131	2
C₃₁×#3054	2
C₃₁×#3050	2
C₃₁×#2943	2
C₄×C₃₁×#135	2
C₂×C₃₁×#405	2
#60685	31

Subgroups — C31×#60685 #21517444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60685 #21517444