Subgroups — C₃₁×#61947 #21518706

Order 7936.

2458 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3391	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3358	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3294	C₂	1	6	yes	maximal, complemented
6	D₄×D₈×C₃₁	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3245	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3205	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3100	C₂	1	3	yes	maximal, complemented
10	C₂×C₃₁×#459	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	—	2	1	no	—
13	C₂²×#41×C₃₁	—	2	1	no	—
14	C₂²×D₈×C₃₁	C₂²	1	6	yes	complemented
15	C₃₁×#550	—	2	1	no	—
16	C₃₁×#550	C₂²	1	2	yes	complemented
17	C₃₁×#550	C₂²	1	1	yes	characteristic, complemented
18	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#548	—	2	1	no	—
20	C₃₁×#547	C₂²	1	3	yes	complemented
21	C₃₁×#546	—	2	3	no	—
22	D₄²×C₃₁	—	2	3	no	—
23	D₄²×C₃₁	C₂²	1	3	yes	complemented
24	D₄²×C₃₁	C₂²	1	6	yes	complemented
25	C₃₁×#535	—	2	3	no	—
26	C₃₁×#532	C₂²	1	3	yes	complemented
27	C₃₁×#532	C₂²	1	3	yes	complemented
28	C₂×C₃₁×#127	C₂²	1	3	yes	complemented
29	C₃₁×#525	—	2	3	no	—
30	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	complemented
32	C₃₁×#493	C₂²	1	6	yes	complemented
33	C₃₁×#486	C₂²	1	6	yes	complemented
34	C₃₁×#480	C₂²	1	6	yes	complemented
35	C₃₁×#474	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#474	C₂²	1	3	yes	complemented
37	C₃₁×#474	C₂²	1	6	yes	complemented
38	C₃₁×#466	C₂²	1	12	yes	complemented
39	C₃₁×#459	C₂²	1	3	yes	complemented
40	C₃₁×#459	C₂²	1	3	yes	complemented
41	C₃₁×#459	C₂²	1	6	yes	complemented
42	C₃₁×#459	C₂²	1	6	yes	complemented
43	C₃₁×#459	C₂²	1	6	yes	complemented
44	C₃₁×#459	C₂²	1	6	yes	complemented
45	C₃₁×#447	C₂²	1	6	yes	complemented
46	C₃₁×#447	C₂²	1	12	yes	complemented
47	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
48	C₄×D₈×C₃₁	C₂²	1	6	yes	complemented
49	C₄×D₈×C₃₁	C₂²	1	6	yes	complemented
50	C₈×D₄×C₃₁	C₂²	1	6	yes	complemented
51	C₃₁×#431	C₂²	1	3	yes	complemented
52	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	complemented
53	C₂×C₃₁×#102	C₂²	1	6	yes	complemented
54	C₃₁×H(5,𝔽₂)	—	4	8	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂×#41×C₃₁	—	2	2	no	—
57	C₂×#41×C₃₁	—	2	3	no	—
58	C₂×#41×C₃₁	—	2	3	no	—
59	C₂×#41×C₃₁	—	2	3	no	—
60	C₂×#41×C₃₁	—	4	8	no	—
61	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	—	2	1	no	—
63	C₂²×D₄×C₃₁	—	2	6	no	—
64	C₂²×D₄×C₃₁	—	2	3	no	—
65	C₂²×D₄×C₃₁	—	2	3	no	—
66	C₂²×D₄×C₃₁	—	2	3	no	—
67	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
69	C₂³×C₄×C₃₁	—	2	3	no	—
70	C₂×D₈×C₃₁	—	4	12	no	—
71	C₂×D₈×C₃₁	C₂³	1	6	yes	—
72	C₂×D₈×C₃₁	C₂³	1	6	yes	—
73	C₂×D₈×C₃₁	C₂³	1	12	yes	—
74	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
75	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
77	C₃₁×#127	—	2	3	no	—
78	C₃₁×#127	—	2	3	no	—
79	C₃₁×#127	C₂³	1	3	yes	—
80	C₃₁×#127	C₂³	1	6	yes	—
81	C₃₁×#127	C₂³	1	3	yes	—
82	C₃₁×#124	—	2	3	no	—
83	C₃₁×#124	—	2	3	no	—
84	C₃₁×#123	—	2	6	no	—
85	C₃₁×C₂²⋊Q₈	—	2	3	no	—
86	C₃₁×C₂²⋊Q₈	—	2	3	no	—
87	C₃₁×#121	—	2	3	no	—
88	C₃₁×#121	—	2	3	no	—
89	C₃₁×#121	—	2	3	no	—
90	C₃₁×#121	—	2	3	no	—
91	C₃₁×#121	—	2	6	no	—
92	C₃₁×#121	—	2	6	no	—
93	C₃₁×#121	C₂³	1	6	yes	—
94	C₃₁×#121	C₂³	1	6	yes	—
95	C₃₁×#120	—	2	6	no	—
96	C₃₁×#120	—	2	12	no	—
97	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
98	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×D₄×C₃₁	—	2	3	no	—
100	C₄×D₄×C₃₁	—	2	3	no	—
101	C₄×D₄×C₃₁	—	2	3	no	—
102	C₄×D₄×C₃₁	—	2	3	no	—
103	C₄×D₄×C₃₁	D₄	1	3	yes	—
104	C₄×D₄×C₃₁	C₂³	1	3	yes	—
105	C₄×D₄×C₃₁	C₂³	1	3	yes	—
106	C₄×D₄×C₃₁	C₂³	1	6	yes	—
107	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
108	C₄×D₄×C₃₁	C₂³	1	3	yes	—
109	C₄×D₄×C₃₁	C₂³	1	3	yes	—
110	C₃₁×#117	D₄	1	3	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
113	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
114	C₃₁×#107	C₂³	1	2	yes	—
115	C₃₁×#107	C₂³	1	6	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
117	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
118	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₃₁×#102	C₂³	1	6	yes	—
120	C₃₁×#102	C₂³	1	12	yes	—
121	C₃₁×#102	C₂³	1	2	yes	—
122	C₃₁×#102	C₂³	1	6	yes	—
123	C₃₁×#102	C₂³	1	6	yes	—
124	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
125	C₄×C₈×C₃₁	C₂³	1	6	yes	—
126	C₂⁴×C₃₁	—	2	3	no	—
127	C₂⁴×C₃₁	—	2	6	no	—
128	#41×C₃₁	—	4	4	no	—
129	#41×C₃₁	—	4	8	no	—
130	#41×C₃₁	—	4	12	no	—
131	#41×C₃₁	—	4	12	no	—
132	#41×C₃₁	—	4	12	no	—
133	#41×C₃₁	D₈	1	8	yes	complemented
134	C₂×Q₈×C₃₁	—	2	3	no	—
135	C₂×Q₈×C₃₁	—	2	1	no	—
136	C₂×Q₈×C₃₁	—	4	4	no	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×D₄×C₃₁	—	2	6	no	—
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	6	no	—
141	C₂×D₄×C₃₁	—	2	3	no	—
142	C₂×D₄×C₃₁	—	2	3	no	—
143	C₂×D₄×C₃₁	—	2	3	no	—
144	C₂×D₄×C₃₁	—	2	6	no	—
145	C₂×D₄×C₃₁	—	2	3	no	—
146	C₂×D₄×C₃₁	—	2	3	no	—
147	C₂×D₄×C₃₁	—	2	6	no	—
148	C₂×D₄×C₃₁	—	2	6	no	—
149	C₂×D₄×C₃₁	—	4	3	no	—
150	C₂×D₄×C₃₁	—	4	6	no	—
151	C₂×D₄×C₃₁	—	4	6	no	—
152	C₂×D₄×C₃₁	—	4	12	no	—
153	C₂×D₄×C₃₁	—	4	24	no	—
154	C₂×D₄×C₃₁	—	4	12	no	—
155	C₂×D₄×C₃₁	—	8	6	no	—
156	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
157	C₂×D₄×C₃₁	C₂⁴	1	6	yes	—
158	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
159	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	—	2	3	no	—
161	C₂²×C₄×C₃₁	—	2	1	no	—
162	C₂²×C₄×C₃₁	—	2	3	no	—
163	C₂²×C₄×C₃₁	—	2	6	no	—
164	C₂²×C₄×C₃₁	—	2	3	no	—
165	C₂²×C₄×C₃₁	—	2	3	no	—
166	C₂²×C₄×C₃₁	—	4	12	no	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
168	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
170	D₈×C₃₁	—	2	12	no	—
171	D₈×C₃₁	—	4	12	no	—
172	D₈×C₃₁	—	4	6	no	—
173	C₂×C₈×C₃₁	—	4	6	no	—
174	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
175	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
176	C₄⋊C₄×C₃₁	—	2	3	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	3	no	—
179	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
180	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
181	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
183	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂²⋊C₄×C₃₁	—	2	1	no	—
185	C₂²⋊C₄×C₃₁	—	2	3	no	—
186	C₂²⋊C₄×C₃₁	—	2	3	no	—
187	C₂²⋊C₄×C₃₁	—	2	3	no	—
188	C₂²⋊C₄×C₃₁	—	2	6	no	—
189	C₂²⋊C₄×C₃₁	—	2	6	no	—
190	C₂²⋊C₄×C₃₁	—	2	6	no	—
191	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
192	C₄²×C₃₁	C₂×D₄	1	3	yes	—
193	C₄²×C₃₁	C₂×D₄	1	3	yes	—
194	C₄²×C₃₁	C₂⁴	1	3	yes	—
195	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
196	#61947	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	1	no	—
198	C₂³×C₃₁	—	2	3	no	—
199	C₂³×C₃₁	—	2	6	no	—
200	C₂³×C₃₁	—	4	6	no	—
201	C₂³×C₃₁	—	4	6	no	—
202	C₂³×C₃₁	—	4	12	no	—
203	C₂³×C₃₁	—	8	3	no	—
204	C₂³×C₃₁	—	8	6	no	—
205	C₂³×C₃₁	—	8	6	no	—
206	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
207	Q₈×C₃₁	—	4	6	no	—
208	Q₈×C₃₁	—	4	2	no	—
209	Q₈×C₃₁	C₂×D₈	1	4	yes	complemented
210	D₄×C₃₁	—	2	6	no	—
211	D₄×C₃₁	—	2	3	no	—
212	D₄×C₃₁	—	2	1	no	—
213	D₄×C₃₁	—	2	3	no	—
214	D₄×C₃₁	—	2	6	no	—
215	D₄×C₃₁	—	4	6	no	—
216	D₄×C₃₁	—	4	6	no	—
217	D₄×C₃₁	—	4	12	no	—
218	D₄×C₃₁	—	4	12	no	—
219	D₄×C₃₁	—	4	6	no	—
220	D₄×C₃₁	—	4	12	no	—
221	D₄×C₃₁	—	4	6	no	—
222	D₄×C₃₁	—	8	3	no	—
223	D₄×C₃₁	—	8	6	no	—
224	D₄×C₃₁	—	8	6	no	—
225	D₄×C₃₁	C₂×D₈	1	12	yes	complemented
226	D₄×C₃₁	H(5,𝔽₂)	1	2	yes	complemented
227	C₂×C₄×C₃₁	—	2	3	no	—
228	C₂×C₄×C₃₁	—	2	3	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	3	no	—
232	C₂×C₄×C₃₁	—	4	3	no	—
233	C₂×C₄×C₃₁	—	4	6	no	—
234	C₂×C₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	4	6	no	—
236	C₂×C₄×C₃₁	—	4	6	no	—
237	C₂×C₄×C₃₁	—	4	6	no	—
238	C₂×C₄×C₃₁	—	4	12	no	—
239	C₂×C₄×C₃₁	C₂×D₈	1	12	yes	complemented
240	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
245	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
246	C₈×C₃₁	—	2	6	no	—
247	C₈×C₃₁	—	2	2	no	—
248	#3498	C₂×C₃₁	1	1	yes	characteristic, complemented
249	D₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#3391	C₂×C₃₁	1	2	yes	complemented
251	#3358	C₂×C₃₁	1	2	yes	complemented
252	#3294	C₂×C₃₁	1	6	yes	complemented
253	D₄×D₈	C₂×C₃₁	1	6	yes	complemented
254	#3245	C₂×C₃₁	1	3	yes	complemented
255	#3205	C₂×C₃₁	1	3	yes	complemented
256	#3100	C₂×C₃₁	1	3	yes	complemented
257	C₂×#459	C₂×C₃₁	1	3	yes	complemented
258	#2988	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂²×C₃₁	—	2	3	no	—
260	C₂²×C₃₁	—	2	3	no	—
261	C₂²×C₃₁	—	2	2	no	—
262	C₂²×C₃₁	—	4	1	no	—
263	C₂²×C₃₁	—	4	1	no	—
264	C₂²×C₃₁	—	4	6	no	—
265	C₂²×C₃₁	—	4	3	no	—
266	C₂²×C₃₁	—	4	3	no	—
267	C₂²×C₃₁	—	4	6	no	—
268	C₂²×C₃₁	—	4	3	no	—
269	C₂²×C₃₁	—	4	6	no	—
270	C₂²×C₃₁	—	8	12	no	—
271	C₂²×C₃₁	—	16	12	no	—
272	C₂²×C₃₁	C₂²×D₈	1	6	yes	—
273	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	2	3	no	—
276	C₄×C₃₁	—	2	3	no	—
277	C₄×C₃₁	—	4	1	no	—
278	C₄×C₃₁	—	4	2	no	—
279	C₄×C₃₁	—	4	3	no	—
280	C₄×C₃₁	C₂²×D₈	1	2	yes	—
281	C₄×C₃₁	C₂²×D₈	1	6	yes	—
282	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
283	C₄×C₃₁	#585	1	1	yes	characteristic
284	C₂×H(5,𝔽₂)	—	2	1	no	—
285	C₂²×#41	—	2	1	no	—
286	C₂²×D₈	C₂²×C₃₁	1	6	yes	complemented
287	#550	—	2	1	no	—
288	#550	C₂²×C₃₁	1	2	yes	complemented
289	#550	C₂²×C₃₁	1	1	yes	characteristic, complemented
290	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
291	#548	—	2	1	no	—
292	#547	C₂²×C₃₁	1	3	yes	complemented
293	#546	—	2	3	no	—
294	D₄²	—	2	3	no	—
295	D₄²	C₂²×C₃₁	1	3	yes	complemented
296	D₄²	C₂²×C₃₁	1	6	yes	complemented
297	#535	—	2	3	no	—
298	#532	C₂²×C₃₁	1	3	yes	complemented
299	#532	C₂²×C₃₁	1	3	yes	complemented
300	C₂×#127	C₂²×C₃₁	1	3	yes	complemented
301	#525	—	2	3	no	—
302	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	complemented
304	#493	C₂²×C₃₁	1	6	yes	complemented
305	#486	C₂²×C₃₁	1	6	yes	complemented
306	#480	C₂²×C₃₁	1	6	yes	complemented
307	#474	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	#474	C₂²×C₃₁	1	3	yes	complemented
309	#474	C₂²×C₃₁	1	6	yes	complemented
310	#466	C₂²×C₃₁	1	12	yes	complemented
311	#459	C₂²×C₃₁	1	3	yes	complemented
312	#459	C₂²×C₃₁	1	3	yes	complemented
313	#459	C₂²×C₃₁	1	6	yes	complemented
314	#459	C₂²×C₃₁	1	6	yes	complemented
315	#459	C₂²×C₃₁	1	6	yes	complemented
316	#459	C₂²×C₃₁	1	6	yes	complemented
317	#447	C₂²×C₃₁	1	6	yes	complemented
318	#447	C₂²×C₃₁	1	12	yes	complemented
319	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
320	C₄×D₈	C₂²×C₃₁	1	6	yes	complemented
321	C₄×D₈	C₂²×C₃₁	1	6	yes	complemented
322	C₈×D₄	C₂²×C₃₁	1	6	yes	complemented
323	#431	C₂²×C₃₁	1	3	yes	complemented
324	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
325	C₂×#102	C₂²×C₃₁	1	6	yes	complemented
326	C₂×C₃₁	—	2	6	no	—
327	C₂×C₃₁	—	4	2	no	—
328	C₂×C₃₁	—	8	3	no	—
329	C₂×C₃₁	—	8	6	no	—
330	C₂×C₃₁	#3488	1	1	yes	characteristic
331	C₂×C₃₁	C₂³×D₈	1	1	yes	characteristic
332	C₂×C₃₁	#3585	1	1	yes	characteristic
333	H(5,𝔽₂)	—	4	8	no	—
334	C₂×#41	—	2	1	no	—
335	C₂×#41	—	2	2	no	—
336	C₂×#41	—	2	3	no	—
337	C₂×#41	—	2	3	no	—
338	C₂×#41	—	2	3	no	—
339	C₂×#41	—	4	8	no	—
340	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
341	C₂²×Q₈	—	2	1	no	—
342	C₂²×D₄	—	2	6	no	—
343	C₂²×D₄	—	2	3	no	—
344	C₂²×D₄	—	2	3	no	—
345	C₂²×D₄	—	2	3	no	—
346	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
347	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
348	C₂³×C₄	—	2	3	no	—
349	C₂×D₈	—	4	12	no	—
350	C₂×D₈	C₂³×C₃₁	1	6	yes	—
351	C₂×D₈	C₂³×C₃₁	1	12	yes	—
352	C₂×D₈	C₂³×C₃₁	1	6	yes	—
353	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
354	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
355	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
356	#127	—	2	3	no	—
357	#127	—	2	3	no	—
358	#127	C₂³×C₃₁	1	3	yes	—
359	#127	C₂³×C₃₁	1	6	yes	—
360	#127	C₂³×C₃₁	1	3	yes	—
361	#124	—	2	3	no	—
362	#124	—	2	3	no	—
363	#123	—	2	6	no	—
364	C₂²⋊Q₈	—	2	3	no	—
365	C₂²⋊Q₈	—	2	3	no	—
366	#121	—	2	3	no	—
367	#121	—	2	3	no	—
368	#121	—	2	3	no	—
369	#121	—	2	3	no	—
370	#121	—	2	6	no	—
371	#121	—	2	6	no	—
372	#121	C₂³×C₃₁	1	6	yes	—
373	#121	C₂³×C₃₁	1	6	yes	—
374	#120	—	2	6	no	—
375	#120	—	2	12	no	—
376	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
377	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₄×D₄	—	2	3	no	—
379	C₄×D₄	—	2	3	no	—
380	C₄×D₄	—	2	3	no	—
381	C₄×D₄	—	2	3	no	—
382	C₄×D₄	D₄×C₃₁	1	3	yes	—
383	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
384	C₄×D₄	C₂³×C₃₁	1	3	yes	—
385	C₄×D₄	C₂³×C₃₁	1	3	yes	—
386	C₄×D₄	C₂³×C₃₁	1	3	yes	—
387	C₄×D₄	C₂³×C₃₁	1	6	yes	—
388	C₄×D₄	C₂³×C₃₁	1	3	yes	—
389	#117	D₄×C₃₁	1	3	yes	—
390	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
391	C₂×C₂²⋊C₄	—	2	3	no	—
392	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
393	#107	C₂³×C₃₁	1	2	yes	—
394	#107	C₂³×C₃₁	1	6	yes	—
395	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
396	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
397	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
398	#102	C₂³×C₃₁	1	6	yes	—
399	#102	C₂³×C₃₁	1	12	yes	—
400	#102	C₂³×C₃₁	1	2	yes	—
401	#102	C₂³×C₃₁	1	6	yes	—
402	#102	C₂³×C₃₁	1	6	yes	—
403	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
404	C₄×C₈	C₂³×C₃₁	1	6	yes	—
405	C₃₁	#61947	1	1	yes	characteristic, direct product factor, Sylow
406	C₂⁴	—	2	3	no	—
407	C₂⁴	—	2	6	no	—
408	#41	—	4	4	no	—
409	#41	—	4	8	no	—
410	#41	—	4	12	no	—
411	#41	—	4	12	no	—
412	#41	—	4	12	no	—
413	#41	D₈×C₃₁	1	8	yes	complemented
414	C₂×Q₈	—	2	3	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	4	4	no	—
417	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
418	C₂×D₄	—	2	6	no	—
419	C₂×D₄	—	2	6	no	—
420	C₂×D₄	—	2	6	no	—
421	C₂×D₄	—	2	3	no	—
422	C₂×D₄	—	2	3	no	—
423	C₂×D₄	—	2	3	no	—
424	C₂×D₄	—	2	6	no	—
425	C₂×D₄	—	2	3	no	—
426	C₂×D₄	—	2	3	no	—
427	C₂×D₄	—	2	6	no	—
428	C₂×D₄	—	2	6	no	—
429	C₂×D₄	—	4	3	no	—
430	C₂×D₄	—	4	6	no	—
431	C₂×D₄	—	4	6	no	—
432	C₂×D₄	—	4	12	no	—
433	C₂×D₄	—	4	24	no	—
434	C₂×D₄	—	4	12	no	—
435	C₂×D₄	—	8	6	no	—
436	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
437	C₂×D₄	C₂⁴×C₃₁	1	6	yes	—
438	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
439	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	—	2	3	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	3	no	—
443	C₂²×C₄	—	2	6	no	—
444	C₂²×C₄	—	2	3	no	—
445	C₂²×C₄	—	2	3	no	—
446	C₂²×C₄	—	4	12	no	—
447	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
448	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
450	D₈	—	2	12	no	—
451	D₈	—	4	12	no	—
452	D₈	—	4	6	no	—
453	C₂×C₈	—	4	6	no	—
454	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
455	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
456	C₄⋊C₄	—	2	3	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	3	no	—
459	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
460	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
461	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
462	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
463	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
464	C₂²⋊C₄	—	2	1	no	—
465	C₂²⋊C₄	—	2	3	no	—
466	C₂²⋊C₄	—	2	3	no	—
467	C₂²⋊C₄	—	2	3	no	—
468	C₂²⋊C₄	—	2	6	no	—
469	C₂²⋊C₄	—	2	6	no	—
470	C₂²⋊C₄	—	2	6	no	—
471	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
472	C₄²	C₂×D₄×C₃₁	1	3	yes	—
473	C₄²	C₂×D₄×C₃₁	1	3	yes	—
474	C₄²	C₂⁴×C₃₁	1	3	yes	—
475	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
476	C₂³	—	2	1	no	—
477	C₂³	—	2	3	no	—
478	C₂³	—	2	6	no	—
479	C₂³	—	4	6	no	—
480	C₂³	—	4	6	no	—
481	C₂³	—	4	12	no	—
482	C₂³	—	8	3	no	—
483	C₂³	—	8	6	no	—
484	C₂³	—	8	6	no	—
485	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
486	Q₈	—	4	6	no	—
487	Q₈	—	4	2	no	—
488	Q₈	C₂×D₈×C₃₁	1	4	yes	complemented
489	D₄	—	2	6	no	—
490	D₄	—	2	3	no	—
491	D₄	—	2	1	no	—
492	D₄	—	2	3	no	—
493	D₄	—	2	6	no	—
494	D₄	—	4	6	no	—
495	D₄	—	4	6	no	—
496	D₄	—	4	12	no	—
497	D₄	—	4	12	no	—
498	D₄	—	4	6	no	—
499	D₄	—	4	12	no	—
500	D₄	—	4	6	no	—
501	D₄	—	8	3	no	—
502	D₄	—	8	6	no	—
503	D₄	—	8	6	no	—
504	D₄	C₂×D₈×C₃₁	1	12	yes	complemented
505	D₄	C₃₁×H(5,𝔽₂)	1	2	yes	complemented
506	C₂×C₄	—	2	3	no	—
507	C₂×C₄	—	2	3	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	3	no	—
511	C₂×C₄	—	4	3	no	—
512	C₂×C₄	—	4	6	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	6	no	—
515	C₂×C₄	—	4	6	no	—
516	C₂×C₄	—	4	6	no	—
517	C₂×C₄	—	4	12	no	—
518	C₂×C₄	C₂×D₈×C₃₁	1	12	yes	complemented
519	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
521	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
522	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
524	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
525	C₈	—	2	6	no	—
526	C₈	—	2	2	no	—
527	C₂²	—	2	3	no	—
528	C₂²	—	2	3	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	4	1	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	3	no	—
533	C₂²	—	4	6	no	—
534	C₂²	—	4	3	no	—
535	C₂²	—	4	6	no	—
536	C₂²	—	4	6	no	—
537	C₂²	—	4	3	no	—
538	C₂²	—	8	12	no	—
539	C₂²	—	16	12	no	—
540	C₂²	C₂²×D₈×C₃₁	1	6	yes	—
541	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
542	C₄	—	2	3	no	—
543	C₄	—	2	1	no	—
544	C₄	—	2	3	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	3	no	—
548	C₄	C₂²×D₈×C₃₁	1	2	yes	—
549	C₄	C₂²×D₈×C₃₁	1	6	yes	—
550	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
551	C₄	C₃₁×#585	1	1	yes	characteristic
552	C₂	—	2	6	no	—
553	C₂	—	4	2	no	—
554	C₂	—	8	3	no	—
555	C₂	—	8	6	no	—
556	C₂	C₃₁×#3488	1	1	yes	characteristic
557	C₂	C₂³×D₈×C₃₁	1	1	yes	characteristic
558	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
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×#3498	C₂	semidirect
2. D₄×#41×C₃₁	C₂	semidirect
3. C₃₁×#3391	C₂	semidirect
4. C₃₁×#3358	C₂	semidirect
5. C₃₁×#3294	C₂	semidirect
6. D₄×D₈×C₃₁	C₂	semidirect
7. C₃₁×#3245	C₂	semidirect
8. C₃₁×#3205	C₂	semidirect
9. C₃₁×#3100	C₂	semidirect
10. C₂×C₃₁×#459	C₂	semidirect
11. C₃₁×#2988	C₂	semidirect
14. C₂²×D₈×C₃₁	C₂²	semidirect
16. C₃₁×#550	C₂²	semidirect
17. C₃₁×#550	C₂²	semidirect
18. D₄×Q₈×C₃₁	C₂²	semidirect
20. C₃₁×#547	C₂²	semidirect
23. D₄²×C₃₁	C₂²	semidirect
24. D₄²×C₃₁	C₂²	semidirect
26. C₃₁×#532	C₂²	semidirect
27. C₃₁×#532	C₂²	semidirect
28. C₂×C₃₁×#127	C₂²	semidirect
31. C₂×C₄×D₄×C₃₁	C₂²	semidirect
32. C₃₁×#493	C₂²	semidirect
33. C₃₁×#486	C₂²	semidirect
34. C₃₁×#480	C₂²	semidirect
35. C₃₁×#474	C₂²	semidirect
36. C₃₁×#474	C₂²	semidirect
37. C₃₁×#474	C₂²	semidirect
38. C₃₁×#466	C₂²	semidirect
39. C₃₁×#459	C₂²	semidirect
40. C₃₁×#459	C₂²	semidirect
41. C₃₁×#459	C₂²	semidirect
42. C₃₁×#459	C₂²	semidirect
43. C₃₁×#459	C₂²	semidirect
44. C₃₁×#459	C₂²	semidirect
45. C₃₁×#447	C₂²	semidirect
46. C₃₁×#447	C₂²	semidirect
47. C₈×Q₈×C₃₁	C₂²	semidirect
48. C₄×D₈×C₃₁	C₂²	semidirect
49. C₄×D₈×C₃₁	C₂²	semidirect
50. C₈×D₄×C₃₁	C₂²	semidirect
51. C₃₁×#431	C₂²	semidirect
52. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
53. C₂×C₃₁×#102	C₂²	semidirect
133. #41×C₃₁	D₈	semidirect
196. #61947	C₃₁	direct (N × H)
209. Q₈×C₃₁	C₂×D₈	semidirect
225. D₄×C₃₁	C₂×D₈	semidirect
226. D₄×C₃₁	H(5,𝔽₂)	semidirect
239. C₂×C₄×C₃₁	C₂×D₈	semidirect
248. #3498	C₂×C₃₁	semidirect
249. D₄×#41	C₂×C₃₁	semidirect
250. #3391	C₂×C₃₁	semidirect
251. #3358	C₂×C₃₁	semidirect
252. #3294	C₂×C₃₁	semidirect
253. D₄×D₈	C₂×C₃₁	semidirect
254. #3245	C₂×C₃₁	semidirect
255. #3205	C₂×C₃₁	semidirect
256. #3100	C₂×C₃₁	semidirect
257. C₂×#459	C₂×C₃₁	semidirect
258. #2988	C₂×C₃₁	semidirect
286. C₂²×D₈	C₂²×C₃₁	semidirect
288. #550	C₂²×C₃₁	semidirect
289. #550	C₂²×C₃₁	semidirect
290. D₄×Q₈	C₂²×C₃₁	semidirect
292. #547	C₂²×C₃₁	semidirect
295. D₄²	C₂²×C₃₁	semidirect
296. D₄²	C₂²×C₃₁	semidirect
298. #532	C₂²×C₃₁	semidirect
299. #532	C₂²×C₃₁	semidirect
300. C₂×#127	C₂²×C₃₁	semidirect
303. C₂×C₄×D₄	C₂²×C₃₁	semidirect
304. #493	C₂²×C₃₁	semidirect
305. #486	C₂²×C₃₁	semidirect
306. #480	C₂²×C₃₁	semidirect
307. #474	C₂²×C₃₁	semidirect
308. #474	C₂²×C₃₁	semidirect
309. #474	C₂²×C₃₁	semidirect
310. #466	C₂²×C₃₁	semidirect
311. #459	C₂²×C₃₁	semidirect
312. #459	C₂²×C₃₁	semidirect
313. #459	C₂²×C₃₁	semidirect
314. #459	C₂²×C₃₁	semidirect
315. #459	C₂²×C₃₁	semidirect
316. #459	C₂²×C₃₁	semidirect
317. #447	C₂²×C₃₁	semidirect
318. #447	C₂²×C₃₁	semidirect
319. C₈×Q₈	C₂²×C₃₁	semidirect
320. C₄×D₈	C₂²×C₃₁	semidirect
321. C₄×D₈	C₂²×C₃₁	semidirect
322. C₈×D₄	C₂²×C₃₁	semidirect
323. #431	C₂²×C₃₁	semidirect
324. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
325. C₂×#102	C₂²×C₃₁	semidirect
405. C₃₁	#61947	direct (N × H)
413. #41	D₈×C₃₁	semidirect
488. Q₈	C₂×D₈×C₃₁	semidirect
504. D₄	C₂×D₈×C₃₁	semidirect
505. D₄	C₃₁×H(5,𝔽₂)	semidirect
518. C₂×C₄	C₂×D₈×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3498	2
D₄×#41×C₃₁	2
C₃₁×#3391	2
C₃₁×#3358	2
C₃₁×#3294	2
D₄×D₈×C₃₁	2
C₃₁×#3245	2
C₃₁×#3205	2
C₃₁×#3100	2
C₂×C₃₁×#459	2
C₃₁×#2988	2
#61947	31

Subgroups — C31×#61947 #21518706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61947 #21518706