Subgroups — C₃₁×#36707 #21493466

Order 7936.

2782 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#537	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2847	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2692	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2676	C₂	1	8	yes	maximal, complemented
6	C₃₁×#2618	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2539	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2436	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2406	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2404	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2313	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#537	—	2	8	no	—
16	C₂×C₃₁×#124	C₂²	1	4	yes	complemented
17	C₂×C₃₁×#123	C₂²	1	4	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	—	2	8	no	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	8	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
21	C₂×C₃₁×#121	C₂²	1	2	yes	—
22	C₂×C₃₁×#121	C₂²	1	2	yes	—
23	C₂×C₃₁×#121	C₂²	1	4	yes	—
24	C₂×C₃₁×#121	C₂²	1	4	yes	complemented
25	C₂×C₃₁×#120	C₂²	1	4	yes	—
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#399	C₂²	1	4	yes	—
32	C₃₁×#398	C₂²	1	4	yes	complemented
33	C₃₁×#396	C₂²	1	8	yes	complemented
34	C₃₁×#395	C₂²	1	4	yes	complemented
35	C₃₁×#394	C₂²	1	4	yes	complemented
36	C₃₁×#394	C₂²	1	4	yes	complemented
37	C₃₁×#393	C₂²	1	4	yes	—
38	C₃₁×#393	C₂²	1	8	yes	complemented
39	C₃₁×#392	C₂²	1	4	yes	—
40	C₃₁×#392	C₂²	1	4	yes	—
41	C₃₁×#391	C₂²	1	4	yes	complemented
42	C₃₁×#390	C₂²	1	8	yes	—
43	C₃₁×#387	C₂²	1	8	yes	complemented
44	C₃₁×#386	C₂²	1	4	yes	—
45	C₃₁×#386	C₂²	1	8	yes	—
46	C₃₁×#386	C₂²	1	4	yes	—
47	C₃₁×#386	C₂²	1	8	yes	—
48	C₃₁×#385	C₂²	1	8	yes	complemented
49	C₃₁×#381	C₂²	1	2	yes	—
50	C₃₁×#380	C₂²	1	2	yes	—
51	C₃₁×#380	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
53	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×#41×C₃₁	—	2	8	no	—
55	C₂×#41×C₃₁	—	4	8	no	—
56	C₂²×Q₈×C₃₁	—	2	4	no	—
57	C₂²×Q₈×C₃₁	—	4	4	no	—
58	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
59	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
61	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
62	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
63	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
64	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
65	C₂³×C₄×C₃₁	—	2	4	no	—
66	C₂³×C₄×C₃₁	—	4	4	no	—
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#124	—	4	8	no	—
76	C₃₁×#123	—	4	8	no	—
77	C₃₁×C₂²⋊Q₈	—	2	32	no	—
78	C₃₁×C₂²⋊Q₈	—	2	16	no	—
79	C₃₁×#121	—	2	8	no	—
80	C₃₁×#121	—	2	8	no	—
81	C₃₁×#121	—	2	16	no	—
82	C₃₁×#121	—	4	8	no	—
83	C₃₁×#120	—	4	8	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	complemented
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
95	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
105	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
106	C₃₁×#95	C₂³	1	8	yes	—
107	C₃₁×#95	C₂³	1	4	yes	—
108	C₃₁×#95	C₂³	1	8	yes	—
109	C₃₁×#95	C₂³	1	4	yes	—
110	C₃₁×#95	C₂³	1	4	yes	—
111	C₃₁×#95	C₂³	1	4	yes	—
112	C₂⁴×C₃₁	—	2	2	no	—
113	C₂⁴×C₃₁	—	2	4	no	—
114	C₂⁴×C₃₁	—	2	4	no	—
115	C₂⁴×C₃₁	—	4	2	no	—
116	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
117	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
118	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
119	#41×C₃₁	—	4	32	no	—
120	C₂×Q₈×C₃₁	—	2	8	no	—
121	C₂×Q₈×C₃₁	—	2	4	no	—
122	C₂×Q₈×C₃₁	—	2	8	no	—
123	C₂×Q₈×C₃₁	—	4	4	no	—
124	C₂×Q₈×C₃₁	—	4	4	no	—
125	C₂×Q₈×C₃₁	—	4	16	no	—
126	C₂×D₄×C₃₁	—	2	8	no	—
127	C₂×D₄×C₃₁	—	2	8	no	—
128	C₂×D₄×C₃₁	—	2	8	no	—
129	C₂×D₄×C₃₁	—	2	16	no	—
130	C₂×D₄×C₃₁	—	2	4	no	—
131	C₂×D₄×C₃₁	—	2	4	no	—
132	C₂×D₄×C₃₁	—	2	4	no	—
133	C₂×D₄×C₃₁	—	4	4	no	—
134	C₂×D₄×C₃₁	—	4	8	no	—
135	C₂×D₄×C₃₁	—	4	4	no	—
136	C₂×D₄×C₃₁	—	4	8	no	—
137	C₂×D₄×C₃₁	—	4	4	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	—	2	4	no	—
144	C₂²×C₄×C₃₁	—	2	8	no	—
145	C₂²×C₄×C₃₁	—	2	8	no	—
146	C₂²×C₄×C₃₁	—	2	8	no	—
147	C₂²×C₄×C₃₁	—	4	2	no	—
148	C₂²×C₄×C₃₁	—	4	4	no	—
149	C₂²×C₄×C₃₁	—	4	2	no	—
150	C₂²×C₄×C₃₁	—	4	2	no	—
151	C₂²×C₄×C₃₁	—	4	4	no	—
152	C₂²×C₄×C₃₁	—	4	4	no	—
153	C₂²×C₄×C₃₁	—	4	16	no	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
165	C₄⋊C₄×C₃₁	—	2	8	no	—
166	C₄⋊C₄×C₃₁	—	2	8	no	—
167	C₄⋊C₄×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	4	4	no	—
170	C₄⋊C₄×C₃₁	—	4	8	no	—
171	C₄⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
172	C₂²⋊C₄×C₃₁	—	2	8	no	—
173	C₂²⋊C₄×C₃₁	—	2	8	no	—
174	C₂²⋊C₄×C₃₁	—	2	8	no	—
175	C₂²⋊C₄×C₃₁	—	2	8	no	—
176	C₂²⋊C₄×C₃₁	—	2	8	no	—
177	C₂²⋊C₄×C₃₁	—	2	16	no	—
178	C₂²⋊C₄×C₃₁	—	4	8	no	—
179	C₂²⋊C₄×C₃₁	—	4	8	no	—
180	C₂²⋊C₄×C₃₁	—	4	8	no	—
181	C₄²×C₃₁	—	4	4	no	—
182	#36707	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	2	no	—
184	C₂³×C₃₁	—	2	2	no	—
185	C₂³×C₃₁	—	2	8	no	—
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	—	2	4	no	—
188	C₂³×C₃₁	—	2	4	no	—
189	C₂³×C₃₁	—	2	2	no	—
190	C₂³×C₃₁	—	2	4	no	—
191	C₂³×C₃₁	—	2	8	no	—
192	C₂³×C₃₁	—	2	4	no	—
193	C₂³×C₃₁	—	2	4	no	—
194	C₂³×C₃₁	—	2	8	no	—
195	C₂³×C₃₁	—	4	16	no	—
196	C₂³×C₃₁	—	4	4	no	—
197	C₂³×C₃₁	—	4	4	no	—
198	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
199	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
200	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
201	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
202	Q₈×C₃₁	—	4	16	no	—
203	Q₈×C₃₁	—	4	8	no	—
204	D₄×C₃₁	—	4	8	no	—
205	D₄×C₃₁	—	4	16	no	—
206	D₄×C₃₁	—	8	4	no	—
207	D₄×C₃₁	—	8	4	no	—
208	D₄×C₃₁	—	8	16	no	—
209	C₂×C₄×C₃₁	—	2	8	no	—
210	C₂×C₄×C₃₁	—	2	8	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	8	no	—
220	C₂×C₄×C₃₁	—	2	8	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	8	no	—
226	C₂×C₄×C₃₁	—	4	4	no	—
227	C₂×C₄×C₃₁	—	4	4	no	—
228	C₂×C₄×C₃₁	—	4	16	no	—
229	C₂×C₄×C₃₁	—	4	16	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	—	4	4	no	—
232	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
235	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
236	C₂×#537	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#2847	C₂×C₃₁	1	2	yes	complemented
239	#2692	C₂×C₃₁	1	8	yes	complemented
240	#2676	C₂×C₃₁	1	8	yes	complemented
241	#2618	C₂×C₃₁	1	2	yes	complemented
242	#2539	C₂×C₃₁	1	2	yes	complemented
243	#2436	C₂×C₃₁	1	2	yes	complemented
244	#2406	C₂×C₃₁	1	2	yes	complemented
245	#2404	C₂×C₃₁	1	2	yes	complemented
246	#2313	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂²×C₃₁	—	2	2	no	—
248	C₂²×C₃₁	—	2	2	no	—
249	C₂²×C₃₁	—	2	8	no	—
250	C₂²×C₃₁	—	2	4	no	—
251	C₂²×C₃₁	—	2	8	no	—
252	C₂²×C₃₁	—	4	16	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	4	4	no	—
256	C₂²×C₃₁	—	4	4	no	—
257	C₂²×C₃₁	—	4	4	no	—
258	C₂²×C₃₁	—	4	4	no	—
259	C₂²×C₃₁	—	4	8	no	—
260	C₂²×C₃₁	—	4	8	no	—
261	C₂²×C₃₁	#535	1	4	yes	—
262	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
263	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
264	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
265	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
266	C₄×C₃₁	—	2	4	no	—
267	C₄×C₃₁	—	2	4	no	—
268	C₄×C₃₁	—	2	4	no	—
269	C₄×C₃₁	—	2	4	no	—
270	C₄×C₃₁	—	4	2	no	—
271	C₄×C₃₁	—	4	4	no	—
272	C₄×C₃₁	—	4	2	no	—
273	C₄×C₃₁	—	4	8	no	—
274	C₂²×#41	C₂²×C₃₁	1	2	yes	—
275	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
276	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
277	#537	—	2	8	no	—
278	C₂×#124	C₂²×C₃₁	1	4	yes	complemented
279	C₂×#123	C₂²×C₃₁	1	4	yes	—
280	C₂×C₂²⋊Q₈	—	2	8	no	—
281	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	8	yes	—
282	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
283	C₂×#121	C₂²×C₃₁	1	2	yes	—
284	C₂×#121	C₂²×C₃₁	1	2	yes	—
285	C₂×#121	C₂²×C₃₁	1	4	yes	—
286	C₂×#121	C₂²×C₃₁	1	4	yes	complemented
287	C₂×#120	C₂²×C₃₁	1	4	yes	—
288	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
289	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
290	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
291	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
292	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
293	#399	C₂²×C₃₁	1	4	yes	—
294	#398	C₂²×C₃₁	1	4	yes	complemented
295	#396	C₂²×C₃₁	1	8	yes	complemented
296	#395	C₂²×C₃₁	1	4	yes	complemented
297	#394	C₂²×C₃₁	1	4	yes	complemented
298	#394	C₂²×C₃₁	1	4	yes	complemented
299	#393	C₂²×C₃₁	1	4	yes	—
300	#393	C₂²×C₃₁	1	8	yes	complemented
301	#392	C₂²×C₃₁	1	4	yes	—
302	#392	C₂²×C₃₁	1	4	yes	—
303	#391	C₂²×C₃₁	1	4	yes	complemented
304	#390	C₂²×C₃₁	1	8	yes	—
305	#387	C₂²×C₃₁	1	8	yes	complemented
306	#386	C₂²×C₃₁	1	4	yes	—
307	#386	C₂²×C₃₁	1	8	yes	—
308	#386	C₂²×C₃₁	1	4	yes	—
309	#386	C₂²×C₃₁	1	8	yes	—
310	#385	C₂²×C₃₁	1	8	yes	complemented
311	#381	C₂²×C₃₁	1	2	yes	—
312	#380	C₂²×C₃₁	1	2	yes	—
313	#380	C₂²×C₃₁	1	4	yes	—
314	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
315	C₂×C₃₁	—	2	4	no	—
316	C₂×C₃₁	—	4	4	no	—
317	C₂×C₃₁	—	8	4	no	—
318	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
319	C₂×C₃₁	#3484	1	1	yes	characteristic
320	C₂×C₃₁	#3486	1	1	yes	characteristic
321	C₂×C₃₁	#3536	1	4	yes	—
322	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
323	C₂×#41	—	2	8	no	—
324	C₂×#41	—	4	8	no	—
325	C₂²×Q₈	—	2	4	no	—
326	C₂²×Q₈	—	4	4	no	—
327	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
328	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
329	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
330	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
331	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
332	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
333	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
334	C₂³×C₄	—	2	4	no	—
335	C₂³×C₄	—	4	4	no	—
336	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
337	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
339	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
342	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
344	#124	—	4	8	no	—
345	#123	—	4	8	no	—
346	C₂²⋊Q₈	—	2	32	no	—
347	C₂²⋊Q₈	—	2	16	no	—
348	#121	—	2	8	no	—
349	#121	—	2	8	no	—
350	#121	—	2	16	no	—
351	#121	—	4	8	no	—
352	#120	—	4	8	no	—
353	C₂×C₄⋊C₄	—	2	4	no	—
354	C₂×C₄⋊C₄	—	2	4	no	—
355	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	complemented
356	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
357	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
358	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
359	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
361	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
363	C₂×C₂²⋊C₄	—	2	4	no	—
364	C₂×C₂²⋊C₄	—	2	4	no	—
365	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
369	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
370	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
371	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
373	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
374	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
375	#95	C₂³×C₃₁	1	4	yes	—
376	#95	C₂³×C₃₁	1	8	yes	—
377	#95	C₂³×C₃₁	1	4	yes	—
378	#95	C₂³×C₃₁	1	4	yes	—
379	#95	C₂³×C₃₁	1	4	yes	—
380	#95	C₂³×C₃₁	1	8	yes	—
381	C₃₁	#36707	1	1	yes	characteristic, direct product factor, Sylow
382	C₂⁴	—	2	2	no	—
383	C₂⁴	—	2	4	no	—
384	C₂⁴	—	2	4	no	—
385	C₂⁴	—	4	2	no	—
386	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
387	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
388	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
389	#41	—	4	32	no	—
390	C₂×Q₈	—	2	8	no	—
391	C₂×Q₈	—	2	4	no	—
392	C₂×Q₈	—	2	8	no	—
393	C₂×Q₈	—	4	4	no	—
394	C₂×Q₈	—	4	4	no	—
395	C₂×Q₈	—	4	16	no	—
396	C₂×D₄	—	2	8	no	—
397	C₂×D₄	—	2	8	no	—
398	C₂×D₄	—	2	8	no	—
399	C₂×D₄	—	2	16	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	4	4	no	—
404	C₂×D₄	—	4	8	no	—
405	C₂×D₄	—	4	4	no	—
406	C₂×D₄	—	4	8	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	8	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	4	2	no	—
418	C₂²×C₄	—	4	4	no	—
419	C₂²×C₄	—	4	2	no	—
420	C₂²×C₄	—	4	2	no	—
421	C₂²×C₄	—	4	4	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	—	4	16	no	—
424	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
425	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
426	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
428	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
429	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
430	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
432	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
433	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	2	8	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	4	no	—
439	C₄⋊C₄	—	4	4	no	—
440	C₄⋊C₄	—	4	8	no	—
441	C₄⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	—	2	8	no	—
445	C₂²⋊C₄	—	2	8	no	—
446	C₂²⋊C₄	—	2	8	no	—
447	C₂²⋊C₄	—	2	16	no	—
448	C₂²⋊C₄	—	4	8	no	—
449	C₂²⋊C₄	—	4	8	no	—
450	C₂²⋊C₄	—	4	8	no	—
451	C₄²	—	4	4	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	2	8	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	8	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	4	no	—
463	C₂³	—	2	8	no	—
464	C₂³	—	4	16	no	—
465	C₂³	—	4	4	no	—
466	C₂³	—	4	4	no	—
467	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
468	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
469	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
470	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
471	Q₈	—	4	16	no	—
472	Q₈	—	4	8	no	—
473	D₄	—	4	8	no	—
474	D₄	—	4	16	no	—
475	D₄	—	8	4	no	—
476	D₄	—	8	4	no	—
477	D₄	—	8	16	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	2	8	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	8	no	—
489	C₂×C₄	—	2	8	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	4	4	no	—
496	C₂×C₄	—	4	4	no	—
497	C₂×C₄	—	4	16	no	—
498	C₂×C₄	—	4	16	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
502	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
503	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
504	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	2	8	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	2	8	no	—
510	C₂²	—	4	16	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	4	8	no	—
518	C₂²	—	4	8	no	—
519	C₂²	C₃₁×#535	1	4	yes	—
520	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
521	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
522	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
523	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
524	C₄	—	2	4	no	—
525	C₄	—	2	4	no	—
526	C₄	—	2	4	no	—
527	C₄	—	2	4	no	—
528	C₄	—	4	4	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	8	no	—
532	C₂	—	2	4	no	—
533	C₂	—	4	4	no	—
534	C₂	—	8	4	no	—
535	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
536	C₂	C₃₁×#3484	1	1	yes	characteristic
537	C₂	C₃₁×#3486	1	1	yes	characteristic
538	C₂	C₃₁×#3536	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#537	C₂	semidirect
2. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#2847	C₂	semidirect
4. C₃₁×#2692	C₂	semidirect
5. C₃₁×#2676	C₂	semidirect
6. C₃₁×#2618	C₂	semidirect
7. C₃₁×#2539	C₂	semidirect
8. C₃₁×#2436	C₂	semidirect
9. C₃₁×#2406	C₂	semidirect
10. C₃₁×#2404	C₂	semidirect
11. C₃₁×#2313	C₂	semidirect
16. C₂×C₃₁×#124	C₂²	semidirect
20. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
24. C₂×C₃₁×#121	C₂²	semidirect
32. C₃₁×#398	C₂²	semidirect
33. C₃₁×#396	C₂²	semidirect
34. C₃₁×#395	C₂²	semidirect
35. C₃₁×#394	C₂²	semidirect
36. C₃₁×#394	C₂²	semidirect
38. C₃₁×#393	C₂²	semidirect
41. C₃₁×#391	C₂²	semidirect
43. C₃₁×#387	C₂²	semidirect
48. C₃₁×#385	C₂²	semidirect
86. C₂×C₄⋊C₄×C₃₁	D₄	semidirect
182. #36707	C₃₁	direct (N × H)
236. C₂×#537	C₂×C₃₁	semidirect
237. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
238. #2847	C₂×C₃₁	semidirect
239. #2692	C₂×C₃₁	semidirect
240. #2676	C₂×C₃₁	semidirect
241. #2618	C₂×C₃₁	semidirect
242. #2539	C₂×C₃₁	semidirect
243. #2436	C₂×C₃₁	semidirect
244. #2406	C₂×C₃₁	semidirect
245. #2404	C₂×C₃₁	semidirect
246. #2313	C₂×C₃₁	semidirect
278. C₂×#124	C₂²×C₃₁	semidirect
282. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
286. C₂×#121	C₂²×C₃₁	semidirect
294. #398	C₂²×C₃₁	semidirect
295. #396	C₂²×C₃₁	semidirect
296. #395	C₂²×C₃₁	semidirect
297. #394	C₂²×C₃₁	semidirect
298. #394	C₂²×C₃₁	semidirect
300. #393	C₂²×C₃₁	semidirect
303. #391	C₂²×C₃₁	semidirect
305. #387	C₂²×C₃₁	semidirect
310. #385	C₂²×C₃₁	semidirect
355. C₂×C₄⋊C₄	D₄×C₃₁	semidirect
381. C₃₁	#36707	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#537	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₃₁×#2847	2
C₃₁×#2692	2
C₃₁×#2676	2
C₃₁×#2618	2
C₃₁×#2539	2
C₃₁×#2436	2
C₃₁×#2406	2
C₃₁×#2404	2
C₃₁×#2313	2
#36707	31

Subgroups — C31×#36707 #21493466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36707 #21493466