Subgroups — C₃₁×#34123 #21490882

Order 7936.

2598 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2356	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2353	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2349	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2346	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2343	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2339	C₂	1	2	yes	maximal
8	C₃₁×#2295	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2290	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2283	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄²×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
13	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×#376	C₂	1	2	yes	maximal, complemented
15	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×#41×C₃₁	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₃₁×#117	C₂²	1	2	yes	—
25	C₂×C₃₁×#117	C₂²	1	2	yes	—
26	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
28	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#391	C₂²	1	4	yes	—
31	C₃₁×#389	C₂²	1	4	yes	—
32	C₃₁×#388	C₂²	1	8	yes	—
33	C₃₁×#387	C₂²	1	8	yes	—
34	C₃₁×#387	C₂²	1	8	yes	—
35	C₃₁×#386	C₂²	1	8	yes	—
36	C₃₁×#385	C₂²	1	8	yes	—
37	C₃₁×#383	C₂²	1	2	yes	—
38	C₃₁×#383	C₂²	1	4	yes	—
39	C₃₁×#382	C₂²	1	4	yes	—
40	C₃₁×#382	C₂²	1	2	yes	—
41	C₃₁×#381	C₂²	1	4	yes	—
42	C₃₁×#381	C₂²	1	2	yes	—
43	C₃₁×#380	C₂²	1	2	yes	—
44	C₃₁×#380	C₂²	1	4	yes	—
45	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
46	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
49	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
51	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
52	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
53	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
54	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
55	C₃₁×#376	C₂²	1	4	yes	—
56	C₃₁×#376	C₂²	1	4	yes	—
57	C₃₁×#376	C₂²	1	1	yes	characteristic
58	C₃₁×#376	C₂²	1	2	yes	—
59	C₃₁×#376	C₂²	1	4	yes	—
60	C₃₁×#376	C₂²	1	1	yes	characteristic
61	C₂×C₃₁×#95	C₂²	1	4	yes	—
62	C₂×C₃₁×#95	C₂²	1	4	yes	—
63	C₄³×C₃₁	C₂²	1	2	yes	—
64	C₂×#41×C₃₁	—	2	8	no	—
65	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
77	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
78	C₄×D₄×C₃₁	—	2	16	no	—
79	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
81	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
83	C₃₁×#117	—	2	8	no	—
84	C₃₁×#117	C₂×C₄	1	16	yes	complemented
85	C₃₁×#117	C₂×C₄	1	8	yes	complemented
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
99	C₂×C₄²×C₃₁	—	2	16	no	—
100	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
101	C₂×C₄²×C₃₁	C₂×C₄	1	16	yes	complemented
102	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
103	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
104	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
105	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
106	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
115	C₃₁×#95	C₂³	1	2	yes	—
116	C₃₁×#95	C₂³	1	2	yes	—
117	C₃₁×#95	C₂³	1	4	yes	—
118	C₃₁×#95	C₂³	1	4	yes	—
119	C₃₁×#95	C₂³	1	8	yes	—
120	C₃₁×#95	C₂³	1	8	yes	—
121	C₃₁×#95	C₂³	1	2	yes	—
122	C₃₁×#95	C₂³	1	2	yes	—
123	C₃₁×#95	C₂³	1	4	yes	—
124	C₃₁×#95	C₂³	1	8	yes	—
125	C₃₁×#95	C₂³	1	4	yes	—
126	C₃₁×#95	C₂³	1	8	yes	—
127	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
128	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
129	#41×C₃₁	—	2	32	no	—
130	C₂×Q₈×C₃₁	—	2	4	no	—
131	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
132	C₂×D₄×C₃₁	—	2	8	no	—
133	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
134	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
135	C₂×D₄×C₃₁	#41	1	8	yes	—
136	C₂²×C₄×C₃₁	—	2	8	no	—
137	C₂²×C₄×C₃₁	—	2	8	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	16	no	—
141	C₂²×C₄×C₃₁	—	2	16	no	—
142	C₂²×C₄×C₃₁	—	4	2	no	—
143	C₂²×C₄×C₃₁	—	4	8	no	—
144	C₂²×C₄×C₃₁	—	4	4	no	—
145	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
146	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
148	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
150	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
163	C₄⋊C₄×C₃₁	—	2	8	no	—
164	C₄⋊C₄×C₃₁	—	2	8	no	—
165	C₄⋊C₄×C₃₁	—	4	8	no	—
166	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
167	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
168	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
169	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
170	C₂²⋊C₄×C₃₁	—	2	16	no	—
171	C₂²⋊C₄×C₃₁	—	2	8	no	—
172	C₂²⋊C₄×C₃₁	—	4	4	no	—
173	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
174	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
175	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
176	C₄²×C₃₁	—	2	16	no	—
177	C₄²×C₃₁	—	2	8	no	—
178	C₄²×C₃₁	—	2	4	no	—
179	C₄²×C₃₁	—	2	8	no	—
180	C₄²×C₃₁	—	4	4	no	—
181	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
182	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
183	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
184	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
185	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
186	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
187	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
188	C₄²×C₃₁	#41	1	8	yes	complemented
189	#34123	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	4	no	—
191	C₂³×C₃₁	—	2	4	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
196	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
197	C₂³×C₃₁	C₂×#41	1	8	yes	—
198	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
199	Q₈×C₃₁	—	2	8	no	—
200	D₄×C₃₁	—	2	16	no	—
201	D₄×C₃₁	#117	1	16	yes	complemented
202	C₂×C₄×C₃₁	—	2	16	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	8	no	—
213	C₂×C₄×C₃₁	—	2	16	no	—
214	C₂×C₄×C₃₁	—	2	8	no	—
215	C₂×C₄×C₃₁	—	2	8	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	8	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	8	no	—
221	C₂×C₄×C₃₁	—	4	8	no	—
222	C₂×C₄×C₃₁	—	4	4	no	—
223	C₂×C₄×C₃₁	—	4	32	no	—
224	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
225	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
226	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
228	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
229	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
230	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
231	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
232	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
233	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
234	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
235	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
236	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
237	#2356	C₂×C₃₁	1	2	yes	complemented
238	#2353	C₂×C₃₁	1	4	yes	complemented
239	#2349	C₂×C₃₁	1	2	yes	complemented
240	#2346	C₂×C₃₁	1	2	yes	complemented
241	#2343	C₂×C₃₁	1	4	yes	complemented
242	#2339	C₂×C₃₁	1	2	yes	—
243	#2295	C₂×C₃₁	1	2	yes	complemented
244	#2290	C₂×C₃₁	1	4	yes	complemented
245	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#2283	C₂×C₃₁	1	2	yes	complemented
247	C₄²×D₄	C₂×C₃₁	1	2	yes	complemented
248	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂×#376	C₂×C₃₁	1	2	yes	complemented
250	C₂²×C₃₁	—	2	4	no	—
251	C₂²×C₃₁	—	2	4	no	—
252	C₂²×C₃₁	—	2	16	no	—
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	4	4	no	—
257	C₂²×C₃₁	C₂×#117	1	8	yes	—
258	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
259	C₂²×C₃₁	C₂²×#41	1	2	yes	—
260	C₂²×C₃₁	C₂²×#41	1	2	yes	—
261	C₂²×C₃₁	#585	1	2	yes	—
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	2	4	no	—
264	C₄×C₃₁	—	2	4	no	—
265	C₄×C₃₁	—	2	2	no	—
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	2	2	no	—
268	C₄×C₃₁	—	2	4	no	—
269	C₄×C₃₁	—	2	8	no	—
270	C₄×C₃₁	—	4	4	no	—
271	C₄×C₃₁	—	4	8	no	—
272	C₄×C₃₁	C₂×#117	1	4	yes	—
273	C₄×C₃₁	C₄×#41	1	4	yes	—
274	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
275	C₄×#41	C₄×C₃₁	1	16	yes	complemented
276	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
279	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
282	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
283	C₂×#117	C₂²×C₃₁	1	2	yes	—
284	C₂×#117	C₂²×C₃₁	1	2	yes	—
285	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
286	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
287	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
288	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
289	#391	C₂²×C₃₁	1	4	yes	—
290	#389	C₂²×C₃₁	1	4	yes	—
291	#388	C₂²×C₃₁	1	8	yes	—
292	#387	C₂²×C₃₁	1	8	yes	—
293	#387	C₂²×C₃₁	1	8	yes	—
294	#386	C₂²×C₃₁	1	8	yes	—
295	#385	C₂²×C₃₁	1	8	yes	—
296	#383	C₂²×C₃₁	1	2	yes	—
297	#383	C₂²×C₃₁	1	4	yes	—
298	#382	C₂²×C₃₁	1	4	yes	—
299	#382	C₂²×C₃₁	1	2	yes	—
300	#381	C₂²×C₃₁	1	4	yes	—
301	#381	C₂²×C₃₁	1	2	yes	—
302	#380	C₂²×C₃₁	1	2	yes	—
303	#380	C₂²×C₃₁	1	4	yes	—
304	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
305	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
306	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
307	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
308	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
309	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
310	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
311	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
312	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
313	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
314	#376	C₂²×C₃₁	1	4	yes	—
315	#376	C₂²×C₃₁	1	4	yes	—
316	#376	C₂²×C₃₁	1	1	yes	characteristic
317	#376	C₂²×C₃₁	1	2	yes	—
318	#376	C₂²×C₃₁	1	4	yes	—
319	#376	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×#95	C₂²×C₃₁	1	4	yes	—
321	C₂×#95	C₂²×C₃₁	1	4	yes	—
322	C₄³	C₂²×C₃₁	1	2	yes	—
323	C₂×C₃₁	—	2	8	no	—
324	C₂×C₃₁	—	4	2	no	—
325	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
326	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
327	C₂×C₃₁	#3428	1	1	yes	characteristic
328	C₂×C₃₁	#3475	1	4	yes	—
329	C₂×#41	—	2	8	no	—
330	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
331	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
332	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
333	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
334	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
335	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
337	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
339	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
340	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
342	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
343	C₄×D₄	—	2	16	no	—
344	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
345	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
346	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
347	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
348	#117	—	2	8	no	—
349	#117	C₂×C₄×C₃₁	1	16	yes	complemented
350	#117	C₂×C₄×C₃₁	1	8	yes	complemented
351	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
352	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
354	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
362	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
364	C₂×C₄²	—	2	16	no	—
365	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
366	C₂×C₄²	C₂×C₄×C₃₁	1	16	yes	complemented
367	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
368	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
369	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
370	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
371	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
372	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
374	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
376	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
377	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
380	#95	C₂³×C₃₁	1	2	yes	—
381	#95	C₂³×C₃₁	1	2	yes	—
382	#95	C₂³×C₃₁	1	4	yes	—
383	#95	C₂³×C₃₁	1	4	yes	—
384	#95	C₂³×C₃₁	1	8	yes	—
385	#95	C₂³×C₃₁	1	8	yes	—
386	#95	C₂³×C₃₁	1	2	yes	—
387	#95	C₂³×C₃₁	1	2	yes	—
388	#95	C₂³×C₃₁	1	4	yes	—
389	#95	C₂³×C₃₁	1	8	yes	—
390	#95	C₂³×C₃₁	1	4	yes	—
391	#95	C₂³×C₃₁	1	8	yes	—
392	C₃₁	#34123	1	1	yes	characteristic, direct product factor, Sylow
393	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
394	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
395	#41	—	2	32	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
398	C₂×D₄	—	2	8	no	—
399	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
400	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
401	C₂×D₄	#41×C₃₁	1	8	yes	—
402	C₂²×C₄	—	2	8	no	—
403	C₂²×C₄	—	2	8	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	16	no	—
407	C₂²×C₄	—	2	16	no	—
408	C₂²×C₄	—	4	2	no	—
409	C₂²×C₄	—	4	8	no	—
410	C₂²×C₄	—	4	4	no	—
411	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
412	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
413	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
414	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
415	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
416	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
417	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
418	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
419	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
422	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
425	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
426	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
428	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
429	C₄⋊C₄	—	2	8	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	4	8	no	—
432	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
433	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
434	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
435	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
436	C₂²⋊C₄	—	2	16	no	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	—	4	4	no	—
439	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
440	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
441	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
442	C₄²	—	2	16	no	—
443	C₄²	—	2	8	no	—
444	C₄²	—	2	4	no	—
445	C₄²	—	2	8	no	—
446	C₄²	—	4	4	no	—
447	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
448	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
449	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
450	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
451	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
452	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
453	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
454	C₄²	#41×C₃₁	1	8	yes	complemented
455	C₂³	—	2	4	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
461	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
462	C₂³	C₂×#41×C₃₁	1	8	yes	—
463	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
464	Q₈	—	2	8	no	—
465	D₄	—	2	16	no	—
466	D₄	C₃₁×#117	1	16	yes	complemented
467	C₂×C₄	—	2	16	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	8	no	—
478	C₂×C₄	—	2	16	no	—
479	C₂×C₄	—	2	8	no	—
480	C₂×C₄	—	2	8	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	8	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	8	no	—
486	C₂×C₄	—	4	8	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	32	no	—
489	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
490	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
491	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
492	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
493	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
494	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
495	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
496	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
497	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
498	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
499	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
500	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	16	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	4	4	no	—
508	C₂²	C₂×C₃₁×#117	1	8	yes	—
509	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
510	C₂²	C₂²×#41×C₃₁	1	2	yes	—
511	C₂²	C₂²×#41×C₃₁	1	2	yes	—
512	C₂²	C₃₁×#585	1	2	yes	—
513	C₄	—	2	2	no	—
514	C₄	—	2	4	no	—
515	C₄	—	2	4	no	—
516	C₄	—	2	2	no	—
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	2	4	no	—
520	C₄	—	2	8	no	—
521	C₄	—	4	8	no	—
522	C₄	—	4	4	no	—
523	C₄	C₂×C₃₁×#117	1	4	yes	—
524	C₄	C₄×#41×C₃₁	1	4	yes	—
525	C₂	—	2	8	no	—
526	C₂	—	4	2	no	—
527	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
528	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
529	C₂	C₃₁×#3428	1	1	yes	characteristic
530	C₂	C₃₁×#3475	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2356	C₂	semidirect
3. C₃₁×#2353	C₂	semidirect
4. C₃₁×#2349	C₂	semidirect
5. C₃₁×#2346	C₂	semidirect
6. C₃₁×#2343	C₂	semidirect
8. C₃₁×#2295	C₂	semidirect
9. C₃₁×#2290	C₂	semidirect
10. C₃₁×#2283	C₂	semidirect
11. C₃₁×#2283	C₂	semidirect
12. C₄²×D₄×C₃₁	C₂	semidirect
13. C₄×C₃₁×#117	C₂	semidirect
14. C₂×C₃₁×#376	C₂	semidirect
16. C₄×#41×C₃₁	C₄	semidirect
76. C₄×Q₈×C₃₁	C₂×C₄	semidirect
77. C₄×Q₈×C₃₁	C₂×C₄	semidirect
81. C₄×D₄×C₃₁	C₂×C₄	semidirect
82. C₄×D₄×C₃₁	C₂×C₄	semidirect
84. C₃₁×#117	C₂×C₄	semidirect
85. C₃₁×#117	C₂×C₄	semidirect
100. C₂×C₄²×C₃₁	C₂×C₄	semidirect
101. C₂×C₄²×C₃₁	C₂×C₄	semidirect
188. C₄²×C₃₁	#41	semidirect
189. #34123	C₃₁	direct (N × H)
201. D₄×C₃₁	#117	semidirect
237. #2356	C₂×C₃₁	semidirect
238. #2353	C₂×C₃₁	semidirect
239. #2349	C₂×C₃₁	semidirect
240. #2346	C₂×C₃₁	semidirect
241. #2343	C₂×C₃₁	semidirect
243. #2295	C₂×C₃₁	semidirect
244. #2290	C₂×C₃₁	semidirect
245. #2283	C₂×C₃₁	semidirect
246. #2283	C₂×C₃₁	semidirect
247. C₄²×D₄	C₂×C₃₁	semidirect
248. C₄×#117	C₂×C₃₁	semidirect
249. C₂×#376	C₂×C₃₁	semidirect
275. C₄×#41	C₄×C₃₁	semidirect
341. C₄×Q₈	C₂×C₄×C₃₁	semidirect
342. C₄×Q₈	C₂×C₄×C₃₁	semidirect
346. C₄×D₄	C₂×C₄×C₃₁	semidirect
347. C₄×D₄	C₂×C₄×C₃₁	semidirect
349. #117	C₂×C₄×C₃₁	semidirect
350. #117	C₂×C₄×C₃₁	semidirect
365. C₂×C₄²	C₂×C₄×C₃₁	semidirect
366. C₂×C₄²	C₂×C₄×C₃₁	semidirect
392. C₃₁	#34123	direct (N × H)
454. C₄²	#41×C₃₁	semidirect
466. D₄	C₃₁×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₃₁	2
C₃₁×#2356	2
C₃₁×#2353	2
C₃₁×#2349	2
C₃₁×#2346	2
C₃₁×#2343	2
C₃₁×#2339	2
C₃₁×#2295	2
C₃₁×#2290	2
C₃₁×#2283	2
C₄²×D₄×C₃₁	2
C₄×C₃₁×#117	2
C₂×C₃₁×#376	2
#34123	31

Subgroups — C31×#34123 #21490882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34123 #21490882