Subgroups — C₃₁×#34247 #21491006

Order 7936.

2230 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#126	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2373	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2373	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2345	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2344	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2342	C₂	1	2	yes	maximal
7	C₃₁×#2342	C₂	1	2	yes	maximal
8	C₃₁×#2334	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2313	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2312	C₂	1	1	yes	maximal, characteristic
11	C₄×C₃₁×#126	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2274	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2273	C₂	1	1	yes	maximal, characteristic
15	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#126	C₄	1	16	yes	complemented
17	C₂×C₃₁×#126	C₂²	1	2	yes	—
18	C₂×C₃₁×#126	C₂²	1	2	yes	—
19	C₂×C₃₁×#126	C₂²	1	2	yes	—
20	C₂×C₃₁×#126	C₂²	1	2	yes	—
21	C₂×C₃₁×#117	C₂²	1	2	yes	—
22	C₂×C₃₁×#117	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#389	C₂²	1	4	yes	complemented
32	C₃₁×#389	C₂²	1	4	yes	complemented
33	C₃₁×#388	C₂²	1	4	yes	—
34	C₃₁×#388	C₂²	1	4	yes	—
35	C₃₁×#388	C₂²	1	4	yes	—
36	C₃₁×#388	C₂²	1	4	yes	—
37	C₃₁×#388	C₂²	1	4	yes	—
38	C₃₁×#388	C₂²	1	4	yes	—
39	C₃₁×#387	C₂²	1	4	yes	complemented
40	C₃₁×#387	C₂²	1	4	yes	complemented
41	C₃₁×#387	C₂²	1	4	yes	complemented
42	C₃₁×#387	C₂²	1	4	yes	complemented
43	C₃₁×#387	C₂²	1	4	yes	complemented
44	C₃₁×#387	C₂²	1	4	yes	complemented
45	C₃₁×#385	C₂²	1	4	yes	—
46	C₃₁×#385	C₂²	1	4	yes	—
47	C₃₁×#383	C₂²	1	2	yes	complemented
48	C₃₁×#382	C₂²	1	2	yes	complemented
49	C₃₁×#381	C₂²	1	2	yes	—
50	C₃₁×#381	C₂²	1	2	yes	—
51	C₃₁×#380	C₂²	1	2	yes	—
52	C₃₁×#380	C₂²	1	2	yes	—
53	C₃₁×#380	C₂²	1	2	yes	—
54	C₃₁×#379	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
59	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
60	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
62	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
66	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
68	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
69	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
71	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
72	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
73	C₃₁×#376	C₂²	1	2	yes	complemented
74	C₃₁×#376	C₂²	1	2	yes	complemented
75	C₄³×C₃₁	C₂²	1	2	yes	complemented
76	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
80	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#126	C₂×C₄	1	16	yes	complemented
85	C₃₁×#126	C₂×C₄	1	16	yes	complemented
86	C₃₁×#126	C₂×C₄	1	16	yes	complemented
87	C₃₁×#126	C₂×C₄	1	16	yes	complemented
88	C₃₁×#117	—	2	8	no	—
89	C₃₁×#117	—	2	8	no	—
90	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
91	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
92	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
105	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
122	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
125	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
126	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
127	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
128	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
129	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
130	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
131	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
135	C₃₁×#95	C₂³	1	2	yes	—
136	C₃₁×#95	C₂³	1	4	yes	—
137	C₃₁×#95	C₂³	1	4	yes	—
138	C₃₁×#95	C₂³	1	2	yes	—
139	C₃₁×#95	C₂³	1	4	yes	—
140	C₃₁×#95	C₂³	1	2	yes	—
141	C₃₁×#95	C₂³	1	4	yes	—
142	C₃₁×#95	C₂³	1	4	yes	—
143	C₃₁×#95	C₂³	1	2	yes	—
144	C₃₁×#95	C₂³	1	4	yes	—
145	C₃₁×#95	C₂³	1	4	yes	—
146	C₃₁×#95	C₂³	1	4	yes	—
147	C₃₁×#95	C₂³	1	4	yes	—
148	C₃₁×#95	C₂³	1	4	yes	—
149	C₂⁴×C₃₁	—	2	4	no	—
150	C₂⁴×C₃₁	—	4	2	no	—
151	C₂⁴×C₃₁	—	4	2	no	—
152	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
153	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
154	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	—	2	4	no	—
156	C₂²×C₄×C₃₁	—	2	4	no	—
157	C₂²×C₄×C₃₁	—	2	8	no	—
158	C₂²×C₄×C₃₁	—	4	2	no	—
159	C₂²×C₄×C₃₁	—	4	2	no	—
160	C₂²×C₄×C₃₁	—	4	2	no	—
161	C₂²×C₄×C₃₁	—	4	2	no	—
162	C₂²×C₄×C₃₁	—	4	4	no	—
163	C₂²×C₄×C₃₁	—	4	4	no	—
164	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
165	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
183	C₄⋊C₄×C₃₁	—	2	8	no	—
184	C₄⋊C₄×C₃₁	—	2	8	no	—
185	C₄⋊C₄×C₃₁	—	4	4	no	—
186	C₄⋊C₄×C₃₁	—	4	4	no	—
187	C₄⋊C₄×C₃₁	—	4	4	no	—
188	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
191	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
192	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
193	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
194	C₂²⋊C₄×C₃₁	—	2	4	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₃₁	—	2	8	no	—
198	C₂²⋊C₄×C₃₁	—	4	4	no	—
199	C₂²⋊C₄×C₃₁	—	4	4	no	—
200	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
201	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
202	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
203	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
204	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
205	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
206	C₂²⋊C₄×C₃₁	#41	1	8	yes	—
207	C₄²×C₃₁	—	2	8	no	—
208	C₄²×C₃₁	—	2	8	no	—
209	C₄²×C₃₁	—	4	2	no	—
210	C₄²×C₃₁	—	4	4	no	—
211	C₄²×C₃₁	—	4	4	no	—
212	C₄²×C₃₁	—	4	4	no	—
213	C₄²×C₃₁	—	4	2	no	—
214	C₄²×C₃₁	—	4	2	no	—
215	C₄²×C₃₁	—	4	2	no	—
216	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
217	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
218	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
219	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
220	#34247	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂³×C₃₁	—	2	2	no	—
222	C₂³×C₃₁	—	2	2	no	—
223	C₂³×C₃₁	—	2	2	no	—
224	C₂³×C₃₁	—	2	2	no	—
225	C₂³×C₃₁	—	2	4	no	—
226	C₂³×C₃₁	—	2	4	no	—
227	C₂³×C₃₁	—	2	8	no	—
228	C₂³×C₃₁	—	4	4	no	—
229	C₂³×C₃₁	—	4	4	no	—
230	C₂³×C₃₁	—	8	8	no	—
231	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
232	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
233	C₂³×C₃₁	C₂×#41	1	4	yes	—
234	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
235	C₂×C₄×C₃₁	—	2	8	no	—
236	C₂×C₄×C₃₁	—	2	8	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	2	4	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	2	4	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	2	4	no	—
258	C₂×C₄×C₃₁	—	4	4	no	—
259	C₂×C₄×C₃₁	—	4	4	no	—
260	C₂×C₄×C₃₁	—	4	4	no	—
261	C₂×C₄×C₃₁	—	4	4	no	—
262	C₂×C₄×C₃₁	—	8	8	no	—
263	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
264	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
269	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
270	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
271	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
272	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
273	C₂²×#126	C₂×C₃₁	1	1	yes	characteristic
274	#2373	C₂×C₃₁	1	4	yes	complemented
275	#2373	C₂×C₃₁	1	4	yes	complemented
276	#2345	C₂×C₃₁	1	2	yes	complemented
277	#2344	C₂×C₃₁	1	2	yes	complemented
278	#2342	C₂×C₃₁	1	2	yes	—
279	#2342	C₂×C₃₁	1	2	yes	—
280	#2334	C₂×C₃₁	1	4	yes	complemented
281	#2313	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#2312	C₂×C₃₁	1	1	yes	characteristic
283	C₄×#126	C₂×C₃₁	1	4	yes	complemented
284	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
285	#2274	C₂×C₃₁	1	1	yes	characteristic, complemented
286	#2273	C₂×C₃₁	1	1	yes	characteristic
287	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	2	2	no	—
290	C₂²×C₃₁	—	2	8	no	—
291	C₂²×C₃₁	—	2	4	no	—
292	C₂²×C₃₁	—	2	4	no	—
293	C₂²×C₃₁	—	2	4	no	—
294	C₂²×C₃₁	—	4	4	no	—
295	C₂²×C₃₁	—	4	4	no	—
296	C₂²×C₃₁	—	8	8	no	—
297	C₂²×C₃₁	C₄×#41	1	4	yes	—
298	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
299	C₂²×C₃₁	C₂²×#41	1	2	yes	—
300	C₂²×C₃₁	#585	1	2	yes	—
301	C₂²×C₃₁	#585	1	2	yes	—
302	C₄×C₃₁	—	2	2	no	—
303	C₄×C₃₁	—	2	4	no	—
304	C₄×C₃₁	—	2	2	no	—
305	C₄×C₃₁	—	2	2	no	—
306	C₄×C₃₁	—	2	2	no	—
307	C₄×C₃₁	—	2	2	no	—
308	C₄×C₃₁	—	2	4	no	—
309	C₄×C₃₁	—	2	4	no	—
310	C₄×C₃₁	—	2	2	no	—
311	C₄×C₃₁	—	2	8	no	—
312	C₄×C₃₁	—	4	4	no	—
313	C₄×C₃₁	—	4	4	no	—
314	C₄×C₃₁	—	4	4	no	—
315	C₂×#126	C₄×C₃₁	1	16	yes	complemented
316	C₂×#126	C₂²×C₃₁	1	2	yes	—
317	C₂×#126	C₂²×C₃₁	1	2	yes	—
318	C₂×#126	C₂²×C₃₁	1	2	yes	—
319	C₂×#126	C₂²×C₃₁	1	2	yes	—
320	C₂×#117	C₂²×C₃₁	1	2	yes	—
321	C₂×#117	C₂²×C₃₁	1	2	yes	—
322	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
323	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
324	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
325	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
326	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
327	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
328	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
329	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
330	#389	C₂²×C₃₁	1	4	yes	complemented
331	#389	C₂²×C₃₁	1	4	yes	complemented
332	#388	C₂²×C₃₁	1	4	yes	—
333	#388	C₂²×C₃₁	1	4	yes	—
334	#388	C₂²×C₃₁	1	4	yes	—
335	#388	C₂²×C₃₁	1	4	yes	—
336	#388	C₂²×C₃₁	1	4	yes	—
337	#388	C₂²×C₃₁	1	4	yes	—
338	#387	C₂²×C₃₁	1	4	yes	complemented
339	#387	C₂²×C₃₁	1	4	yes	complemented
340	#387	C₂²×C₃₁	1	4	yes	complemented
341	#387	C₂²×C₃₁	1	4	yes	complemented
342	#387	C₂²×C₃₁	1	4	yes	complemented
343	#387	C₂²×C₃₁	1	4	yes	complemented
344	#385	C₂²×C₃₁	1	4	yes	—
345	#385	C₂²×C₃₁	1	4	yes	—
346	#383	C₂²×C₃₁	1	2	yes	complemented
347	#382	C₂²×C₃₁	1	2	yes	complemented
348	#381	C₂²×C₃₁	1	2	yes	—
349	#381	C₂²×C₃₁	1	2	yes	—
350	#380	C₂²×C₃₁	1	2	yes	—
351	#380	C₂²×C₃₁	1	2	yes	—
352	#380	C₂²×C₃₁	1	2	yes	—
353	#379	C₂²×C₃₁	1	2	yes	—
354	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
355	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
356	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
357	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
358	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
359	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
360	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
361	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
362	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
363	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
364	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
365	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
366	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
367	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
368	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
369	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
370	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
371	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
372	#376	C₂²×C₃₁	1	2	yes	complemented
373	#376	C₂²×C₃₁	1	2	yes	complemented
374	C₄³	C₂²×C₃₁	1	2	yes	complemented
375	C₂×C₃₁	—	2	4	no	—
376	C₂×C₃₁	—	4	4	no	—
377	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
378	C₂×C₃₁	#3428	1	1	yes	characteristic
379	C₂×C₃₁	#3428	1	1	yes	characteristic
380	C₂×C₃₁	#3521	1	4	yes	—
381	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
382	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
383	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
384	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
385	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
389	#126	C₂×C₄×C₃₁	1	16	yes	complemented
390	#126	C₂×C₄×C₃₁	1	16	yes	complemented
391	#126	C₂×C₄×C₃₁	1	16	yes	complemented
392	#126	C₂×C₄×C₃₁	1	16	yes	complemented
393	#117	—	2	8	no	—
394	#117	—	2	8	no	—
395	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
396	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
397	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
398	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
401	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
402	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
403	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
409	C₂×C₂²⋊C₄	—	2	8	no	—
410	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
411	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
412	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
426	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
427	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
428	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
429	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
430	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
431	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
432	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
433	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
434	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
435	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
436	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
440	#95	C₂³×C₃₁	1	2	yes	—
441	#95	C₂³×C₃₁	1	4	yes	—
442	#95	C₂³×C₃₁	1	4	yes	—
443	#95	C₂³×C₃₁	1	2	yes	—
444	#95	C₂³×C₃₁	1	4	yes	—
445	#95	C₂³×C₃₁	1	2	yes	—
446	#95	C₂³×C₃₁	1	4	yes	—
447	#95	C₂³×C₃₁	1	4	yes	—
448	#95	C₂³×C₃₁	1	2	yes	—
449	#95	C₂³×C₃₁	1	4	yes	—
450	#95	C₂³×C₃₁	1	4	yes	—
451	#95	C₂³×C₃₁	1	4	yes	—
452	#95	C₂³×C₃₁	1	4	yes	—
453	#95	C₂³×C₃₁	1	4	yes	—
454	C₃₁	#34247	1	1	yes	characteristic, direct product factor, Sylow
455	C₂⁴	—	2	4	no	—
456	C₂⁴	—	4	2	no	—
457	C₂⁴	—	4	2	no	—
458	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
459	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
460	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	4	2	no	—
466	C₂²×C₄	—	4	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	4	no	—
469	C₂²×C₄	—	4	4	no	—
470	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
471	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
472	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
473	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
474	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
475	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
476	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
478	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
487	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
488	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
489	C₄⋊C₄	—	2	8	no	—
490	C₄⋊C₄	—	2	8	no	—
491	C₄⋊C₄	—	4	4	no	—
492	C₄⋊C₄	—	4	4	no	—
493	C₄⋊C₄	—	4	4	no	—
494	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
495	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
496	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
497	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
498	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
499	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	4	4	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
507	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
508	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
509	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
510	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
511	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
512	C₂²⋊C₄	#41×C₃₁	1	8	yes	—
513	C₄²	—	2	8	no	—
514	C₄²	—	2	8	no	—
515	C₄²	—	4	2	no	—
516	C₄²	—	4	4	no	—
517	C₄²	—	4	4	no	—
518	C₄²	—	4	4	no	—
519	C₄²	—	4	2	no	—
520	C₄²	—	4	2	no	—
521	C₄²	—	4	2	no	—
522	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
523	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
524	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
525	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	4	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	2	8	no	—
533	C₂³	—	4	4	no	—
534	C₂³	—	4	4	no	—
535	C₂³	—	8	8	no	—
536	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
537	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
538	C₂³	C₂×#41×C₃₁	1	4	yes	—
539	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
540	C₂×C₄	—	2	8	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	—	4	4	no	—
565	C₂×C₄	—	4	4	no	—
566	C₂×C₄	—	4	4	no	—
567	C₂×C₄	—	8	8	no	—
568	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
569	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
570	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
571	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
572	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
573	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
574	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
575	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
576	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
577	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	2	8	no	—
581	C₂²	—	2	4	no	—
582	C₂²	—	2	4	no	—
583	C₂²	—	2	4	no	—
584	C₂²	—	4	4	no	—
585	C₂²	—	4	4	no	—
586	C₂²	—	8	8	no	—
587	C₂²	C₄×#41×C₃₁	1	4	yes	—
588	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
589	C₂²	C₂²×#41×C₃₁	1	2	yes	—
590	C₂²	C₃₁×#585	1	2	yes	—
591	C₂²	C₃₁×#585	1	2	yes	—
592	C₄	—	2	2	no	—
593	C₄	—	2	4	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	4	no	—
599	C₄	—	2	4	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	8	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	4	no	—
604	C₄	—	4	4	no	—
605	C₂	—	2	4	no	—
606	C₂	—	4	4	no	—
607	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
608	C₂	C₃₁×#3428	1	1	yes	characteristic
609	C₂	C₃₁×#3428	1	1	yes	characteristic
610	C₂	C₃₁×#3521	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2373	C₂	semidirect
3. C₃₁×#2373	C₂	semidirect
4. C₃₁×#2345	C₂	semidirect
5. C₃₁×#2344	C₂	semidirect
8. C₃₁×#2334	C₂	semidirect
9. C₃₁×#2313	C₂	semidirect
11. C₄×C₃₁×#126	C₂	semidirect
12. C₃₁×#2283	C₂	semidirect
13. C₃₁×#2274	C₂	semidirect
15. C₄×C₃₁×#117	C₂	semidirect
16. C₂×C₃₁×#126	C₄	semidirect
31. C₃₁×#389	C₂²	semidirect
32. C₃₁×#389	C₂²	semidirect
39. C₃₁×#387	C₂²	semidirect
40. C₃₁×#387	C₂²	semidirect
41. C₃₁×#387	C₂²	semidirect
42. C₃₁×#387	C₂²	semidirect
43. C₃₁×#387	C₂²	semidirect
44. C₃₁×#387	C₂²	semidirect
47. C₃₁×#383	C₂²	semidirect
48. C₃₁×#382	C₂²	semidirect
55. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
56. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
57. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
58. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
59. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
60. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
61. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
73. C₃₁×#376	C₂²	semidirect
74. C₃₁×#376	C₂²	semidirect
75. C₄³×C₃₁	C₂²	semidirect
84. C₃₁×#126	C₂×C₄	semidirect
85. C₃₁×#126	C₂×C₄	semidirect
86. C₃₁×#126	C₂×C₄	semidirect
87. C₃₁×#126	C₂×C₄	semidirect
90. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
91. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
92. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
121. C₂×C₄²×C₃₁	C₂×C₄	semidirect
220. #34247	C₃₁	direct (N × H)
274. #2373	C₂×C₃₁	semidirect
275. #2373	C₂×C₃₁	semidirect
276. #2345	C₂×C₃₁	semidirect
277. #2344	C₂×C₃₁	semidirect
280. #2334	C₂×C₃₁	semidirect
281. #2313	C₂×C₃₁	semidirect
283. C₄×#126	C₂×C₃₁	semidirect
284. #2283	C₂×C₃₁	semidirect
285. #2274	C₂×C₃₁	semidirect
287. C₄×#117	C₂×C₃₁	semidirect
315. C₂×#126	C₄×C₃₁	semidirect
330. #389	C₂²×C₃₁	semidirect
331. #389	C₂²×C₃₁	semidirect
338. #387	C₂²×C₃₁	semidirect
339. #387	C₂²×C₃₁	semidirect
340. #387	C₂²×C₃₁	semidirect
341. #387	C₂²×C₃₁	semidirect
342. #387	C₂²×C₃₁	semidirect
343. #387	C₂²×C₃₁	semidirect
346. #383	C₂²×C₃₁	semidirect
347. #382	C₂²×C₃₁	semidirect
354. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
355. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
356. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
357. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
358. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
359. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
360. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
372. #376	C₂²×C₃₁	semidirect
373. #376	C₂²×C₃₁	semidirect
374. C₄³	C₂²×C₃₁	semidirect
389. #126	C₂×C₄×C₃₁	semidirect
390. #126	C₂×C₄×C₃₁	semidirect
391. #126	C₂×C₄×C₃₁	semidirect
392. #126	C₂×C₄×C₃₁	semidirect
395. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
396. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
397. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
426. C₂×C₄²	C₂×C₄×C₃₁	semidirect
454. C₃₁	#34247	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#126	2
C₃₁×#2373	2
C₃₁×#2345	2
C₃₁×#2344	2
C₃₁×#2342	2
C₃₁×#2334	2
C₃₁×#2313	2
C₃₁×#2312	2
C₄×C₃₁×#126	2
C₃₁×#2283	2
C₃₁×#2274	2
C₃₁×#2273	2
C₄×C₃₁×#117	2
#34247	31

Subgroups — C31×#34247 #21491006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34247 #21491006