Subgroups — C₃₁×#34126 #21490885

Order 7936.

3142 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2357	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×#2345	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2342	C₂	1	4	yes	maximal, complemented
7	D₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal
8	C₃₁×#2293	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2290	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2280	C₂	1	2	yes	maximal, complemented
12	C₄²×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
13	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
15	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₃₁	C₄	1	16	yes	complemented
18	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#117	C₂²	1	2	yes	—
26	C₂×C₃₁×#117	C₂²	1	4	yes	complemented
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
29	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₃₁	C₂²	1	4	yes	complemented
32	C₃₁×#390	C₂²	1	4	yes	—
33	C₃₁×#389	C₂²	1	4	yes	complemented
34	C₃₁×#388	C₂²	1	8	yes	complemented
35	C₃₁×#388	C₂²	1	8	yes	complemented
36	C₃₁×#387	C₂²	1	8	yes	complemented
37	C₃₁×#386	C₂²	1	8	yes	—
38	C₃₁×#385	C₂²	1	8	yes	complemented
39	C₃₁×#383	C₂²	1	2	yes	complemented
40	C₃₁×#382	C₂²	1	2	yes	complemented
41	C₃₁×#381	C₂²	1	2	yes	—
42	C₃₁×#381	C₂²	1	4	yes	complemented
43	C₃₁×#380	C₂²	1	2	yes	—
44	C₃₁×#380	C₂²	1	4	yes	—
45	C₃₁×#380	C₂²	1	4	yes	complemented
46	C₃₁×#379	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
48	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
49	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
50	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
51	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
52	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
53	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
54	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
55	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
56	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
57	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
58	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
59	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
60	C₃₁×#376	C₂²	1	2	yes	complemented
61	C₃₁×#376	C₂²	1	2	yes	complemented
62	C₂×C₃₁×#95	C₂²	1	4	yes	complemented
63	C₄³×C₃₁	C₂²	1	2	yes	complemented
64	C₂⁵×C₃₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₃₁	—	2	8	no	—
66	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₃₁	—	4	8	no	—
71	C₂³×C₄×C₃₁	C₂×C₄	1	16	yes	complemented
72	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
78	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
80	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
81	C₄×D₄×C₃₁	—	2	16	no	—
82	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
83	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
84	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
85	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
86	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
87	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
88	C₃₁×#117	—	2	8	no	—
89	C₃₁×#117	—	4	8	no	—
90	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
98	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	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	2	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄²×C₃₁	—	4	8	no	—
110	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
111	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
112	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
113	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
115	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₃₁×#95	C₂³	1	2	yes	—
122	C₃₁×#95	C₂³	1	4	yes	—
123	C₃₁×#95	C₂³	1	8	yes	—
124	C₃₁×#95	C₂³	1	2	yes	—
125	C₃₁×#95	C₂³	1	4	yes	—
126	C₃₁×#95	C₂³	1	4	yes	—
127	C₃₁×#95	C₂³	1	4	yes	—
128	C₃₁×#95	C₂³	1	8	yes	—
129	C₃₁×#95	C₂³	1	4	yes	—
130	C₂⁴×C₃₁	—	2	8	no	—
131	C₂⁴×C₃₁	—	2	8	no	—
132	C₂⁴×C₃₁	—	4	4	no	—
133	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
134	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
135	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
136	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
137	C₂×D₄×C₃₁	—	2	8	no	—
138	C₂×D₄×C₃₁	—	2	32	no	—
139	C₂×D₄×C₃₁	—	2	4	no	—
140	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
141	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
142	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
143	C₂×D₄×C₃₁	#41	1	8	yes	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	2	8	no	—
147	C₂²×C₄×C₃₁	—	2	8	no	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	16	no	—
150	C₂²×C₄×C₃₁	—	2	16	no	—
151	C₂²×C₄×C₃₁	—	4	32	no	—
152	C₂²×C₄×C₃₁	—	4	8	no	—
153	C₂²×C₄×C₃₁	—	4	4	no	—
154	C₂²×C₄×C₃₁	—	4	4	no	—
155	C₂²×C₄×C₃₁	—	4	4	no	—
156	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
159	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
160	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
161	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
162	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
163	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂⁴	1	2	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	C₄⋊C₄×C₃₁	—	2	8	no	—
176	C₄⋊C₄×C₃₁	—	2	8	no	—
177	C₄⋊C₄×C₃₁	—	4	8	no	—
178	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
179	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
180	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
181	C₂²⋊C₄×C₃₁	—	2	8	no	—
182	C₂²⋊C₄×C₃₁	—	2	4	no	—
183	C₂²⋊C₄×C₃₁	—	2	16	no	—
184	C₂²⋊C₄×C₃₁	—	2	16	no	—
185	C₂²⋊C₄×C₃₁	—	4	8	no	—
186	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
187	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
188	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
189	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
190	C₂²⋊C₄×C₃₁	#41	1	8	yes	—
191	C₄²×C₃₁	—	2	8	no	—
192	C₄²×C₃₁	—	2	8	no	—
193	C₄²×C₃₁	—	2	4	no	—
194	C₄²×C₃₁	—	2	8	no	—
195	C₄²×C₃₁	—	2	4	no	—
196	C₄²×C₃₁	—	4	8	no	—
197	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
198	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
199	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
200	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
201	#34126	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	2	4	no	—
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	2	16	no	—
207	C₂³×C₃₁	—	2	16	no	—
208	C₂³×C₃₁	—	2	4	no	—
209	C₂³×C₃₁	—	2	4	no	—
210	C₂³×C₃₁	—	2	4	no	—
211	C₂³×C₃₁	—	4	8	no	—
212	C₂³×C₃₁	—	4	32	no	—
213	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
214	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
215	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
216	C₂³×C₃₁	C₂×#41	1	8	yes	—
217	C₂³×C₃₁	C₂×#41	1	4	yes	—
218	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
219	D₄×C₃₁	—	2	16	no	—
220	D₄×C₃₁	—	2	8	no	—
221	D₄×C₃₁	#117	1	16	yes	complemented
222	C₂×C₄×C₃₁	—	2	8	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	8	no	—
232	C₂×C₄×C₃₁	—	2	8	no	—
233	C₂×C₄×C₃₁	—	2	16	no	—
234	C₂×C₄×C₃₁	—	2	16	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	8	no	—
238	C₂×C₄×C₃₁	—	2	8	no	—
239	C₂×C₄×C₃₁	—	2	8	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	4	8	no	—
242	C₂×C₄×C₃₁	—	4	8	no	—
243	C₂×C₄×C₃₁	—	4	8	no	—
244	C₂×C₄×C₃₁	—	4	32	no	—
245	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
246	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
247	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
248	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
249	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
250	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
251	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
254	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
255	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
256	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
257	#2357	C₂×C₃₁	1	2	yes	complemented
258	#2353	C₂×C₃₁	1	4	yes	complemented
259	#2349	C₂×C₃₁	1	2	yes	complemented
260	#2345	C₂×C₃₁	1	2	yes	complemented
261	#2342	C₂×C₃₁	1	4	yes	complemented
262	D₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	—
263	#2293	C₂×C₃₁	1	2	yes	complemented
264	#2290	C₂×C₃₁	1	4	yes	complemented
265	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#2280	C₂×C₃₁	1	2	yes	complemented
267	C₄²×D₄	C₂×C₃₁	1	2	yes	complemented
268	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
269	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
270	C₂²×C₃₁	—	2	2	no	—
271	C₂²×C₃₁	—	2	4	no	—
272	C₂²×C₃₁	—	2	2	no	—
273	C₂²×C₃₁	—	2	4	no	—
274	C₂²×C₃₁	—	2	16	no	—
275	C₂²×C₃₁	—	2	8	no	—
276	C₂²×C₃₁	—	2	4	no	—
277	C₂²×C₃₁	—	2	4	no	—
278	C₂²×C₃₁	—	2	4	no	—
279	C₂²×C₃₁	—	4	32	no	—
280	C₂²×C₃₁	—	4	8	no	—
281	C₂²×C₃₁	C₂×#117	1	8	yes	—
282	C₂²×C₃₁	C₄×#41	1	4	yes	—
283	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
284	C₂²×C₃₁	C₂²×#41	1	2	yes	—
285	C₂²×C₃₁	C₂²×#41	1	2	yes	—
286	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
287	C₄×C₃₁	—	2	2	no	—
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	2	2	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	4	no	—
292	C₄×C₃₁	—	2	4	no	—
293	C₄×C₃₁	—	2	8	no	—
294	C₄×C₃₁	—	4	4	no	—
295	C₄×C₃₁	—	4	8	no	—
296	C₄×C₃₁	C₂×#117	1	4	yes	—
297	C₄×C₃₁	#518	1	4	yes	—
298	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
299	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
300	C₂×C₄×D₄	C₄×C₃₁	1	16	yes	complemented
301	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
302	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
305	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
307	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
308	C₂×#117	C₂²×C₃₁	1	2	yes	—
309	C₂×#117	C₂²×C₃₁	1	4	yes	complemented
310	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
311	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
312	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
313	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
314	C₂²×C₄²	C₂²×C₃₁	1	4	yes	complemented
315	#390	C₂²×C₃₁	1	4	yes	—
316	#389	C₂²×C₃₁	1	4	yes	complemented
317	#388	C₂²×C₃₁	1	8	yes	complemented
318	#388	C₂²×C₃₁	1	8	yes	complemented
319	#387	C₂²×C₃₁	1	8	yes	complemented
320	#386	C₂²×C₃₁	1	8	yes	—
321	#385	C₂²×C₃₁	1	8	yes	complemented
322	#383	C₂²×C₃₁	1	2	yes	complemented
323	#382	C₂²×C₃₁	1	2	yes	complemented
324	#381	C₂²×C₃₁	1	2	yes	—
325	#381	C₂²×C₃₁	1	4	yes	complemented
326	#380	C₂²×C₃₁	1	2	yes	—
327	#380	C₂²×C₃₁	1	4	yes	—
328	#380	C₂²×C₃₁	1	4	yes	complemented
329	#379	C₂²×C₃₁	1	4	yes	—
330	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
331	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
332	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
333	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
334	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
335	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
336	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
337	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
338	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
339	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
340	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
341	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
342	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
343	#376	C₂²×C₃₁	1	2	yes	complemented
344	#376	C₂²×C₃₁	1	2	yes	complemented
345	C₂×#95	C₂²×C₃₁	1	4	yes	complemented
346	C₄³	C₂²×C₃₁	1	2	yes	complemented
347	C₂×C₃₁	—	2	4	no	—
348	C₂×C₃₁	—	2	8	no	—
349	C₂×C₃₁	—	4	4	no	—
350	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
351	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
352	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
353	C₂×C₃₁	#3481	1	4	yes	—
354	C₂⁵	C₂³×C₃₁	1	2	yes	—
355	C₂²×D₄	—	2	8	no	—
356	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
357	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
358	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
359	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
360	C₂³×C₄	—	4	8	no	—
361	C₂³×C₄	C₂×C₄×C₃₁	1	16	yes	complemented
362	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
363	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
364	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
365	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
366	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
367	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
368	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
369	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
370	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
371	C₄×D₄	—	2	16	no	—
372	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
373	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
374	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
375	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
376	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
377	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
378	#117	—	2	8	no	—
379	#117	—	4	8	no	—
380	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
381	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
382	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
383	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
384	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×C₂²⋊C₄	—	2	16	no	—
388	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
389	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
399	C₂×C₄²	—	4	8	no	—
400	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
401	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
402	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
403	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
404	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
405	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
407	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
408	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
411	#95	C₂³×C₃₁	1	2	yes	—
412	#95	C₂³×C₃₁	1	4	yes	—
413	#95	C₂³×C₃₁	1	8	yes	—
414	#95	C₂³×C₃₁	1	2	yes	—
415	#95	C₂³×C₃₁	1	4	yes	—
416	#95	C₂³×C₃₁	1	4	yes	—
417	#95	C₂³×C₃₁	1	4	yes	—
418	#95	C₂³×C₃₁	1	8	yes	—
419	#95	C₂³×C₃₁	1	4	yes	—
420	C₃₁	#34126	1	1	yes	characteristic, direct product factor, Sylow
421	C₂⁴	—	2	8	no	—
422	C₂⁴	—	2	8	no	—
423	C₂⁴	—	4	4	no	—
424	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
425	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
426	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
427	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	2	32	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
432	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
433	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
434	C₂×D₄	#41×C₃₁	1	8	yes	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	8	no	—
438	C₂²×C₄	—	2	8	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	2	16	no	—
441	C₂²×C₄	—	2	16	no	—
442	C₂²×C₄	—	4	32	no	—
443	C₂²×C₄	—	4	8	no	—
444	C₂²×C₄	—	4	4	no	—
445	C₂²×C₄	—	4	4	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
448	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
449	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
450	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
451	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
452	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
453	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
454	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
455	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
456	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
457	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
464	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
465	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	—	4	8	no	—
469	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
470	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
471	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	16	no	—
475	C₂²⋊C₄	—	2	16	no	—
476	C₂²⋊C₄	—	4	8	no	—
477	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
478	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
479	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
480	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
481	C₂²⋊C₄	#41×C₃₁	1	8	yes	—
482	C₄²	—	2	8	no	—
483	C₄²	—	2	8	no	—
484	C₄²	—	2	4	no	—
485	C₄²	—	2	8	no	—
486	C₄²	—	2	4	no	—
487	C₄²	—	4	8	no	—
488	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
489	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
490	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
491	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
492	C₂³	—	2	4	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	4	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	16	no	—
497	C₂³	—	2	16	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	4	8	no	—
502	C₂³	—	4	32	no	—
503	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
504	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
505	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
506	C₂³	C₂×#41×C₃₁	1	8	yes	—
507	C₂³	C₂×#41×C₃₁	1	4	yes	—
508	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
509	D₄	—	2	16	no	—
510	D₄	—	2	8	no	—
511	D₄	C₃₁×#117	1	16	yes	complemented
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	8	no	—
522	C₂×C₄	—	2	8	no	—
523	C₂×C₄	—	2	16	no	—
524	C₂×C₄	—	2	16	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	8	no	—
528	C₂×C₄	—	2	8	no	—
529	C₂×C₄	—	2	8	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	4	8	no	—
532	C₂×C₄	—	4	8	no	—
533	C₂×C₄	—	4	8	no	—
534	C₂×C₄	—	4	32	no	—
535	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
536	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
537	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
538	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
539	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
540	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
541	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
542	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
543	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
544	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
545	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	4	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	16	no	—
551	C₂²	—	2	8	no	—
552	C₂²	—	2	4	no	—
553	C₂²	—	2	4	no	—
554	C₂²	—	2	4	no	—
555	C₂²	—	4	32	no	—
556	C₂²	—	4	8	no	—
557	C₂²	C₂×C₃₁×#117	1	8	yes	—
558	C₂²	C₄×#41×C₃₁	1	4	yes	—
559	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
560	C₂²	C₂²×#41×C₃₁	1	2	yes	—
561	C₂²	C₂²×#41×C₃₁	1	2	yes	—
562	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
563	C₄	—	2	2	no	—
564	C₄	—	2	2	no	—
565	C₄	—	2	2	no	—
566	C₄	—	2	2	no	—
567	C₄	—	2	4	no	—
568	C₄	—	2	4	no	—
569	C₄	—	2	8	no	—
570	C₄	—	4	8	no	—
571	C₄	—	4	4	no	—
572	C₄	C₂×C₃₁×#117	1	4	yes	—
573	C₄	C₃₁×#518	1	4	yes	—
574	C₂	—	2	4	no	—
575	C₂	—	2	8	no	—
576	C₂	—	4	4	no	—
577	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
578	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
579	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
580	C₂	C₃₁×#3481	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	C₂³×D₄×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×#2357	C₂	semidirect
3. C₃₁×#2353	C₂	semidirect
4. C₃₁×#2349	C₂	semidirect
5. C₃₁×#2345	C₂	semidirect
6. C₃₁×#2342	C₂	semidirect
8. C₃₁×#2293	C₂	semidirect
9. C₃₁×#2290	C₂	semidirect
10. C₃₁×#2283	C₂	semidirect
11. C₃₁×#2280	C₂	semidirect
12. C₄²×D₄×C₃₁	C₂	semidirect
13. C₄×C₃₁×#117	C₂	semidirect
14. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
17. C₂×C₄×D₄×C₃₁	C₄	semidirect
26. C₂×C₃₁×#117	C₂²	semidirect
31. C₂²×C₄²×C₃₁	C₂²	semidirect
33. C₃₁×#389	C₂²	semidirect
34. C₃₁×#388	C₂²	semidirect
35. C₃₁×#388	C₂²	semidirect
36. C₃₁×#387	C₂²	semidirect
38. C₃₁×#385	C₂²	semidirect
39. C₃₁×#383	C₂²	semidirect
40. C₃₁×#382	C₂²	semidirect
42. C₃₁×#381	C₂²	semidirect
45. C₃₁×#380	C₂²	semidirect
47. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
48. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
49. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
50. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
56. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
57. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
58. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
59. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
60. C₃₁×#376	C₂²	semidirect
61. C₃₁×#376	C₂²	semidirect
62. C₂×C₃₁×#95	C₂²	semidirect
63. C₄³×C₃₁	C₂²	semidirect
71. C₂³×C₄×C₃₁	C₂×C₄	semidirect
84. C₄×D₄×C₃₁	C₂×C₄	semidirect
85. C₄×D₄×C₃₁	C₂×C₄	semidirect
86. C₄×D₄×C₃₁	C₂×C₄	semidirect
87. C₄×D₄×C₃₁	C₂×C₄	semidirect
90. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
98. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
110. C₂×C₄²×C₃₁	C₂×C₄	semidirect
201. #34126	C₃₁	direct (N × H)
221. D₄×C₃₁	#117	semidirect
257. #2357	C₂×C₃₁	semidirect
258. #2353	C₂×C₃₁	semidirect
259. #2349	C₂×C₃₁	semidirect
260. #2345	C₂×C₃₁	semidirect
261. #2342	C₂×C₃₁	semidirect
263. #2293	C₂×C₃₁	semidirect
264. #2290	C₂×C₃₁	semidirect
265. #2283	C₂×C₃₁	semidirect
266. #2280	C₂×C₃₁	semidirect
267. C₄²×D₄	C₂×C₃₁	semidirect
268. C₄×#117	C₂×C₃₁	semidirect
269. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
300. C₂×C₄×D₄	C₄×C₃₁	semidirect
309. C₂×#117	C₂²×C₃₁	semidirect
314. C₂²×C₄²	C₂²×C₃₁	semidirect
316. #389	C₂²×C₃₁	semidirect
317. #388	C₂²×C₃₁	semidirect
318. #388	C₂²×C₃₁	semidirect
319. #387	C₂²×C₃₁	semidirect
321. #385	C₂²×C₃₁	semidirect
322. #383	C₂²×C₃₁	semidirect
323. #382	C₂²×C₃₁	semidirect
325. #381	C₂²×C₃₁	semidirect
328. #380	C₂²×C₃₁	semidirect
330. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
331. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
332. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
333. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
339. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
340. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
341. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
342. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
343. #376	C₂²×C₃₁	semidirect
344. #376	C₂²×C₃₁	semidirect
345. C₂×#95	C₂²×C₃₁	semidirect
346. C₄³	C₂²×C₃₁	semidirect
361. C₂³×C₄	C₂×C₄×C₃₁	semidirect
374. C₄×D₄	C₂×C₄×C₃₁	semidirect
375. C₄×D₄	C₂×C₄×C₃₁	semidirect
376. C₄×D₄	C₂×C₄×C₃₁	semidirect
377. C₄×D₄	C₂×C₄×C₃₁	semidirect
380. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
388. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
400. C₂×C₄²	C₂×C₄×C₃₁	semidirect
420. C₃₁	#34126	direct (N × H)
511. 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
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2357	2
C₃₁×#2353	2
C₃₁×#2349	2
C₃₁×#2345	2
C₃₁×#2342	2
D₄×C₂²⋊C₄×C₃₁	2
C₃₁×#2293	2
C₃₁×#2290	2
C₃₁×#2283	2
C₃₁×#2280	2
C₄²×D₄×C₃₁	2
C₄×C₃₁×#117	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
#34126	31

Subgroups — C31×#34126 #21490885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34126 #21490885