Subgroups — C₃₁×#22650 #21479409

Order 7936.

1096 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	2	yes	maximal
2	C₂×C₃₁×#423	C₂	1	2	yes	maximal, complemented
3	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1939	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1851	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1797	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
10	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#436	—	2	8	no	—
15	C₃₁×#423	—	2	8	no	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
28	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
29	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₄³×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#336	C₂²	1	4	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
37	C₂×M₁₆×C₃₁	—	2	8	no	—
38	C₂×M₁₆×C₃₁	—	2	8	no	—
39	C₂×M₁₆×C₃₁	—	4	4	no	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₄×D₄×C₃₁	—	2	4	no	—
44	C₄×D₄×C₃₁	—	2	4	no	—
45	C₄×D₄×C₃₁	D₄	1	4	yes	—
46	C₄×D₄×C₃₁	Q₈	1	4	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₃₁	—	2	4	no	—
54	C₂×C₄²×C₃₁	—	2	4	no	—
55	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊C₈	—	2	8	no	—
66	C₃₁×C₄⋊C₈	—	2	4	no	—
67	C₃₁×C₄⋊C₈	—	2	4	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	—	4	2	no	—
70	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
71	C₃₁×C₂²⋊C₈	—	2	8	no	—
72	C₄×C₈×C₃₁	—	2	4	no	—
73	C₃₁×#95	C₂×C₄	1	2	yes	—
74	C₃₁×#95	C₂×C₄	1	2	yes	—
75	C₃₁×#95	C₂×C₄	1	2	yes	—
76	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	4	no	—
85	C₂²×C₄×C₃₁	—	2	4	no	—
86	C₂²×C₄×C₃₁	—	2	4	no	—
87	C₂²×C₄×C₃₁	—	2	4	no	—
88	C₂²×C₄×C₃₁	—	2	4	no	—
89	C₂²×C₄×C₃₁	—	4	2	no	—
90	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
100	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
101	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	M₁₆×C₃₁	—	4	16	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	4	no	—
107	C₂×C₈×C₃₁	—	2	4	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	4	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
112	C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
114	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	complemented
115	C₂²⋊C₄×C₃₁	—	2	4	no	—
116	C₂²⋊C₄×C₃₁	—	2	4	no	—
117	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
118	C₄²×C₃₁	—	2	4	no	—
119	C₄²×C₃₁	—	2	4	no	—
120	C₄²×C₃₁	—	2	2	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	1	no	—
125	C₄²×C₃₁	—	2	1	no	—
126	C₄²×C₃₁	—	4	1	no	—
127	C₄²×C₃₁	M₁₆	1	4	yes	complemented
128	C₄²×C₃₁	C₂×D₄	1	2	yes	—
129	C₄²×C₃₁	C₂×D₄	1	2	yes	—
130	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
131	C₄²×C₃₁	#41	1	2	yes	—
132	#22650	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	—	2	2	no	—
135	C₂³×C₃₁	—	2	2	no	—
136	C₂³×C₃₁	—	2	2	no	—
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
140	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
141	D₄×C₃₁	—	2	8	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	4	no	—
153	C₂×C₄×C₃₁	—	2	4	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	4	8	no	—
163	C₂×C₄×C₃₁	—	4	8	no	—
164	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
166	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
169	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
170	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
171	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
172	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
173	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
174	C₂×C₄×C₃₁	#131	1	2	yes	—
175	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
181	C₈×C₃₁	—	2	4	no	—
182	C₈×C₃₁	—	4	4	no	—
183	C₈×C₃₁	—	4	2	no	—
184	C₂×#436	C₂×C₃₁	1	2	yes	—
185	C₂×#423	C₂×C₃₁	1	2	yes	complemented
186	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
187	#1939	C₂×C₃₁	1	2	yes	complemented
188	#1851	C₂×C₃₁	1	4	yes	complemented
189	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1797	C₂×C₃₁	1	2	yes	complemented
191	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
192	C₂²×C₃₁	—	2	4	no	—
193	C₂²×C₃₁	—	2	4	no	—
194	C₂²×C₃₁	—	2	4	no	—
195	C₂²×C₃₁	—	4	2	no	—
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
200	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×C₃₁	#520	1	1	yes	characteristic
202	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
203	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
204	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
205	C₂²×C₃₁	#568	1	1	yes	characteristic
206	C₄×C₃₁	—	2	4	no	—
207	C₄×C₃₁	—	2	2	no	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	4	no	—
210	C₄×C₃₁	—	2	2	no	—
211	C₄×C₃₁	—	4	2	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	#423	1	4	yes	complemented
214	C₄×C₃₁	#436	1	4	yes	complemented
215	C₄×C₃₁	D₄²	1	1	yes	characteristic
216	C₄×C₃₁	#547	1	1	yes	characteristic
217	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
218	C₄×C₃₁	#554	1	1	yes	characteristic
219	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
220	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
222	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
223	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
224	#436	—	2	8	no	—
225	#423	—	2	8	no	—
226	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
229	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
230	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
231	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
233	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
234	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
235	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
236	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
237	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
238	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
239	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
240	C₄³	C₂²×C₃₁	1	1	yes	characteristic
241	#336	C₂²×C₃₁	1	4	yes	—
242	#336	C₂²×C₃₁	1	2	yes	—
243	C₂×C₃₁	—	4	4	no	—
244	C₂×C₃₁	D₄×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₃₁	C₂×#423	1	1	yes	characteristic
246	C₂×C₃₁	#2906	1	1	yes	characteristic
247	C₂×C₃₁	C₂×#436	1	1	yes	characteristic
248	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
249	C₂×C₃₁	#2989	1	1	yes	characteristic
250	C₂×C₃₁	#2991	1	1	yes	characteristic
251	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
252	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
253	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
254	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
255	C₂×M₁₆	—	2	8	no	—
256	C₂×M₁₆	—	2	8	no	—
257	C₂×M₁₆	—	4	4	no	—
258	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
259	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
260	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
261	C₄×D₄	—	2	4	no	—
262	C₄×D₄	—	2	4	no	—
263	C₄×D₄	D₄×C₃₁	1	4	yes	—
264	C₄×D₄	Q₈×C₃₁	1	4	yes	—
265	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
268	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
269	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
270	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
271	C₂×C₄²	—	2	4	no	—
272	C₂×C₄²	—	2	4	no	—
273	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
276	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
277	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
281	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
283	C₄⋊C₈	—	2	8	no	—
284	C₄⋊C₈	—	2	4	no	—
285	C₄⋊C₈	—	2	4	no	—
286	C₄⋊C₈	—	2	2	no	—
287	C₄⋊C₈	—	4	2	no	—
288	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
289	C₂²⋊C₈	—	2	8	no	—
290	C₄×C₈	—	2	4	no	—
291	#95	C₂×C₄×C₃₁	1	2	yes	—
292	#95	C₂×C₄×C₃₁	1	2	yes	—
293	#95	C₂×C₄×C₃₁	1	2	yes	—
294	C₃₁	#22650	1	1	yes	characteristic, direct product factor, Sylow
295	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	4	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	4	no	—
307	C₂²×C₄	—	2	4	no	—
308	C₂²×C₄	—	4	2	no	—
309	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
311	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
313	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
316	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
319	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
320	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
321	M₁₆	—	4	16	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	4	no	—
326	C₂×C₈	—	2	4	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	4	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
333	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	complemented
334	C₂²⋊C₄	—	2	4	no	—
335	C₂²⋊C₄	—	2	4	no	—
336	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
337	C₄²	—	2	4	no	—
338	C₄²	—	2	4	no	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	4	1	no	—
346	C₄²	M₁₆×C₃₁	1	4	yes	complemented
347	C₄²	C₂×D₄×C₃₁	1	2	yes	—
348	C₄²	C₂×D₄×C₃₁	1	2	yes	—
349	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
350	C₄²	#41×C₃₁	1	2	yes	—
351	C₂³	—	2	2	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	2	2	no	—
357	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
358	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
359	D₄	—	2	8	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	8	no	—
381	C₂×C₄	—	4	8	no	—
382	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
383	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
384	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
385	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
386	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
387	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
388	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
389	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
390	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
391	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
392	C₂×C₄	C₃₁×#131	1	2	yes	—
393	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
398	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
399	C₈	—	2	4	no	—
400	C₈	—	4	4	no	—
401	C₈	—	4	2	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	4	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
410	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
411	C₂²	C₃₁×#520	1	1	yes	characteristic
412	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
413	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
414	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
415	C₂²	C₃₁×#568	1	1	yes	characteristic
416	C₄	—	2	4	no	—
417	C₄	—	2	2	no	—
418	C₄	—	2	2	no	—
419	C₄	—	2	4	no	—
420	C₄	—	2	2	no	—
421	C₄	—	4	2	no	—
422	C₄	—	4	1	no	—
423	C₄	C₃₁×#423	1	4	yes	complemented
424	C₄	C₃₁×#436	1	4	yes	complemented
425	C₄	D₄²×C₃₁	1	1	yes	characteristic
426	C₄	C₃₁×#547	1	1	yes	characteristic
427	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
428	C₄	C₃₁×#554	1	1	yes	characteristic
429	C₂	—	4	4	no	—
430	C₂	D₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
431	C₂	C₂×C₃₁×#423	1	1	yes	characteristic
432	C₂	C₃₁×#2906	1	1	yes	characteristic
433	C₂	C₂×C₃₁×#436	1	1	yes	characteristic
434	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
435	C₂	C₃₁×#2989	1	1	yes	characteristic
436	C₂	C₃₁×#2991	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#423	C₂	semidirect
4. C₃₁×#1939	C₂	semidirect
5. C₃₁×#1851	C₂	semidirect
6. C₃₁×#1842	C₂	semidirect
7. C₃₁×#1797	C₂	semidirect
8. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
70. C₃₁×C₄⋊C₈	D₄	semidirect
114. C₄⋊C₄×C₃₁	M₁₆	semidirect
127. C₄²×C₃₁	M₁₆	semidirect
132. #22650	C₃₁	direct (N × H)
185. C₂×#423	C₂×C₃₁	semidirect
187. #1939	C₂×C₃₁	semidirect
188. #1851	C₂×C₃₁	semidirect
189. #1842	C₂×C₃₁	semidirect
190. #1797	C₂×C₃₁	semidirect
191. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
213. C₄×C₃₁	#423	semidirect
214. C₄×C₃₁	#436	semidirect
288. C₄⋊C₈	D₄×C₃₁	semidirect
294. C₃₁	#22650	direct (N × H)
333. C₄⋊C₄	M₁₆×C₃₁	semidirect
346. C₄²	M₁₆×C₃₁	semidirect
423. C₄	C₃₁×#423	semidirect
424. C₄	C₃₁×#436	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#436	2
C₂×C₃₁×#423	2
C₄²×D₄×C₃₁	2
C₃₁×#1939	2
C₃₁×#1851	2
C₃₁×#1842	2
C₃₁×#1797	2
C₄×C₃₁×C₄⋊C₈	2
#22650	31

Subgroups — C31×#22650 #21479409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22650 #21479409