Subgroups — C₃₁×#22702 #21479461

Order 7936.

888 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#435	C₂	1	2	yes	maximal
2	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1940	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1850	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1798	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1729	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	2	yes	—
14	C₃₁×#435	—	2	8	no	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×#97	C₂²	1	2	yes	—
22	C₃₁×#390	C₄	1	1	yes	characteristic
23	C₃₁×#390	C₄	1	1	yes	characteristic
24	C₃₁×#389	C₄	1	1	yes	characteristic
25	C₃₁×#389	C₄	1	1	yes	characteristic
26	C₃₁×#388	C₄	1	2	yes	—
27	C₃₁×#388	C₄	1	2	yes	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₄³×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#336	C₂²	1	1	yes	characteristic
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	1	yes	characteristic
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂×M₁₆×C₃₁	—	2	4	no	—
41	C₂×M₁₆×C₃₁	—	2	4	no	—
42	C₂×M₁₆×C₃₁	—	4	2	no	—
43	C₂²×C₈×C₃₁	—	2	4	no	—
44	C₂²×C₈×C₃₁	—	2	4	no	—
45	C₂²×C₈×C₃₁	—	4	2	no	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₄×D₄×C₃₁	D₄	1	4	yes	—
51	C₄×D₄×C₃₁	Q₈	1	4	yes	—
52	C₃₁×#117	—	4	2	no	—
53	C₃₁×#117	—	4	2	no	—
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	—	2	4	no	—
73	C₃₁×C₄⋊C₈	—	2	4	no	—
74	C₃₁×C₄⋊C₈	—	4	2	no	—
75	C₃₁×C₄⋊C₈	—	4	2	no	—
76	C₃₁×C₂²⋊C₈	—	2	4	no	—
77	C₃₁×C₂²⋊C₈	—	2	4	no	—
78	C₃₁×#97	—	2	4	no	—
79	C₃₁×#95	C₂×C₄	1	2	yes	—
80	C₃₁×#95	C₂×C₄	1	2	yes	—
81	C₃₁×#95	C₂×C₄	1	2	yes	—
82	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
83	C₂×D₄×C₃₁	—	4	1	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	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	2	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	8	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	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	4	8	no	—
115	C₂×C₈×C₃₁	#41	1	4	yes	—
116	C₄⋊C₄×C₃₁	—	4	1	no	—
117	C₄⋊C₄×C₃₁	—	4	1	no	—
118	C₄⋊C₄×C₃₁	—	4	2	no	—
119	C₄⋊C₄×C₃₁	—	4	2	no	—
120	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
121	C₂²⋊C₄×C₃₁	—	2	4	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₂²⋊C₄×C₃₁	—	4	2	no	—
124	C₂²⋊C₄×C₃₁	—	4	2	no	—
125	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	4	1	no	—
132	C₄²×C₃₁	—	4	1	no	—
133	C₄²×C₃₁	—	4	1	no	—
134	C₄²×C₃₁	C₂×D₄	1	2	yes	—
135	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
136	#22702	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	—	2	2	no	—
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	2	2	no	—
143	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
144	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
145	D₄×C₃₁	—	4	4	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	4	no	—
157	C₂×C₄×C₃₁	—	2	4	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
174	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
175	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
176	C₂×C₄×C₃₁	#131	1	2	yes	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
180	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
181	C₈×C₃₁	—	2	4	no	—
182	C₈×C₃₁	—	4	2	no	—
183	C₈×C₃₁	—	4	2	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₂×#435	C₂×C₃₁	1	2	yes	—
186	#2293	C₂×C₃₁	1	1	yes	characteristic
187	#1940	C₂×C₃₁	1	2	yes	complemented
188	#1851	C₂×C₃₁	1	2	yes	complemented
189	#1850	C₂×C₃₁	1	2	yes	complemented
190	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1798	C₂×C₃₁	1	2	yes	complemented
192	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1729	C₂×C₃₁	1	2	yes	complemented
194	C₂²×C₃₁	—	2	4	no	—
195	C₂²×C₃₁	—	2	4	no	—
196	C₂²×C₃₁	—	2	4	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	4	2	no	—
200	C₂²×C₃₁	—	4	2	no	—
201	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
202	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
203	C₂²×C₃₁	#518	1	1	yes	characteristic
204	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
205	C₂²×C₃₁	#568	1	1	yes	characteristic
206	C₂²×C₃₁	#568	1	1	yes	characteristic
207	C₂²×C₃₁	#568	1	1	yes	characteristic
208	C₄×C₃₁	—	2	4	no	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	2	2	no	—
211	C₄×C₃₁	—	2	2	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	—	4	2	no	—
215	C₄×C₃₁	—	4	1	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	#546	1	2	yes	—
218	C₄×C₃₁	#554	1	2	yes	—
219	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
220	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
221	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×#117	C₂²×C₃₁	1	2	yes	—
223	#435	—	2	8	no	—
224	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
225	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
226	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
228	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
229	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
230	C₂×#97	C₂²×C₃₁	1	2	yes	—
231	#390	C₄×C₃₁	1	1	yes	characteristic
232	#390	C₄×C₃₁	1	1	yes	characteristic
233	#389	C₄×C₃₁	1	1	yes	characteristic
234	#389	C₄×C₃₁	1	1	yes	characteristic
235	#388	C₄×C₃₁	1	2	yes	—
236	#388	C₄×C₃₁	1	2	yes	—
237	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
238	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
239	C₄³	C₂²×C₃₁	1	1	yes	characteristic
240	#336	C₂²×C₃₁	1	1	yes	characteristic
241	#336	C₂²×C₃₁	1	2	yes	—
242	#336	C₂²×C₃₁	1	1	yes	characteristic
243	#336	C₂²×C₃₁	1	1	yes	characteristic
244	#336	C₂²×C₃₁	1	2	yes	—
245	#336	C₂²×C₃₁	1	2	yes	—
246	#336	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₃₁	—	4	4	no	—
248	C₂×C₃₁	#2349	1	1	yes	characteristic
249	C₂×C₃₁	#2905	1	1	yes	characteristic
250	C₂×C₃₁	#2906	1	1	yes	characteristic
251	C₂×C₃₁	#2967	1	1	yes	characteristic
252	C₂×C₃₁	#2968	1	1	yes	characteristic
253	C₂×C₃₁	#2980	1	1	yes	characteristic
254	C₂×C₃₁	#2981	1	1	yes	characteristic
255	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
256	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
257	C₂×M₁₆	—	2	4	no	—
258	C₂×M₁₆	—	2	4	no	—
259	C₂×M₁₆	—	4	2	no	—
260	C₂²×C₈	—	2	4	no	—
261	C₂²×C₈	—	2	4	no	—
262	C₂²×C₈	—	4	2	no	—
263	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
264	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
265	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
266	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
267	C₄×D₄	D₄×C₃₁	1	4	yes	—
268	C₄×D₄	Q₈×C₃₁	1	4	yes	—
269	#117	—	4	2	no	—
270	#117	—	4	2	no	—
271	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
272	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
273	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
276	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
277	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
278	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
279	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
280	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
281	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
282	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
287	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
289	C₄⋊C₈	—	2	4	no	—
290	C₄⋊C₈	—	2	4	no	—
291	C₄⋊C₈	—	4	2	no	—
292	C₄⋊C₈	—	4	2	no	—
293	C₂²⋊C₈	—	2	4	no	—
294	C₂²⋊C₈	—	2	4	no	—
295	#97	—	2	4	no	—
296	#95	C₂×C₄×C₃₁	1	2	yes	—
297	#95	C₂×C₄×C₃₁	1	2	yes	—
298	#95	C₂×C₄×C₃₁	1	2	yes	—
299	C₃₁	#22702	1	1	yes	characteristic, direct product factor, Sylow
300	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
301	C₂×D₄	—	4	1	no	—
302	C₂×D₄	—	4	1	no	—
303	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
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	1	yes	characteristic
313	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
315	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
318	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
319	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
320	M₁₆	—	4	8	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	4	8	no	—
333	C₂×C₈	#41×C₃₁	1	4	yes	—
334	C₄⋊C₄	—	4	1	no	—
335	C₄⋊C₄	—	4	1	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
339	C₂²⋊C₄	—	2	4	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	4	1	no	—
350	C₄²	—	4	1	no	—
351	C₄²	—	4	1	no	—
352	C₄²	C₂×D₄×C₃₁	1	2	yes	—
353	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
354	C₂³	—	2	2	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	2	2	no	—
357	C₂³	—	2	2	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
361	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
362	D₄	—	4	4	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
391	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
392	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
393	C₂×C₄	C₃₁×#131	1	2	yes	—
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	2	yes	—
397	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
398	C₈	—	2	4	no	—
399	C₈	—	4	2	no	—
400	C₈	—	4	2	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	C₄×#41×C₃₁	1	1	yes	characteristic
411	C₂²	C₃₁×#518	1	1	yes	characteristic
412	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
413	C₂²	C₃₁×#568	1	1	yes	characteristic
414	C₂²	C₃₁×#568	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	2	no	—
420	C₄	—	4	1	no	—
421	C₄	—	4	1	no	—
422	C₄	—	4	2	no	—
423	C₄	—	4	1	no	—
424	C₄	—	4	2	no	—
425	C₄	C₃₁×#546	1	2	yes	—
426	C₄	C₃₁×#554	1	2	yes	—
427	C₂	—	4	4	no	—
428	C₂	C₃₁×#2349	1	1	yes	characteristic
429	C₂	C₃₁×#2905	1	1	yes	characteristic
430	C₂	C₃₁×#2906	1	1	yes	characteristic
431	C₂	C₃₁×#2967	1	1	yes	characteristic
432	C₂	C₃₁×#2968	1	1	yes	characteristic
433	C₂	C₃₁×#2980	1	1	yes	characteristic
434	C₂	C₃₁×#2981	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
3. C₃₁×#1940	C₂	semidirect
4. C₃₁×#1851	C₂	semidirect
5. C₃₁×#1850	C₂	semidirect
6. C₃₁×#1831	C₂	semidirect
7. C₃₁×#1798	C₂	semidirect
8. C₃₁×#1767	C₂	semidirect
9. C₃₁×#1729	C₂	semidirect
136. #22702	C₃₁	direct (N × H)
187. #1940	C₂×C₃₁	semidirect
188. #1851	C₂×C₃₁	semidirect
189. #1850	C₂×C₃₁	semidirect
190. #1831	C₂×C₃₁	semidirect
191. #1798	C₂×C₃₁	semidirect
192. #1767	C₂×C₃₁	semidirect
193. #1729	C₂×C₃₁	semidirect
299. C₃₁	#22702	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×#435	2
C₃₁×#2293	2
C₃₁×#1940	2
C₃₁×#1851	2
C₃₁×#1850	2
C₃₁×#1831	2
C₃₁×#1798	2
C₃₁×#1767	2
C₃₁×#1729	2
#22702	31

Subgroups — C31×#22702 #21479461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22702 #21479461