Subgroups — C₃₁×#21899 #21478658

Order 7936.

1242 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1986	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1980	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1896	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1806	C₂	1	4	yes	maximal, complemented
8	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
9	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
10	C₃₁×#533	C₂²	1	1	yes	characteristic
11	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#529	—	2	1	no	—
15	C₃₁×#529	—	2	1	no	—
16	C₃₁×#529	—	2	1	no	—
17	C₃₁×#529	—	2	1	no	—
18	C₃₁×#529	C₄	1	2	yes	complemented
19	C₃₁×#529	C₄	1	2	yes	complemented
20	C₃₁×#525	—	2	1	no	—
21	C₃₁×#525	—	2	1	no	—
22	C₃₁×#525	C₂²	1	2	yes	—
23	C₄×#41×C₃₁	—	2	1	no	—
24	C₄×#41×C₃₁	—	2	1	no	—
25	C₄×#41×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
27	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#432	—	2	1	no	—
30	C₃₁×#432	—	2	1	no	—
31	C₃₁×#431	—	2	1	no	—
32	C₃₁×#431	—	2	1	no	—
33	C₃₁×#423	C₂²	1	2	yes	—
34	C₂×C₃₁×#104	C₂²	1	8	yes	—
35	C₃₁×#407	C₂²	1	2	yes	—
36	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
41	C₂×C₃₁×#95	—	2	1	no	—
42	C₃₁×#339	C₂²	1	8	yes	—
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂×#41×C₃₁	—	2	1	no	—
45	C₂×#41×C₃₁	C₂³	1	2	yes	—
46	C₂³×C₄×C₃₁	—	2	2	no	—
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
52	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
53	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
54	C₃₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
55	C₃₁×#126	—	2	2	no	—
56	C₃₁×#126	—	2	2	no	—
57	C₃₁×#125	—	2	1	no	—
58	C₃₁×#125	—	2	1	no	—
59	C₃₁×#125	C₂×C₄	1	2	yes	complemented
60	C₃₁×#124	—	2	1	no	—
61	C₃₁×#124	—	2	1	no	—
62	C₃₁×#124	C₂×C₄	1	2	yes	complemented
63	C₃₁×#123	—	2	2	no	—
64	C₃₁×#123	—	2	2	no	—
65	C₃₁×#123	—	2	2	no	—
66	C₃₁×C₂²⋊Q₈	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
71	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
72	C₃₁×#121	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	C₂×C₄	1	2	yes	—
77	C₃₁×#121	C₂×C₄	1	2	yes	—
78	C₃₁×#120	—	2	2	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×Q₈×C₃₁	—	2	1	no	—
81	C₄×Q₈×C₃₁	—	2	1	no	—
82	C₄×Q₈×C₃₁	—	2	1	no	—
83	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
84	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	2	1	no	—
92	C₄×D₄×C₃₁	—	2	1	no	—
93	C₄×D₄×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
96	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
97	C₃₁×#117	—	2	1	no	—
98	C₃₁×#117	—	2	1	no	—
99	C₃₁×#117	—	2	2	no	—
100	C₃₁×#117	—	2	2	no	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₄²×C₃₁	—	2	2	no	—
107	C₂×C₄²×C₃₁	—	2	2	no	—
108	C₂×C₄²×C₃₁	—	4	4	no	—
109	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₃₁×C₄⋊C₈	—	2	2	no	—
123	C₃₁×C₄⋊C₈	—	2	2	no	—
124	C₃₁×#104	—	4	8	no	—
125	C₃₁×C₂²⋊C₈	—	2	2	no	—
126	C₃₁×C₂²⋊C₈	—	2	2	no	—
127	C₃₁×#97	—	2	2	no	—
128	C₄×C₈×C₃₁	—	2	2	no	—
129	C₃₁×#95	—	2	2	no	—
130	C₃₁×#95	—	2	2	no	—
131	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
132	#41×C₃₁	—	2	4	no	—
133	#41×C₃₁	—	2	4	no	—
134	#41×C₃₁	—	4	4	no	—
135	#41×C₃₁	—	4	4	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	2	8	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	2	no	—
153	C₂²×C₄×C₃₁	—	4	2	no	—
154	C₂²×C₄×C₃₁	—	4	4	no	—
155	C₂²×C₄×C₃₁	—	4	4	no	—
156	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	M₁₆×C₃₁	—	2	4	no	—
175	M₁₆×C₃₁	—	2	4	no	—
176	C₂×C₈×C₃₁	—	2	4	no	—
177	C₄⋊C₄×C₃₁	—	2	8	no	—
178	C₄⋊C₄×C₃₁	—	2	8	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	2	2	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
191	C₂²⋊C₄×C₃₁	—	2	2	no	—
192	C₂²⋊C₄×C₃₁	—	2	2	no	—
193	C₂²⋊C₄×C₃₁	—	2	2	no	—
194	C₂²⋊C₄×C₃₁	—	2	2	no	—
195	C₂²⋊C₄×C₃₁	—	2	2	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	2	no	—
199	C₂²⋊C₄×C₃₁	—	2	2	no	—
200	C₄²×C₃₁	—	2	2	no	—
201	C₄²×C₃₁	—	2	2	no	—
202	C₄²×C₃₁	—	2	2	no	—
203	C₄²×C₃₁	—	2	8	no	—
204	C₄²×C₃₁	—	4	4	no	—
205	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
209	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
214	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
215	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
216	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
217	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
221	#21899	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂³×C₃₁	—	2	1	no	—
223	C₂³×C₃₁	—	2	1	no	—
224	C₂³×C₃₁	—	2	2	no	—
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
230	Q₈×C₃₁	—	2	4	no	—
231	Q₈×C₃₁	—	4	2	no	—
232	Q₈×C₃₁	—	4	2	no	—
233	D₄×C₃₁	—	2	4	no	—
234	D₄×C₃₁	—	4	2	no	—
235	D₄×C₃₁	—	4	2	no	—
236	D₄×C₃₁	—	4	2	no	—
237	D₄×C₃₁	—	4	4	no	—
238	D₄×C₃₁	—	4	4	no	—
239	D₄×C₃₁	—	4	2	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	8	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	4	1	no	—
258	C₂×C₄×C₃₁	—	4	1	no	—
259	C₂×C₄×C₃₁	—	4	1	no	—
260	C₂×C₄×C₃₁	—	4	1	no	—
261	C₂×C₄×C₃₁	—	4	4	no	—
262	C₂×C₄×C₃₁	—	4	16	no	—
263	C₂×C₄×C₃₁	—	4	4	no	—
264	C₂×C₄×C₃₁	—	4	4	no	—
265	C₂×C₄×C₃₁	—	4	4	no	—
266	C₂×C₄×C₃₁	—	4	4	no	—
267	C₂×C₄×C₃₁	#104	1	4	yes	complemented
268	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
281	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
282	C₈×C₃₁	—	4	4	no	—
283	#3444	C₂×C₃₁	1	1	yes	characteristic
284	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
285	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
286	#1986	C₂×C₃₁	1	2	yes	complemented
287	#1980	C₂×C₃₁	1	2	yes	complemented
288	#1896	C₂×C₃₁	1	4	yes	complemented
289	#1806	C₂×C₃₁	1	4	yes	complemented
290	C₂²×C₃₁	—	2	4	no	—
291	C₂²×C₃₁	—	2	4	no	—
292	C₂²×C₃₁	—	2	1	no	—
293	C₂²×C₃₁	—	2	1	no	—
294	C₂²×C₃₁	—	2	1	no	—
295	C₂²×C₃₁	—	2	1	no	—
296	C₂²×C₃₁	—	2	1	no	—
297	C₂²×C₃₁	—	2	1	no	—
298	C₂²×C₃₁	—	4	2	no	—
299	C₂²×C₃₁	—	4	2	no	—
300	C₂²×C₃₁	—	4	2	no	—
301	C₂²×C₃₁	C₂×#104	1	2	yes	—
302	C₂²×C₃₁	#421	1	2	yes	—
303	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
304	C₂²×C₃₁	#535	1	1	yes	characteristic
305	C₂²×C₃₁	#536	1	1	yes	characteristic
306	C₄×C₃₁	—	2	1	no	—
307	C₄×C₃₁	—	2	1	no	—
308	C₄×C₃₁	—	2	1	no	—
309	C₄×C₃₁	—	2	1	no	—
310	C₄×C₃₁	—	2	1	no	—
311	C₄×C₃₁	—	2	1	no	—
312	C₄×C₃₁	—	2	1	no	—
313	C₄×C₃₁	—	2	1	no	—
314	C₄×C₃₁	—	4	8	no	—
315	C₄×C₃₁	—	4	2	no	—
316	C₄×C₃₁	—	4	2	no	—
317	C₄×C₃₁	—	4	2	no	—
318	C₄×C₃₁	C₂×#104	1	2	yes	—
319	C₄×C₃₁	C₂×#104	1	2	yes	—
320	C₄×C₃₁	#520	1	2	yes	—
321	C₄×C₃₁	#537	1	2	yes	—
322	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
323	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
324	#533	C₂²×C₃₁	1	1	yes	characteristic
325	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
326	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
327	#532	C₂²×C₃₁	1	1	yes	characteristic
328	#529	—	2	1	no	—
329	#529	—	2	1	no	—
330	#529	—	2	1	no	—
331	#529	—	2	1	no	—
332	#529	C₄×C₃₁	1	2	yes	complemented
333	#529	C₄×C₃₁	1	2	yes	complemented
334	#525	—	2	1	no	—
335	#525	—	2	1	no	—
336	#525	C₂²×C₃₁	1	2	yes	—
337	C₄×#41	—	2	1	no	—
338	C₄×#41	—	2	1	no	—
339	C₄×#41	C₂²×C₃₁	1	2	yes	—
340	C₂²×C₄⋊C₄	—	2	1	no	—
341	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
342	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
343	#432	—	2	1	no	—
344	#432	—	2	1	no	—
345	#431	—	2	1	no	—
346	#431	—	2	1	no	—
347	#423	C₂²×C₃₁	1	2	yes	—
348	C₂×#104	C₂²×C₃₁	1	8	yes	—
349	#407	C₂²×C₃₁	1	2	yes	—
350	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
351	C₄×C₄⋊C₄	—	2	1	no	—
352	C₄×C₄⋊C₄	—	2	1	no	—
353	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
354	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
355	C₂×#95	—	2	1	no	—
356	#339	C₂²×C₃₁	1	8	yes	—
357	C₂×C₃₁	—	2	1	no	—
358	C₂×C₃₁	—	2	1	no	—
359	C₂×C₃₁	—	2	2	no	—
360	C₂×C₃₁	—	2	2	no	—
361	C₂×C₃₁	—	8	2	no	—
362	C₂×C₃₁	#2311	1	1	yes	characteristic
363	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
364	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
365	C₂×#41	—	2	1	no	—
366	C₂×#41	—	2	1	no	—
367	C₂×#41	C₂³×C₃₁	1	2	yes	—
368	C₂³×C₄	—	2	2	no	—
369	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
370	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
371	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
372	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
373	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
374	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
375	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
376	#127	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
377	#126	—	2	2	no	—
378	#126	—	2	2	no	—
379	#125	—	2	1	no	—
380	#125	—	2	1	no	—
381	#125	C₂×C₄×C₃₁	1	2	yes	complemented
382	#124	—	2	1	no	—
383	#124	—	2	1	no	—
384	#124	C₂×C₄×C₃₁	1	2	yes	complemented
385	#123	—	2	2	no	—
386	#123	—	2	2	no	—
387	#123	—	2	2	no	—
388	C₂²⋊Q₈	—	2	1	no	—
389	C₂²⋊Q₈	—	2	1	no	—
390	C₂²⋊Q₈	—	2	1	no	—
391	C₂²⋊Q₈	—	2	1	no	—
392	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
393	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
394	#121	—	2	1	no	—
395	#121	—	2	1	no	—
396	#121	—	2	1	no	—
397	#121	—	2	1	no	—
398	#121	C₂×C₄×C₃₁	1	2	yes	—
399	#121	C₂×C₄×C₃₁	1	2	yes	—
400	#120	—	2	2	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	—	2	1	no	—
404	C₄×Q₈	—	2	1	no	—
405	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
406	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
418	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
419	#117	—	2	1	no	—
420	#117	—	2	1	no	—
421	#117	—	2	2	no	—
422	#117	—	2	2	no	—
423	#117	—	2	2	no	—
424	#117	C₂³×C₃₁	1	2	yes	—
425	C₂×C₄⋊C₄	—	2	8	no	—
426	C₂×C₄⋊C₄	—	2	2	no	—
427	C₂×C₄⋊C₄	—	2	2	no	—
428	C₂×C₄²	—	2	2	no	—
429	C₂×C₄²	—	2	2	no	—
430	C₂×C₄²	—	4	4	no	—
431	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
435	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
436	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
439	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
440	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
442	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
444	C₄⋊C₈	—	2	2	no	—
445	C₄⋊C₈	—	2	2	no	—
446	#104	—	4	8	no	—
447	C₂²⋊C₈	—	2	2	no	—
448	C₂²⋊C₈	—	2	2	no	—
449	#97	—	2	2	no	—
450	C₄×C₈	—	2	2	no	—
451	#95	—	2	2	no	—
452	#95	—	2	2	no	—
453	C₃₁	#21899	1	1	yes	characteristic, direct product factor, Sylow
454	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
455	#41	—	2	4	no	—
456	#41	—	2	4	no	—
457	#41	—	4	4	no	—
458	#41	—	4	4	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	8	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	4	2	no	—
477	C₂²×C₄	—	4	4	no	—
478	C₂²×C₄	—	4	4	no	—
479	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
483	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
488	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
490	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
497	M₁₆	—	2	4	no	—
498	M₁₆	—	2	4	no	—
499	C₂×C₈	—	2	4	no	—
500	C₄⋊C₄	—	2	8	no	—
501	C₄⋊C₄	—	2	8	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
513	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	2	no	—
525	C₄²	—	2	2	no	—
526	C₄²	—	2	8	no	—
527	C₄²	—	4	4	no	—
528	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
529	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
530	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
531	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
532	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
533	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
534	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
535	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
536	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
537	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
538	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
539	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
540	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
541	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
542	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
543	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	2	no	—
549	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
550	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
551	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
552	Q₈	—	2	4	no	—
553	Q₈	—	4	2	no	—
554	Q₈	—	4	2	no	—
555	D₄	—	2	4	no	—
556	D₄	—	4	2	no	—
557	D₄	—	4	4	no	—
558	D₄	—	4	4	no	—
559	D₄	—	4	2	no	—
560	D₄	—	4	2	no	—
561	D₄	—	4	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	8	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	—	4	16	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	C₃₁×#104	1	4	yes	complemented
590	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
592	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
593	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
594	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
595	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
596	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
597	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
598	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
599	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
600	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
601	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
602	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
603	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
604	C₈	—	4	4	no	—
605	C₂²	—	2	4	no	—
606	C₂²	—	2	4	no	—
607	C₂²	—	2	1	no	—
608	C₂²	—	2	1	no	—
609	C₂²	—	2	1	no	—
610	C₂²	—	2	1	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	2	1	no	—
613	C₂²	—	4	2	no	—
614	C₂²	—	4	2	no	—
615	C₂²	—	4	2	no	—
616	C₂²	C₂×C₃₁×#104	1	2	yes	—
617	C₂²	C₃₁×#421	1	2	yes	—
618	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
619	C₂²	C₃₁×#535	1	1	yes	characteristic
620	C₂²	C₃₁×#536	1	1	yes	characteristic
621	C₄	—	2	1	no	—
622	C₄	—	2	1	no	—
623	C₄	—	2	1	no	—
624	C₄	—	2	1	no	—
625	C₄	—	2	1	no	—
626	C₄	—	2	1	no	—
627	C₄	—	2	1	no	—
628	C₄	—	2	1	no	—
629	C₄	—	4	8	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	2	no	—
632	C₄	—	4	2	no	—
633	C₄	C₂×C₃₁×#104	1	2	yes	—
634	C₄	C₂×C₃₁×#104	1	2	yes	—
635	C₄	C₃₁×#520	1	2	yes	—
636	C₄	C₃₁×#537	1	2	yes	—
637	C₂	—	2	1	no	—
638	C₂	—	2	2	no	—
639	C₂	—	2	2	no	—
640	C₂	—	2	1	no	—
641	C₂	—	8	2	no	—
642	C₂	C₃₁×#2311	1	1	yes	characteristic
643	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic
644	C₂	C₂×C₃₁×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2920	C₂	semidirect
3. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
4. C₃₁×#1986	C₂	semidirect
5. C₃₁×#1980	C₂	semidirect
6. C₃₁×#1896	C₂	semidirect
7. C₃₁×#1806	C₂	semidirect
8. C₃₁×#533	C₄	semidirect
9. C₃₁×#533	C₄	semidirect
11. C₃₁×#532	C₄	semidirect
12. C₃₁×#532	C₄	semidirect
18. C₃₁×#529	C₄	semidirect
19. C₃₁×#529	C₄	semidirect
51. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
52. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
53. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
54. C₃₁×#127	C₂×C₄	semidirect
59. C₃₁×#125	C₂×C₄	semidirect
62. C₃₁×#124	C₂×C₄	semidirect
83. C₄×Q₈×C₃₁	C₂×C₄	semidirect
84. C₄×Q₈×C₃₁	C₂×C₄	semidirect
95. C₄×D₄×C₃₁	C₂×C₄	semidirect
96. C₄×D₄×C₃₁	C₂×C₄	semidirect
221. #21899	C₃₁	direct (N × H)
267. C₂×C₄×C₃₁	#104	semidirect
284. #2920	C₂×C₃₁	semidirect
285. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
286. #1986	C₂×C₃₁	semidirect
287. #1980	C₂×C₃₁	semidirect
288. #1896	C₂×C₃₁	semidirect
289. #1806	C₂×C₃₁	semidirect
322. #533	C₄×C₃₁	semidirect
323. #533	C₄×C₃₁	semidirect
325. #532	C₄×C₃₁	semidirect
326. #532	C₄×C₃₁	semidirect
332. #529	C₄×C₃₁	semidirect
333. #529	C₄×C₃₁	semidirect
373. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
374. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
375. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
376. #127	C₂×C₄×C₃₁	semidirect
381. #125	C₂×C₄×C₃₁	semidirect
384. #124	C₂×C₄×C₃₁	semidirect
405. C₄×Q₈	C₂×C₄×C₃₁	semidirect
406. C₄×Q₈	C₂×C₄×C₃₁	semidirect
417. C₄×D₄	C₂×C₄×C₃₁	semidirect
418. C₄×D₄	C₂×C₄×C₃₁	semidirect
453. C₃₁	#21899	direct (N × H)
589. C₂×C₄	C₃₁×#104	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3444	2
C₃₁×#2920	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1986	2
C₃₁×#1980	2
C₃₁×#1896	2
C₃₁×#1806	2
#21899	31

Subgroups — C31×#21899 #21478658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21899 #21478658