Subgroups — C₃₁×#17548 #21474307

Order 7936.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3520	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2974	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2275	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1509	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1508	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1506	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1504	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1503	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1501	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1500	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#328	C₂	1	2	yes	maximal, complemented
12	C₃₁×#556	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#554	—	2	1	no	—
14	C₃₁×#552	C₄	1	2	yes	complemented
15	C₃₁×#551	—	2	2	no	—
16	D₄×Q₈×C₃₁	C₄	1	1	yes	characteristic, complemented
17	C₃₁×#548	C₄	1	2	yes	complemented
18	C₃₁×#547	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#547	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#546	—	2	1	no	—
21	C₃₁×#542	—	2	1	no	—
22	C₃₁×#539	—	2	2	no	—
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#529	C₂²	1	1	yes	characteristic
25	C₃₁×#525	C₂²	1	2	yes	—
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₃₁×#520	—	2	1	no	—
28	C₂×C₄×D₄×C₃₁	—	2	1	no	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#436	—	2	1	no	—
31	C₃₁×#435	—	2	2	no	—
32	C₈×D₄×C₃₁	—	2	1	no	—
33	C₃₁×#432	C₂²	1	1	yes	characteristic
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#407	C₂²	1	2	yes	—
36	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
40	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
41	C₃₁×#376	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#95	C₂²	1	2	yes	—
43	C₃₁×#328	C₂²	1	1	yes	characteristic
44	C₃₁×#328	C₂²	1	2	yes	—
45	C₃₁×#328	C₂²	1	2	yes	—
46	C₃₁×#328	C₂²	1	2	yes	—
47	C₃₁×#328	C₂²	1	1	yes	characteristic
48	C₃₁×#328	C₂²	1	2	yes	—
49	C₃₁×#328	C₂²	1	2	yes	—
50	C₃₁×#327	C₂²	1	1	yes	characteristic
51	C₃₁×#327	C₂²	1	1	yes	characteristic
52	C₃₁×#327	C₂²	1	2	yes	—
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	C₂×C₄	1	2	yes	complemented
55	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
56	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₃₁	—	2	2	no	—
58	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
64	C₃₁×#126	—	2	2	no	—
65	C₃₁×#126	—	2	2	no	—
66	C₃₁×#126	—	2	1	no	—
67	C₃₁×#125	—	2	2	no	—
68	C₃₁×#125	—	2	1	no	—
69	C₃₁×#125	C₂×C₄	1	1	yes	characteristic, complemented
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#124	—	2	2	no	—
72	C₃₁×#124	C₂×C₄	1	1	yes	characteristic, complemented
73	C₃₁×#123	—	2	2	no	—
74	C₃₁×#123	—	2	2	no	—
75	C₃₁×#123	—	2	1	no	—
76	C₃₁×#123	—	2	2	no	—
77	C₃₁×#123	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	—	2	2	no	—
80	C₃₁×C₂²⋊Q₈	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	—	2	1	no	—
83	C₃₁×C₂²⋊Q₈	—	2	2	no	—
84	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
85	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
86	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
87	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
88	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
89	C₃₁×#121	—	2	2	no	—
90	C₃₁×#121	—	2	1	no	—
91	C₃₁×#121	C₂×C₄	1	2	yes	complemented
92	C₃₁×#121	C₂³	1	1	yes	characteristic
93	C₃₁×#120	—	2	2	no	—
94	C₄×Q₈×C₃₁	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
97	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
98	C₄×D₄×C₃₁	—	2	8	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	2	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	—	2	2	no	—
104	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
105	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
107	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
108	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
109	C₄×D₄×C₃₁	D₄	1	2	yes	—
110	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
111	C₃₁×#117	—	2	1	no	—
112	C₃₁×#117	—	2	2	no	—
113	C₃₁×#117	—	2	1	no	—
114	C₃₁×#117	C₂×C₄	1	2	yes	complemented
115	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	complemented
119	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
120	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
123	C₂×C₄²×C₃₁	—	2	1	no	—
124	C₂×C₄²×C₃₁	—	2	1	no	—
125	C₂×C₄²×C₃₁	—	2	2	no	—
126	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
127	C₃₁×C₄⋊C₈	—	2	1	no	—
128	C₃₁×C₄⋊C₈	—	2	1	no	—
129	C₃₁×C₂²⋊C₈	—	2	2	no	—
130	C₃₁×C₂²⋊C₈	—	2	2	no	—
131	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
132	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
133	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
134	C₃₁×#97	—	2	1	no	—
135	C₄×C₈×C₃₁	—	2	1	no	—
136	C₃₁×#95	—	2	2	no	—
137	C₃₁×#95	—	2	1	no	—
138	C₃₁×#95	C₂³	1	1	yes	characteristic
139	C₃₁×#95	C₂³	1	2	yes	—
140	C₃₁×#95	C₂³	1	1	yes	characteristic
141	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
142	#41×C₃₁	—	4	4	no	—
143	#41×C₃₁	—	8	2	no	—
144	C₂×Q₈×C₃₁	—	2	1	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	—	8	2	no	—
147	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	2	no	—
151	C₂×D₄×C₃₁	—	2	1	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
154	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
155	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	1	no	—
159	C₂²×C₄×C₃₁	—	2	1	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	8	no	—
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	2	1	no	—
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	—	8	2	no	—
166	C₂²×C₄×C₃₁	—	8	2	no	—
167	C₂²×C₄×C₃₁	—	8	2	no	—
168	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	2	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	M₁₆×C₃₁	—	8	2	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	2	2	no	—
180	C₂×C₈×C₃₁	—	8	2	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	2	2	no	—
195	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
197	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
200	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
201	C₂²⋊C₄×C₃₁	—	2	8	no	—
202	C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂²⋊C₄×C₃₁	—	2	2	no	—
206	C₂²⋊C₄×C₃₁	—	2	1	no	—
207	C₂²⋊C₄×C₃₁	—	2	2	no	—
208	C₂²⋊C₄×C₃₁	—	2	1	no	—
209	C₂²⋊C₄×C₃₁	—	2	1	no	—
210	C₂²⋊C₄×C₃₁	—	2	2	no	—
211	C₂²⋊C₄×C₃₁	—	2	1	no	—
212	C₂²⋊C₄×C₃₁	—	2	1	no	—
213	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
214	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
215	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
216	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
217	C₄²×C₃₁	—	2	4	no	—
218	C₄²×C₃₁	—	2	2	no	—
219	C₄²×C₃₁	—	2	2	no	—
220	C₄²×C₃₁	—	2	1	no	—
221	C₄²×C₃₁	—	8	1	no	—
222	C₄²×C₃₁	—	8	2	no	—
223	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
224	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
225	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
227	#17548	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂³×C₃₁	—	2	2	no	—
229	C₂³×C₃₁	—	2	2	no	—
230	C₂³×C₃₁	—	2	4	no	—
231	C₂³×C₃₁	—	2	1	no	—
232	C₂³×C₃₁	—	4	2	no	—
233	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
234	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
235	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
236	Q₈×C₃₁	—	4	1	no	—
237	Q₈×C₃₁	—	4	2	no	—
238	Q₈×C₃₁	—	4	2	no	—
239	Q₈×C₃₁	—	8	2	no	—
240	D₄×C₃₁	—	2	4	no	—
241	D₄×C₃₁	—	2	2	no	—
242	D₄×C₃₁	—	4	1	no	—
243	D₄×C₃₁	—	4	1	no	—
244	D₄×C₃₁	—	4	1	no	—
245	D₄×C₃₁	—	4	4	no	—
246	D₄×C₃₁	—	8	2	no	—
247	D₄×C₃₁	#104	1	4	yes	—
248	C₂×C₄×C₃₁	—	2	1	no	—
249	C₂×C₄×C₃₁	—	2	1	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	4	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	1	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	1	no	—
259	C₂×C₄×C₃₁	—	4	2	no	—
260	C₂×C₄×C₃₁	—	4	2	no	—
261	C₂×C₄×C₃₁	—	4	2	no	—
262	C₂×C₄×C₃₁	—	4	1	no	—
263	C₂×C₄×C₃₁	—	4	2	no	—
264	C₂×C₄×C₃₁	—	4	2	no	—
265	C₂×C₄×C₃₁	—	4	1	no	—
266	C₂×C₄×C₃₁	—	4	2	no	—
267	C₂×C₄×C₃₁	—	4	1	no	—
268	C₂×C₄×C₃₁	—	4	2	no	—
269	C₂×C₄×C₃₁	—	4	1	no	—
270	C₂×C₄×C₃₁	—	4	1	no	—
271	C₂×C₄×C₃₁	—	4	4	no	—
272	C₂×C₄×C₃₁	—	4	2	no	—
273	C₂×C₄×C₃₁	—	8	2	no	—
274	C₂×C₄×C₃₁	—	8	2	no	—
275	C₂×C₄×C₃₁	—	8	8	no	—
276	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
278	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
281	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
283	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
284	C₈×C₃₁	—	4	1	no	—
285	C₈×C₃₁	—	4	2	no	—
286	C₈×C₃₁	—	4	1	no	—
287	#3520	C₂×C₃₁	1	1	yes	characteristic
288	#2974	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#2275	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#1509	C₂×C₃₁	1	1	yes	characteristic, complemented
291	#1508	C₂×C₃₁	1	2	yes	complemented
292	#1506	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#1504	C₂×C₃₁	1	2	yes	complemented
294	#1503	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#1501	C₂×C₃₁	1	2	yes	complemented
296	#1500	C₂×C₃₁	1	1	yes	characteristic, complemented
297	C₂×#328	C₂×C₃₁	1	2	yes	complemented
298	C₂²×C₃₁	—	2	1	no	—
299	C₂²×C₃₁	—	2	2	no	—
300	C₂²×C₃₁	—	2	2	no	—
301	C₂²×C₃₁	—	2	1	no	—
302	C₂²×C₃₁	—	2	2	no	—
303	C₂²×C₃₁	—	2	2	no	—
304	C₂²×C₃₁	—	2	2	no	—
305	C₂²×C₃₁	—	4	4	no	—
306	C₂²×C₃₁	—	4	4	no	—
307	C₂²×C₃₁	—	4	1	no	—
308	C₂²×C₃₁	—	4	1	no	—
309	C₂²×C₃₁	—	4	1	no	—
310	C₂²×C₃₁	C₂×#104	1	4	yes	—
311	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
312	C₂²×C₃₁	#577	1	1	yes	characteristic
313	C₂²×C₃₁	#578	1	1	yes	characteristic
314	C₄×C₃₁	—	2	1	no	—
315	C₄×C₃₁	—	2	1	no	—
316	C₄×C₃₁	—	2	2	no	—
317	C₄×C₃₁	—	2	2	no	—
318	C₄×C₃₁	—	2	1	no	—
319	C₄×C₃₁	—	2	1	no	—
320	C₄×C₃₁	—	2	1	no	—
321	C₄×C₃₁	—	4	2	no	—
322	C₄×C₃₁	—	4	4	no	—
323	C₄×C₃₁	—	4	2	no	—
324	C₄×C₃₁	—	4	2	no	—
325	C₄×C₃₁	—	4	1	no	—
326	C₄×C₃₁	—	4	1	no	—
327	C₄×C₃₁	—	4	1	no	—
328	C₄×C₃₁	—	4	2	no	—
329	C₄×C₃₁	C₂×#104	1	2	yes	—
330	#556	C₄×C₃₁	1	1	yes	characteristic, complemented
331	#554	—	2	1	no	—
332	#552	C₄×C₃₁	1	2	yes	complemented
333	#551	—	2	2	no	—
334	D₄×Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
335	#548	C₄×C₃₁	1	2	yes	complemented
336	#547	C₄×C₃₁	1	1	yes	characteristic, complemented
337	#547	C₄×C₃₁	1	1	yes	characteristic, complemented
338	#546	—	2	1	no	—
339	#542	—	2	1	no	—
340	#539	—	2	2	no	—
341	#533	C₂²×C₃₁	1	1	yes	characteristic
342	#529	C₂²×C₃₁	1	1	yes	characteristic
343	#525	C₂²×C₃₁	1	2	yes	—
344	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
345	#520	—	2	1	no	—
346	C₂×C₄×D₄	—	2	1	no	—
347	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
348	#436	—	2	1	no	—
349	#435	—	2	2	no	—
350	C₈×D₄	—	2	1	no	—
351	#432	C₂²×C₃₁	1	1	yes	characteristic
352	#431	C₂²×C₃₁	1	1	yes	characteristic
353	#407	C₂²×C₃₁	1	2	yes	—
354	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
355	C₄×C₄⋊C₄	—	2	1	no	—
356	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
357	C₄×C₂²⋊C₄	—	2	2	no	—
358	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
359	#376	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×#95	C₂²×C₃₁	1	2	yes	—
361	#328	C₂²×C₃₁	1	1	yes	characteristic
362	#328	C₂²×C₃₁	1	2	yes	—
363	#328	C₂²×C₃₁	1	2	yes	—
364	#328	C₂²×C₃₁	1	2	yes	—
365	#328	C₂²×C₃₁	1	1	yes	characteristic
366	#328	C₂²×C₃₁	1	2	yes	—
367	#328	C₂²×C₃₁	1	2	yes	—
368	#327	C₂²×C₃₁	1	1	yes	characteristic
369	#327	C₂²×C₃₁	1	1	yes	characteristic
370	#327	C₂²×C₃₁	1	2	yes	—
371	C₂×C₃₁	—	2	1	no	—
372	C₂×C₃₁	—	2	1	no	—
373	C₂×C₃₁	—	2	4	no	—
374	C₂×C₃₁	—	4	2	no	—
375	C₂×C₃₁	—	8	1	no	—
376	C₂×C₃₁	#2883	1	1	yes	characteristic
377	C₂×C₃₁	#2898	1	1	yes	characteristic
378	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
379	C₂×#41	—	2	1	no	—
380	C₂×#41	C₂×C₄×C₃₁	1	2	yes	complemented
381	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
382	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
383	C₂³×C₄	—	2	2	no	—
384	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
385	C₂×M₁₆	—	2	2	no	—
386	C₂²×C₈	—	2	2	no	—
387	C₄⋊Q₈	—	2	1	no	—
388	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
389	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
390	#126	—	2	2	no	—
391	#126	—	2	2	no	—
392	#126	—	2	1	no	—
393	#125	—	2	2	no	—
394	#125	—	2	1	no	—
395	#125	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
396	#124	—	2	1	no	—
397	#124	—	2	2	no	—
398	#124	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
399	#123	—	2	2	no	—
400	#123	—	2	2	no	—
401	#123	—	2	1	no	—
402	#123	—	2	2	no	—
403	#123	—	2	2	no	—
404	C₂²⋊Q₈	—	2	2	no	—
405	C₂²⋊Q₈	—	2	2	no	—
406	C₂²⋊Q₈	—	2	1	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	—	2	1	no	—
409	C₂²⋊Q₈	—	2	2	no	—
410	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
411	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
412	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
413	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
414	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
415	#121	—	2	2	no	—
416	#121	—	2	1	no	—
417	#121	C₂×C₄×C₃₁	1	2	yes	complemented
418	#121	C₂³×C₃₁	1	1	yes	characteristic
419	#120	—	2	2	no	—
420	C₄×Q₈	—	2	1	no	—
421	C₄×Q₈	—	2	1	no	—
422	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
423	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
424	C₄×D₄	—	2	8	no	—
425	C₄×D₄	—	2	1	no	—
426	C₄×D₄	—	2	2	no	—
427	C₄×D₄	—	2	2	no	—
428	C₄×D₄	—	2	1	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
431	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
432	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
433	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
434	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
435	C₄×D₄	D₄×C₃₁	1	2	yes	—
436	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
437	#117	—	2	1	no	—
438	#117	—	2	2	no	—
439	#117	—	2	1	no	—
440	#117	C₂×C₄×C₃₁	1	2	yes	complemented
441	C₂×C₄⋊C₄	—	2	1	no	—
442	C₂×C₄⋊C₄	—	2	2	no	—
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	complemented
445	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
446	C₂×C₂²⋊C₄	—	2	2	no	—
447	C₂×C₂²⋊C₄	—	2	2	no	—
448	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
449	C₂×C₄²	—	2	1	no	—
450	C₂×C₄²	—	2	1	no	—
451	C₂×C₄²	—	2	2	no	—
452	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
453	C₄⋊C₈	—	2	1	no	—
454	C₄⋊C₈	—	2	1	no	—
455	C₂²⋊C₈	—	2	2	no	—
456	C₂²⋊C₈	—	2	2	no	—
457	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
458	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
459	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
460	#97	—	2	1	no	—
461	C₄×C₈	—	2	1	no	—
462	#95	—	2	2	no	—
463	#95	—	2	1	no	—
464	#95	C₂³×C₃₁	1	1	yes	characteristic
465	#95	C₂³×C₃₁	1	2	yes	—
466	#95	C₂³×C₃₁	1	1	yes	characteristic
467	C₃₁	#17548	1	1	yes	characteristic, direct product factor, Sylow
468	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
469	#41	—	4	4	no	—
470	#41	—	8	2	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	8	2	no	—
474	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
475	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	2	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
481	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
482	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
483	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	8	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	8	2	no	—
493	C₂²×C₄	—	8	2	no	—
494	C₂²×C₄	—	8	2	no	—
495	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
496	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
497	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
500	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
501	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
502	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
503	M₁₆	—	8	2	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	8	2	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
523	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
524	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
525	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
526	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
527	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
528	C₂²⋊C₄	—	2	8	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	1	no	—
534	C₂²⋊C₄	—	2	2	no	—
535	C₂²⋊C₄	—	2	1	no	—
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	1	no	—
539	C₂²⋊C₄	—	2	1	no	—
540	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
541	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
542	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
543	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
544	C₄²	—	2	4	no	—
545	C₄²	—	2	2	no	—
546	C₄²	—	2	2	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	8	1	no	—
549	C₄²	—	8	2	no	—
550	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
551	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
552	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
553	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
554	C₂³	—	2	4	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	4	2	no	—
559	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
560	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
561	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
562	Q₈	—	4	1	no	—
563	Q₈	—	4	2	no	—
564	Q₈	—	4	2	no	—
565	Q₈	—	8	2	no	—
566	D₄	—	2	4	no	—
567	D₄	—	2	2	no	—
568	D₄	—	4	1	no	—
569	D₄	—	4	4	no	—
570	D₄	—	4	1	no	—
571	D₄	—	4	1	no	—
572	D₄	—	8	2	no	—
573	D₄	C₃₁×#104	1	4	yes	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	4	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	8	2	no	—
600	C₂×C₄	—	8	2	no	—
601	C₂×C₄	—	8	8	no	—
602	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
603	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
604	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
605	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
606	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
607	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
608	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
610	C₈	—	4	1	no	—
611	C₈	—	4	2	no	—
612	C₈	—	4	1	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	2	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	4	4	no	—
621	C₂²	—	4	4	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	1	no	—
624	C₂²	—	4	1	no	—
625	C₂²	C₂×C₃₁×#104	1	4	yes	—
626	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
627	C₂²	C₃₁×#577	1	1	yes	characteristic
628	C₂²	C₃₁×#578	1	1	yes	characteristic
629	C₄	—	2	1	no	—
630	C₄	—	2	1	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	1	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	4	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	1	no	—
641	C₄	—	4	1	no	—
642	C₄	—	4	1	no	—
643	C₄	—	4	2	no	—
644	C₄	C₂×C₃₁×#104	1	2	yes	—
645	C₂	—	2	1	no	—
646	C₂	—	2	1	no	—
647	C₂	—	2	4	no	—
648	C₂	—	4	2	no	—
649	C₂	—	8	1	no	—
650	C₂	C₃₁×#2883	1	1	yes	characteristic
651	C₂	C₃₁×#2898	1	1	yes	characteristic
652	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3520	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2974	C₂	semidirect
3. C₃₁×#2275	C₂	semidirect
4. C₃₁×#1509	C₂	semidirect
5. C₃₁×#1508	C₂	semidirect
6. C₃₁×#1506	C₂	semidirect
7. C₃₁×#1504	C₂	semidirect
8. C₃₁×#1503	C₂	semidirect
9. C₃₁×#1501	C₂	semidirect
10. C₃₁×#1500	C₂	semidirect
11. C₂×C₃₁×#328	C₂	semidirect
12. C₃₁×#556	C₄	semidirect
14. C₃₁×#552	C₄	semidirect
16. D₄×Q₈×C₃₁	C₄	semidirect
17. C₃₁×#548	C₄	semidirect
18. C₃₁×#547	C₄	semidirect
19. C₃₁×#547	C₄	semidirect
54. C₂×#41×C₃₁	C₂×C₄	semidirect
55. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
62. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
63. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
69. C₃₁×#125	C₂×C₄	semidirect
72. C₃₁×#124	C₂×C₄	semidirect
84. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
85. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
86. C₃₁×C₂²⋊Q₈	C₂×C₄	semidirect
91. C₃₁×#121	C₂×C₄	semidirect
96. C₄×Q₈×C₃₁	C₂×C₄	semidirect
97. C₄×Q₈×C₃₁	C₂×C₄	semidirect
107. C₄×D₄×C₃₁	C₂×C₄	semidirect
114. C₃₁×#117	C₂×C₄	semidirect
118. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
227. #17548	C₃₁	direct (N × H)
288. #2974	C₂×C₃₁	semidirect
289. #2275	C₂×C₃₁	semidirect
290. #1509	C₂×C₃₁	semidirect
291. #1508	C₂×C₃₁	semidirect
292. #1506	C₂×C₃₁	semidirect
293. #1504	C₂×C₃₁	semidirect
294. #1503	C₂×C₃₁	semidirect
295. #1501	C₂×C₃₁	semidirect
296. #1500	C₂×C₃₁	semidirect
297. C₂×#328	C₂×C₃₁	semidirect
330. #556	C₄×C₃₁	semidirect
332. #552	C₄×C₃₁	semidirect
334. D₄×Q₈	C₄×C₃₁	semidirect
335. #548	C₄×C₃₁	semidirect
336. #547	C₄×C₃₁	semidirect
337. #547	C₄×C₃₁	semidirect
380. C₂×#41	C₂×C₄×C₃₁	semidirect
381. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
388. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
389. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
395. #125	C₂×C₄×C₃₁	semidirect
398. #124	C₂×C₄×C₃₁	semidirect
410. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
411. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
412. C₂²⋊Q₈	C₂×C₄×C₃₁	semidirect
417. #121	C₂×C₄×C₃₁	semidirect
422. C₄×Q₈	C₂×C₄×C₃₁	semidirect
423. C₄×Q₈	C₂×C₄×C₃₁	semidirect
433. C₄×D₄	C₂×C₄×C₃₁	semidirect
440. #117	C₂×C₄×C₃₁	semidirect
444. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
467. C₃₁	#17548	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3520	2
C₃₁×#2974	2
C₃₁×#2275	2
C₃₁×#1509	2
C₃₁×#1508	2
C₃₁×#1506	2
C₃₁×#1504	2
C₃₁×#1503	2
C₃₁×#1501	2
C₃₁×#1500	2
C₂×C₃₁×#328	2
#17548	31

Subgroups — C31×#17548 #21474307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17548 #21474307