Subgroups — C₃₁×#17544 #21474303

Order 7936.

1364 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1507	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1505	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1502	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₃₁	—	2	1	no	—
8	C₃₁×#532	—	2	1	no	—
9	C₃₁×#532	—	2	1	no	—
10	C₃₁×#532	—	2	1	no	—
11	C₃₁×#532	—	2	1	no	—
12	C₃₁×#532	C₄	1	2	yes	complemented
13	C₃₁×#532	C₄	1	2	yes	complemented
14	C₃₁×#532	C₂²	1	2	yes	—
15	C₃₁×#532	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
17	C₂×C₃₁×#127	C₄	1	1	yes	characteristic, complemented
18	C₂×C₃₁×#124	—	2	1	no	—
19	C₂×C₃₁×#121	—	2	1	no	—
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₃₁	—	2	1	no	—
22	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	complemented
23	C₂×C₃₁×#117	—	2	1	no	—
24	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#432	—	2	1	no	—
26	C₃₁×#432	C₂²	1	2	yes	complemented
27	C₃₁×#431	—	2	1	no	—
28	C₃₁×#431	C₂²	1	2	yes	complemented
29	C₃₁×#423	—	2	1	no	—
30	C₃₁×#407	C₂²	1	2	yes	—
31	C₄×M₁₆×C₃₁	—	2	1	no	—
32	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
34	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
35	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#376	C₂²	1	2	yes	complemented
37	C₄³×C₃₁	—	2	1	no	—
38	C₃₁×#327	C₂²	1	16	yes	complemented
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×#41×C₃₁	—	2	2	no	—
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂×#41×C₃₁	—	2	2	no	—
43	C₂×#41×C₃₁	—	4	8	no	—
44	C₂²×Q₈×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
50	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₃₁×C₄⋊Q₈	—	2	1	no	—
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
57	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
58	C₃₁×#127	—	2	1	no	—
59	C₃₁×#127	—	2	1	no	—
60	C₃₁×#127	C₂×C₄	1	2	yes	—
61	C₃₁×#127	C₂×C₄	1	2	yes	—
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×#124	—	2	2	no	—
64	C₃₁×#124	—	2	2	no	—
65	C₃₁×#124	—	2	2	no	—
66	C₃₁×#121	—	2	2	no	—
67	C₃₁×#121	—	2	2	no	—
68	C₃₁×#121	—	2	2	no	—
69	C₃₁×#121	—	2	2	no	—
70	C₃₁×#121	—	2	2	no	—
71	C₃₁×#121	—	2	2	no	—
72	C₃₁×#121	C₂×C₄	1	4	yes	—
73	C₃₁×#121	C₂³	1	4	yes	—
74	C₄×D₄×C₃₁	—	2	2	no	—
75	C₄×D₄×C₃₁	—	2	2	no	—
76	C₄×D₄×C₃₁	—	2	2	no	—
77	C₄×D₄×C₃₁	—	2	2	no	—
78	C₄×D₄×C₃₁	—	2	2	no	—
79	C₄×D₄×C₃₁	—	2	2	no	—
80	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
81	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
82	C₃₁×#117	—	2	8	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₄²×C₃₁	—	2	2	no	—
90	C₂×C₄²×C₃₁	—	2	2	no	—
91	C₂×C₄²×C₃₁	—	2	2	no	—
92	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₃₁×C₄⋊C₈	—	2	2	no	—
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×C₂²⋊C₈	—	2	2	no	—
107	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
108	C₃₁×#97	—	2	2	no	—
109	C₄×C₈×C₃₁	—	2	2	no	—
110	C₃₁×#95	—	2	2	no	—
111	C₃₁×#95	C₂³	1	4	yes	—
112	C₂⁴×C₃₁	—	2	2	no	—
113	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
114	#41×C₃₁	—	4	8	no	—
115	#41×C₃₁	—	4	8	no	—
116	#41×C₃₁	—	4	8	no	—
117	#41×C₃₁	—	4	8	no	—
118	C₂×Q₈×C₃₁	—	2	2	no	—
119	C₂×Q₈×C₃₁	—	2	2	no	—
120	C₂×Q₈×C₃₁	—	4	4	no	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	4	4	no	—
129	C₂×D₄×C₃₁	—	4	4	no	—
130	C₂×D₄×C₃₁	—	4	8	no	—
131	C₂×D₄×C₃₁	—	4	4	no	—
132	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	4	2	no	—
140	C₂²×C₄×C₃₁	—	4	8	no	—
141	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
159	M₁₆×C₃₁	—	4	4	no	—
160	C₂×C₈×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	8	no	—
162	C₄⋊C₄×C₃₁	—	2	2	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
166	C₂²⋊C₄×C₃₁	—	2	8	no	—
167	C₂²⋊C₄×C₃₁	—	2	4	no	—
168	C₂²⋊C₄×C₃₁	—	2	4	no	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₄²×C₃₁	—	2	8	no	—
172	C₄²×C₃₁	—	4	4	no	—
173	C₄²×C₃₁	—	4	4	no	—
174	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
179	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
182	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
184	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
185	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	#17544	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	4	no	—
194	C₂³×C₃₁	—	4	1	no	—
195	C₂³×C₃₁	—	4	4	no	—
196	C₂³×C₃₁	—	4	4	no	—
197	C₂³×C₃₁	—	4	4	no	—
198	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
200	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
201	Q₈×C₃₁	—	4	4	no	—
202	Q₈×C₃₁	—	4	4	no	—
203	D₄×C₃₁	—	4	4	no	—
204	D₄×C₃₁	—	4	4	no	—
205	D₄×C₃₁	—	4	8	no	—
206	D₄×C₃₁	—	4	8	no	—
207	D₄×C₃₁	—	4	4	no	—
208	D₄×C₃₁	—	4	4	no	—
209	D₄×C₃₁	—	4	4	no	—
210	D₄×C₃₁	—	4	4	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	8	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	4	8	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	—	4	4	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	—	4	8	no	—
234	C₂×C₄×C₃₁	#104	1	4	yes	—
235	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
246	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
248	C₈×C₃₁	—	4	4	no	—
249	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
250	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1507	C₂×C₃₁	1	4	yes	complemented
253	#1505	C₂×C₃₁	1	4	yes	complemented
254	#1502	C₂×C₃₁	1	4	yes	complemented
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	2	1	no	—
257	C₂²×C₃₁	—	2	1	no	—
258	C₂²×C₃₁	—	2	1	no	—
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	1	no	—
262	C₂²×C₃₁	—	4	4	no	—
263	C₂²×C₃₁	—	4	4	no	—
264	C₂²×C₃₁	—	4	4	no	—
265	C₂²×C₃₁	—	4	4	no	—
266	C₂²×C₃₁	—	8	8	no	—
267	C₂²×C₃₁	C₂×#104	1	2	yes	—
268	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
269	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
270	C₄×C₃₁	—	2	1	no	—
271	C₄×C₃₁	—	2	1	no	—
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	2	1	no	—
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	2	1	no	—
276	C₄×C₃₁	—	2	1	no	—
277	C₄×C₃₁	—	4	8	no	—
278	C₄×C₃₁	—	4	4	no	—
279	C₄×C₃₁	#410	1	2	yes	—
280	C₄×C₃₁	#418	1	2	yes	—
281	C₄×C₃₁	#418	1	2	yes	—
282	C₄×C₃₁	C₂×#104	1	2	yes	—
283	C₄×C₃₁	C₂×#104	1	2	yes	—
284	C₂²×#41	—	2	1	no	—
285	#532	—	2	1	no	—
286	#532	—	2	1	no	—
287	#532	—	2	1	no	—
288	#532	—	2	1	no	—
289	#532	C₄×C₃₁	1	2	yes	complemented
290	#532	C₄×C₃₁	1	2	yes	complemented
291	#532	C₂²×C₃₁	1	2	yes	—
292	#532	C₂²×C₃₁	1	2	yes	—
293	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic, complemented
294	C₂×#127	C₄×C₃₁	1	1	yes	characteristic, complemented
295	C₂×#124	—	2	1	no	—
296	C₂×#121	—	2	1	no	—
297	C₂×#121	C₂²×C₃₁	1	2	yes	—
298	C₂×C₄×D₄	—	2	1	no	—
299	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	complemented
300	C₂×#117	—	2	1	no	—
301	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
302	#432	—	2	1	no	—
303	#432	C₂²×C₃₁	1	2	yes	complemented
304	#431	—	2	1	no	—
305	#431	C₂²×C₃₁	1	2	yes	complemented
306	#423	—	2	1	no	—
307	#407	C₂²×C₃₁	1	2	yes	—
308	C₄×M₁₆	—	2	1	no	—
309	C₄×C₄⋊C₄	—	2	1	no	—
310	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
311	C₄×C₂²⋊C₄	—	2	1	no	—
312	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
313	#376	C₂²×C₃₁	1	2	yes	complemented
314	C₄³	—	2	1	no	—
315	#327	C₂²×C₃₁	1	16	yes	complemented
316	C₂×C₃₁	—	2	2	no	—
317	C₂×C₃₁	—	2	2	no	—
318	C₂×C₃₁	—	4	1	no	—
319	C₂×C₃₁	—	8	4	no	—
320	C₂×C₃₁	C₂×#410	1	1	yes	characteristic
321	C₂×C₃₁	C₂×#418	1	1	yes	characteristic
322	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
323	C₂×#41	—	2	2	no	—
324	C₂×#41	—	2	2	no	—
325	C₂×#41	—	2	2	no	—
326	C₂×#41	—	2	2	no	—
327	C₂×#41	—	4	8	no	—
328	C₂²×Q₈	—	2	1	no	—
329	C₂²×D₄	—	2	2	no	—
330	C₂²×D₄	—	2	1	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
333	C₂³×C₄	—	2	2	no	—
334	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
335	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
336	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
337	C₂×M₁₆	—	2	2	no	—
338	C₄⋊Q₈	—	2	1	no	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
341	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
342	#127	—	2	1	no	—
343	#127	—	2	1	no	—
344	#127	C₂×C₄×C₃₁	1	2	yes	—
345	#127	C₂×C₄×C₃₁	1	2	yes	—
346	#124	—	2	2	no	—
347	#124	—	2	2	no	—
348	#124	—	2	2	no	—
349	#124	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	—	2	2	no	—
354	#121	—	2	2	no	—
355	#121	—	2	2	no	—
356	#121	C₂×C₄×C₃₁	1	4	yes	—
357	#121	C₂³×C₃₁	1	4	yes	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
365	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
366	#117	—	2	8	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₄²	—	2	2	no	—
374	C₂×C₄²	—	2	2	no	—
375	C₂×C₄²	—	2	2	no	—
376	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
379	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
388	C₄⋊C₈	—	2	2	no	—
389	C₄⋊C₈	—	2	2	no	—
390	C₂²⋊C₈	—	2	2	no	—
391	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
392	#97	—	2	2	no	—
393	C₄×C₈	—	2	2	no	—
394	#95	—	2	2	no	—
395	#95	C₂³×C₃₁	1	4	yes	—
396	C₃₁	#17544	1	1	yes	characteristic, direct product factor, Sylow
397	C₂⁴	—	2	2	no	—
398	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
399	#41	—	4	8	no	—
400	#41	—	4	8	no	—
401	#41	—	4	8	no	—
402	#41	—	4	8	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	4	4	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	4	4	no	—
414	C₂×D₄	—	4	4	no	—
415	C₂×D₄	—	4	8	no	—
416	C₂×D₄	—	4	4	no	—
417	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	4	2	no	—
425	C₂²×C₄	—	4	8	no	—
426	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
427	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
428	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
444	M₁₆	—	4	4	no	—
445	C₂×C₈	—	2	4	no	—
446	C₄⋊C₄	—	2	8	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	4	no	—
456	C₄²	—	2	8	no	—
457	C₄²	—	4	4	no	—
458	C₄²	—	4	4	no	—
459	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
460	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
461	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
462	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
464	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
467	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
468	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
469	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
470	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
471	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₂³	—	2	2	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	4	1	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	4	4	no	—
481	C₂³	—	4	4	no	—
482	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
483	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
484	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
485	Q₈	—	4	4	no	—
486	Q₈	—	4	4	no	—
487	D₄	—	4	4	no	—
488	D₄	—	4	4	no	—
489	D₄	—	4	8	no	—
490	D₄	—	4	8	no	—
491	D₄	—	4	4	no	—
492	D₄	—	4	4	no	—
493	D₄	—	4	4	no	—
494	D₄	—	4	4	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	C₃₁×#104	1	4	yes	—
519	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
520	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
521	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
524	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
525	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
526	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
527	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
530	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
531	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
532	C₈	—	4	4	no	—
533	C₂²	—	2	1	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	8	8	no	—
545	C₂²	C₂×C₃₁×#104	1	2	yes	—
546	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
547	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	2	1	no	—
555	C₄	—	4	8	no	—
556	C₄	—	4	4	no	—
557	C₄	C₃₁×#410	1	2	yes	—
558	C₄	C₃₁×#418	1	2	yes	—
559	C₄	C₃₁×#418	1	2	yes	—
560	C₄	C₂×C₃₁×#104	1	2	yes	—
561	C₄	C₂×C₃₁×#104	1	2	yes	—
562	C₂	—	2	2	no	—
563	C₂	—	2	2	no	—
564	C₂	—	4	1	no	—
565	C₂	—	8	4	no	—
566	C₂	C₂×C₃₁×#410	1	1	yes	characteristic
567	C₂	C₂×C₃₁×#418	1	1	yes	characteristic
568	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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. C₄×C₃₁×#117	C₂	semidirect
4. C₃₁×#1507	C₂	semidirect
5. C₃₁×#1505	C₂	semidirect
6. C₃₁×#1502	C₂	semidirect
12. C₃₁×#532	C₄	semidirect
13. C₃₁×#532	C₄	semidirect
16. C₂×C₃₁×C₄⋊Q₈	C₄	semidirect
17. C₂×C₃₁×#127	C₄	semidirect
22. C₂×C₄×D₄×C₃₁	C₄	semidirect
26. C₃₁×#432	C₂²	semidirect
28. C₃₁×#431	C₂²	semidirect
33. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
36. C₃₁×#376	C₂²	semidirect
38. C₃₁×#327	C₂²	semidirect
190. #17544	C₃₁	direct (N × H)
250. #2920	C₂×C₃₁	semidirect
251. C₄×#117	C₂×C₃₁	semidirect
252. #1507	C₂×C₃₁	semidirect
253. #1505	C₂×C₃₁	semidirect
254. #1502	C₂×C₃₁	semidirect
289. #532	C₄×C₃₁	semidirect
290. #532	C₄×C₃₁	semidirect
293. C₂×C₄⋊Q₈	C₄×C₃₁	semidirect
294. C₂×#127	C₄×C₃₁	semidirect
299. C₂×C₄×D₄	C₄×C₃₁	semidirect
303. #432	C₂²×C₃₁	semidirect
305. #431	C₂²×C₃₁	semidirect
310. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
313. #376	C₂²×C₃₁	semidirect
315. #327	C₂²×C₃₁	semidirect
396. C₃₁	#17544	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×#532	2
C₃₁×#2920	2
C₄×C₃₁×#117	2
C₃₁×#1507	2
C₃₁×#1505	2
C₃₁×#1502	2
#17544	31

Subgroups — C31×#17544 #21474303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17544 #21474303