Subgroups — C₂×C₃₁×#1855 #21477609

Order 7936.

1470 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#108	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#407	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1855	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×#341	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₁	—	2	2	no	—
9	C₂²×M₁₆×C₃₁	—	2	1	no	—
10	C₂²×M₁₆×C₃₁	—	2	8	no	—
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂³×C₈×C₃₁	—	2	1	no	—
16	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₃₁×#108	—	2	8	no	—
18	C₂×C₃₁×#108	C₂²	1	8	yes	complemented
19	C₃₁×#407	—	2	2	no	—
20	C₃₁×#407	C₂²	1	4	yes	—
21	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
22	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
23	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
24	C₂×C₃₁×C₂²⋊C₈	C₄	1	4	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
27	C₃₁×#341	C₂²	1	8	yes	complemented
28	C₂⁵×C₃₁	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	—	2	4	no	—
30	C₂³×C₄×C₃₁	—	2	4	no	—
31	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₃₁	D₄	1	2	yes	—
35	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	Q₈	1	2	yes	—
39	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₃₁	—	2	16	no	—
42	C₂×M₁₆×C₃₁	—	2	8	no	—
43	C₂×M₁₆×C₃₁	—	2	4	no	—
44	C₂×M₁₆×C₃₁	—	2	4	no	—
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	—	2	16	no	—
47	C₂×M₁₆×C₃₁	—	2	1	no	—
48	C₂×M₁₆×C₃₁	—	2	1	no	—
49	C₂×M₁₆×C₃₁	—	2	4	no	—
50	C₂×M₁₆×C₃₁	—	2	4	no	—
51	C₂×M₁₆×C₃₁	D₄	1	8	yes	—
52	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
53	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
54	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
55	C₂²×C₈×C₃₁	—	2	8	no	—
56	C₂²×C₈×C₃₁	—	2	4	no	—
57	C₂²×C₈×C₃₁	—	2	4	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	2	4	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	—	2	1	no	—
62	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
63	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
64	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
65	C₃₁×#108	—	2	16	no	—
66	C₃₁×C₂²⋊C₈	—	2	4	no	—
67	C₃₁×C₂²⋊C₈	—	2	4	no	—
68	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
69	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
70	C₂⁴×C₃₁	—	2	4	no	—
71	C₂⁴×C₃₁	—	2	4	no	—
72	C₂⁴×C₃₁	—	4	2	no	—
73	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
74	C₂⁴×C₃₁	C₄⋊C₄	1	2	yes	—
75	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
76	C₂⁴×C₃₁	C₂×Q₈	1	2	yes	—
77	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	—	2	4	no	—
79	C₂²×C₄×C₃₁	—	2	4	no	—
80	C₂²×C₄×C₃₁	—	2	8	no	—
81	C₂²×C₄×C₃₁	—	2	4	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	4	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	8	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	4	no	—
89	C₂²×C₄×C₃₁	—	2	4	no	—
90	C₂²×C₄×C₃₁	—	2	4	no	—
91	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
98	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	#41	1	4	yes	—
110	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
111	M₁₆×C₃₁	—	2	16	no	—
112	M₁₆×C₃₁	—	2	8	no	—
113	M₁₆×C₃₁	—	2	4	no	—
114	M₁₆×C₃₁	—	2	4	no	—
115	M₁₆×C₃₁	—	2	4	no	—
116	M₁₆×C₃₁	—	4	2	no	—
117	M₁₆×C₃₁	—	4	2	no	—
118	M₁₆×C₃₁	C₂×D₄	1	8	yes	—
119	C₂×C₈×C₃₁	—	2	16	no	—
120	C₂×C₈×C₃₁	—	2	8	no	—
121	C₂×C₈×C₃₁	—	2	8	no	—
122	C₂×C₈×C₃₁	—	2	8	no	—
123	C₂×C₈×C₃₁	—	2	4	no	—
124	C₂×C₈×C₃₁	—	2	4	no	—
125	C₂×C₈×C₃₁	—	2	4	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	4	2	no	—
129	C₂×C₈×C₃₁	—	4	2	no	—
130	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
131	C₂×#1855	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₃₁	—	2	2	no	—
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	—	2	2	no	—
135	C₂³×C₃₁	—	2	2	no	—
136	C₂³×C₃₁	—	2	4	no	—
137	C₂³×C₃₁	—	2	4	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	—	2	4	no	—
141	C₂³×C₃₁	—	2	4	no	—
142	C₂³×C₃₁	—	2	2	no	—
143	C₂³×C₃₁	—	4	2	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	—	4	16	no	—
146	C₂³×C₃₁	#108	1	4	yes	complemented
147	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
148	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
150	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
152	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₃₁	—	2	4	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	4	no	—
158	C₂×C₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	4	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
168	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
169	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
172	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
173	C₂×C₄×C₃₁	#120	1	2	yes	—
174	C₂×C₄×C₃₁	#120	1	2	yes	—
175	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
176	C₂×C₄×C₃₁	#123	1	2	yes	—
177	C₂×C₄×C₃₁	#123	1	2	yes	—
178	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
182	C₈×C₃₁	—	2	8	no	—
183	C₈×C₃₁	—	2	8	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	—	4	2	no	—
186	C₂³×M₁₆	C₂×C₃₁	1	1	yes	characteristic
187	C₂²×#108	C₂×C₃₁	1	2	yes	complemented
188	C₂×#407	C₂×C₃₁	1	1	yes	characteristic
189	C₂²×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
190	#1855	C₂×C₃₁	1	8	yes	direct product factor
191	C₂×#341	C₂×C₃₁	1	2	yes	complemented
192	C₂²×C₃₁	—	2	8	no	—
193	C₂²×C₃₁	—	2	8	no	—
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	—	2	1	no	—
196	C₂²×C₃₁	—	2	1	no	—
197	C₂²×C₃₁	—	2	2	no	—
198	C₂²×C₃₁	—	2	1	no	—
199	C₂²×C₃₁	—	2	2	no	—
200	C₂²×C₃₁	—	2	2	no	—
201	C₂²×C₃₁	—	2	2	no	—
202	C₂²×C₃₁	—	2	2	no	—
203	C₂²×C₃₁	—	4	8	no	—
204	C₂²×C₃₁	—	4	8	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	#385	1	2	yes	—
207	C₂²×C₃₁	#385	1	2	yes	—
208	C₂²×C₃₁	C₂×#108	1	2	yes	—
209	C₂²×C₃₁	C₂×#108	1	4	yes	complemented
210	C₂²×C₃₁	#430	1	2	yes	—
211	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
212	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
213	C₄×C₃₁	—	2	2	no	—
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	#385	1	2	yes	—
216	C₄×C₃₁	#385	1	2	yes	—
217	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
218	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
219	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
220	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
221	C₂²×M₁₆	—	2	1	no	—
222	C₂²×M₁₆	—	2	2	no	—
223	C₂²×M₁₆	—	2	8	no	—
224	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
225	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
226	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
227	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
228	C₂³×C₈	—	2	1	no	—
229	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
230	C₂×#108	—	2	8	no	—
231	C₂×#108	C₂²×C₃₁	1	8	yes	complemented
232	#407	—	2	2	no	—
233	#407	C₂²×C₃₁	1	4	yes	—
234	C₂×C₂²⋊C₈	—	2	1	no	—
235	C₂×C₂²⋊C₈	—	2	1	no	—
236	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
237	C₂×C₂²⋊C₈	C₄×C₃₁	1	4	yes	—
238	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
239	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
240	#341	C₂²×C₃₁	1	8	yes	complemented
241	C₂×C₃₁	—	2	2	no	—
242	C₂×C₃₁	—	2	2	no	—
243	C₂×C₃₁	—	2	4	no	—
244	C₂×C₃₁	—	4	2	no	—
245	C₂×C₃₁	#1855	1	4	yes	direct product factor
246	C₂×C₃₁	C₂×#385	1	1	yes	characteristic
247	C₂×C₃₁	C₂²×#108	1	1	yes	characteristic
248	C₂×C₃₁	C₂×#430	1	1	yes	characteristic
249	C₂⁵	Q₈×C₃₁	1	1	yes	characteristic
250	C₂³×C₄	—	2	4	no	—
251	C₂³×C₄	—	2	4	no	—
252	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
253	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
254	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
255	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
256	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
257	C₂³×C₄	D₄×C₃₁	1	2	yes	—
258	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
259	C₂³×C₄	Q₈×C₃₁	1	2	yes	—
260	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
261	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
262	C₂×M₁₆	—	2	16	no	—
263	C₂×M₁₆	—	2	8	no	—
264	C₂×M₁₆	—	2	4	no	—
265	C₂×M₁₆	—	2	4	no	—
266	C₂×M₁₆	—	2	2	no	—
267	C₂×M₁₆	—	2	16	no	—
268	C₂×M₁₆	—	2	1	no	—
269	C₂×M₁₆	—	2	1	no	—
270	C₂×M₁₆	—	2	4	no	—
271	C₂×M₁₆	—	2	4	no	—
272	C₂×M₁₆	D₄×C₃₁	1	8	yes	—
273	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
274	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
275	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
276	C₂²×C₈	—	2	8	no	—
277	C₂²×C₈	—	2	4	no	—
278	C₂²×C₈	—	2	2	no	—
279	C₂²×C₈	—	2	4	no	—
280	C₂²×C₈	—	2	4	no	—
281	C₂²×C₈	—	2	1	no	—
282	C₂²×C₈	—	2	1	no	—
283	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
284	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
285	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
286	#108	—	2	16	no	—
287	C₂²⋊C₈	—	2	4	no	—
288	C₂²⋊C₈	—	2	4	no	—
289	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
290	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
291	C₃₁	C₂×#1855	1	1	yes	characteristic, direct product factor, Sylow
292	C₂⁴	—	2	4	no	—
293	C₂⁴	—	2	4	no	—
294	C₂⁴	—	4	2	no	—
295	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
296	C₂⁴	C₄⋊C₄×C₃₁	1	2	yes	—
297	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
298	C₂⁴	C₂×Q₈×C₃₁	1	2	yes	—
299	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	2	8	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	8	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	4	no	—
313	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
314	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
317	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
318	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
319	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
320	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
321	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
322	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
325	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
326	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
329	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
330	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	#41×C₃₁	1	4	yes	—
332	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
333	M₁₆	—	2	16	no	—
334	M₁₆	—	2	8	no	—
335	M₁₆	—	2	4	no	—
336	M₁₆	—	2	4	no	—
337	M₁₆	—	2	4	no	—
338	M₁₆	—	4	2	no	—
339	M₁₆	—	4	2	no	—
340	M₁₆	C₂×D₄×C₃₁	1	8	yes	—
341	C₂×C₈	—	2	16	no	—
342	C₂×C₈	—	2	8	no	—
343	C₂×C₈	—	2	8	no	—
344	C₂×C₈	—	2	8	no	—
345	C₂×C₈	—	2	4	no	—
346	C₂×C₈	—	2	4	no	—
347	C₂×C₈	—	2	4	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	4	2	no	—
351	C₂×C₈	—	4	2	no	—
352	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	2	2	no	—
357	C₂³	—	2	2	no	—
358	C₂³	—	2	4	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	2	4	no	—
362	C₂³	—	2	4	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	4	2	no	—
366	C₂³	—	4	16	no	—
367	C₂³	C₃₁×#108	1	4	yes	complemented
368	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
369	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
370	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
371	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
372	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
373	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
389	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
390	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
393	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
394	C₂×C₄	C₃₁×#120	1	2	yes	—
395	C₂×C₄	C₃₁×#120	1	2	yes	—
396	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
397	C₂×C₄	C₃₁×#123	1	2	yes	—
398	C₂×C₄	C₃₁×#123	1	2	yes	—
399	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
401	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
403	C₈	—	2	8	no	—
404	C₈	—	2	8	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	2	8	no	—
408	C₂²	—	2	8	no	—
409	C₂²	—	2	1	no	—
410	C₂²	—	2	1	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	1	no	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	2	no	—
415	C₂²	—	2	2	no	—
416	C₂²	—	2	2	no	—
417	C₂²	—	2	2	no	—
418	C₂²	—	4	8	no	—
419	C₂²	—	4	8	no	—
420	C₂²	—	4	4	no	—
421	C₂²	C₃₁×#385	1	2	yes	—
422	C₂²	C₃₁×#385	1	2	yes	—
423	C₂²	C₂×C₃₁×#108	1	2	yes	—
424	C₂²	C₂×C₃₁×#108	1	4	yes	complemented
425	C₂²	C₃₁×#430	1	2	yes	—
426	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
427	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
428	C₄	—	2	2	no	—
429	C₄	—	2	2	no	—
430	C₄	C₃₁×#385	1	2	yes	—
431	C₄	C₃₁×#385	1	2	yes	—
432	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
433	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
434	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
435	C₂	—	2	4	no	—
436	C₂	—	2	2	no	—
437	C₂	—	2	2	no	—
438	C₂	—	4	2	no	—
439	C₂	C₃₁×#1855	1	4	yes	direct product factor
440	C₂	C₂×C₃₁×#385	1	1	yes	characteristic
441	C₂	C₂²×C₃₁×#108	1	1	yes	characteristic
442	C₂	C₂×C₃₁×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁵×C₃₁	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#108	C₂	semidirect
5. C₃₁×#1855	C₂	direct (N × H)
6. C₂×C₃₁×#341	C₂	semidirect
18. C₂×C₃₁×#108	C₂²	semidirect
27. C₃₁×#341	C₂²	semidirect
131. C₂×#1855	C₃₁	direct (N × H)
146. C₂³×C₃₁	#108	semidirect
187. C₂²×#108	C₂×C₃₁	semidirect
190. #1855	C₂×C₃₁	direct (N × H)
191. C₂×#341	C₂×C₃₁	semidirect
209. C₂²×C₃₁	C₂×#108	semidirect
231. C₂×#108	C₂²×C₃₁	semidirect
240. #341	C₂²×C₃₁	semidirect
245. C₂×C₃₁	#1855	direct (N × H)
291. C₃₁	C₂×#1855	direct (N × H)
367. C₂³	C₃₁×#108	semidirect
424. C₂²	C₂×C₃₁×#108	semidirect
439. C₂	C₃₁×#1855	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₃₁	2
C₂²×C₃₁×#108	2
C₂×C₃₁×#407	2
C₂²×C₃₁×C₂²⋊C₈	2
C₃₁×#1855	2
C₂×C₃₁×#341	2
C₂×#1855	31

Subgroups — C2×C31×#1855 #21477609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1855 #21477609