Subgroups — C₂×C₃₁×#1757 #21477159

Order 7936.

1474 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#413	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#410	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#405	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1757	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×#344	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#131	—	2	2	no	—
9	C₂×C₃₁×#131	—	2	4	no	—
10	C₂×C₃₁×#131	—	2	2	no	—
11	C₂×C₃₁×#131	C₄	1	4	yes	complemented
12	C₂×C₃₁×#131	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₃₁	C₄	1	2	yes	complemented
14	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₃₁	C₄	1	2	yes	complemented
17	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#117	C₂²	1	2	yes	—
19	C₃₁×#413	C₄	1	4	yes	complemented
20	C₃₁×#413	C₂²	1	4	yes	—
21	C₂×C₃₁×#101	C₄	1	2	yes	complemented
22	C₂×C₃₁×#100	C₄	1	2	yes	complemented
23	C₃₁×#410	C₄	1	4	yes	—
24	C₃₁×#410	C₂²	1	4	yes	—
25	C₂×C₃₁×#99	C₄	1	4	yes	—
26	C₃₁×#405	—	2	4	no	—
27	C₃₁×#405	C₂²	1	8	yes	complemented
28	C₂×C₃₁×#97	—	2	2	no	—
29	C₂×C₄×C₈×C₃₁	—	2	2	no	—
30	C₃₁×#344	C₂²	1	8	yes	complemented
31	C₂×#41×C₃₁	—	2	2	no	—
32	C₂×#41×C₃₁	—	2	2	no	—
33	C₂×#41×C₃₁	—	2	2	no	—
34	C₂×#41×C₃₁	—	2	2	no	—
35	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×#41×C₃₁	C₂³	1	2	yes	—
39	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×#131	—	2	8	no	—
45	C₃₁×#131	—	2	8	no	—
46	C₃₁×#131	—	2	8	no	—
47	C₃₁×#131	—	2	8	no	—
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	4	no	—
54	C₂×M₁₆×C₃₁	—	2	4	no	—
55	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
56	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
57	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
59	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₂²×C₈×C₃₁	—	2	4	no	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	—	2	2	no	—
65	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
68	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
70	C₂²×C₈×C₃₁	D₄	1	2	yes	—
71	C₂²×C₈×C₃₁	D₄	1	4	yes	—
72	C₂²×C₈×C₃₁	D₄	1	2	yes	—
73	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
74	C₃₁×#117	—	2	2	no	—
75	C₃₁×#117	—	2	2	no	—
76	C₃₁×#117	C₂×C₄	1	4	yes	—
77	C₃₁×#117	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
80	C₂×C₄²×C₃₁	—	2	2	no	—
81	C₃₁×#101	C₂×C₄	1	8	yes	complemented
82	C₃₁×#100	C₂×C₄	1	8	yes	complemented
83	C₃₁×#99	C₂×C₄	1	16	yes	—
84	C₃₁×#97	—	2	8	no	—
85	C₄×C₈×C₃₁	—	2	8	no	—
86	C₂⁴×C₃₁	C₄²	1	2	yes	—
87	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	#41×C₃₁	—	2	4	no	—
89	#41×C₃₁	—	2	4	no	—
90	#41×C₃₁	—	2	4	no	—
91	#41×C₃₁	—	2	4	no	—
92	#41×C₃₁	—	2	4	no	—
93	#41×C₃₁	—	2	4	no	—
94	#41×C₃₁	—	2	4	no	—
95	#41×C₃₁	—	2	4	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	C₄²	1	2	yes	—
99	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂×D₄×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	2	4	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	C₄²	1	4	yes	—
105	C₂×D₄×C₃₁	C₄²	1	2	yes	—
106	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
107	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
114	C₂²×C₄×C₃₁	C₄²	1	4	yes	—
115	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
116	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₃₁	#41	1	2	yes	—
124	C₂²×C₄×C₃₁	#41	1	2	yes	—
125	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	M₁₆×C₃₁	—	2	4	no	—
127	M₁₆×C₃₁	—	2	4	no	—
128	M₁₆×C₃₁	—	2	4	no	—
129	M₁₆×C₃₁	—	2	4	no	—
130	M₁₆×C₃₁	—	2	4	no	—
131	M₁₆×C₃₁	—	2	16	no	—
132	M₁₆×C₃₁	C₂²×C₄	1	8	yes	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	4	no	—
136	C₂×C₈×C₃₁	—	2	4	no	—
137	C₂×C₈×C₃₁	—	2	4	no	—
138	C₂×C₈×C₃₁	—	2	8	no	—
139	C₂×C₈×C₃₁	—	2	8	no	—
140	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
142	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
143	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
144	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
145	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
146	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
148	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
149	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
150	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
151	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₂²⋊C₄×C₃₁	—	2	4	no	—
155	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
156	C₄²×C₃₁	—	2	4	no	—
157	C₄²×C₃₁	—	2	4	no	—
158	C₂×#1757	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	2	2	no	—
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	4	no	—
163	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
164	C₂³×C₃₁	C₂×C₄²	1	4	yes	—
165	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
166	C₂³×C₃₁	C₄×D₄	1	2	yes	—
167	C₂³×C₃₁	C₄×D₄	1	2	yes	—
168	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
169	Q₈×C₃₁	—	2	4	no	—
170	Q₈×C₃₁	—	2	4	no	—
171	D₄×C₃₁	—	2	8	no	—
172	D₄×C₃₁	—	2	8	no	—
173	D₄×C₃₁	—	2	4	no	—
174	D₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	2	4	no	—
180	C₂×C₄×C₃₁	—	2	4	no	—
181	C₂×C₄×C₃₁	—	2	8	no	—
182	C₂×C₄×C₃₁	—	4	8	no	—
183	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
184	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
189	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
190	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
191	C₂×C₄×C₃₁	#117	1	2	yes	—
192	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
194	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
195	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
196	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
197	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
198	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
201	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
202	C₈×C₃₁	—	2	4	no	—
203	C₈×C₃₁	—	2	8	no	—
204	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
205	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
206	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
207	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic
208	C₂×#413	C₂×C₃₁	1	1	yes	characteristic
209	C₂×#410	C₂×C₃₁	1	1	yes	characteristic
210	C₂×#405	C₂×C₃₁	1	2	yes	complemented
211	#1757	C₂×C₃₁	1	8	yes	direct product factor
212	C₂×#344	C₂×C₃₁	1	2	yes	complemented
213	C₂²×C₃₁	—	2	4	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	4	no	—
216	C₂²×C₃₁	—	4	1	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	1	no	—
219	C₂²×C₃₁	—	4	4	no	—
220	C₂²×C₃₁	—	4	4	no	—
221	C₂²×C₃₁	—	4	4	no	—
222	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
223	C₂²×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
224	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
226	C₂²×C₃₁	C₂×C₄×D₄	1	2	yes	—
227	C₄×C₃₁	—	2	4	no	—
228	C₄×C₃₁	—	4	8	no	—
229	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
230	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
231	C₄×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
232	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
233	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
234	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
235	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#131	—	2	2	no	—
237	C₂×#131	—	2	2	no	—
238	C₂×#131	—	2	4	no	—
239	C₂×#131	C₄×C₃₁	1	4	yes	complemented
240	C₂×#131	C₂²×C₃₁	1	4	yes	—
241	C₂²×M₁₆	C₄×C₃₁	1	2	yes	complemented
242	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
243	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
244	C₂³×C₈	C₄×C₃₁	1	2	yes	complemented
245	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#117	C₂²×C₃₁	1	2	yes	—
247	#413	C₄×C₃₁	1	4	yes	complemented
248	#413	C₂²×C₃₁	1	4	yes	—
249	C₂×#101	C₄×C₃₁	1	2	yes	complemented
250	C₂×#100	C₄×C₃₁	1	2	yes	complemented
251	#410	C₄×C₃₁	1	4	yes	—
252	#410	C₂²×C₃₁	1	4	yes	—
253	C₂×#99	C₄×C₃₁	1	4	yes	—
254	#405	—	2	4	no	—
255	#405	C₂²×C₃₁	1	8	yes	complemented
256	C₂×#97	—	2	2	no	—
257	C₂×C₄×C₈	—	2	2	no	—
258	#344	C₂²×C₃₁	1	8	yes	complemented
259	C₂×C₃₁	—	2	2	no	—
260	C₂×C₃₁	—	2	1	no	—
261	C₂×C₃₁	—	2	1	no	—
262	C₂×C₃₁	—	2	2	no	—
263	C₂×C₃₁	—	4	4	no	—
264	C₂×C₃₁	#1757	1	2	yes	direct product factor
265	C₂×C₃₁	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×#41	—	2	2	no	—
267	C₂×#41	—	2	2	no	—
268	C₂×#41	—	2	2	no	—
269	C₂×#41	—	2	2	no	—
270	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
271	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
272	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
273	C₂×#41	C₂³×C₃₁	1	2	yes	—
274	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
275	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
276	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
277	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
278	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
279	#131	—	2	8	no	—
280	#131	—	2	8	no	—
281	#131	—	2	8	no	—
282	#131	—	2	8	no	—
283	C₂×M₁₆	—	2	2	no	—
284	C₂×M₁₆	—	2	2	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	—	2	2	no	—
287	C₂×M₁₆	—	2	2	no	—
288	C₂×M₁₆	—	2	4	no	—
289	C₂×M₁₆	—	2	4	no	—
290	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
291	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
292	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
293	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
294	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
295	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
296	C₂²×C₈	—	2	2	no	—
297	C₂²×C₈	—	2	2	no	—
298	C₂²×C₈	—	2	4	no	—
299	C₂²×C₈	—	2	2	no	—
300	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
301	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
303	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
304	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
305	C₂²×C₈	D₄×C₃₁	1	2	yes	—
306	C₂²×C₈	D₄×C₃₁	1	2	yes	—
307	C₂²×C₈	D₄×C₃₁	1	4	yes	—
308	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
309	#117	—	2	2	no	—
310	#117	—	2	2	no	—
311	#117	C₂×C₄×C₃₁	1	4	yes	—
312	#117	C₂³×C₃₁	1	4	yes	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
315	C₂×C₄²	—	2	2	no	—
316	#101	C₂×C₄×C₃₁	1	8	yes	complemented
317	#100	C₂×C₄×C₃₁	1	8	yes	complemented
318	#99	C₂×C₄×C₃₁	1	16	yes	—
319	#97	—	2	8	no	—
320	C₄×C₈	—	2	8	no	—
321	C₃₁	C₂×#1757	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁴	C₄²×C₃₁	1	2	yes	—
323	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
324	#41	—	2	4	no	—
325	#41	—	2	4	no	—
326	#41	—	2	4	no	—
327	#41	—	2	4	no	—
328	#41	—	2	4	no	—
329	#41	—	2	4	no	—
330	#41	—	2	4	no	—
331	#41	—	2	4	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	C₄²×C₃₁	1	2	yes	—
335	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	4	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	C₄²×C₃₁	1	4	yes	—
341	C₂×D₄	C₄²×C₃₁	1	2	yes	—
342	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
343	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
350	C₂²×C₄	C₄²×C₃₁	1	4	yes	—
351	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
352	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
358	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
359	C₂²×C₄	#41×C₃₁	1	2	yes	—
360	C₂²×C₄	#41×C₃₁	1	2	yes	—
361	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
362	M₁₆	—	2	4	no	—
363	M₁₆	—	2	4	no	—
364	M₁₆	—	2	4	no	—
365	M₁₆	—	2	4	no	—
366	M₁₆	—	2	4	no	—
367	M₁₆	—	2	16	no	—
368	M₁₆	C₂²×C₄×C₃₁	1	8	yes	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	4	no	—
372	C₂×C₈	—	2	4	no	—
373	C₂×C₈	—	2	4	no	—
374	C₂×C₈	—	2	8	no	—
375	C₂×C₈	—	2	8	no	—
376	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
377	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
378	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
379	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
380	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
381	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
382	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
383	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
384	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
385	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
386	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
387	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	4	no	—
390	C₂²⋊C₄	—	2	4	no	—
391	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
392	C₄²	—	2	4	no	—
393	C₄²	—	2	4	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	4	4	no	—
398	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
399	C₂³	C₂×C₄²×C₃₁	1	4	yes	—
400	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
401	C₂³	C₄×D₄×C₃₁	1	2	yes	—
402	C₂³	C₄×D₄×C₃₁	1	2	yes	—
403	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
404	Q₈	—	2	4	no	—
405	Q₈	—	2	4	no	—
406	D₄	—	2	8	no	—
407	D₄	—	2	8	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	4	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	8	no	—
417	C₂×C₄	—	4	8	no	—
418	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
419	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
421	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
424	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
425	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
426	C₂×C₄	C₃₁×#117	1	2	yes	—
427	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
428	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
429	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
430	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
431	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
432	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
433	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
436	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
437	C₈	—	2	4	no	—
438	C₈	—	2	8	no	—
439	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
440	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
441	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
442	C₂²	—	2	4	no	—
443	C₂²	—	2	2	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	1	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	4	4	no	—
450	C₂²	—	4	4	no	—
451	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
452	C₂²	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
453	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
454	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
455	C₂²	C₂×C₄×D₄×C₃₁	1	2	yes	—
456	C₄	—	2	4	no	—
457	C₄	—	4	8	no	—
458	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
459	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
460	C₄	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
461	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
462	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
463	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
464	C₂	—	2	2	no	—
465	C₂	—	2	1	no	—
466	C₂	—	2	1	no	—
467	C₂	—	2	2	no	—
468	C₂	—	4	4	no	—
469	C₂	C₃₁×#1757	1	2	yes	direct product factor
470	C₂	C₂×C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₄×C₂²⋊C₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₃₁×#405	C₂	semidirect
5. C₃₁×#1757	C₂	direct (N × H)
6. C₂×C₃₁×#344	C₂	semidirect
11. C₂×C₃₁×#131	C₄	semidirect
13. C₂²×M₁₆×C₃₁	C₄	semidirect
16. C₂³×C₈×C₃₁	C₄	semidirect
19. C₃₁×#413	C₄	semidirect
21. C₂×C₃₁×#101	C₄	semidirect
22. C₂×C₃₁×#100	C₄	semidirect
27. C₃₁×#405	C₂²	semidirect
30. C₃₁×#344	C₂²	semidirect
58. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
69. C₂²×C₈×C₃₁	C₂×C₄	semidirect
81. C₃₁×#101	C₂×C₄	semidirect
82. C₃₁×#100	C₂×C₄	semidirect
158. C₂×#1757	C₃₁	direct (N × H)
210. C₂×#405	C₂×C₃₁	semidirect
211. #1757	C₂×C₃₁	direct (N × H)
212. C₂×#344	C₂×C₃₁	semidirect
239. C₂×#131	C₄×C₃₁	semidirect
241. C₂²×M₁₆	C₄×C₃₁	semidirect
244. C₂³×C₈	C₄×C₃₁	semidirect
247. #413	C₄×C₃₁	semidirect
249. C₂×#101	C₄×C₃₁	semidirect
250. C₂×#100	C₄×C₃₁	semidirect
255. #405	C₂²×C₃₁	semidirect
258. #344	C₂²×C₃₁	semidirect
264. C₂×C₃₁	#1757	direct (N × H)
293. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
304. C₂²×C₈	C₂×C₄×C₃₁	semidirect
316. #101	C₂×C₄×C₃₁	semidirect
317. #100	C₂×C₄×C₃₁	semidirect
321. C₃₁	C₂×#1757	direct (N × H)
469. C₂	C₃₁×#1757	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#131	2
C₂×C₃₁×#413	2
C₂×C₃₁×#410	2
C₂×C₃₁×#405	2
C₃₁×#1757	2
C₂×C₃₁×#344	2
C₂×#1757	31

Subgroups — C2×C31×#1757 #21477159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1757 #21477159