Subgroups — C₂×C₃₁×#1816 #21477436

Order 7936.

1374 conjugacy classes of subgroups.

The subgroup lattice has 456 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	C₂²×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#417	C₂	1	2	yes	maximal
5	C₃₁×#1816	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×#343	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#131	—	2	4	no	—
9	C₂×C₃₁×#131	—	2	2	no	—
10	C₂×C₃₁×#131	C₄	1	8	yes	complemented
11	C₂×C₃₁×#131	C₂²	1	4	yes	—
12	C₂²×M₁₆×C₃₁	—	2	1	no	—
13	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	—	2	1	no	—
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₃₁×#108	—	2	4	no	—
19	C₂×C₃₁×#108	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#107	—	2	1	no	—
21	C₂×C₃₁×#107	—	2	1	no	—
22	C₂×C₃₁×#107	—	2	2	no	—
23	C₂×C₃₁×#107	C₂²	1	4	yes	complemented
24	C₃₁×#417	—	2	4	no	—
25	C₃₁×#417	C₂²	1	8	yes	—
26	C₂×C₃₁×#103	—	2	2	no	—
27	C₂×C₃₁×#102	—	2	2	no	—
28	C₃₁×#343	C₂²	1	8	yes	complemented
29	C₂×#41×C₃₁	—	2	2	no	—
30	C₂×#41×C₃₁	—	2	1	no	—
31	C₂×#41×C₃₁	—	2	1	no	—
32	C₂×#41×C₃₁	—	2	2	no	—
33	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
34	C₂×#41×C₃₁	D₄	1	2	yes	—
35	C₂×#41×C₃₁	Q₈	1	4	yes	—
36	C₂×#41×C₃₁	C₂³	1	2	yes	—
37	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×D₄×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	—	2	1	no	—
41	C₂³×C₄×C₃₁	—	2	2	no	—
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#131	—	2	16	no	—
44	C₃₁×#131	—	2	8	no	—
45	C₃₁×#131	C₂×C₄	1	16	yes	complemented
46	C₂×M₁₆×C₃₁	—	2	2	no	—
47	C₂×M₁₆×C₃₁	—	2	1	no	—
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	8	no	—
50	C₂×M₁₆×C₃₁	—	2	4	no	—
51	C₂×M₁₆×C₃₁	—	2	4	no	—
52	C₂×M₁₆×C₃₁	—	2	4	no	—
53	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
54	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
55	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	8	no	—
58	C₂²×C₈×C₃₁	—	2	4	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
61	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	D₄	1	2	yes	—
64	C₂²×C₈×C₃₁	D₄	1	4	yes	—
65	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
72	C₃₁×#108	—	2	8	no	—
73	C₃₁×#107	—	2	4	no	—
74	C₃₁×#107	—	2	4	no	—
75	C₃₁×#103	—	2	8	no	—
76	C₃₁×#102	—	2	8	no	—
77	C₂⁴×C₃₁	—	2	1	no	—
78	C₂⁴×C₃₁	#41	1	1	yes	characteristic
79	#41×C₃₁	—	2	4	no	—
80	#41×C₃₁	—	2	4	no	—
81	#41×C₃₁	—	2	4	no	—
82	#41×C₃₁	—	2	4	no	—
83	#41×C₃₁	—	2	4	no	—
84	#41×C₃₁	—	2	2	no	—
85	#41×C₃₁	—	2	2	no	—
86	#41×C₃₁	C₄⋊C₄	1	8	yes	—
87	#41×C₃₁	C₂²×C₄	1	4	yes	—
88	#41×C₃₁	C₂×Q₈	1	4	yes	—
89	C₂×Q₈×C₃₁	—	2	1	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
92	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×D₄×C₃₁	—	2	4	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
100	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	4	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	4	2	no	—
108	C₂²×C₄×C₃₁	—	4	2	no	—
109	C₂²×C₄×C₃₁	—	8	4	no	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
117	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	#41	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	M₁₆×C₃₁	—	2	4	no	—
121	M₁₆×C₃₁	—	2	2	no	—
122	M₁₆×C₃₁	—	2	16	no	—
123	M₁₆×C₃₁	—	2	4	no	—
124	M₁₆×C₃₁	—	2	4	no	—
125	M₁₆×C₃₁	—	4	4	no	—
126	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
127	C₂×C₈×C₃₁	—	2	4	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	16	no	—
130	C₂×C₈×C₃₁	—	2	4	no	—
131	C₂×C₈×C₃₁	—	4	4	no	—
132	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
133	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
134	C₂×C₈×C₃₁	C₂²⋊C₄	1	8	yes	—
135	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
136	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
139	C₄⋊C₄×C₃₁	—	2	4	no	—
140	C₄⋊C₄×C₃₁	—	2	4	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₂×#1816	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	2	4	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	#117	1	2	yes	—
151	C₂³×C₃₁	#121	1	1	yes	characteristic
152	C₂³×C₃₁	#121	1	1	yes	characteristic
153	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
154	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
155	Q₈×C₃₁	—	2	2	no	—
156	Q₈×C₃₁	—	2	4	no	—
157	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
158	D₄×C₃₁	—	2	4	no	—
159	D₄×C₃₁	—	2	4	no	—
160	D₄×C₃₁	—	2	8	no	—
161	D₄×C₃₁	—	2	2	no	—
162	D₄×C₃₁	—	2	4	no	—
163	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	8	no	—
169	C₂×C₄×C₃₁	—	4	4	no	—
170	C₂×C₄×C₃₁	—	8	4	no	—
171	C₂×C₄×C₃₁	—	8	4	no	—
172	C₂×C₄×C₃₁	—	8	4	no	—
173	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
174	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
175	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
176	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
177	C₂×C₄×C₃₁	#117	1	2	yes	—
178	C₂×C₄×C₃₁	#121	1	2	yes	—
179	C₂×C₄×C₃₁	#121	1	2	yes	—
180	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
181	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
183	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
184	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
185	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
191	C₈×C₃₁	—	2	4	no	—
192	C₈×C₃₁	—	4	4	no	—
193	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
194	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
195	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic
196	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
197	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂×#417	C₂×C₃₁	1	2	yes	—
199	#1816	C₂×C₃₁	1	8	yes	direct product factor
200	C₂×#343	C₂×C₃₁	1	2	yes	complemented
201	C₂²×C₃₁	—	2	4	no	—
202	C₂²×C₃₁	—	2	4	no	—
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	1	no	—
205	C₂²×C₃₁	—	2	1	no	—
206	C₂²×C₃₁	—	2	1	no	—
207	C₂²×C₃₁	—	2	4	no	—
208	C₂²×C₃₁	—	4	8	no	—
209	C₂²×C₃₁	—	4	2	no	—
210	C₂²×C₃₁	—	4	2	no	—
211	C₂²×C₃₁	#380	1	2	yes	—
212	C₂²×C₃₁	#380	1	2	yes	—
213	C₂²×C₃₁	#472	1	2	yes	—
214	C₂²×C₃₁	#473	1	2	yes	—
215	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
216	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
217	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
218	C₄×C₃₁	—	2	4	no	—
219	C₄×C₃₁	—	8	4	no	—
220	C₄×C₃₁	#380	1	2	yes	—
221	C₄×C₃₁	#380	1	2	yes	—
222	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
223	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
224	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
225	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#131	—	2	4	no	—
228	C₂×#131	—	2	2	no	—
229	C₂×#131	C₄×C₃₁	1	8	yes	complemented
230	C₂×#131	C₂²×C₃₁	1	4	yes	—
231	C₂²×M₁₆	—	2	1	no	—
232	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
233	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
234	C₂³×C₈	—	2	1	no	—
235	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
237	C₂×#108	—	2	4	no	—
238	C₂×#108	C₂²×C₃₁	1	4	yes	complemented
239	C₂×#107	—	2	1	no	—
240	C₂×#107	—	2	1	no	—
241	C₂×#107	—	2	2	no	—
242	C₂×#107	C₂²×C₃₁	1	4	yes	complemented
243	#417	—	2	4	no	—
244	#417	C₂²×C₃₁	1	8	yes	—
245	C₂×#103	—	2	2	no	—
246	C₂×#102	—	2	2	no	—
247	#343	C₂²×C₃₁	1	8	yes	complemented
248	C₂×C₃₁	—	2	1	no	—
249	C₂×C₃₁	—	2	1	no	—
250	C₂×C₃₁	—	2	2	no	—
251	C₂×C₃₁	—	4	4	no	—
252	C₂×C₃₁	#1816	1	4	yes	direct product factor
253	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
254	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
255	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
256	C₂×#41	—	2	2	no	—
257	C₂×#41	—	2	1	no	—
258	C₂×#41	—	2	1	no	—
259	C₂×#41	—	2	2	no	—
260	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
261	C₂×#41	D₄×C₃₁	1	2	yes	—
262	C₂×#41	Q₈×C₃₁	1	4	yes	—
263	C₂×#41	C₂³×C₃₁	1	2	yes	—
264	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
265	C₂²×D₄	—	2	1	no	—
266	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
267	C₂³×C₄	—	2	1	no	—
268	C₂³×C₄	—	2	2	no	—
269	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
270	#131	—	2	16	no	—
271	#131	—	2	8	no	—
272	#131	C₂×C₄×C₃₁	1	16	yes	complemented
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	—	2	1	no	—
275	C₂×M₁₆	—	2	2	no	—
276	C₂×M₁₆	—	2	8	no	—
277	C₂×M₁₆	—	2	4	no	—
278	C₂×M₁₆	—	2	4	no	—
279	C₂×M₁₆	—	2	4	no	—
280	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
281	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
282	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
283	C₂²×C₈	—	2	1	no	—
284	C₂²×C₈	—	2	8	no	—
285	C₂²×C₈	—	2	4	no	—
286	C₂²×C₈	—	2	2	no	—
287	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
288	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
289	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
290	C₂²×C₈	D₄×C₃₁	1	2	yes	—
291	C₂²×C₈	D₄×C₃₁	1	4	yes	—
292	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
299	#108	—	2	8	no	—
300	#107	—	2	4	no	—
301	#107	—	2	4	no	—
302	#103	—	2	8	no	—
303	#102	—	2	8	no	—
304	C₃₁	C₂×#1816	1	1	yes	characteristic, direct product factor, Sylow
305	C₂⁴	—	2	1	no	—
306	C₂⁴	#41×C₃₁	1	1	yes	characteristic
307	#41	—	2	4	no	—
308	#41	—	2	4	no	—
309	#41	—	2	4	no	—
310	#41	—	2	4	no	—
311	#41	—	2	4	no	—
312	#41	—	2	2	no	—
313	#41	—	2	2	no	—
314	#41	C₄⋊C₄×C₃₁	1	8	yes	—
315	#41	C₂²×C₄×C₃₁	1	4	yes	—
316	#41	C₂×Q₈×C₃₁	1	4	yes	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
320	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
321	C₂×D₄	—	2	4	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	4	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
328	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	4	2	no	—
336	C₂²×C₄	—	4	2	no	—
337	C₂²×C₄	—	8	4	no	—
338	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
339	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
345	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	#41×C₃₁	1	2	yes	—
347	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
348	M₁₆	—	2	4	no	—
349	M₁₆	—	2	2	no	—
350	M₁₆	—	2	16	no	—
351	M₁₆	—	2	4	no	—
352	M₁₆	—	2	4	no	—
353	M₁₆	—	4	4	no	—
354	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
355	C₂×C₈	—	2	4	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	16	no	—
358	C₂×C₈	—	2	4	no	—
359	C₂×C₈	—	4	4	no	—
360	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
361	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
362	C₂×C₈	C₂²⋊C₄×C₃₁	1	8	yes	—
363	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
364	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
365	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
366	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
367	C₄⋊C₄	—	2	4	no	—
368	C₄⋊C₄	—	2	4	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₂³	—	2	2	no	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	2	4	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	2	no	—
377	C₂³	C₃₁×#117	1	2	yes	—
378	C₂³	C₃₁×#121	1	1	yes	characteristic
379	C₂³	C₃₁×#121	1	1	yes	characteristic
380	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
381	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
382	Q₈	—	2	2	no	—
383	Q₈	—	2	4	no	—
384	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
385	D₄	—	2	4	no	—
386	D₄	—	2	4	no	—
387	D₄	—	2	8	no	—
388	D₄	—	2	2	no	—
389	D₄	—	2	4	no	—
390	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	8	no	—
396	C₂×C₄	—	4	4	no	—
397	C₂×C₄	—	8	4	no	—
398	C₂×C₄	—	8	4	no	—
399	C₂×C₄	—	8	4	no	—
400	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
401	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
402	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
403	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
404	C₂×C₄	C₃₁×#117	1	2	yes	—
405	C₂×C₄	C₃₁×#121	1	2	yes	—
406	C₂×C₄	C₃₁×#121	1	2	yes	—
407	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
408	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
409	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
410	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
411	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
412	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
418	C₈	—	2	4	no	—
419	C₈	—	4	4	no	—
420	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
421	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
422	C₂²	—	2	4	no	—
423	C₂²	—	2	4	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	2	1	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	2	4	no	—
429	C₂²	—	4	8	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	C₃₁×#380	1	2	yes	—
433	C₂²	C₃₁×#380	1	2	yes	—
434	C₂²	C₃₁×#472	1	2	yes	—
435	C₂²	C₃₁×#473	1	2	yes	—
436	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
437	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
438	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
439	C₄	—	2	4	no	—
440	C₄	—	8	4	no	—
441	C₄	C₃₁×#380	1	2	yes	—
442	C₄	C₃₁×#380	1	2	yes	—
443	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
444	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
445	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
446	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
447	C₂	—	2	1	no	—
448	C₂	—	2	1	no	—
449	C₂	—	2	2	no	—
450	C₂	—	4	4	no	—
451	C₂	C₃₁×#1816	1	4	yes	direct product factor
452	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
453	C₂	C₂×C₃₁×#472	1	1	yes	characteristic
454	C₂	C₂×C₃₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×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
3. C₂²×C₃₁×#107	C₂	semidirect
5. C₃₁×#1816	C₂	direct (N × H)
6. C₂×C₃₁×#343	C₂	semidirect
10. C₂×C₃₁×#131	C₄	semidirect
19. C₂×C₃₁×#108	C₂²	semidirect
23. C₂×C₃₁×#107	C₂²	semidirect
28. C₃₁×#343	C₂²	semidirect
45. C₃₁×#131	C₂×C₄	semidirect
143. C₂×#1816	C₃₁	direct (N × H)
196. C₂²×#108	C₂×C₃₁	semidirect
197. C₂²×#107	C₂×C₃₁	semidirect
199. #1816	C₂×C₃₁	direct (N × H)
200. C₂×#343	C₂×C₃₁	semidirect
229. C₂×#131	C₄×C₃₁	semidirect
238. C₂×#108	C₂²×C₃₁	semidirect
242. C₂×#107	C₂²×C₃₁	semidirect
247. #343	C₂²×C₃₁	semidirect
252. C₂×C₃₁	#1816	direct (N × H)
272. #131	C₂×C₄×C₃₁	semidirect
304. C₃₁	C₂×#1816	direct (N × H)
451. C₂	C₃₁×#1816	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
C₂²×C₃₁×#108	2
C₂²×C₃₁×#107	2
C₂×C₃₁×#417	2
C₃₁×#1816	2
C₂×C₃₁×#343	2
C₂×#1816	31

Subgroups — C2×C31×#1816 #21477436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1816 #21477436