Subgroups — C₂×C₃₁×#1574 #21474564

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1574	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#335	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#334	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₃₁	—	2	1	no	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#431	—	2	2	no	—
10	C₃₁×#431	C₄	1	4	yes	—
11	C₃₁×#431	C₂²	1	4	yes	—
12	C₃₁×#431	C₂²	1	4	yes	—
13	C₃₁×#423	—	2	2	no	—
14	C₃₁×#423	C₂²	1	4	yes	—
15	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
16	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
21	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₃₁	—	2	1	no	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#335	C₂²	1	8	yes	complemented
25	C₃₁×#334	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	—	4	4	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	—	2	1	no	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×C₄⋊C₈	—	2	4	no	—
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
49	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
50	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
51	C₃₁×C₂²⋊C₈	—	2	4	no	—
52	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₃₁	—	2	4	no	—
54	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₃₁	C₂³	1	4	yes	—
56	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
73	M₁₆×C₃₁	—	4	8	no	—
74	C₂×C₈×C₃₁	—	2	8	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	4	4	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
82	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
83	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
84	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
85	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
87	C₄²×C₃₁	C₂×D₄	1	2	yes	—
88	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
90	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
92	C₂×#1574	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	—	2	2	no	—
94	C₂³×C₃₁	—	2	2	no	—
95	C₂³×C₃₁	—	2	2	no	—
96	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	#108	1	4	yes	—
109	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₃₁	#131	1	4	yes	—
116	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₃₁	#137	1	2	yes	—
118	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
121	C₈×C₃₁	—	2	8	no	—
122	C₈×C₃₁	—	4	2	no	—
123	C₈×C₃₁	—	4	4	no	—
124	C₈×C₃₁	—	4	2	no	—
125	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
126	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
127	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
128	#1574	C₂×C₃₁	1	8	yes	direct product factor
129	C₂×#335	C₂×C₃₁	1	2	yes	complemented
130	C₂×#334	C₂×C₃₁	1	2	yes	complemented
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	2	2	no	—
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	4	4	no	—
135	C₂²×C₃₁	#424	1	2	yes	—
136	C₂²×C₃₁	#428	1	2	yes	—
137	C₂²×C₃₁	C₂×#108	1	2	yes	—
138	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₃₁	—	2	2	no	—
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	#424	1	2	yes	—
144	C₄×C₃₁	#428	1	2	yes	—
145	C₄×C₃₁	C₂×#108	1	4	yes	—
146	C₄×C₃₁	C₂×#131	1	2	yes	—
147	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
149	C₂²×M₁₆	—	2	1	no	—
150	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
151	#431	—	2	2	no	—
152	#431	C₄×C₃₁	1	4	yes	—
153	#431	C₂²×C₃₁	1	4	yes	—
154	#431	C₂²×C₃₁	1	4	yes	—
155	#423	—	2	2	no	—
156	#423	C₂²×C₃₁	1	4	yes	—
157	C₂×C₄⋊C₈	—	2	1	no	—
158	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×C₂²⋊C₈	—	2	1	no	—
163	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
164	C₂×C₄×C₈	—	2	1	no	—
165	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
166	#335	C₂²×C₃₁	1	8	yes	complemented
167	#334	C₂²×C₃₁	1	8	yes	complemented
168	C₂×C₃₁	—	4	2	no	—
169	C₂×C₃₁	#1574	1	4	yes	direct product factor
170	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
171	C₂×C₃₁	C₂×#428	1	1	yes	characteristic
172	C₂×C₃₁	C₂²×#108	1	1	yes	characteristic
173	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
175	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	—	4	4	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
182	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
184	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄²	D₄×C₃₁	1	2	yes	—
186	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
187	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
188	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
195	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
196	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
197	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
198	C₂²⋊C₈	—	2	4	no	—
199	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
202	C₄×C₈	C₂³×C₃₁	1	4	yes	—
203	C₃₁	C₂×#1574	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
212	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
214	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
221	M₁₆	—	4	8	no	—
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
230	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
231	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
232	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
233	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
234	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
235	C₄²	C₂×D₄×C₃₁	1	2	yes	—
236	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
238	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
239	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
244	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	C₃₁×#108	1	4	yes	—
256	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
262	C₂×C₄	C₃₁×#131	1	4	yes	—
263	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
264	C₂×C₄	C₃₁×#137	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	4	no	—
271	C₈	—	4	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₃₁×#424	1	2	yes	—
277	C₂²	C₃₁×#428	1	2	yes	—
278	C₂²	C₂×C₃₁×#108	1	2	yes	—
279	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	C₃₁×#424	1	2	yes	—
285	C₄	C₃₁×#428	1	2	yes	—
286	C₄	C₂×C₃₁×#108	1	4	yes	—
287	C₄	C₂×C₃₁×#131	1	2	yes	—
288	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	C₃₁×#1574	1	4	yes	direct product factor
292	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
293	C₂	C₂×C₃₁×#428	1	1	yes	characteristic
294	C₂	C₂²×C₃₁×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1574	C₂	direct (N × H)
5. C₂×C₃₁×#335	C₂	semidirect
6. C₂×C₃₁×#334	C₂	semidirect
24. C₃₁×#335	C₂²	semidirect
25. C₃₁×#334	C₂²	semidirect
92. C₂×#1574	C₃₁	direct (N × H)
128. #1574	C₂×C₃₁	direct (N × H)
129. C₂×#335	C₂×C₃₁	semidirect
130. C₂×#334	C₂×C₃₁	semidirect
166. #335	C₂²×C₃₁	semidirect
167. #334	C₂²×C₃₁	semidirect
169. C₂×C₃₁	#1574	direct (N × H)
203. C₃₁	C₂×#1574	direct (N × H)
291. C₂	C₃₁×#1574	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×#431	2
C₂×C₃₁×#423	2
C₃₁×#1574	2
C₂×C₃₁×#335	2
C₂×C₃₁×#334	2
C₂×#1574	31

Subgroups — C2×C31×#1574 #21474564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1574 #21474564