Subgroups — C₂×C₃₁×#1673 #21475862

Order 7936.

790 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#461	C₂	1	2	yes	maximal
3	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1673	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#332	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆×C₃₁	—	2	2	no	—
8	C₂³×C₈×C₃₁	—	2	1	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#500	—	2	2	no	—
12	C₃₁×#500	C₂²	1	4	yes	complemented
13	C₃₁×#461	—	2	4	no	—
14	C₃₁×#461	C₂²	1	8	yes	—
15	C₈×D₄×C₃₁	—	2	2	no	—
16	C₈×D₄×C₃₁	C₂²	1	4	yes	—
17	C₂×C₃₁×#107	—	2	1	no	—
18	C₂×C₃₁×#107	—	2	1	no	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	—	2	2	no	—
22	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#335	C₂²	1	4	yes	complemented
25	C₃₁×#332	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	—	2	1	no	—
29	C₂×SD₁₆×C₃₁	—	2	8	no	—
30	C₂×SD₁₆×C₃₁	—	4	8	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₂²×C₈×C₃₁	—	2	4	no	—
34	C₂²×C₈×C₃₁	—	4	4	no	—
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	—	2	2	no	—
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₄×D₄×C₃₁	—	2	1	no	—
42	C₄×D₄×C₃₁	—	2	1	no	—
43	C₄×D₄×C₃₁	D₄	1	2	yes	—
44	C₄×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×#107	—	2	4	no	—
51	C₃₁×#107	—	2	4	no	—
52	C₃₁×C₄⋊C₈	—	2	1	no	—
53	C₃₁×C₄⋊C₈	—	2	2	no	—
54	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
55	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
56	C₃₁×#103	—	2	8	no	—
57	C₃₁×C₂²⋊C₈	—	2	4	no	—
58	C₄×C₈×C₃₁	D₄	1	2	yes	—
59	C₄×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂⁴×C₃₁	—	2	1	no	—
61	C₂×Q₈×C₃₁	—	2	4	no	—
62	C₂×Q₈×C₃₁	—	4	2	no	—
63	C₂×Q₈×C₃₁	—	4	4	no	—
64	C₂×Q₈×C₃₁	—	4	4	no	—
65	C₂×D₄×C₃₁	—	2	1	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	D₈	1	4	yes	—
68	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	—	2	1	no	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	4	2	no	—
73	C₂²×C₄×C₃₁	—	4	2	no	—
74	C₂²×C₄×C₃₁	—	4	2	no	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
78	SD₁₆×C₃₁	—	4	16	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	4	4	no	—
83	C₂×C₈×C₃₁	—	4	2	no	—
84	C₂×C₈×C₃₁	—	4	8	no	—
85	C₂×C₈×C₃₁	SD₁₆	1	4	yes	complemented
86	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₃₁	—	2	1	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	4	2	no	—
92	C₄⋊C₄×C₃₁	—	4	2	no	—
93	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
94	C₂²⋊C₄×C₃₁	—	2	2	no	—
95	C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₃₁	#41	1	1	yes	characteristic
99	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₂×#1673	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₃₁	—	2	2	no	—
102	C₂³×C₃₁	—	2	4	no	—
103	C₂³×C₃₁	—	4	2	no	—
104	C₂³×C₃₁	—	4	2	no	—
105	C₂³×C₃₁	#121	1	1	yes	characteristic
106	Q₈×C₃₁	—	4	4	no	—
107	Q₈×C₃₁	—	4	4	no	—
108	Q₈×C₃₁	—	8	4	no	—
109	D₄×C₃₁	—	2	2	no	—
110	D₄×C₃₁	—	2	4	no	—
111	D₄×C₃₁	C₂×D₈	1	4	yes	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	4	no	—
119	C₂×C₄×C₃₁	—	8	4	no	—
120	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₃₁	#135	1	2	yes	—
126	C₂×C₄×C₃₁	#137	1	2	yes	—
127	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
130	C₈×C₃₁	—	2	2	no	—
131	C₈×C₃₁	—	4	4	no	—
132	C₈×C₃₁	—	4	2	no	—
133	C₈×C₃₁	C₂×SD₁₆	1	4	yes	complemented
134	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
135	C₂×#461	C₂×C₃₁	1	2	yes	—
136	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
137	#1673	C₂×C₃₁	1	8	yes	direct product factor
138	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
139	C₂×#332	C₂×C₃₁	1	2	yes	complemented
140	C₂²×C₃₁	—	2	4	no	—
141	C₂²×C₃₁	—	4	8	no	—
142	C₂²×C₃₁	—	4	2	no	—
143	C₂²×C₃₁	—	4	2	no	—
144	C₂²×C₃₁	#461	1	2	yes	—
145	C₂²×C₃₁	#466	1	2	yes	—
146	C₂²×C₃₁	#473	1	2	yes	—
147	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
148	C₄×C₃₁	—	2	1	no	—
149	C₄×C₃₁	—	2	1	no	—
150	C₄×C₃₁	—	4	2	no	—
151	C₄×C₃₁	—	8	4	no	—
152	C₄×C₃₁	#461	1	2	yes	—
153	C₄×C₃₁	#466	1	2	yes	—
154	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
156	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
157	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
158	C₂²×SD₁₆	—	2	2	no	—
159	C₂³×C₈	—	2	1	no	—
160	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
161	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
162	#500	—	2	2	no	—
163	#500	C₂²×C₃₁	1	4	yes	complemented
164	#461	—	2	4	no	—
165	#461	C₂²×C₃₁	1	8	yes	—
166	C₈×D₄	—	2	2	no	—
167	C₈×D₄	C₂²×C₃₁	1	4	yes	—
168	C₂×#107	—	2	1	no	—
169	C₂×#107	—	2	1	no	—
170	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
172	C₂×#103	—	2	2	no	—
173	C₂×C₂²⋊C₈	—	2	1	no	—
174	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
175	#335	C₂²×C₃₁	1	4	yes	complemented
176	#332	C₂²×C₃₁	1	8	yes	complemented
177	C₂×C₃₁	—	4	4	no	—
178	C₂×C₃₁	#1673	1	4	yes	direct product factor
179	C₂×C₃₁	C₂×#461	1	1	yes	characteristic
180	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
181	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
182	C₂²×Q₈	—	2	2	no	—
183	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
184	C₂³×C₄	—	2	1	no	—
185	C₂×SD₁₆	—	2	8	no	—
186	C₂×SD₁₆	—	4	8	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	4	4	no	—
191	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	2	no	—
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	D₄×C₃₁	1	2	yes	—
200	C₄×D₄	C₂³×C₃₁	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#107	—	2	4	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
211	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
212	#103	—	2	8	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	C₄×C₈	D₄×C₃₁	1	2	yes	—
215	C₄×C₈	C₂³×C₃₁	1	2	yes	—
216	C₃₁	C₂×#1673	1	1	yes	characteristic, direct product factor, Sylow
217	C₂⁴	—	2	1	no	—
218	C₂×Q₈	—	2	4	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂×Q₈	—	4	4	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	D₈×C₃₁	1	4	yes	—
225	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
235	SD₁₆	—	4	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	8	no	—
242	C₂×C₈	SD₁₆×C₃₁	1	4	yes	complemented
243	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
254	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₄²	#41×C₃₁	1	1	yes	characteristic
256	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
257	C₂³	—	2	4	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	C₃₁×#121	1	1	yes	characteristic
262	Q₈	—	4	4	no	—
263	Q₈	—	4	4	no	—
264	Q₈	—	8	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂×D₈×C₃₁	1	4	yes	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
277	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
278	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
279	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
280	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
281	C₂×C₄	C₃₁×#135	1	2	yes	—
282	C₂×C₄	C₃₁×#137	1	2	yes	—
283	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	4	4	no	—
288	C₈	—	4	2	no	—
289	C₈	C₂×SD₁₆×C₃₁	1	4	yes	complemented
290	C₂²	—	2	4	no	—
291	C₂²	—	4	8	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	C₃₁×#461	1	2	yes	—
295	C₂²	C₃₁×#466	1	2	yes	—
296	C₂²	C₃₁×#473	1	2	yes	—
297	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₃₁×#461	1	2	yes	—
303	C₄	C₃₁×#466	1	2	yes	—
304	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
305	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
306	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
307	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
308	C₂	—	4	4	no	—
309	C₂	C₃₁×#1673	1	4	yes	direct product factor
310	C₂	C₂×C₃₁×#461	1	1	yes	characteristic
311	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
312	C₂	C₂×C₃₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#500	C₂	semidirect
4. C₃₁×#1673	C₂	direct (N × H)
5. C₂×C₃₁×#335	C₂	semidirect
6. C₂×C₃₁×#332	C₂	semidirect
12. C₃₁×#500	C₂²	semidirect
24. C₃₁×#335	C₂²	semidirect
25. C₃₁×#332	C₂²	semidirect
85. C₂×C₈×C₃₁	SD₁₆	semidirect
100. C₂×#1673	C₃₁	direct (N × H)
133. C₈×C₃₁	C₂×SD₁₆	semidirect
134. C₂×#500	C₂×C₃₁	semidirect
137. #1673	C₂×C₃₁	direct (N × H)
138. C₂×#335	C₂×C₃₁	semidirect
139. C₂×#332	C₂×C₃₁	semidirect
163. #500	C₂²×C₃₁	semidirect
175. #335	C₂²×C₃₁	semidirect
176. #332	C₂²×C₃₁	semidirect
178. C₂×C₃₁	#1673	direct (N × H)
216. C₃₁	C₂×#1673	direct (N × H)
242. C₂×C₈	SD₁₆×C₃₁	semidirect
289. C₈	C₂×SD₁₆×C₃₁	semidirect
309. C₂	C₃₁×#1673	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×#500	2
C₂×C₃₁×#461	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#1673	2
C₂×C₃₁×#335	2
C₂×C₃₁×#332	2
C₂×#1673	31

Subgroups — C2×C31×#1673 #21475862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1673 #21475862