Subgroups — C₂×C₃₁×#1811 #21477425

Order 7936.

1438 conjugacy classes of subgroups.

The subgroup lattice has 318 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	2	yes	maximal
4	C₃₁×#1811	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#341	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₃₁	—	2	8	no	—
8	C₂²×M₁₆×C₃₁	C₄	1	4	yes	—
9	C₂²×M₁₆×C₃₁	C₄	1	4	yes	—
10	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#108	—	2	8	no	—
15	C₂×C₃₁×#108	C₂²	1	8	yes	complemented
16	C₃₁×#407	—	2	4	no	—
17	C₃₁×#407	C₂²	1	8	yes	—
18	C₂×C₃₁×C₂²⋊C₈	—	2	4	no	—
19	C₃₁×#341	C₂²	1	8	yes	complemented
20	C₂⁵×C₃₁	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	—	2	4	no	—
22	C₂³×C₄×C₃₁	—	2	4	no	—
23	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₃₁	D₄	1	2	yes	—
26	C₂³×C₄×C₃₁	D₄	1	2	yes	—
27	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	Q₈	1	2	yes	—
29	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
30	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₃₁	—	2	4	no	—
32	C₂×M₁₆×C₃₁	—	2	16	no	—
33	C₂×M₁₆×C₃₁	—	2	16	no	—
34	C₂×M₁₆×C₃₁	—	2	16	no	—
35	C₂×M₁₆×C₃₁	—	2	8	no	—
36	C₂×M₁₆×C₃₁	—	2	8	no	—
37	C₂×M₁₆×C₃₁	C₂×C₄	1	16	yes	—
38	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
39	C₂²×C₈×C₃₁	—	2	4	no	—
40	C₂²×C₈×C₃₁	—	2	4	no	—
41	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₃₁	D₄	1	8	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₃₁×#108	—	2	16	no	—
49	C₃₁×C₂²⋊C₈	—	2	16	no	—
50	C₂⁴×C₃₁	—	2	4	no	—
51	C₂⁴×C₃₁	—	4	4	no	—
52	C₂⁴×C₃₁	C₄⋊C₄	1	2	yes	—
53	C₂⁴×C₃₁	C₄⋊C₄	1	2	yes	—
54	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂⁴×C₃₁	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	8	no	—
62	C₂²×C₄×C₃₁	—	2	8	no	—
63	C₂²×C₄×C₃₁	—	2	8	no	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	2	4	no	—
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	#41	1	4	yes	—
80	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₃₁	—	2	32	no	—
82	M₁₆×C₃₁	—	2	8	no	—
83	M₁₆×C₃₁	—	2	8	no	—
84	M₁₆×C₃₁	—	4	8	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	—	2	4	no	—
88	C₂×C₈×C₃₁	—	2	8	no	—
89	C₂×C₈×C₃₁	—	4	8	no	—
90	C₂×C₈×C₃₁	C₂²⋊C₄	1	8	yes	—
91	C₂×C₈×C₃₁	C₂²⋊C₄	1	8	yes	—
92	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
95	C₂×#1811	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₃₁	—	2	4	no	—
97	C₂³×C₃₁	—	2	4	no	—
98	C₂³×C₃₁	—	2	4	no	—
99	C₂³×C₃₁	—	2	4	no	—
100	C₂³×C₃₁	—	2	8	no	—
101	C₂³×C₃₁	—	4	4	no	—
102	C₂³×C₃₁	—	4	4	no	—
103	C₂³×C₃₁	—	4	16	no	—
104	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
105	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
108	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	2	4	no	—
114	C₂×C₄×C₃₁	—	2	8	no	—
115	C₂×C₄×C₃₁	—	2	8	no	—
116	C₂×C₄×C₃₁	—	2	8	no	—
117	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#117	1	2	yes	—
123	C₂×C₄×C₃₁	#117	1	2	yes	—
124	C₂×C₄×C₃₁	#121	1	4	yes	—
125	C₂×C₄×C₃₁	#121	1	4	yes	—
126	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
127	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
131	C₈×C₃₁	—	2	4	no	—
132	C₈×C₃₁	—	4	8	no	—
133	C₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
134	C₂³×M₁₆	C₂×C₃₁	1	1	yes	characteristic
135	C₂²×#108	C₂×C₃₁	1	2	yes	complemented
136	C₂×#407	C₂×C₃₁	1	2	yes	—
137	#1811	C₂×C₃₁	1	8	yes	direct product factor
138	C₂×#341	C₂×C₃₁	1	2	yes	complemented
139	C₂²×C₃₁	—	2	8	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	2	2	no	—
142	C₂²×C₃₁	—	2	8	no	—
143	C₂²×C₃₁	—	2	4	no	—
144	C₂²×C₃₁	—	4	8	no	—
145	C₂²×C₃₁	—	4	8	no	—
146	C₂²×C₃₁	—	4	8	no	—
147	C₂²×C₃₁	#380	1	2	yes	—
148	C₂²×C₃₁	#380	1	2	yes	—
149	C₂²×C₃₁	#430	1	4	yes	—
150	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
152	C₄×C₃₁	—	2	4	no	—
153	C₄×C₃₁	#380	1	2	yes	—
154	C₄×C₃₁	#380	1	2	yes	—
155	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
157	C₄×C₃₁	C₂×#121	1	2	yes	—
158	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
159	C₂²×M₁₆	—	2	8	no	—
160	C₂²×M₁₆	C₄×C₃₁	1	4	yes	—
161	C₂²×M₁₆	C₄×C₃₁	1	4	yes	—
162	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
163	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
164	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×#108	—	2	8	no	—
167	C₂×#108	C₂²×C₃₁	1	8	yes	complemented
168	#407	—	2	4	no	—
169	#407	C₂²×C₃₁	1	8	yes	—
170	C₂×C₂²⋊C₈	—	2	4	no	—
171	#341	C₂²×C₃₁	1	8	yes	complemented
172	C₂×C₃₁	—	2	2	no	—
173	C₂×C₃₁	—	2	2	no	—
174	C₂×C₃₁	—	4	4	no	—
175	C₂×C₃₁	#1811	1	4	yes	direct product factor
176	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
177	C₂×C₃₁	C₂×#430	1	2	yes	—
178	C₂⁵	Q₈×C₃₁	1	1	yes	characteristic
179	C₂³×C₄	—	2	4	no	—
180	C₂³×C₄	—	2	4	no	—
181	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
182	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
183	C₂³×C₄	D₄×C₃₁	1	2	yes	—
184	C₂³×C₄	D₄×C₃₁	1	2	yes	—
185	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
186	C₂³×C₄	Q₈×C₃₁	1	2	yes	—
187	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
188	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	—	2	16	no	—
191	C₂×M₁₆	—	2	16	no	—
192	C₂×M₁₆	—	2	16	no	—
193	C₂×M₁₆	—	2	8	no	—
194	C₂×M₁₆	—	2	8	no	—
195	C₂×M₁₆	C₂×C₄×C₃₁	1	16	yes	—
196	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
197	C₂²×C₈	—	2	4	no	—
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
200	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
201	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
202	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
203	C₂²×C₈	D₄×C₃₁	1	8	yes	—
204	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
205	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
206	#108	—	2	16	no	—
207	C₂²⋊C₈	—	2	16	no	—
208	C₃₁	C₂×#1811	1	1	yes	characteristic, direct product factor, Sylow
209	C₂⁴	—	2	4	no	—
210	C₂⁴	—	4	4	no	—
211	C₂⁴	C₄⋊C₄×C₃₁	1	2	yes	—
212	C₂⁴	C₄⋊C₄×C₃₁	1	2	yes	—
213	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
214	C₂⁴	C₂×Q₈×C₃₁	1	2	yes	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
226	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
227	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
228	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
233	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
234	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
237	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	#41×C₃₁	1	4	yes	—
239	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
240	M₁₆	—	2	32	no	—
241	M₁₆	—	2	8	no	—
242	M₁₆	—	2	8	no	—
243	M₁₆	—	4	8	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	8	no	—
248	C₂×C₈	—	4	8	no	—
249	C₂×C₈	C₂²⋊C₄×C₃₁	1	8	yes	—
250	C₂×C₈	C₂²⋊C₄×C₃₁	1	8	yes	—
251	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
252	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
253	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	8	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	4	16	no	—
262	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
263	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
264	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
265	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
266	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
276	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
277	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
278	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₃₁×#117	1	2	yes	—
281	C₂×C₄	C₃₁×#117	1	2	yes	—
282	C₂×C₄	C₃₁×#121	1	4	yes	—
283	C₂×C₄	C₃₁×#121	1	4	yes	—
284	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
285	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
287	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
289	C₈	—	2	4	no	—
290	C₈	—	4	8	no	—
291	C₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	8	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	8	no	—
299	C₂²	—	4	8	no	—
300	C₂²	C₃₁×#380	1	2	yes	—
301	C₂²	C₃₁×#380	1	2	yes	—
302	C₂²	C₃₁×#430	1	4	yes	—
303	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
304	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
305	C₄	—	2	4	no	—
306	C₄	C₃₁×#380	1	2	yes	—
307	C₄	C₃₁×#380	1	2	yes	—
308	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
309	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
310	C₄	C₂×C₃₁×#121	1	2	yes	—
311	C₂	—	2	2	no	—
312	C₂	—	2	2	no	—
313	C₂	—	4	4	no	—
314	C₂	C₃₁×#1811	1	4	yes	direct product factor
315	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
316	C₂	C₂×C₃₁×#430	1	2	yes	—

	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
4. C₃₁×#1811	C₂	direct (N × H)
5. C₂×C₃₁×#341	C₂	semidirect
15. C₂×C₃₁×#108	C₂²	semidirect
19. C₃₁×#341	C₂²	semidirect
95. C₂×#1811	C₃₁	direct (N × H)
135. C₂²×#108	C₂×C₃₁	semidirect
137. #1811	C₂×C₃₁	direct (N × H)
138. C₂×#341	C₂×C₃₁	semidirect
167. C₂×#108	C₂²×C₃₁	semidirect
171. #341	C₂²×C₃₁	semidirect
175. C₂×C₃₁	#1811	direct (N × H)
208. C₃₁	C₂×#1811	direct (N × H)
314. C₂	C₃₁×#1811	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
C₃₁×#1811	2
C₂×C₃₁×#341	2
C₂×#1811	31

Subgroups — C2×C31×#1811 #21477425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1811 #21477425