Subgroups — C₂×C₃₁×#1807 #21477415

Order 7936.

1790 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#106	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₁×#380	C₂	1	2	yes	maximal
4	C₃₁×#1807	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#340	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	—	2	8	no	—
8	C₂³×C₈×C₃₁	C₄	1	8	yes	complemented
9	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₃₁	C₂²	1	4	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
14	C₂×C₃₁×#106	—	2	8	no	—
15	C₂×C₃₁×#106	C₂²	1	8	yes	complemented
16	C₃₁×#380	—	2	4	no	—
17	C₃₁×#380	C₂²	1	8	yes	—
18	C₂×C₃₁×#95	—	2	2	no	—
19	C₃₁×#340	C₂²	1	8	yes	complemented
20	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	—	2	4	no	—
22	C₂³×C₄×C₃₁	—	2	4	no	—
23	C₂³×C₄×C₃₁	—	2	4	no	—
24	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₃₁	D₄	1	2	yes	—
26	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	Q₈	1	4	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
29	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	—	2	4	no	—
31	C₂²×C₈×C₃₁	—	2	32	no	—
32	C₂²×C₈×C₃₁	—	2	16	no	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	16	yes	complemented
36	C₂²×C₈×C₃₁	D₄	1	8	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
45	C₃₁×#106	—	2	16	no	—
46	C₃₁×#95	—	2	8	no	—
47	C₂⁴×C₃₁	—	2	4	no	—
48	C₂⁴×C₃₁	—	2	4	no	—
49	C₂⁴×C₃₁	C₄⋊C₄	1	4	yes	—
50	C₂⁴×C₃₁	SD₁₆	1	8	yes	—
51	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
52	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	—	2	8	no	—
55	C₂²×C₄×C₃₁	—	2	8	no	—
56	C₂²×C₄×C₃₁	—	2	8	no	—
57	C₂²×C₄×C₃₁	—	2	16	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	8	no	—
61	C₂²×C₄×C₃₁	—	4	8	no	—
62	C₂²×C₄×C₃₁	—	4	8	no	—
63	C₂²×C₄×C₃₁	—	4	8	no	—
64	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₃₁	#41	1	4	yes	—
74	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₃₁	—	2	8	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	32	no	—
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
82	C₄⋊C₄×C₃₁	—	2	8	no	—
83	C₄⋊C₄×C₃₁	—	2	8	no	—
84	C₄⋊C₄×C₃₁	—	4	8	no	—
85	C₂²⋊C₄×C₃₁	—	4	16	no	—
86	C₂×#1807	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	—	2	4	no	—
88	C₂³×C₃₁	—	2	4	no	—
89	C₂³×C₃₁	—	2	8	no	—
90	C₂³×C₃₁	—	2	8	no	—
91	C₂³×C₃₁	—	2	32	no	—
92	C₂³×C₃₁	#106	1	16	yes	complemented
93	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
94	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₃₁	#121	1	4	yes	—
96	C₂³×C₃₁	C₂×SD₁₆	1	8	yes	—
97	C₂³×C₃₁	C₂×SD₁₆	1	4	yes	—
98	C₂³×C₃₁	#135	1	4	yes	—
99	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	8	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	2	8	no	—
105	C₂×C₄×C₃₁	—	2	16	no	—
106	C₂×C₄×C₃₁	—	4	8	no	—
107	C₂×C₄×C₃₁	—	4	8	no	—
108	C₂×C₄×C₃₁	—	4	8	no	—
109	C₂×C₄×C₃₁	—	8	16	no	—
110	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₃₁	#117	1	4	yes	—
114	C₂×C₄×C₃₁	#121	1	4	yes	—
115	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
116	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
118	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
120	C₈×C₃₁	—	2	4	no	—
121	C₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	complemented
122	C₂⁴×C₈	C₂×C₃₁	1	1	yes	characteristic
123	C₂²×#106	C₂×C₃₁	1	2	yes	complemented
124	C₂×#380	C₂×C₃₁	1	2	yes	—
125	#1807	C₂×C₃₁	1	8	yes	direct product factor
126	C₂×#340	C₂×C₃₁	1	2	yes	complemented
127	C₂²×C₃₁	—	2	32	no	—
128	C₂²×C₃₁	—	2	4	no	—
129	C₂²×C₃₁	—	2	4	no	—
130	C₂²×C₃₁	—	2	16	no	—
131	C₂²×C₃₁	#380	1	4	yes	—
132	C₂²×C₃₁	C₂×#106	1	4	yes	—
133	C₂²×C₃₁	C₂×#106	1	8	yes	complemented
134	C₂²×C₃₁	#427	1	4	yes	—
135	C₂²×C₃₁	#465	1	16	yes	—
136	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₃₁	C₂×#121	1	2	yes	—
138	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
139	C₂²×C₃₁	C₂×#135	1	2	yes	—
140	C₄×C₃₁	—	2	4	no	—
141	C₄×C₃₁	—	8	8	no	—
142	C₄×C₃₁	#380	1	4	yes	—
143	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
145	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
146	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
147	C₂³×C₈	—	2	8	no	—
148	C₂³×C₈	C₄×C₃₁	1	8	yes	complemented
149	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₂³×C₈	C₂²×C₃₁	1	4	yes	—
151	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
152	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
153	C₂²×C₂²⋊C₄	—	2	2	no	—
154	C₂×#106	—	2	8	no	—
155	C₂×#106	C₂²×C₃₁	1	8	yes	complemented
156	#380	—	2	4	no	—
157	#380	C₂²×C₃₁	1	8	yes	—
158	C₂×#95	—	2	2	no	—
159	#340	C₂²×C₃₁	1	8	yes	complemented
160	C₂×C₃₁	—	2	8	no	—
161	C₂×C₃₁	#1807	1	8	yes	direct product factor
162	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
163	C₂×C₃₁	C₂²×#106	1	1	yes	characteristic
164	C₂×C₃₁	C₂×#427	1	1	yes	characteristic
165	C₂×C₃₁	C₂×#465	1	4	yes	—
166	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
167	C₂³×C₄	—	2	4	no	—
168	C₂³×C₄	—	2	4	no	—
169	C₂³×C₄	—	2	4	no	—
170	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
171	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
172	C₂³×C₄	D₄×C₃₁	1	2	yes	—
173	C₂³×C₄	Q₈×C₃₁	1	4	yes	—
174	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
175	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	—	2	32	no	—
178	C₂²×C₈	—	2	16	no	—
179	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
180	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
181	C₂²×C₈	C₂×C₄×C₃₁	1	16	yes	complemented
182	C₂²×C₈	D₄×C₃₁	1	8	yes	—
183	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
184	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
185	C₂×C₄⋊C₄	—	2	8	no	—
186	C₂×C₄⋊C₄	—	2	8	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
189	C₂×C₂²⋊C₄	—	2	8	no	—
190	C₂×C₂²⋊C₄	—	4	8	no	—
191	#106	—	2	16	no	—
192	#95	—	2	8	no	—
193	C₃₁	C₂×#1807	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	C₄⋊C₄×C₃₁	1	4	yes	—
197	C₂⁴	SD₁₆×C₃₁	1	8	yes	—
198	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
199	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	16	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	4	8	no	—
209	C₂²×C₄	—	4	8	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
212	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
213	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
214	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
219	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
220	C₂²×C₄	#41×C₃₁	1	4	yes	—
221	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	32	no	—
225	C₂×C₈	C₂²⋊C₄×C₃₁	1	16	yes	complemented
226	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
227	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
228	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	4	8	no	—
232	C₂²⋊C₄	—	4	16	no	—
233	C₂³	—	2	8	no	—
234	C₂³	—	2	32	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	8	no	—
238	C₂³	C₃₁×#106	1	16	yes	complemented
239	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
240	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
241	C₂³	C₃₁×#121	1	4	yes	—
242	C₂³	C₂×SD₁₆×C₃₁	1	8	yes	—
243	C₂³	C₂×SD₁₆×C₃₁	1	4	yes	—
244	C₂³	C₃₁×#135	1	4	yes	—
245	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	16	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	8	16	no	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
257	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
259	C₂×C₄	C₃₁×#117	1	4	yes	—
260	C₂×C₄	C₃₁×#121	1	4	yes	—
261	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
262	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
264	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	complemented
268	C₂²	—	2	32	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	16	no	—
272	C₂²	C₃₁×#380	1	4	yes	—
273	C₂²	C₂×C₃₁×#106	1	4	yes	—
274	C₂²	C₂×C₃₁×#106	1	8	yes	complemented
275	C₂²	C₃₁×#427	1	4	yes	—
276	C₂²	C₃₁×#465	1	16	yes	—
277	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
278	C₂²	C₂×C₃₁×#121	1	2	yes	—
279	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
280	C₂²	C₂×C₃₁×#135	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	8	8	no	—
283	C₄	C₃₁×#380	1	4	yes	—
284	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
285	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
286	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
287	C₂	—	2	8	no	—
288	C₂	C₃₁×#1807	1	8	yes	direct product factor
289	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
290	C₂	C₂²×C₃₁×#106	1	1	yes	characteristic
291	C₂	C₂×C₃₁×#427	1	1	yes	characteristic
292	C₂	C₂×C₃₁×#465	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#106	C₂	semidirect
4. C₃₁×#1807	C₂	direct (N × H)
5. C₂×C₃₁×#340	C₂	semidirect
8. C₂³×C₈×C₃₁	C₄	semidirect
15. C₂×C₃₁×#106	C₂²	semidirect
19. C₃₁×#340	C₂²	semidirect
35. C₂²×C₈×C₃₁	C₂×C₄	semidirect
78. C₂×C₈×C₃₁	C₂²⋊C₄	semidirect
86. C₂×#1807	C₃₁	direct (N × H)
92. C₂³×C₃₁	#106	semidirect
121. C₈×C₃₁	C₂×C₂²⋊C₄	semidirect
123. C₂²×#106	C₂×C₃₁	semidirect
125. #1807	C₂×C₃₁	direct (N × H)
126. C₂×#340	C₂×C₃₁	semidirect
133. C₂²×C₃₁	C₂×#106	semidirect
148. C₂³×C₈	C₄×C₃₁	semidirect
155. C₂×#106	C₂²×C₃₁	semidirect
159. #340	C₂²×C₃₁	semidirect
161. C₂×C₃₁	#1807	direct (N × H)
181. C₂²×C₈	C₂×C₄×C₃₁	semidirect
193. C₃₁	C₂×#1807	direct (N × H)
225. C₂×C₈	C₂²⋊C₄×C₃₁	semidirect
238. C₂³	C₃₁×#106	semidirect
267. C₈	C₂×C₂²⋊C₄×C₃₁	semidirect
274. C₂²	C₂×C₃₁×#106	semidirect
288. C₂	C₃₁×#1807	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₃₁	2
C₂²×C₃₁×#106	2
C₂×C₃₁×#380	2
C₃₁×#1807	2
C₂×C₃₁×#340	2
C₂×#1807	31

Subgroups — C2×C31×#1807 #21477415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1807 #21477415