Subgroups — C₂×C₃₁×#2863 #21493392

Order 7936.

2142 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2863	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃₁×#391	C₂	1	3	yes	maximal
3	C₂×C₃₁×#389	C₂	1	3	yes	maximal
4	C₂×C₃₁×#387	C₂	1	6	yes	maximal
5	C₂×C₃₁×#383	C₂	1	2	yes	maximal
6	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
9	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
11	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#391	C₂²	1	24	yes	—
14	C₃₁×#389	C₂²	1	24	yes	—
15	C₃₁×#387	C₂²	1	48	yes	—
16	C₃₁×#383	C₂²	1	16	yes	—
17	C₂×C₃₁×#95	C₂²	1	6	yes	—
18	C₂×C₃₁×#95	C₂²	1	6	yes	—
19	C₂×C₃₁×#95	C₂²	1	6	yes	—
20	C₄³×C₃₁	C₂²	1	8	yes	—
21	C₂²×Q₈×C₃₁	—	2	12	no	—
22	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
24	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
25	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
26	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
27	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
30	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
31	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
32	C₂×C₄²×C₃₁	—	2	12	no	—
33	C₂×C₄²×C₃₁	—	2	12	no	—
34	C₂×C₄²×C₃₁	Q₈	1	8	yes	complemented
35	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
37	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
38	C₃₁×#95	C₂³	1	24	yes	—
39	C₃₁×#95	C₂³	1	24	yes	—
40	C₃₁×#95	C₂³	1	24	yes	—
41	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
42	C₂×Q₈×C₃₁	—	2	24	no	—
43	C₂×Q₈×C₃₁	—	4	16	no	—
44	C₂²×C₄×C₃₁	—	2	6	no	—
45	C₂²×C₄×C₃₁	—	2	6	no	—
46	C₂²×C₄×C₃₁	—	2	6	no	—
47	C₂²×C₄×C₃₁	—	2	6	no	—
48	C₂²×C₄×C₃₁	—	2	12	no	—
49	C₂²×C₄×C₃₁	—	2	24	no	—
50	C₂²×C₄×C₃₁	—	2	12	no	—
51	C₂²×C₄×C₃₁	C₂×Q₈	1	12	yes	—
52	C₂²×C₄×C₃₁	#41	1	12	yes	—
53	C₂²×C₄×C₃₁	#41	1	12	yes	—
54	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
55	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
56	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
57	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
58	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
59	C₄⋊C₄×C₃₁	—	2	48	no	—
60	C₄⋊C₄×C₃₁	—	2	24	no	—
61	C₄²×C₃₁	—	2	24	no	—
62	C₄²×C₃₁	—	2	24	no	—
63	C₄²×C₃₁	C₂×Q₈	1	16	yes	complemented
64	C₂×#2863	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₃₁	C₂²×Q₈	1	2	yes	—
66	C₂³×C₃₁	C₂×#41	1	6	yes	—
67	C₂³×C₃₁	C₂×#41	1	6	yes	—
68	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
69	Q₈×C₃₁	—	4	16	no	—
70	C₂×C₄×C₃₁	—	2	24	no	—
71	C₂×C₄×C₃₁	—	2	24	no	—
72	C₂×C₄×C₃₁	—	2	6	no	—
73	C₂×C₄×C₃₁	—	2	6	no	—
74	C₂×C₄×C₃₁	—	2	6	no	—
75	C₂×C₄×C₃₁	—	2	6	no	—
76	C₂×C₄×C₃₁	—	2	12	no	—
77	C₂×C₄×C₃₁	—	2	24	no	—
78	C₂×C₄×C₃₁	—	2	12	no	—
79	C₂×C₄×C₃₁	—	4	8	no	—
80	C₂×C₄×C₃₁	—	4	24	no	—
81	C₂×C₄×C₃₁	#126	1	16	yes	—
82	C₂×C₄×C₃₁	C₂²×Q₈	1	12	yes	—
83	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
84	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
85	#2863	C₂×C₃₁	1	16	yes	direct product factor
86	C₂×#391	C₂×C₃₁	1	3	yes	—
87	C₂×#389	C₂×C₃₁	1	3	yes	—
88	C₂×#387	C₂×C₃₁	1	6	yes	—
89	C₂×#383	C₂×C₃₁	1	2	yes	—
90	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
91	C₂²×C₃₁	C₂×#126	1	4	yes	—
92	C₂²×C₃₁	#529	1	12	yes	—
93	C₂²×C₃₁	#533	1	12	yes	—
94	C₂²×C₃₁	C₂³×Q₈	1	1	yes	characteristic
95	C₂²×C₃₁	C₂²×#41	1	3	yes	—
96	C₂²×C₃₁	C₂²×#41	1	3	yes	—
97	C₄×C₃₁	—	2	12	no	—
98	C₄×C₃₁	—	2	12	no	—
99	C₄×C₃₁	—	4	4	no	—
100	C₄×C₃₁	—	4	12	no	—
101	C₄×C₃₁	C₂×#126	1	8	yes	—
102	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
103	C₂²×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
104	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
105	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
106	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
107	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
108	#391	C₂²×C₃₁	1	24	yes	—
109	#389	C₂²×C₃₁	1	24	yes	—
110	#387	C₂²×C₃₁	1	48	yes	—
111	#383	C₂²×C₃₁	1	16	yes	—
112	C₂×#95	C₂²×C₃₁	1	6	yes	—
113	C₂×#95	C₂²×C₃₁	1	6	yes	—
114	C₂×#95	C₂²×C₃₁	1	6	yes	—
115	C₄³	C₂²×C₃₁	1	8	yes	—
116	C₂×C₃₁	#2863	1	8	yes	direct product factor
117	C₂×C₃₁	C₂²×#126	1	1	yes	characteristic
118	C₂×C₃₁	C₂×#529	1	3	yes	—
119	C₂×C₃₁	C₂×#533	1	3	yes	—
120	C₂²×Q₈	—	2	12	no	—
121	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
122	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
123	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
124	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
125	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
126	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
127	C₂×C₄⋊C₄	—	2	24	no	—
128	C₂×C₄⋊C₄	—	2	12	no	—
129	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
130	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
131	C₂×C₄²	—	2	12	no	—
132	C₂×C₄²	—	2	12	no	—
133	C₂×C₄²	Q₈×C₃₁	1	8	yes	complemented
134	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
135	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
136	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
137	#95	C₂³×C₃₁	1	24	yes	—
138	#95	C₂³×C₃₁	1	24	yes	—
139	#95	C₂³×C₃₁	1	24	yes	—
140	C₃₁	C₂×#2863	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
142	C₂×Q₈	—	2	24	no	—
143	C₂×Q₈	—	4	16	no	—
144	C₂²×C₄	—	2	6	no	—
145	C₂²×C₄	—	2	6	no	—
146	C₂²×C₄	—	2	6	no	—
147	C₂²×C₄	—	2	6	no	—
148	C₂²×C₄	—	2	12	no	—
149	C₂²×C₄	—	2	24	no	—
150	C₂²×C₄	—	2	12	no	—
151	C₂²×C₄	C₂×Q₈×C₃₁	1	12	yes	—
152	C₂²×C₄	#41×C₃₁	1	12	yes	—
153	C₂²×C₄	#41×C₃₁	1	12	yes	—
154	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
155	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
156	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
157	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
158	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
159	C₄⋊C₄	—	2	48	no	—
160	C₄⋊C₄	—	2	24	no	—
161	C₄²	—	2	24	no	—
162	C₄²	—	2	24	no	—
163	C₄²	C₂×Q₈×C₃₁	1	16	yes	complemented
164	C₂³	C₂²×Q₈×C₃₁	1	2	yes	—
165	C₂³	C₂×#41×C₃₁	1	6	yes	—
166	C₂³	C₂×#41×C₃₁	1	6	yes	—
167	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
168	Q₈	—	4	16	no	—
169	C₂×C₄	—	2	24	no	—
170	C₂×C₄	—	2	24	no	—
171	C₂×C₄	—	2	6	no	—
172	C₂×C₄	—	2	6	no	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	—	2	12	no	—
176	C₂×C₄	—	2	24	no	—
177	C₂×C₄	—	2	12	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	24	no	—
180	C₂×C₄	C₃₁×#126	1	16	yes	—
181	C₂×C₄	C₂²×Q₈×C₃₁	1	12	yes	—
182	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
183	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
184	C₂²	C₂×C₃₁×#126	1	4	yes	—
185	C₂²	C₃₁×#529	1	12	yes	—
186	C₂²	C₃₁×#533	1	12	yes	—
187	C₂²	C₂³×Q₈×C₃₁	1	1	yes	characteristic
188	C₂²	C₂²×#41×C₃₁	1	3	yes	—
189	C₂²	C₂²×#41×C₃₁	1	3	yes	—
190	C₄	—	2	12	no	—
191	C₄	—	2	12	no	—
192	C₄	—	4	4	no	—
193	C₄	—	4	12	no	—
194	C₄	C₂×C₃₁×#126	1	8	yes	—
195	C₂	C₃₁×#2863	1	8	yes	direct product factor
196	C₂	C₂²×C₃₁×#126	1	1	yes	characteristic
197	C₂	C₂×C₃₁×#529	1	3	yes	—
198	C₂	C₂×C₃₁×#533	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2863	C₂	direct (N × H)
34. C₂×C₄²×C₃₁	Q₈	semidirect
63. C₄²×C₃₁	C₂×Q₈	semidirect
64. C₂×#2863	C₃₁	direct (N × H)
85. #2863	C₂×C₃₁	direct (N × H)
116. C₂×C₃₁	#2863	direct (N × H)
133. C₂×C₄²	Q₈×C₃₁	semidirect
140. C₃₁	C₂×#2863	direct (N × H)
163. C₄²	C₂×Q₈×C₃₁	semidirect
195. C₂	C₃₁×#2863	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2863	2
C₂×C₃₁×#391	2
C₂×C₃₁×#389	2
C₂×C₃₁×#387	2
C₂×C₃₁×#383	2
C₂×C₄³×C₃₁	2
C₂×#2863	31

Subgroups — C2×C31×#2863 #21493392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2863 #21493392