Subgroups — C₃₁×#8781 #21465540

Order 7936.

310 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1564	C₂	1	2	yes	maximal
3	C₃₁×#1281	C₂	1	2	yes	maximal
4	C₃₁×#1274	C₂	1	2	yes	maximal
5	C₃₁×#431	C₂²	1	4	yes	—
6	C₂×C₃₁×#97	C₄	1	1	yes	characteristic
7	C₂×C₃₁×#97	C₄	1	1	yes	characteristic
8	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#376	C₂²	1	1	yes	characteristic
13	C₃₁×#335	—	2	2	no	—
14	C₃₁×#334	—	2	2	no	—
15	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
22	C₃₁×C₄⋊C₈	—	2	4	no	—
23	C₃₁×C₂²⋊C₈	—	2	4	no	—
24	C₃₁×#97	—	2	2	no	—
25	C₃₁×#97	—	2	2	no	—
26	C₃₁×#97	C₈	1	4	yes	complemented
27	C₄×C₈×C₃₁	—	2	2	no	—
28	C₄×C₈×C₃₁	—	2	2	no	—
29	C₄×C₈×C₃₁	—	2	4	no	—
30	C₄×C₈×C₃₁	D₄	1	2	yes	—
31	C₄×C₈×C₃₁	Q₈	1	2	yes	—
32	C₃₁×#95	—	2	2	no	—
33	C₂²×C₄×C₃₁	—	2	1	no	—
34	C₂²×C₄×C₃₁	—	2	1	no	—
35	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	#41	1	2	yes	—
37	C₂×C₈×C₃₁	—	2	1	no	—
38	C₂×C₈×C₃₁	—	2	1	no	—
39	C₂×C₈×C₃₁	—	2	1	no	—
40	C₂×C₈×C₃₁	—	2	1	no	—
41	C₂×C₈×C₃₁	—	2	1	no	—
42	C₂×C₈×C₃₁	—	2	1	no	—
43	C₂×C₈×C₃₁	—	4	4	no	—
44	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
49	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
50	C₄²×C₃₁	—	2	1	no	—
51	C₄²×C₃₁	—	2	1	no	—
52	C₄²×C₃₁	C₂×C₈	1	2	yes	—
53	C₄²×C₃₁	M₁₆	1	2	yes	—
54	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₃₁	#41	1	2	yes	—
57	#8781	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₃₁	#117	1	1	yes	characteristic
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	4	2	no	—
62	C₂×C₄×C₃₁	—	4	2	no	—
63	C₂×C₄×C₃₁	—	4	2	no	—
64	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
66	C₂×C₄×C₃₁	#124	1	2	yes	—
67	C₂×C₄×C₃₁	#125	1	2	yes	—
68	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₃₁	—	2	2	no	—
73	C₈×C₃₁	—	2	2	no	—
74	C₈×C₃₁	—	2	2	no	—
75	C₈×C₃₁	—	8	4	no	—
76	C₈×C₃₁	C₄⋊C₈	1	4	yes	—
77	#1726	C₂×C₃₁	1	1	yes	characteristic
78	#1564	C₂×C₃₁	1	2	yes	—
79	#1281	C₂×C₃₁	1	2	yes	—
80	#1274	C₂×C₃₁	1	2	yes	—
81	C₂²×C₃₁	—	2	2	no	—
82	C₂²×C₃₁	#382	1	1	yes	characteristic
83	C₂²×C₃₁	#431	1	1	yes	characteristic
84	C₂²×C₃₁	#432	1	1	yes	characteristic
85	C₄×C₃₁	—	2	2	no	—
86	C₄×C₃₁	—	2	2	no	—
87	C₄×C₃₁	—	4	2	no	—
88	C₄×C₃₁	—	4	2	no	—
89	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₄×C₃₁	#423	1	1	yes	characteristic
91	C₄×C₃₁	#431	1	1	yes	characteristic
92	C₄×C₃₁	#432	1	1	yes	characteristic
93	#431	C₂²×C₃₁	1	4	yes	—
94	C₂×#97	C₄×C₃₁	1	1	yes	characteristic
95	C₂×#97	C₄×C₃₁	1	1	yes	characteristic
96	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
97	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
98	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
100	#376	C₂²×C₃₁	1	1	yes	characteristic
101	#335	—	2	2	no	—
102	#334	—	2	2	no	—
103	C₂×C₃₁	—	2	1	no	—
104	C₂×C₃₁	—	2	1	no	—
105	C₂×C₃₁	#1831	1	1	yes	characteristic
106	C₂×C₃₁	#1833	1	2	yes	—
107	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
110	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
113	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
114	C₄⋊C₈	—	2	4	no	—
115	C₂²⋊C₈	—	2	4	no	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	#97	C₈×C₃₁	1	4	yes	complemented
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	4	no	—
122	C₄×C₈	D₄×C₃₁	1	2	yes	—
123	C₄×C₈	Q₈×C₃₁	1	2	yes	—
124	#95	—	2	2	no	—
125	C₃₁	#8781	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
129	C₂²×C₄	#41×C₃₁	1	2	yes	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	4	no	—
137	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
138	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
139	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
140	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
141	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
142	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₂×C₈×C₃₁	1	2	yes	—
146	C₄²	M₁₆×C₃₁	1	2	yes	—
147	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
148	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
149	C₄²	#41×C₃₁	1	2	yes	—
150	C₂³	C₃₁×#117	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
157	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
158	C₂×C₄	C₃₁×#124	1	2	yes	—
159	C₂×C₄	C₃₁×#125	1	2	yes	—
160	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
161	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
163	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	8	4	no	—
168	C₈	C₃₁×C₄⋊C₈	1	4	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	C₃₁×#382	1	1	yes	characteristic
171	C₂²	C₃₁×#431	1	1	yes	characteristic
172	C₂²	C₃₁×#432	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
178	C₄	C₃₁×#423	1	1	yes	characteristic
179	C₄	C₃₁×#431	1	1	yes	characteristic
180	C₄	C₃₁×#432	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	C₃₁×#1831	1	1	yes	characteristic
184	C₂	C₃₁×#1833	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#117
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#382
prime radical	C₂³×C₃₁	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
26. C₃₁×#97	C₈	semidirect
57. #8781	C₃₁	direct (N × H)
118. #97	C₈×C₃₁	semidirect
125. C₃₁	#8781	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×#1726	2
C₃₁×#1564	2
C₃₁×#1281	2
C₃₁×#1274	2
#8781	31

Subgroups — C31×#8781 #21465540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8781 #21465540