Subgroups — C₃₁×#13169 #21469928

Order 7936.

286 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#368	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1945	C₂	1	2	yes	maximal
5	C₃₁×#1382	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#108	C₂²	1	2	yes	—
9	C₃₁×#427	C₂²	1	2	yes	—
10	C₃₁×#424	—	2	2	no	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#368	—	2	2	no	—
13	C₃₁×#367	C₄	1	2	yes	complemented
14	C₃₁×#366	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₃₁	—	2	2	no	—
16	C₃₁×#339	—	2	2	no	—
17	C₂×M₁₆×C₃₁	—	2	2	no	—
18	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
20	C₃₁×#117	—	2	2	no	—
21	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₃₁	—	2	1	no	—
23	C₂×C₁₆×C₃₁	—	2	1	no	—
24	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
27	C₂×C₁₆×C₃₁	Q₈	1	2	yes	—
28	C₂×C₁₆×C₃₁	Q₈	1	2	yes	—
29	C₃₁×#108	—	2	2	no	—
30	C₃₁×#108	—	2	2	no	—
31	C₃₁×#107	C₂×C₄	1	4	yes	—
32	C₃₁×#106	—	2	2	no	—
33	C₃₁×C₄⋊C₈	—	4	4	no	—
34	C₄×C₈×C₃₁	—	2	1	no	—
35	C₄×C₈×C₃₁	—	2	1	no	—
36	C₂²×C₄×C₃₁	—	2	1	no	—
37	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₃₁	—	4	2	no	—
39	M₁₆×C₃₁	—	4	2	no	—
40	C₂×C₈×C₃₁	—	2	1	no	—
41	C₂×C₈×C₃₁	—	2	1	no	—
42	C₂×C₈×C₃₁	—	4	2	no	—
43	C₂×C₈×C₃₁	D₈	1	2	yes	—
44	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
49	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₂²⋊C₄×C₃₁	—	4	2	no	—
52	C₄²×C₃₁	—	2	1	no	—
53	C₄²×C₃₁	—	4	2	no	—
54	C₄²×C₃₁	Q₁₆	1	2	yes	—
55	C₁₆×C₃₁	—	2	2	no	—
56	C₁₆×C₃₁	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₃₁	C₄⋊C₄	1	2	yes	—
58	#13169	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	#135	1	1	yes	characteristic
60	C₂×C₄×C₃₁	—	2	4	no	—
61	C₂×C₄×C₃₁	—	2	2	no	—
62	C₂×C₄×C₃₁	—	4	4	no	—
63	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
65	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
69	C₈×C₃₁	—	2	2	no	—
70	C₈×C₃₁	—	8	2	no	—
71	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
74	C₈×C₃₁	#125	1	1	yes	characteristic
75	C₂×#368	C₂×C₃₁	1	1	yes	characteristic
76	#2158	C₂×C₃₁	1	1	yes	characteristic
77	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
78	#1945	C₂×C₃₁	1	2	yes	—
79	#1382	C₂×C₃₁	1	2	yes	—
80	C₂²×C₃₁	—	2	2	no	—
81	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
82	C₂²×C₃₁	#467	1	1	yes	characteristic
83	C₂²×C₃₁	#499	1	1	yes	characteristic
84	C₄×C₃₁	—	2	4	no	—
85	C₄×C₃₁	—	8	4	no	—
86	C₄×C₃₁	#389	1	1	yes	characteristic
87	C₄×C₃₁	C₄×Q₁₆	1	1	yes	characteristic
88	C₄×C₃₁	#466	1	1	yes	characteristic
89	C₄×C₃₁	#500	1	1	yes	characteristic
90	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
91	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
92	C₂×#108	C₂²×C₃₁	1	2	yes	—
93	#427	C₂²×C₃₁	1	2	yes	—
94	#424	—	2	2	no	—
95	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
96	#368	—	2	2	no	—
97	#367	C₄×C₃₁	1	2	yes	complemented
98	#366	C₄×C₃₁	1	2	yes	complemented
99	C₄×C₁₆	—	2	2	no	—
100	#339	—	2	2	no	—
101	C₂×C₃₁	—	2	1	no	—
102	C₂×C₃₁	—	2	1	no	—
103	C₂×C₃₁	#1942	1	1	yes	characteristic
104	C₂×C₃₁	#2178	1	2	yes	—
105	C₂×M₁₆	—	2	2	no	—
106	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
108	#117	—	2	2	no	—
109	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
113	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
114	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
115	C₂×C₁₆	Q₈×C₃₁	1	2	yes	—
116	C₂×C₁₆	Q₈×C₃₁	1	2	yes	—
117	#108	—	2	2	no	—
118	#108	—	2	2	no	—
119	#107	C₂×C₄×C₃₁	1	4	yes	—
120	#106	—	2	2	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₃₁	#13169	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
127	M₁₆	—	4	2	no	—
128	M₁₆	—	4	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	D₈×C₃₁	1	2	yes	—
133	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
134	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
135	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
136	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
137	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
138	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	4	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	4	2	no	—
143	C₄²	Q₁₆×C₃₁	1	2	yes	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₄⋊C₄×C₃₁	1	2	yes	—
146	C₁₆	C₄⋊C₄×C₃₁	1	2	yes	—
147	C₂³	C₃₁×#135	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
153	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
155	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
156	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	8	2	no	—
159	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
160	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
161	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
162	C₈	C₃₁×#125	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
165	C₂²	C₃₁×#467	1	1	yes	characteristic
166	C₂²	C₃₁×#499	1	1	yes	characteristic
167	C₄	—	2	4	no	—
168	C₄	—	8	4	no	—
169	C₄	C₃₁×#389	1	1	yes	characteristic
170	C₄	C₄×Q₁₆×C₃₁	1	1	yes	characteristic
171	C₄	C₃₁×#466	1	1	yes	characteristic
172	C₄	C₃₁×#500	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	C₃₁×#1942	1	1	yes	characteristic
176	C₂	C₃₁×#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×D₈
prime radical	C₂³×C₃₁	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₃₁×#367	C₄	semidirect
14. C₃₁×#366	C₄	semidirect
58. #13169	C₃₁	direct (N × H)
97. #367	C₄×C₃₁	semidirect
98. #366	C₄×C₃₁	semidirect
124. C₃₁	#13169	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×#368	2
C₃₁×#2158	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1945	2
C₃₁×#1382	2
#13169	31

Subgroups — C31×#13169 #21469928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13169 #21469928