Subgroups — C₃₁×#1736 #16841669

Order 3968.

448 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#340	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₃₁	—	2	1	no	—
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₃₁	—	2	2	no	—
8	C₂×M₁₆×C₃₁	—	2	2	no	—
9	C₂×M₁₆×C₃₁	C₄	1	4	yes	complemented
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#117	—	2	2	no	—
12	C₃₁×#117	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
15	C₂×C₄⋊C₄×C₃₁	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
19	C₂×C₄²×C₃₁	—	2	1	no	—
20	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	—	2	4	no	—
22	C₂²×C₄×C₃₁	—	2	2	no	—
23	C₂²×C₄×C₃₁	—	2	2	no	—
24	C₂²×C₄×C₃₁	—	4	4	no	—
25	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	D₄	1	2	yes	—
30	C₂²×C₄×C₃₁	D₄	1	2	yes	—
31	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	M₁₆×C₃₁	—	2	8	no	—
34	C₂×C₈×C₃₁	—	2	2	no	—
35	C₂×C₈×C₃₁	—	2	2	no	—
36	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
37	C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
41	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	complemented
42	C₂²⋊C₄×C₃₁	—	2	4	no	—
43	C₄²×C₃₁	—	2	4	no	—
44	C₂³×C₃₁	—	2	2	no	—
45	C₂³×C₃₁	—	2	2	no	—
46	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
49	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	—	2	4	no	—
51	C₂×C₄×C₃₁	—	2	4	no	—
52	C₂×C₄×C₃₁	—	4	8	no	—
53	C₂×C₄×C₃₁	—	4	2	no	—
54	C₂×C₄×C₃₁	—	4	4	no	—
55	C₂×C₄×C₃₁	—	4	2	no	—
56	C₂×C₄×C₃₁	C₄²	1	4	yes	—
57	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
60	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
62	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
63	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
69	C₈×C₃₁	—	2	4	no	—
70	#1736	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₃₁	—	2	4	no	—
72	C₂²×C₃₁	—	2	2	no	—
73	C₂²×C₃₁	—	2	2	no	—
74	C₂²×C₃₁	—	2	4	no	—
75	C₂²×C₃₁	#95	1	4	yes	—
76	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
80	C₂²×C₃₁	#137	1	2	yes	—
81	C₄×C₃₁	—	4	4	no	—
82	C₄×C₃₁	—	4	4	no	—
83	C₄×C₃₁	#95	1	4	yes	—
84	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
85	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×M₁₆	C₂×C₃₁	1	1	yes	characteristic
89	C₂×#117	C₂×C₃₁	1	1	yes	characteristic
90	C₂²×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
91	#340	C₂×C₃₁	1	4	yes	complemented
92	C₂×C₃₁	—	2	4	no	—
93	C₂×C₃₁	C₂×#95	1	1	yes	characteristic
94	C₂×C₃₁	#417	1	2	yes	—
95	C₂×C₃₁	#418	1	1	yes	characteristic
96	C₂×C₃₁	#419	1	1	yes	characteristic
97	C₂×C₃₁	#428	1	2	yes	—
98	C₂³×C₄	—	2	1	no	—
99	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	C₄×C₃₁	1	4	yes	complemented
103	C₂²×C₈	C₂²×C₃₁	1	2	yes	—
104	#117	—	2	2	no	—
105	#117	C₄×C₃₁	1	4	yes	complemented
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	C₄×C₃₁	1	4	yes	complemented
109	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	—	2	1	no	—
113	C₃₁	#1736	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴	D₄×C₃₁	1	1	yes	characteristic
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	C₂×C₄×C₃₁	1	2	yes	—
120	C₂²×C₄	C₂×C₄×C₃₁	1	2	yes	—
121	C₂²×C₄	C₂×C₄×C₃₁	1	2	yes	—
122	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
123	C₂²×C₄	D₄×C₃₁	1	2	yes	—
124	C₂²×C₄	D₄×C₃₁	1	2	yes	—
125	C₂²×C₄	Q₈×C₃₁	1	2	yes	—
126	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
135	C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	complemented
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂²⋊C₄×C₃₁	1	2	yes	—
141	C₂³	C₂²⋊C₄×C₃₁	1	2	yes	—
142	C₂³	C₄⋊C₄×C₃₁	1	2	yes	—
143	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	4	8	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	C₄²×C₃₁	1	4	yes	—
151	C₂×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
152	C₂×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
153	C₂×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
154	C₂×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
155	C₂×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
156	C₂×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
157	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
158	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
159	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
160	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
161	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
162	C₂×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	C₃₁×#95	1	4	yes	—
169	C₂²	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₂²	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
171	C₂²	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
172	C₂²	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
173	C₂²	C₃₁×#137	1	2	yes	—
174	C₄	—	4	4	no	—
175	C₄	—	4	4	no	—
176	C₄	C₃₁×#95	1	4	yes	—
177	C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
178	C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
179	C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
180	C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
181	C₂	—	2	4	no	—
182	C₂	C₂×C₃₁×#95	1	1	yes	characteristic
183	C₂	C₃₁×#417	1	2	yes	—
184	C₂	C₃₁×#418	1	1	yes	characteristic
185	C₂	C₃₁×#419	1	1	yes	characteristic
186	C₂	C₃₁×#428	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#340	C₂	semidirect
9. C₂×M₁₆×C₃₁	C₄	semidirect
12. C₃₁×#117	C₄	semidirect
15. C₂×C₄⋊C₄×C₃₁	C₄	semidirect
36. C₂×C₈×C₃₁	C₂×C₄	semidirect
41. C₄⋊C₄×C₃₁	C₂×C₄	semidirect
70. #1736	C₃₁	direct (N × H)
91. #340	C₂×C₃₁	semidirect
102. C₂×M₁₆	C₄×C₃₁	semidirect
105. #117	C₄×C₃₁	semidirect
108. C₂×C₄⋊C₄	C₄×C₃₁	semidirect
113. C₃₁	#1736	direct (N × H)
130. C₂×C₈	C₂×C₄×C₃₁	semidirect
135. C₄⋊C₄	C₂×C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₃₁	2
C₂×C₃₁×#117	2
C₂²×C₄⋊C₄×C₃₁	2
C₃₁×#340	2
#1736	31

Subgroups — C31×#1736 #16841669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#1736 #16841669