Subgroups — C₃₁×#2546 #16842479

Order 3968.

540 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#398	C₂	1	4	yes	maximal
5	C₃₁×#391	C₂	1	2	yes	maximal
6	C₃₁×#389	C₂	1	4	yes	maximal
7	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
9	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×C₄⋊Q₈	—	2	8	no	—
11	C₃₁×C₄⋊Q₈	—	2	4	no	—
12	C₄×Q₈×C₃₁	—	2	4	no	—
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#95	C₂²	1	2	yes	—
23	C₃₁×#95	C₂²	1	8	yes	—
24	C₂×Q₈×C₃₁	—	2	2	no	—
25	C₂×Q₈×C₃₁	—	2	4	no	—
26	C₂×Q₈×C₃₁	—	2	4	no	—
27	C₂×Q₈×C₃₁	—	2	2	no	—
28	C₂×Q₈×C₃₁	—	2	4	no	—
29	C₂×Q₈×C₃₁	D₄	1	4	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
31	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
35	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₄⋊C₄×C₃₁	—	2	16	no	—
40	C₄⋊C₄×C₃₁	—	2	4	no	—
41	C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₄⋊C₄×C₃₁	Q₈	1	8	yes	complemented
43	C₄²×C₃₁	—	2	4	no	—
44	C₄²×C₃₁	—	2	4	no	—
45	C₄²×C₃₁	D₄	1	4	yes	—
46	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
47	Q₈×C₃₁	—	2	8	no	—
48	Q₈×C₃₁	—	4	8	no	—
49	C₂×C₄×C₃₁	—	2	4	no	—
50	C₂×C₄×C₃₁	—	2	2	no	—
51	C₂×C₄×C₃₁	—	2	2	no	—
52	C₂×C₄×C₃₁	—	2	2	no	—
53	C₂×C₄×C₃₁	—	2	4	no	—
54	C₂×C₄×C₃₁	—	2	2	no	—
55	C₂×C₄×C₃₁	—	2	4	no	—
56	C₂×C₄×C₃₁	—	2	4	no	—
57	C₂×C₄×C₃₁	—	2	4	no	—
58	C₂×C₄×C₃₁	—	2	4	no	—
59	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₃₁	C₂×Q₈	1	8	yes	—
65	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₃₁	#41	1	2	yes	—
67	#2546	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
70	C₂²×C₃₁	C₂²×Q₈	1	2	yes	—
71	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
72	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
73	C₄×C₃₁	—	2	4	no	—
74	C₄×C₃₁	—	2	2	no	—
75	C₄×C₃₁	—	2	8	no	—
76	C₄×C₃₁	—	4	4	no	—
77	C₄×C₃₁	—	4	2	no	—
78	C₄×C₃₁	#121	1	4	yes	—
79	C₄×C₃₁	C₄⋊Q₈	1	4	yes	complemented
80	C₂×C₄⋊Q₈	C₂×C₃₁	1	2	yes	—
81	C₂×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
82	C₂×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
83	#398	C₂×C₃₁	1	4	yes	—
84	#391	C₂×C₃₁	1	2	yes	—
85	#389	C₂×C₃₁	1	4	yes	—
86	C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
87	C₂×C₃₁	C₂×#121	1	1	yes	characteristic
88	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₃₁	#536	1	1	yes	characteristic
90	C₂×C₃₁	D₄×Q₈	1	2	yes	—
91	C₂×C₃₁	#557	1	2	yes	—
92	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
93	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
94	C₄⋊Q₈	—	2	8	no	—
95	C₄⋊Q₈	—	2	4	no	—
96	C₄×Q₈	—	2	4	no	—
97	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
101	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
103	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
104	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
105	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
106	#95	C₂²×C₃₁	1	2	yes	—
107	#95	C₂²×C₃₁	1	8	yes	—
108	C₃₁	#2546	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂×Q₈	—	2	2	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	D₄×C₃₁	1	4	yes	—
115	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
116	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
117	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
119	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
120	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
121	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	16	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	Q₈×C₃₁	1	8	yes	complemented
128	C₄²	—	2	4	no	—
129	C₄²	—	2	4	no	—
130	C₄²	D₄×C₃₁	1	4	yes	—
131	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
132	Q₈	—	2	8	no	—
133	Q₈	—	4	8	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
145	C₂×C₄	C₂×D₄×C₃₁	1	4	yes	—
146	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
147	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
148	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
149	C₂×C₄	C₂×Q₈×C₃₁	1	8	yes	—
150	C₂×C₄	C₂×Q₈×C₃₁	1	4	yes	—
151	C₂×C₄	#41×C₃₁	1	2	yes	—
152	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
153	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
154	C₂²	C₂²×Q₈×C₃₁	1	2	yes	—
155	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
156	C₂²	C₃₁×2₋¹⁺⁴	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	8	no	—
160	C₄	—	4	4	no	—
161	C₄	—	4	2	no	—
162	C₄	C₃₁×#121	1	4	yes	—
163	C₄	C₃₁×C₄⋊Q₈	1	4	yes	complemented
164	C₂	C₂×C₃₁×#121	1	1	yes	characteristic
165	C₂	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
166	C₂	C₃₁×#536	1	1	yes	characteristic
167	C₂	D₄×Q₈×C₃₁	1	2	yes	—
168	C₂	C₃₁×#557	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
42. C₄⋊C₄×C₃₁	Q₈	semidirect
67. #2546	C₃₁	direct (N × H)
79. C₄×C₃₁	C₄⋊Q₈	semidirect
108. C₃₁	#2546	direct (N × H)
127. C₄⋊C₄	Q₈×C₃₁	semidirect
163. C₄	C₃₁×C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×C₄⋊Q₈	2
C₂×C₄×Q₈×C₃₁	2
C₃₁×#398	2
C₃₁×#391	2
C₃₁×#389	2
C₄×C₄⋊C₄×C₃₁	2
#2546	31

Subgroups — C31×#2546 #16842479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2546 #16842479