Subgroups — D₄×Q₁₆×C₃₁ #16843219

Order 3968.

472 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈×C₃₁	C₂	1	2	yes	maximal
3	C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#467	C₂	1	2	yes	maximal, complemented
5	C₃₁×#462	C₂	1	2	yes	maximal, complemented
6	C₃₁×#451	C₂	1	4	yes	maximal, complemented
7	C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₃₁	C₂²	1	4	yes	—
10	C₂×Q₁₆×C₃₁	—	2	8	no	—
11	C₂×Q₁₆×C₃₁	C₂²	1	4	yes	—
12	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×C₄⋊Q₈	—	2	2	no	—
17	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₃₁×C₂²⋊Q₈	—	2	4	no	—
19	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
21	C₄×D₄×C₃₁	—	2	2	no	—
22	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₃₁×#103	C₂²	1	2	yes	—
26	C₃₁×#103	C₂²	1	4	yes	—
27	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×Q₈×C₃₁	—	2	4	no	—
30	C₂×Q₈×C₃₁	—	2	8	no	—
31	C₂×Q₈×C₃₁	—	4	4	no	—
32	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
33	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
34	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	—	2	4	no	—
36	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
37	Q₁₆×C₃₁	—	2	8	no	—
38	Q₁₆×C₃₁	—	2	4	no	—
39	Q₁₆×C₃₁	—	4	2	no	—
40	Q₁₆×C₃₁	D₄	1	4	yes	direct product factor
41	C₂×C₈×C₃₁	—	2	4	no	—
42	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₄⋊C₄×C₃₁	—	2	4	no	—
49	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
51	C₂²⋊C₄×C₃₁	—	2	4	no	—
52	C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
53	C₄²×C₃₁	—	2	2	no	—
54	C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₂³×C₃₁	C₂×D₄	1	2	yes	—
56	Q₈×C₃₁	—	2	2	no	—
57	Q₈×C₃₁	—	2	4	no	—
58	Q₈×C₃₁	—	2	4	no	—
59	Q₈×C₃₁	—	4	8	no	—
60	Q₈×C₃₁	C₂×D₄	1	4	yes	—
61	D₄×C₃₁	Q₁₆	1	4	yes	direct product factor
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	4	no	—
67	C₂×C₄×C₃₁	—	4	8	no	—
68	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	C₈×C₃₁	—	2	2	no	—
73	C₈×C₃₁	—	2	1	no	—
74	C₈×C₃₁	C₂×D₄	1	2	yes	—
75	D₄×Q₁₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₃₁	—	2	2	no	—
77	C₂²×C₃₁	—	2	2	no	—
78	C₂²×C₃₁	C₂×Q₁₆	1	4	yes	complemented
79	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₃₁	—	2	2	no	—
81	C₄×C₃₁	—	2	4	no	—
82	C₄×C₃₁	—	2	1	no	—
83	C₄×C₃₁	—	4	4	no	—
84	C₄×C₃₁	—	4	2	no	—
85	C₄×C₃₁	C₂×Q₁₆	1	2	yes	complemented
86	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂²×Q₁₆	C₂×C₃₁	1	2	yes	complemented
89	D₄×Q₈	C₂×C₃₁	1	2	yes	—
90	#494	C₂×C₃₁	1	1	yes	characteristic, complemented
91	#467	C₂×C₃₁	1	2	yes	complemented
92	#462	C₂×C₃₁	1	2	yes	complemented
93	#451	C₂×C₃₁	1	4	yes	complemented
94	C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
95	C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
96	C₂×C₃₁	—	2	4	no	—
97	C₂×C₃₁	D₄²	1	1	yes	characteristic
98	C₂×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
99	C₂×C₃₁	#578	1	1	yes	characteristic
100	C₂²×Q₈	C₂²×C₃₁	1	4	yes	—
101	C₂×Q₁₆	—	2	8	no	—
102	C₂×Q₁₆	C₂²×C₃₁	1	4	yes	—
103	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
104	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
105	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₃₁	1	2	yes	—
107	C₄⋊Q₈	—	2	2	no	—
108	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
109	C₂²⋊Q₈	—	2	4	no	—
110	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
111	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
112	C₄×D₄	—	2	2	no	—
113	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
114	#107	C₂²×C₃₁	1	1	yes	characteristic
115	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
116	#103	C₂²×C₃₁	1	2	yes	—
117	#103	C₂²×C₃₁	1	4	yes	—
118	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
119	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
120	C₃₁	D₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×Q₈	—	2	4	no	—
122	C₂×Q₈	—	2	8	no	—
123	C₂×Q₈	—	4	4	no	—
124	C₂×Q₈	C₂³×C₃₁	1	4	yes	—
125	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
126	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
129	Q₁₆	—	2	8	no	—
130	Q₁₆	—	2	4	no	—
131	Q₁₆	—	4	2	no	—
132	Q₁₆	D₄×C₃₁	1	4	yes	direct product factor
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
135	C₂×C₈	C₂³×C₃₁	1	2	yes	—
136	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
142	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
145	C₄²	—	2	2	no	—
146	C₄²	C₂³×C₃₁	1	1	yes	characteristic
147	C₂³	C₂×D₄×C₃₁	1	2	yes	—
148	Q₈	—	2	2	no	—
149	Q₈	—	2	4	no	—
150	Q₈	—	2	4	no	—
151	Q₈	—	4	8	no	—
152	Q₈	C₂×D₄×C₃₁	1	4	yes	—
153	D₄	Q₁₆×C₃₁	1	4	yes	direct product factor
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
161	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
162	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
163	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	C₂×D₄×C₃₁	1	2	yes	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	C₂×Q₁₆×C₃₁	1	4	yes	complemented
170	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	4	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	4	no	—
175	C₄	—	4	2	no	—
176	C₄	C₂×Q₁₆×C₃₁	1	2	yes	complemented
177	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
178	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
179	C₂	—	2	4	no	—
180	C₂	D₄²×C₃₁	1	1	yes	characteristic
181	C₂	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
182	C₂	C₃₁×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₁₆×C₃₁	C₂	semidirect
3. C₃₁×#494	C₂	semidirect
4. C₃₁×#467	C₂	semidirect
5. C₃₁×#462	C₂	semidirect
6. C₃₁×#451	C₂	semidirect
7. C₄×Q₁₆×C₃₁	C₂	semidirect
40. Q₁₆×C₃₁	D₄	direct (N × H)
61. D₄×C₃₁	Q₁₆	direct (N × H)
75. D₄×Q₁₆	C₃₁	direct (N × H)
78. C₂²×C₃₁	C₂×Q₁₆	semidirect
85. C₄×C₃₁	C₂×Q₁₆	semidirect
88. C₂²×Q₁₆	C₂×C₃₁	semidirect
90. #494	C₂×C₃₁	semidirect
91. #467	C₂×C₃₁	semidirect
92. #462	C₂×C₃₁	semidirect
93. #451	C₂×C₃₁	semidirect
94. C₄×Q₁₆	C₂×C₃₁	semidirect
120. C₃₁	D₄×Q₁₆	direct (N × H)
132. Q₁₆	D₄×C₃₁	direct (N × H)
153. D₄	Q₁₆×C₃₁	direct (N × H)
169. C₂²	C₂×Q₁₆×C₃₁	semidirect
176. C₄	C₂×Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₁₆×C₃₁	2
D₄×Q₈×C₃₁	2
C₃₁×#494	2
C₃₁×#467	2
C₃₁×#462	2
C₃₁×#451	2
C₄×Q₁₆×C₃₁	2
C₈×D₄×C₃₁	2
D₄×Q₁₆	31

Subgroups — D4×Q16×C31 #16843219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Q₁₆×C₃₁ #16843219