Subgroups — C₄×M₁₆×D₃₁ #16854116

Order 3968.

656 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₃₁	C₂	1	2	yes	maximal
2	C₂×C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#92474	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521001	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₃₁	C₂	1	2	yes	maximal, complemented
6	C₄×#92469	C₂	1	2	yes	maximal, complemented
7	#97×D₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×C₈×D₃₁	C₂	1	2	yes	maximal, complemented
9	C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₃₁	C₄	1	2	yes	—
11	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₃₁	C₄	1	2	yes	—
15	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
16	C₂×#92474	C₂²	1	2	yes	—
17	M₁₆×D₃₁	C₄	1	16	yes	direct product factor
18	C₂×#92469	C₂²	1	4	yes	—
19	C₂×C₈×D₃₁	C₂²	1	4	yes	—
20	C₄²×D₃₁	C₄	1	2	yes	—
21	C₄²×D₃₁	C₄	1	2	yes	—
22	C₄²×D₃₁	C₂²	1	1	yes	characteristic
23	C₄²×D₃₁	C₂²	1	1	yes	characteristic
24	C₄²×D₃₁	C₂²	1	1	yes	characteristic
25	C₄²×D₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#97	C₂²	1	2	yes	complemented
27	C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
28	#11758403	C₂²	1	4	yes	complemented
29	C₈×Dic₃₁	C₂²	1	4	yes	complemented
30	#11758391	C₂²	1	2	yes	complemented
31	C₄×C₃₁⋊C₈	C₂²	1	2	yes	complemented
32	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
36	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₃₁	—	2	8	no	—
38	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₃₁	C₂×C₄	1	8	yes	—
44	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
47	M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
48	C₂×C₈×C₃₁	C₂³	1	4	yes	—
49	C₄²×C₃₁	C₂×C₄	1	2	yes	—
50	C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
53	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
54	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
55	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
56	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
57	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
58	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
59	#92474	C₂×C₄	1	8	yes	complemented
60	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
61	#92469	C₂×C₄	1	16	yes	complemented
62	C₈×D₃₁	C₂×C₄	1	16	yes	complemented
63	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×D₃₁	—	2	4	no	—
65	C₂²×D₃₁	C₄²	1	4	yes	—
66	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	—	2	4	no	—
69	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₃₁	—	2	4	no	—
75	C₂×Dic₃₁	C₄²	1	4	yes	—
76	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
80	C₄×D₃₁	—	2	4	no	—
81	C₄×D₃₁	—	2	4	no	—
82	C₄×D₃₁	—	2	4	no	—
83	C₄×D₃₁	C₄²	1	4	yes	—
84	C₄×D₃₁	C₄²	1	4	yes	—
85	C₄×D₃₁	M₁₆	1	8	yes	direct product factor
86	C₄×D₃₁	C₂²×C₄	1	4	yes	—
87	C₄×D₃₁	C₂²×C₄	1	4	yes	—
88	C₈×C₃₁	C₂²×C₄	1	8	yes	complemented
89	C₃₁⋊C₈	C₂²×C₄	1	8	yes	complemented
90	C₂×C₄×M₁₆	—	31	1	no	maximal, Sylow
91	C₂²×C₃₁	—	2	2	no	—
92	C₂²×C₃₁	C₂×C₄²	1	2	yes	—
93	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₃₁	—	2	2	no	—
95	C₂×D₃₁	—	2	8	no	—
96	C₂×D₃₁	C₂×C₄²	1	2	yes	—
97	C₂×D₃₁	C₂×C₄²	1	2	yes	—
98	C₂×D₃₁	C₂×M₁₆	1	4	yes	—
99	C₄×C₃₁	—	2	2	no	—
100	C₄×C₃₁	C₂×C₄²	1	2	yes	—
101	C₄×C₃₁	C₂×M₁₆	1	4	yes	complemented
102	C₄×C₃₁	C₂³×C₄	1	2	yes	—
103	Dic₃₁	—	2	2	no	—
104	Dic₃₁	C₂×C₄²	1	2	yes	—
105	Dic₃₁	C₂×C₄²	1	2	yes	—
106	Dic₃₁	C₂×M₁₆	1	4	yes	complemented
107	C₂²×M₁₆	—	31	2	no	—
108	C₂²×C₄²	—	31	1	no	—
109	C₄×M₁₆	—	31	2	no	—
110	C₄×M₁₆	—	31	1	no	—
111	C₄×M₁₆	—	31	2	no	—
112	C₄×M₁₆	—	31	2	no	—
113	C₄×M₁₆	D₃₁	1	1	yes	characteristic, direct product factor
114	C₂×#97	—	31	2	no	—
115	C₂×C₄×C₈	—	31	2	no	—
116	C₂×C₃₁	—	2	2	no	—
117	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
118	C₂×C₃₁	C₂²×M₁₆	1	2	yes	—
119	D₃₁	—	2	2	no	—
120	D₃₁	C₄×M₁₆	1	4	yes	direct product factor
121	C₂³×C₄	—	31	1	no	—
122	C₂³×C₄	—	31	2	no	—
123	C₂×M₁₆	—	31	2	no	—
124	C₂×M₁₆	—	31	16	no	—
125	C₂×M₁₆	—	31	4	no	—
126	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
127	C₂²×C₈	—	31	4	no	—
128	C₂×C₄²	—	31	2	no	—
129	C₂×C₄²	—	31	1	no	—
130	C₂×C₄²	—	31	2	no	—
131	C₂×C₄²	—	31	2	no	—
132	C₂×C₄²	—	31	1	no	—
133	C₂×C₄²	—	31	1	no	—
134	C₂×C₄²	—	31	1	no	—
135	C₂×C₄²	—	31	1	no	—
136	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
137	#97	—	31	4	no	—
138	#97	—	31	2	no	—
139	#97	C₂×D₃₁	1	2	yes	complemented
140	C₄×C₈	—	31	4	no	—
141	C₄×C₈	—	31	2	no	—
142	C₄×C₈	C₂×D₃₁	1	2	yes	complemented
143	C₃₁	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
144	C₂⁴	—	31	1	no	—
145	C₂²×C₄	—	31	1	no	—
146	C₂²×C₄	—	31	1	no	—
147	C₂²×C₄	—	31	1	no	—
148	C₂²×C₄	—	31	2	no	—
149	C₂²×C₄	—	31	8	no	—
150	C₂²×C₄	—	31	1	no	—
151	C₂²×C₄	—	31	1	no	—
152	C₂²×C₄	—	31	2	no	—
153	C₂²×C₄	—	31	2	no	—
154	C₂²×C₄	—	31	2	no	—
155	C₂²×C₄	—	31	2	no	—
156	C₂²×C₄	—	62	8	no	—
157	C₂²×C₄	C₄×D₃₁	1	2	yes	—
158	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
159	M₁₆	—	31	8	no	—
160	M₁₆	—	31	16	no	—
161	M₁₆	C₄×D₃₁	1	8	yes	direct product factor
162	C₂×C₈	—	31	4	no	—
163	C₂×C₈	—	31	16	no	—
164	C₂×C₈	C₂²×D₃₁	1	4	yes	—
165	C₄²	—	31	2	no	—
166	C₄²	—	31	2	no	—
167	C₄²	—	31	2	no	—
168	C₄²	—	31	2	no	—
169	C₄²	—	31	2	no	—
170	C₄²	—	31	1	no	—
171	C₄²	—	31	1	no	—
172	C₄²	C₄×D₃₁	1	2	yes	—
173	C₄²	C₂²×D₃₁	1	1	yes	characteristic
174	C₄²	C₂²×D₃₁	1	1	yes	characteristic
175	C₂³	—	31	1	no	—
176	C₂³	—	31	4	no	—
177	C₂³	—	31	1	no	—
178	C₂³	—	62	4	no	—
179	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	31	4	no	—
182	C₂×C₄	—	31	1	no	—
183	C₂×C₄	—	31	1	no	—
184	C₂×C₄	—	31	2	no	—
185	C₂×C₄	—	31	4	no	—
186	C₂×C₄	—	31	4	no	—
187	C₂×C₄	—	31	8	no	—
188	C₂×C₄	—	31	4	no	—
189	C₂×C₄	—	31	4	no	—
190	C₂×C₄	—	31	2	no	—
191	C₂×C₄	—	62	4	no	—
192	C₂×C₄	—	62	4	no	—
193	C₂×C₄	—	62	4	no	—
194	C₂×C₄	—	62	4	no	—
195	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
196	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
198	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
199	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
200	C₈	—	31	8	no	—
201	C₈	C₂×C₄×D₃₁	1	8	yes	complemented
202	C₂²	—	2	2	no	—
203	C₂²	—	31	2	no	—
204	C₂²	—	31	4	no	—
205	C₂²	—	31	2	no	—
206	C₂²	—	62	8	no	—
207	C₂²	—	62	2	no	—
208	C₂²	C₄²×D₃₁	1	2	yes	—
209	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	31	2	no	—
212	C₄	—	31	2	no	—
213	C₄	—	31	4	no	—
214	C₄	—	62	2	no	—
215	C₄	C₄²×D₃₁	1	2	yes	—
216	C₄	M₁₆×D₃₁	1	4	yes	direct product factor
217	C₄	C₂²×C₄×D₃₁	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	31	4	no	—
220	C₂	—	62	2	no	—
221	C₂	C₂×C₄²×D₃₁	1	1	yes	characteristic
222	C₂	C₂×M₁₆×D₃₁	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×M₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂³×C₄
prime radical	C₂³×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₄×M₁₆	D₃₁
hypocenter	C₃₁	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#92474	C₂	semidirect
4. #14521001	C₂	semidirect
5. M₁₆×Dic₃₁	C₂	semidirect
6. C₄×#92469	C₂	semidirect
7. #97×D₃₁	C₂	semidirect
8. C₄×C₈×D₃₁	C₂	semidirect
9. C₄×M₁₆×C₃₁	C₂	semidirect
17. M₁₆×D₃₁	C₄	direct (N × H)
26. C₃₁×#97	C₂²	semidirect
27. C₄×C₈×C₃₁	C₂²	semidirect
28. #11758403	C₂²	semidirect
29. C₈×Dic₃₁	C₂²	semidirect
30. #11758391	C₂²	semidirect
31. C₄×C₃₁⋊C₈	C₂²	semidirect
47. M₁₆×C₃₁	C₂×C₄	semidirect
59. #92474	C₂×C₄	semidirect
61. #92469	C₂×C₄	semidirect
62. C₈×D₃₁	C₂×C₄	semidirect
85. C₄×D₃₁	M₁₆	direct (N × H)
88. C₈×C₃₁	C₂²×C₄	semidirect
89. C₃₁⋊C₈	C₂²×C₄	semidirect
101. C₄×C₃₁	C₂×M₁₆	semidirect
106. Dic₃₁	C₂×M₁₆	semidirect
113. C₄×M₁₆	D₃₁	direct (N × H)
120. D₃₁	C₄×M₁₆	direct (N × H)
139. #97	C₂×D₃₁	semidirect
142. C₄×C₈	C₂×D₃₁	semidirect
143. C₃₁	C₂×C₄×M₁₆	semidirect
161. M₁₆	C₄×D₃₁	direct (N × H)
201. C₈	C₂×C₄×D₃₁	semidirect
216. C₄	M₁₆×D₃₁	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×M₁₆×D₃₁	2
C₂×C₄²×D₃₁	2
C₄×#92474	2
#14521001	2
M₁₆×Dic₃₁	2
C₄×#92469	2
#97×D₃₁	2
C₄×C₈×D₃₁	2
C₄×M₁₆×C₃₁	2
C₂×C₄×M₁₆	31

Subgroups — C4×M16×D31 #16854116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×M₁₆×D₃₁ #16854116