Subgroups — C₂²×C₁₆×Dic₇₁ #32150074

Order 18176.

1022 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₇₁	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇₁⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₁₆	C₄	1	8	yes	complemented
8	C₂²×C₇₁⋊C₁₆	C₂²	1	6	yes	—
9	C₂×C₈×Dic₇₁	C₄	1	6	yes	—
10	C₂×C₈×Dic₇₁	C₂²	1	6	yes	—
11	C₁₆×Dic₇₁	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₇₁	C₄	1	8	yes	complemented
14	C₂²×C₁₆×C₇₁	C₂²	1	6	yes	—
15	C₂³×Dic₇₁	C₈	1	2	yes	—
16	C₂²×C₇₁⋊C₈	C₈	1	8	yes	—
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	6	yes	—
18	C₂×C₄×Dic₇₁	C₈	1	6	yes	—
19	C₂×C₄×Dic₇₁	C₂×C₄	1	6	yes	—
20	C₂×C₇₁⋊C₁₆	C₂×C₄	1	24	yes	complemented
21	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
22	C₈×Dic₇₁	C₂×C₄	1	12	yes	—
23	C₈×Dic₇₁	C₂³	1	4	yes	—
24	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
29	C₂×C₁₆×C₇₁	C₂×C₄	1	24	yes	complemented
30	C₂×C₁₆×C₇₁	C₂³	1	4	yes	—
31	C₂²×Dic₇₁	C₁₆	1	16	yes	direct product factor
32	C₂²×Dic₇₁	C₂×C₈	1	12	yes	—
33	C₂×C₇₁⋊C₈	C₂×C₈	1	24	yes	—
34	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
35	C₄×Dic₇₁	C₂×C₈	1	12	yes	—
36	C₄×Dic₇₁	C₂²×C₄	1	4	yes	—
37	C₇₁⋊C₁₆	C₂²×C₄	1	8	yes	complemented
38	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₁	C₄²	1	4	yes	—
40	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₇₁	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
43	C₂×C₈×C₇₁	C₄²	1	12	yes	—
44	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
45	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
46	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
47	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
48	C₁₆×C₇₁	C₂²×C₄	1	8	yes	complemented
49	C₂³×C₇₁	C₄×C₈	1	4	yes	—
50	C₂³×C₇₁	C₂×C₁₆	1	8	yes	—
51	C₂³×C₇₁	C₂²×C₈	1	3	yes	—
52	C₂×C₄×C₇₁	C₄×C₈	1	12	yes	—
53	C₂×C₄×C₇₁	C₂×C₄²	1	6	yes	—
54	C₂×C₄×C₇₁	C₂²×C₈	1	6	yes	—
55	C₂×C₄×C₇₁	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
57	C₂×Dic₇₁	C₂×C₁₆	1	48	yes	direct product factor
58	C₂×Dic₇₁	C₂²×C₈	1	8	yes	—
59	C₈×C₇₁	C₂×C₄²	1	6	yes	—
60	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
61	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
62	C₇₁⋊C₈	C₂²×C₈	1	8	yes	—
63	C₂²×C₇₁	C₄×C₁₆	1	16	yes	complemented
64	C₂²×C₇₁	C₂×C₄×C₈	1	6	yes	—
65	C₂²×C₇₁	C₂²×C₁₆	1	12	yes	—
66	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
67	C₄×C₇₁	C₂×C₄×C₈	1	6	yes	—
68	C₄×C₇₁	C₂²×C₄²	1	1	yes	characteristic
69	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
70	Dic₇₁	C₂²×C₁₆	1	16	yes	direct product factor
71	C₂²×C₄×C₁₆	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	C₂×C₄×C₁₆	1	12	yes	complemented
73	C₂×C₇₁	C₂²×C₄×C₈	1	1	yes	characteristic
74	C₂×C₇₁	C₂³×C₁₆	1	2	yes	—
75	C₂³×C₁₆	—	71	1	no	—
76	C₂³×C₁₆	D₇₁	1	1	yes	characteristic
77	C₂²×C₄×C₈	—	71	1	no	—
78	C₂×C₄×C₁₆	—	71	12	no	—
79	C₇₁	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₈	—	71	1	no	—
81	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
82	C₂²×C₄²	—	71	1	no	—
83	C₂²×C₁₆	—	71	8	no	—
84	C₂²×C₁₆	—	71	6	no	—
85	C₂²×C₁₆	Dic₇₁	1	8	yes	direct product factor
86	C₂²×C₁₆	C₂×D₇₁	1	6	yes	—
87	C₂×C₄×C₈	—	71	6	no	—
88	C₂×C₄×C₈	—	71	6	no	—
89	C₄×C₁₆	—	71	16	no	—
90	C₂³×C₄	—	71	2	no	—
91	C₂³×C₄	C₄×D₇₁	1	1	yes	characteristic
92	C₂²×C₈	—	71	8	no	—
93	C₂²×C₈	—	71	6	no	—
94	C₂²×C₈	C₄×D₇₁	1	3	yes	—
95	C₂²×C₈	C₄×D₇₁	1	4	yes	—
96	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
97	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
98	C₂×C₄²	—	71	6	no	—
99	C₂×C₄²	—	71	6	no	—
100	C₂×C₁₆	—	71	24	no	—
101	C₂×C₁₆	—	71	4	no	—
102	C₂×C₁₆	C₂×Dic₇₁	1	24	yes	direct product factor
103	C₂×C₁₆	C₂²×D₇₁	1	4	yes	—
104	C₄×C₈	—	71	12	no	—
105	C₄×C₈	—	71	4	no	—
106	C₂⁴	C₈×D₇₁	1	1	yes	characteristic
107	C₂²×C₄	—	71	16	no	—
108	C₂²×C₄	—	71	12	no	—
109	C₂²×C₄	C₄×Dic₇₁	1	4	yes	—
110	C₂²×C₄	C₈×D₇₁	1	3	yes	—
111	C₂²×C₄	C₈×D₇₁	1	4	yes	—
112	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
113	C₂×C₈	—	71	4	no	—
114	C₂×C₈	—	71	24	no	—
115	C₂×C₈	C₄×Dic₇₁	1	12	yes	—
116	C₂×C₈	C₂×C₄×D₇₁	1	6	yes	—
117	C₂×C₈	C₂×C₄×D₇₁	1	3	yes	—
118	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
119	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
120	C₄²	—	71	4	no	—
121	C₄²	—	71	12	no	—
122	C₁₆	—	71	8	no	—
123	C₁₆	C₂²×Dic₇₁	1	8	yes	direct product factor
124	C₂³	C₈×Dic₇₁	1	4	yes	—
125	C₂³	C₁₆×D₇₁	1	8	yes	—
126	C₂³	C₂×C₈×D₇₁	1	3	yes	—
127	C₂×C₄	—	71	48	no	—
128	C₂×C₄	—	71	8	no	—
129	C₂×C₄	C₈×Dic₇₁	1	12	yes	—
130	C₂×C₄	C₂×C₄×Dic₇₁	1	6	yes	—
131	C₂×C₄	C₂×C₈×D₇₁	1	3	yes	—
132	C₂×C₄	C₂×C₈×D₇₁	1	6	yes	—
133	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
134	C₈	—	71	8	no	—
135	C₈	C₂×C₄×Dic₇₁	1	6	yes	—
136	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
137	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
138	C₂²	C₁₆×Dic₇₁	1	16	yes	direct product factor
139	C₂²	C₂×C₈×Dic₇₁	1	6	yes	—
140	C₂²	C₂×C₁₆×D₇₁	1	12	yes	—
141	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
142	C₄	—	71	16	no	—
143	C₄	C₂×C₈×Dic₇₁	1	6	yes	—
144	C₄	C₂²×C₄×Dic₇₁	1	1	yes	characteristic
145	C₄	C₂²×C₈×D₇₁	1	1	yes	characteristic
146	C₂	C₂×C₁₆×Dic₇₁	1	12	yes	direct product factor
147	C₂	C₂²×C₈×Dic₇₁	1	1	yes	characteristic
148	C₂	C₂²×C₁₆×D₇₁	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂³×C₁₆	D₇₁
commutator, hypocenter	C₇₁	C₂²×C₄×C₁₆
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂³×C₁₆×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₆×Dic₇₁	C₂	direct (N × H)
7. C₂²×C₇₁⋊C₁₆	C₄	semidirect
11. C₁₆×Dic₇₁	C₂²	direct (N × H)
13. C₂²×C₁₆×C₇₁	C₄	semidirect
20. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
29. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
31. C₂²×Dic₇₁	C₁₆	direct (N × H)
37. C₇₁⋊C₁₆	C₂²×C₄	semidirect
48. C₁₆×C₇₁	C₂²×C₄	semidirect
57. C₂×Dic₇₁	C₂×C₁₆	direct (N × H)
63. C₂²×C₇₁	C₄×C₁₆	semidirect
70. Dic₇₁	C₂²×C₁₆	direct (N × H)
72. C₂×C₇₁	C₂×C₄×C₁₆	semidirect
79. C₇₁	C₂²×C₄×C₁₆	semidirect
85. C₂²×C₁₆	Dic₇₁	direct (N × H)
102. C₂×C₁₆	C₂×Dic₇₁	direct (N × H)
123. C₁₆	C₂²×Dic₇₁	direct (N × H)
138. C₂²	C₁₆×Dic₇₁	direct (N × H)
146. C₂	C₂×C₁₆×Dic₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇₁⋊C₁₆	2
C₂²×C₈×Dic₇₁	2
C₂×C₁₆×Dic₇₁	2
C₂³×C₁₆×C₇₁	2
C₂²×C₄×C₁₆	71

Subgroups — C22×C16×Dic71 #32150074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₆×Dic₇₁ #32150074