Subgroups — M₆₄×Dic₇₁ #31917627

Order 18176.

250 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	C₂×#17985833	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586473	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₇₁	C₂	1	2	yes	maximal, complemented
5	C₂×M₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
8	#17985833	C₄	1	4	yes	complemented
9	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
10	C₁₆×Dic₇₁	C₄	1	1	yes	characteristic
11	C₁₆×Dic₇₁	C₄	1	1	yes	characteristic
12	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	M₆₄×C₇₁	C₄	1	4	yes	complemented
16	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
17	C₂²×C₇₁⋊C₈	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₇₁	C₈	1	1	yes	characteristic
19	C₂×C₇₁⋊C₁₆	C₈	1	4	yes	—
20	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₈×Dic₇₁	C₈	1	1	yes	characteristic
23	C₈×Dic₇₁	C₈	1	1	yes	characteristic
24	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
26	C₇₁⋊C₃₂	C₂×C₄	1	4	yes	complemented
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
32	C₃₂×C₇₁	C₂×C₄	1	4	yes	complemented
33	C₂²×Dic₇₁	C₁₆	1	2	yes	—
34	C₂×C₇₁⋊C₈	C₁₆	1	4	yes	—
35	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
36	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
37	C₄×Dic₇₁	C₁₆	1	2	yes	—
38	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
39	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
40	C₇₁⋊C₁₆	C₂×C₈	1	2	yes	—
41	C₇₁⋊C₁₆	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₄²	1	2	yes	—
44	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₁₆×C₇₁	C₄²	1	2	yes	—
48	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
54	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
55	C₂×Dic₇₁	—	2	4	no	—
56	C₂×Dic₇₁	C₂×C₁₆	1	2	yes	—
57	C₂×Dic₇₁	C₂×C₁₆	1	2	yes	—
58	C₈×C₇₁	C₄×C₈	1	2	yes	—
59	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
60	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
61	C₇₁⋊C₈	C₂×C₁₆	1	2	yes	—
62	C₇₁⋊C₈	C₂×C₁₆	1	2	yes	—
63	C₂²×C₇₁	—	2	2	no	—
64	C₂²×C₇₁	C₄×C₁₆	1	2	yes	—
65	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
66	C₄×C₇₁	C₄×C₁₆	1	2	yes	—
67	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
69	Dic₇₁	—	2	2	no	—
70	Dic₇₁	M₆₄	1	4	yes	direct product factor
71	C₄×M₆₄	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	—	2	2	no	—
73	C₂×C₇₁	C₂×C₄×C₁₆	1	1	yes	characteristic
74	C₂×C₇₁	C₂×M₆₄	1	2	yes	—
75	C₂×M₆₄	—	71	1	no	—
76	C₂×M₆₄	D₇₁	1	1	yes	characteristic
77	C₂×C₄×C₁₆	—	71	1	no	—
78	#1397	—	71	2	no	—
79	C₄×C₃₂	—	71	2	no	—
80	C₇₁	C₄×M₆₄	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₁₆	—	71	1	no	—
82	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
83	C₂×C₄×C₈	—	71	1	no	—
84	M₆₄	—	71	4	no	—
85	M₆₄	Dic₇₁	1	4	yes	direct product factor
86	C₂×C₃₂	—	71	2	no	—
87	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
88	C₄×C₁₆	—	71	1	no	—
89	C₄×C₁₆	—	71	1	no	—
90	C₄×C₁₆	—	71	1	no	—
91	C₄×C₁₆	—	71	1	no	—
92	C₂²×C₈	—	71	1	no	—
93	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
94	C₂×C₄²	—	71	1	no	—
95	C₂×C₁₆	—	71	4	no	—
96	C₂×C₁₆	—	71	1	no	—
97	C₂×C₁₆	—	71	1	no	—
98	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
99	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
100	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
101	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
102	C₄×C₈	—	71	1	no	—
103	C₄×C₈	—	71	1	no	—
104	C₄×C₈	—	71	1	no	—
105	C₄×C₈	—	71	1	no	—
106	C₃₂	—	71	4	no	—
107	C₃₂	C₂×Dic₇₁	1	4	yes	complemented
108	C₂²×C₄	—	71	2	no	—
109	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
110	C₂×C₈	—	71	4	no	—
111	C₂×C₈	—	71	1	no	—
112	C₂×C₈	—	71	1	no	—
113	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
114	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
115	C₂×C₈	C₈×D₇₁	1	2	yes	—
116	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
117	C₄²	—	71	2	no	—
118	C₄²	—	71	1	no	—
119	C₄²	—	71	1	no	—
120	C₁₆	—	71	2	no	—
121	C₁₆	—	71	2	no	—
122	C₁₆	C₄×Dic₇₁	1	2	yes	—
123	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
124	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
125	C₂³	C₁₆×D₇₁	1	1	yes	characteristic
126	C₂×C₄	—	71	2	no	—
127	C₂×C₄	—	71	2	no	—
128	C₂×C₄	—	142	4	no	—
129	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
130	C₂×C₄	C₁₆×D₇₁	1	2	yes	—
131	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
132	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
133	C₈	—	71	2	no	—
134	C₈	—	71	2	no	—
135	C₈	C₈×Dic₇₁	1	2	yes	—
136	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
137	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	C₁₆×Dic₇₁	1	2	yes	—
140	C₂²	C₂×C₁₆×D₇₁	1	1	yes	characteristic
141	C₄	—	71	4	no	—
142	C₄	—	142	2	no	—
143	C₄	C₁₆×Dic₇₁	1	2	yes	—
144	C₄	C₂×C₈×Dic₇₁	1	1	yes	characteristic
145	C₄	C₂×C₁₆×D₇₁	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	C₂×C₁₆×Dic₇₁	1	1	yes	characteristic
148	C₂	M₆₄×D₇₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator	C₂×C₇₁	C₂×C₄×C₁₆
Fitting	C₂×M₆₄×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₁₆
prime radical	C₂³×C₇₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂×M₆₄	D₇₁
hypocenter	C₇₁	C₄×M₆₄
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #22586473	C₂	semidirect
4. C₃₂×Dic₇₁	C₂	semidirect
8. #17985833	C₄	semidirect
15. M₆₄×C₇₁	C₄	semidirect
26. C₇₁⋊C₃₂	C₂×C₄	semidirect
32. C₃₂×C₇₁	C₂×C₄	semidirect
70. Dic₇₁	M₆₄	direct (N × H)
80. C₇₁	C₄×M₆₄	semidirect
85. M₆₄	Dic₇₁	direct (N × H)
107. C₃₂	C₂×Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985833	2
C₂×C₁₆×Dic₇₁	2
#22586473	2
C₃₂×Dic₇₁	2
C₂×M₆₄×C₇₁	2
C₄×M₆₄	71

Subgroups — M64×Dic71 #31917627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₆₄×Dic₇₁ #31917627