Subgroups — C₄²×M₁₆×C₆₇ #30699579

Order 17152.

1734 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₆₇	C₂	1	6	yes	maximal
2	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#97×C₆₇	C₂	1	6	yes	maximal, complemented
4	C₄²×C₈×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₆₇	C₂²	1	4	yes	—
6	C₂²×C₄²×C₆₇	C₄	1	4	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
8	C₄×M₁₆×C₆₇	C₄	1	48	yes	direct product factor
9	C₂×#97×C₆₇	C₂²	1	12	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	12	yes	—
11	C₄³×C₆₇	C₄	1	4	yes	—
12	C₄³×C₆₇	C₂²	1	3	yes	—
13	C₄³×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
15	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×M₁₆×C₆₇	C₂×C₄	1	48	yes	—
17	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
18	C₂×C₄²×C₆₇	—	2	16	no	—
19	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
20	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
21	C₂×C₄²×C₆₇	C₂×C₄	1	24	yes	—
22	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
24	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
25	#97×C₆₇	C₂×C₄	1	48	yes	complemented
26	C₄×C₈×C₆₇	C₂×C₄	1	48	yes	complemented
27	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	—	2	24	no	—
29	C₂²×C₄×C₆₇	C₄²	1	24	yes	—
30	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
31	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
33	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
34	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
35	M₁₆×C₆₇	C₄²	1	64	yes	direct product factor
36	C₂×C₈×C₆₇	C₂²×C₄	1	48	yes	—
37	C₄²×C₆₇	—	2	24	no	—
38	C₄²×C₆₇	C₄²	1	24	yes	—
39	C₄²×C₆₇	M₁₆	1	16	yes	direct product factor
40	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
41	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
42	C₂³×C₆₇	—	2	4	no	—
43	C₂³×C₆₇	C₂×C₄²	1	6	yes	—
44	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	—	2	12	no	—
46	C₂×C₄×C₆₇	—	2	48	no	—
47	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
48	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
49	C₂×C₄×C₆₇	C₂×C₄²	1	16	yes	—
50	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
51	C₂×C₄×C₆₇	C₂×M₁₆	1	24	yes	—
52	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
53	C₈×C₆₇	C₂×C₄²	1	32	yes	complemented
54	C₂²×C₆₇	—	2	12	no	—
55	C₂²×C₆₇	C₄³	1	4	yes	—
56	C₂²×C₆₇	C₂²×C₄²	1	3	yes	—
57	C₂²×C₆₇	C₂²×M₁₆	1	4	yes	—
58	C₄×C₆₇	—	2	12	no	—
59	C₄×C₆₇	C₄³	1	4	yes	—
60	C₄×C₆₇	C₄×M₁₆	1	24	yes	direct product factor
61	C₄×C₆₇	C₂²×C₄²	1	4	yes	—
62	C₄²×M₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₆₇	—	2	4	no	—
64	C₂×C₆₇	C₂×C₄³	1	1	yes	characteristic
65	C₂×C₆₇	C₂×C₄×M₁₆	1	6	yes	—
66	C₂×C₄×M₁₆	C₂×C₆₇	1	6	yes	—
67	C₂×C₄³	C₂×C₆₇	1	1	yes	characteristic
68	C₄×#97	C₂×C₆₇	1	6	yes	complemented
69	C₄²×C₈	C₂×C₆₇	1	2	yes	complemented
70	C₆₇	C₄²×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×M₁₆	C₂²×C₆₇	1	4	yes	—
72	C₂²×C₄²	C₄×C₆₇	1	4	yes	—
73	C₂²×C₄²	C₂²×C₆₇	1	3	yes	—
74	C₄×M₁₆	C₄×C₆₇	1	48	yes	direct product factor
75	C₂×#97	C₂²×C₆₇	1	12	yes	—
76	C₂×C₄×C₈	C₂²×C₆₇	1	12	yes	—
77	C₄³	C₄×C₆₇	1	4	yes	—
78	C₄³	C₂²×C₆₇	1	3	yes	—
79	C₄³	C₂²×C₆₇	1	1	yes	characteristic
80	C₂³×C₄	C₂×C₄×C₆₇	1	6	yes	—
81	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
82	C₂×M₁₆	C₂×C₄×C₆₇	1	48	yes	—
83	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
84	C₂×C₄²	—	2	16	no	—
85	C₂×C₄²	C₂×C₄×C₆₇	1	12	yes	—
86	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
87	C₂×C₄²	C₂×C₄×C₆₇	1	24	yes	—
88	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
89	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
90	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
91	#97	C₂×C₄×C₆₇	1	48	yes	complemented
92	C₄×C₈	C₂×C₄×C₆₇	1	48	yes	complemented
93	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
94	C₂²×C₄	—	2	24	no	—
95	C₂²×C₄	C₄²×C₆₇	1	24	yes	—
96	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
97	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
99	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
100	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
101	M₁₆	C₄²×C₆₇	1	64	yes	direct product factor
102	C₂×C₈	C₂²×C₄×C₆₇	1	48	yes	—
103	C₄²	—	2	24	no	—
104	C₄²	C₄²×C₆₇	1	24	yes	—
105	C₄²	M₁₆×C₆₇	1	16	yes	direct product factor
106	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
107	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
108	C₂³	—	2	4	no	—
109	C₂³	C₂×C₄²×C₆₇	1	6	yes	—
110	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
111	C₂×C₄	—	2	12	no	—
112	C₂×C₄	—	2	48	no	—
113	C₂×C₄	C₂×C₄²×C₆₇	1	12	yes	—
114	C₂×C₄	C₂×C₄²×C₆₇	1	16	yes	—
115	C₂×C₄	C₂×C₄²×C₆₇	1	6	yes	—
116	C₂×C₄	C₂×C₄²×C₆₇	1	12	yes	—
117	C₂×C₄	C₂×M₁₆×C₆₇	1	24	yes	—
118	C₂×C₄	C₂³×C₄×C₆₇	1	6	yes	—
119	C₈	C₂×C₄²×C₆₇	1	32	yes	complemented
120	C₂²	—	2	12	no	—
121	C₂²	C₄³×C₆₇	1	4	yes	—
122	C₂²	C₂²×C₄²×C₆₇	1	3	yes	—
123	C₂²	C₂²×M₁₆×C₆₇	1	4	yes	—
124	C₄	—	2	12	no	—
125	C₄	C₄³×C₆₇	1	4	yes	—
126	C₄	C₄×M₁₆×C₆₇	1	24	yes	direct product factor
127	C₄	C₂²×C₄²×C₆₇	1	4	yes	—
128	C₂	—	2	4	no	—
129	C₂	C₂×C₄³×C₆₇	1	1	yes	characteristic
130	C₂	C₂×C₄×M₁₆×C₆₇	1	6	yes	—

	subgroup	G/H
center	C₄³×C₆₇	C₂²
commutator	C₂	C₂×C₄³×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×#97×C₆₇	C₂	semidirect
4. C₄²×C₈×C₆₇	C₂	semidirect
8. C₄×M₁₆×C₆₇	C₄	direct (N × H)
25. #97×C₆₇	C₂×C₄	semidirect
26. C₄×C₈×C₆₇	C₂×C₄	semidirect
35. M₁₆×C₆₇	C₄²	direct (N × H)
39. C₄²×C₆₇	M₁₆	direct (N × H)
53. C₈×C₆₇	C₂×C₄²	semidirect
60. C₄×C₆₇	C₄×M₁₆	direct (N × H)
62. C₄²×M₁₆	C₆₇	direct (N × H)
68. C₄×#97	C₂×C₆₇	semidirect
69. C₄²×C₈	C₂×C₆₇	semidirect
70. C₆₇	C₄²×M₁₆	direct (N × H)
74. C₄×M₁₆	C₄×C₆₇	direct (N × H)
91. #97	C₂×C₄×C₆₇	semidirect
92. C₄×C₈	C₂×C₄×C₆₇	semidirect
101. M₁₆	C₄²×C₆₇	direct (N × H)
105. C₄²	M₁₆×C₆₇	direct (N × H)
119. C₈	C₂×C₄²×C₆₇	semidirect
126. C₄	C₄×M₁₆×C₆₇	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×C₄×M₁₆×C₆₇	2
C₂×C₄³×C₆₇	2
C₄×#97×C₆₇	2
C₄²×C₈×C₆₇	2
C₄²×M₁₆	67

Subgroups — C42×M16×C67 #30699579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×M₁₆×C₆₇ #30699579