Subgroups — C₄×C₁₆×Dic₆₇ #30770649

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₆₇	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
8	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
9	C₄²×Dic₆₇	C₄	1	1	yes	characteristic
10	C₄×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
11	C₁₆×Dic₆₇	C₄	1	16	yes	direct product factor
12	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₆₇	C₄	1	4	yes	complemented
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₇	C₈	1	4	yes	—
17	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₆₇⋊C₁₆	C₂×C₄	1	8	yes	—
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
20	C₄×C₆₇⋊C₈	C₈	1	4	yes	—
21	C₈×Dic₆₇	C₈	1	8	yes	—
22	C₈×Dic₆₇	C₂×C₄	1	8	yes	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	C₂×C₄	1	8	yes	—
27	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×Dic₆₇	C₂×C₈	1	4	yes	—
31	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
32	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
33	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
34	C₄×Dic₆₇	C₁₆	1	16	yes	direct product factor
35	C₄×Dic₆₇	C₄²	1	4	yes	—
36	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
37	C₆₇⋊C₁₆	C₄²	1	8	yes	complemented
38	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₄²	1	2	yes	—
41	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
43	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₆₇	C₄²	1	2	yes	—
46	C₄²×C₆₇	C₂×C₈	1	2	yes	—
47	C₁₆×C₆₇	C₄²	1	8	yes	complemented
48	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
50	C₂×C₄×C₆₇	C₂×C₁₆	1	8	yes	—
51	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
55	C₂×Dic₆₇	C₄×C₈	1	8	yes	—
56	C₂×Dic₆₇	C₂×C₁₆	1	16	yes	—
57	C₈×C₆₇	C₄×C₈	1	4	yes	—
58	C₈×C₆₇	C₂×C₄²	1	2	yes	—
59	C₈×C₆₇	C₂×C₄²	1	2	yes	—
60	C₆₇⋊C₈	C₄×C₈	1	8	yes	—
61	C₂²×C₆₇	C₂×C₄×C₈	1	2	yes	—
62	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂²×C₆₇	C₂²×C₁₆	1	4	yes	—
64	C₄×C₆₇	C₄×C₁₆	1	8	yes	complemented
65	C₄×C₆₇	C₄³	1	1	yes	characteristic
66	C₄×C₆₇	C₂×C₄×C₈	1	2	yes	—
67	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
68	Dic₆₇	C₄×C₁₆	1	16	yes	direct product factor
69	C₄²×C₁₆	—	67	1	no	maximal, Sylow
70	C₂×C₆₇	C₄²×C₈	1	1	yes	characteristic
71	C₂×C₆₇	C₂×C₄×C₁₆	1	4	yes	—
72	C₂×C₆₇	C₂×C₄×C₁₆	1	2	yes	—
73	C₂×C₄×C₁₆	—	67	4	no	—
74	C₂×C₄×C₁₆	—	67	1	no	—
75	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
76	C₄²×C₈	—	67	1	no	—
77	C₆₇	C₄²×C₁₆	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₁₆	—	67	2	no	—
79	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
80	C₂×C₄×C₈	—	67	2	no	—
81	C₂×C₄×C₈	—	67	2	no	—
82	C₂×C₄×C₈	—	67	1	no	—
83	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
84	C₄³	—	67	1	no	—
85	C₄×C₁₆	—	67	16	no	—
86	C₄×C₁₆	—	67	4	no	—
87	C₄×C₁₆	Dic₆₇	1	4	yes	direct product factor
88	C₂²×C₈	—	67	2	no	—
89	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
91	C₂×C₄²	—	67	4	no	—
92	C₂×C₄²	—	67	2	no	—
93	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
94	C₂×C₁₆	—	67	4	no	—
95	C₂×C₁₆	—	67	8	no	—
96	C₂×C₁₆	C₄×D₆₇	1	8	yes	—
97	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
98	C₄×C₈	—	67	8	no	—
99	C₄×C₈	—	67	8	no	—
100	C₄×C₈	—	67	4	no	—
101	C₄×C₈	C₄×D₆₇	1	2	yes	—
102	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
103	C₂²×C₄	—	67	4	no	—
104	C₂²×C₄	C₈×D₆₇	1	2	yes	—
105	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
106	C₂×C₈	—	67	4	no	—
107	C₂×C₈	—	67	4	no	—
108	C₂×C₈	—	67	4	no	—
109	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
110	C₂×C₈	C₈×D₆₇	1	4	yes	—
111	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
112	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
113	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
114	C₄²	—	67	16	no	—
115	C₄²	—	67	4	no	—
116	C₄²	—	67	4	no	—
117	C₄²	C₄×Dic₆₇	1	2	yes	—
118	C₄²	C₈×D₆₇	1	2	yes	—
119	C₁₆	—	67	8	no	—
120	C₁₆	C₄×Dic₆₇	1	8	yes	direct product factor
121	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
122	C₂×C₄	—	67	16	no	—
123	C₂×C₄	—	67	8	no	—
124	C₂×C₄	C₈×Dic₆₇	1	4	yes	—
125	C₂×C₄	C₁₆×D₆₇	1	8	yes	—
126	C₂×C₄	C₄²×D₆₇	1	2	yes	—
127	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
130	C₈	—	67	8	no	—
131	C₈	C₈×Dic₆₇	1	4	yes	—
132	C₈	C₄²×D₆₇	1	2	yes	—
133	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
134	C₂²	C₄×C₈×D₆₇	1	2	yes	—
135	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
136	C₂²	C₂×C₁₆×D₆₇	1	4	yes	—
137	C₄	—	67	16	no	—
138	C₄	C₁₆×Dic₆₇	1	8	yes	direct product factor
139	C₄	C₄²×Dic₆₇	1	1	yes	characteristic
140	C₄	C₄×C₈×D₆₇	1	2	yes	—
141	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
142	C₂	C₄×C₈×Dic₆₇	1	1	yes	characteristic
143	C₂	C₄×C₁₆×D₆₇	1	4	yes	—
144	C₂	C₂×C₁₆×Dic₆₇	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₁₆	D₆₇
commutator, hypocenter	C₆₇	C₄²×C₁₆
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×C₆₇⋊C₁₆	C₄	semidirect
11. C₁₆×Dic₆₇	C₄	direct (N × H)
14. C₄×C₁₆×C₆₇	C₄	semidirect
34. C₄×Dic₆₇	C₁₆	direct (N × H)
37. C₆₇⋊C₁₆	C₄²	semidirect
47. C₁₆×C₆₇	C₄²	semidirect
64. C₄×C₆₇	C₄×C₁₆	semidirect
68. Dic₆₇	C₄×C₁₆	direct (N × H)
77. C₆₇	C₄²×C₁₆	semidirect
87. C₄×C₁₆	Dic₆₇	direct (N × H)
120. C₁₆	C₄×Dic₆₇	direct (N × H)
138. C₄	C₁₆×Dic₆₇	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×C₆₇⋊C₁₆	2
C₂×C₁₆×Dic₆₇	2
C₄×C₈×Dic₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₄²×C₁₆	67

Subgroups — C4×C16×Dic67 #30770649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₆×Dic₆₇ #30770649