Subgroups — C₂²×#346×C₆₇ #30712572

Order 17152.

942 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	C₂³×C₁₆×C₆₇	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₆₇	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₆₇	C₄	1	16	yes	complemented
8	C₂²×C₁₆×C₆₇	C₂²	1	12	yes	—
9	C₂×C₄×C₈×C₆₇	C₄	1	6	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	6	yes	—
11	#346×C₆₇	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₆₇	C₈	1	2	yes	—
13	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₈	1	8	yes	—
15	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
17	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
18	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
19	C₂×C₄²×C₆₇	C₈	1	6	yes	—
20	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
21	C₂×C₁₆×C₆₇	C₂×C₄	1	48	yes	complemented
22	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
23	C₄×C₈×C₆₇	C₂×C₄	1	12	yes	—
24	C₄×C₈×C₆₇	C₂³	1	4	yes	—
25	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₆₇	—	2	8	no	—
27	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
28	C₂²×C₄×C₆₇	C₂×C₈	1	12	yes	—
29	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
30	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
31	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
32	C₂×C₈×C₆₇	C₄²	1	12	yes	—
33	C₂×C₈×C₆₇	C₂×C₈	1	24	yes	—
34	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
35	C₂×C₈×C₆₇	C₂²×C₄	1	12	yes	—
36	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
37	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
38	C₄²×C₆₇	C₂×C₈	1	12	yes	—
39	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
40	C₁₆×C₆₇	C₂²×C₄	1	16	yes	complemented
41	C₂³×C₆₇	C₄×C₈	1	4	yes	—
42	C₂³×C₆₇	M₃₂	1	8	yes	—
43	C₂³×C₆₇	C₂²×C₈	1	3	yes	—
44	C₂×C₄×C₆₇	—	2	24	no	—
45	C₂×C₄×C₆₇	C₄×C₈	1	12	yes	—
46	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
47	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
48	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
49	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
50	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
51	C₈×C₆₇	C₂×C₄²	1	6	yes	—
52	C₈×C₆₇	C₂²×C₈	1	8	yes	—
53	C₈×C₆₇	C₂³×C₄	1	2	yes	—
54	C₂²×C₆₇	#346	1	16	yes	direct product factor
55	C₂²×C₆₇	C₂×C₄×C₈	1	6	yes	—
56	C₂²×C₆₇	C₂×M₃₂	1	12	yes	—
57	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
58	C₄×C₆₇	—	2	8	no	—
59	C₄×C₆₇	C₂×C₄×C₈	1	6	yes	—
60	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
61	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
62	C₂²×#346	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₆₇	C₂×#346	1	12	yes	direct product factor
64	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
65	C₂×C₆₇	C₂²×M₃₂	1	2	yes	—
66	C₂³×C₁₆	C₂×C₆₇	1	2	yes	—
67	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
68	C₂×#346	C₂×C₆₇	1	12	yes	direct product factor
69	C₆₇	C₂²×#346	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³×C₈	C₄×C₆₇	1	1	yes	characteristic
71	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
72	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
73	C₂²×C₁₆	C₄×C₆₇	1	16	yes	complemented
74	C₂²×C₁₆	C₂²×C₆₇	1	12	yes	—
75	C₂×C₄×C₈	C₄×C₆₇	1	6	yes	—
76	C₂×C₄×C₈	C₂²×C₆₇	1	6	yes	—
77	#346	C₂²×C₆₇	1	16	yes	direct product factor
78	C₂³×C₄	C₈×C₆₇	1	2	yes	—
79	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
80	C₂²×C₈	C₈×C₆₇	1	8	yes	—
81	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
82	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	—
83	C₂²×C₈	C₂×C₄×C₆₇	1	6	yes	—
84	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
85	C₂×C₄²	C₈×C₆₇	1	6	yes	—
86	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
87	C₂×C₁₆	C₂×C₄×C₆₇	1	48	yes	complemented
88	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
89	C₄×C₈	C₂×C₄×C₆₇	1	12	yes	—
90	C₄×C₈	C₂³×C₆₇	1	4	yes	—
91	C₂⁴	C₂×C₈×C₆₇	1	1	yes	characteristic
92	C₂²×C₄	—	2	8	no	—
93	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
94	C₂²×C₄	C₂×C₈×C₆₇	1	12	yes	—
95	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
96	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
97	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
98	C₂×C₈	C₄²×C₆₇	1	12	yes	—
99	C₂×C₈	C₂×C₈×C₆₇	1	24	yes	—
100	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
101	C₂×C₈	C₂²×C₄×C₆₇	1	12	yes	—
102	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
103	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
104	C₄²	C₂×C₈×C₆₇	1	12	yes	—
105	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
106	C₁₆	C₂²×C₄×C₆₇	1	16	yes	complemented
107	C₂³	C₄×C₈×C₆₇	1	4	yes	—
108	C₂³	M₃₂×C₆₇	1	8	yes	—
109	C₂³	C₂²×C₈×C₆₇	1	3	yes	—
110	C₂×C₄	—	2	24	no	—
111	C₂×C₄	C₄×C₈×C₆₇	1	12	yes	—
112	C₂×C₄	C₂×C₄²×C₆₇	1	6	yes	—
113	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	—
114	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
115	C₂×C₄	C₂²×C₈×C₆₇	1	3	yes	—
116	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
117	C₈	C₂×C₄²×C₆₇	1	6	yes	—
118	C₈	C₂²×C₈×C₆₇	1	8	yes	—
119	C₈	C₂³×C₄×C₆₇	1	2	yes	—
120	C₂²	#346×C₆₇	1	16	yes	direct product factor
121	C₂²	C₂×C₄×C₈×C₆₇	1	6	yes	—
122	C₂²	C₂×M₃₂×C₆₇	1	12	yes	—
123	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
124	C₄	—	2	8	no	—
125	C₄	C₂×C₄×C₈×C₆₇	1	6	yes	—
126	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
127	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
128	C₂	C₂×#346×C₆₇	1	12	yes	direct product factor
129	C₂	C₂²×C₄×C₈×C₆₇	1	1	yes	characteristic
130	C₂	C₂²×M₃₂×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈×C₆₇	C₂²
commutator	C₂	C₂²×C₄×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle, prime radical	C₂⁴×C₆₇	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#346×C₆₇	C₂	direct (N × H)
7. C₂²×C₁₆×C₆₇	C₄	semidirect
11. #346×C₆₇	C₂²	direct (N × H)
21. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
40. C₁₆×C₆₇	C₂²×C₄	semidirect
54. C₂²×C₆₇	#346	direct (N × H)
62. C₂²×#346	C₆₇	direct (N × H)
63. C₂×C₆₇	C₂×#346	direct (N × H)
68. C₂×#346	C₂×C₆₇	direct (N × H)
69. C₆₇	C₂²×#346	direct (N × H)
73. C₂²×C₁₆	C₄×C₆₇	semidirect
77. #346	C₂²×C₆₇	direct (N × H)
87. C₂×C₁₆	C₂×C₄×C₆₇	semidirect
106. C₁₆	C₂²×C₄×C₆₇	semidirect
120. C₂²	#346×C₆₇	direct (N × H)
128. C₂	C₂×#346×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
C₂³×C₁₆×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂×#346×C₆₇	2
C₂²×#346	67

Subgroups — C22×#346×C67 #30712572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#346×C₆₇ #30712572