Subgroups — C₆₇×#33974 #30713225

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 112 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2258	C₂	1	8	yes	maximal, complemented
3	C₂×M₆₄×C₆₇	C₂	1	6	yes	maximal, complemented
4	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
5	#504×C₆₇	C₄	1	4	yes	—
6	#504×C₆₇	C₂²	1	4	yes	—
7	C₂×M₃₂×C₆₇	C₄	1	3	yes	—
8	C₂²×C₁₆×C₆₇	C₂²	1	3	yes	—
9	M₆₄×C₆₇	C₂²	1	4	yes	—
10	M₆₄×C₆₇	C₂²	1	12	yes	complemented
11	C₂×C₃₂×C₆₇	C₂²	1	12	yes	complemented
12	C₂×#41×C₆₇	C₈	1	1	yes	characteristic
13	#131×C₆₇	C₈	1	4	yes	—
14	#131×C₆₇	C₂×C₄	1	4	yes	—
15	C₂×M₁₆×C₆₇	C₈	1	3	yes	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
17	M₃₂×C₆₇	C₂×C₄	1	12	yes	—
18	C₂×C₁₆×C₆₇	C₂×C₄	1	3	yes	—
19	C₂×C₁₆×C₆₇	C₂×C₄	1	6	yes	—
20	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
21	C₂×C₁₆×C₆₇	C₂³	1	6	yes	—
22	C₃₂×C₆₇	C₂³	1	8	yes	—
23	#41×C₆₇	C₁₆	1	4	yes	—
24	#41×C₆₇	C₂×C₈	1	4	yes	—
25	C₂×Q₈×C₆₇	C₁₆	1	1	yes	characteristic
26	C₂×D₄×C₆₇	C₁₆	1	3	yes	—
27	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
28	M₁₆×C₆₇	C₂×C₈	1	12	yes	—
29	C₂×C₈×C₆₇	C₂×C₈	1	3	yes	—
30	C₂×C₈×C₆₇	C₂×C₈	1	6	yes	—
31	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
33	C₁₆×C₆₇	C₂²×C₄	1	1	yes	characteristic
34	C₁₆×C₆₇	C₂²×C₄	1	6	yes	—
35	C₁₆×C₆₇	C₂⁴	1	1	yes	characteristic
36	C₂³×C₆₇	C₂×C₁₆	1	3	yes	—
37	Q₈×C₆₇	C₂×C₁₆	1	4	yes	—
38	D₄×C₆₇	C₂×C₁₆	1	12	yes	—
39	C₂×C₄×C₆₇	C₂×C₁₆	1	3	yes	—
40	C₂×C₄×C₆₇	C₂×C₁₆	1	6	yes	—
41	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
43	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₆₇	C₂²×C₈	1	6	yes	—
45	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
46	C₂²×C₆₇	—	4	3	no	—
47	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
48	C₂²×C₆₇	C₂²×C₁₆	1	6	yes	—
49	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
50	C₄×C₆₇	C₂²×C₁₆	1	6	yes	—
51	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
52	#33974	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×C₆₇	—	2	1	no	—
54	C₂×C₆₇	—	2	6	no	—
55	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
56	C₂×#504	C₂×C₆₇	1	1	yes	characteristic
57	#2258	C₂×C₆₇	1	8	yes	complemented
58	C₂×M₆₄	C₂×C₆₇	1	6	yes	complemented
59	C₆₇	#33974	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×#131	C₄×C₆₇	1	1	yes	characteristic
61	#504	C₄×C₆₇	1	4	yes	—
62	#504	C₂²×C₆₇	1	4	yes	—
63	C₂×M₃₂	C₄×C₆₇	1	3	yes	—
64	C₂²×C₁₆	C₂²×C₆₇	1	3	yes	—
65	M₆₄	C₂²×C₆₇	1	4	yes	—
66	M₆₄	C₂²×C₆₇	1	12	yes	complemented
67	C₂×C₃₂	C₂²×C₆₇	1	12	yes	complemented
68	C₂×#41	C₈×C₆₇	1	1	yes	characteristic
69	#131	C₈×C₆₇	1	4	yes	—
70	#131	C₂×C₄×C₆₇	1	4	yes	—
71	C₂×M₁₆	C₈×C₆₇	1	3	yes	—
72	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
73	M₃₂	C₂×C₄×C₆₇	1	12	yes	—
74	C₂×C₁₆	C₂×C₄×C₆₇	1	3	yes	—
75	C₂×C₁₆	C₂×C₄×C₆₇	1	6	yes	—
76	C₂×C₁₆	C₂³×C₆₇	1	6	yes	—
77	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
78	C₃₂	C₂³×C₆₇	1	8	yes	—
79	#41	C₁₆×C₆₇	1	4	yes	—
80	#41	C₂×C₈×C₆₇	1	4	yes	—
81	C₂×Q₈	C₁₆×C₆₇	1	1	yes	characteristic
82	C₂×D₄	C₁₆×C₆₇	1	3	yes	—
83	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
84	M₁₆	C₂×C₈×C₆₇	1	12	yes	—
85	C₂×C₈	C₂×C₈×C₆₇	1	3	yes	—
86	C₂×C₈	C₂×C₈×C₆₇	1	6	yes	—
87	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
88	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
89	C₁₆	C₂²×C₄×C₆₇	1	1	yes	characteristic
90	C₁₆	C₂²×C₄×C₆₇	1	6	yes	—
91	C₁₆	C₂⁴×C₆₇	1	1	yes	characteristic
92	C₂³	C₂×C₁₆×C₆₇	1	3	yes	—
93	Q₈	C₂×C₁₆×C₆₇	1	4	yes	—
94	D₄	C₂×C₁₆×C₆₇	1	12	yes	—
95	C₂×C₄	C₂×C₁₆×C₆₇	1	3	yes	—
96	C₂×C₄	C₂×C₁₆×C₆₇	1	6	yes	—
97	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
99	C₈	C₂²×C₈×C₆₇	1	1	yes	characteristic
100	C₈	C₂²×C₈×C₆₇	1	6	yes	—
101	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
102	C₂²	—	4	3	no	—
103	C₂²	C₂²×C₁₆×C₆₇	1	1	yes	characteristic
104	C₂²	C₂²×C₁₆×C₆₇	1	6	yes	—
105	C₄	C₂²×C₁₆×C₆₇	1	1	yes	characteristic
106	C₄	C₂²×C₁₆×C₆₇	1	6	yes	—
107	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
108	C₂	—	2	1	no	—
109	C₂	—	2	6	no	—
110	C₂	C₂³×C₁₆×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₆₇	C₂⁴
commutator	C₂	C₂³×C₁₆×C₆₇
Frattini, cosocle	C₁₆	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂³×C₁₆
prime radical	C₂×#41×C₆₇	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2258	C₂	semidirect
3. C₂×M₆₄×C₆₇	C₂	semidirect
10. M₆₄×C₆₇	C₂²	semidirect
11. C₂×C₃₂×C₆₇	C₂²	semidirect
52. #33974	C₆₇	direct (N × H)
57. #2258	C₂×C₆₇	semidirect
58. C₂×M₆₄	C₂×C₆₇	semidirect
59. C₆₇	#33974	direct (N × H)
66. M₆₄	C₂²×C₆₇	semidirect
67. C₂×C₃₂	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504×C₆₇	2
C₆₇×#2258	2
C₂×M₆₄×C₆₇	2
#33974	67

Subgroups — C67×#33974 #30713225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33974 #30713225