Subgroups — C₂×C₆₇×#2258 #30713224

Order 17152.

470 conjugacy classes of subgroups.

The subgroup lattice has 118 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, direct product factor
3	C₂×M₆₄×C₆₇	C₂	1	3	yes	maximal, complemented
4	C₂²×C₃₂×C₆₇	C₂	1	3	yes	maximal, complemented
5	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
6	#504×C₆₇	C₄	1	4	yes	—
7	#504×C₆₇	C₂²	1	4	yes	—
8	C₂×M₃₂×C₆₇	C₄	1	3	yes	—
9	C₂²×C₁₆×C₆₇	C₂²	1	3	yes	—
10	M₆₄×C₆₇	C₂²	1	12	yes	complemented
11	C₂×C₃₂×C₆₇	C₂²	1	3	yes	—
12	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₆₇	C₂²	1	12	yes	complemented
14	C₂×#41×C₆₇	C₈	1	1	yes	characteristic
15	#131×C₆₇	C₈	1	4	yes	—
16	#131×C₆₇	C₂×C₄	1	4	yes	—
17	C₂×M₁₆×C₆₇	C₈	1	3	yes	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
19	M₃₂×C₆₇	C₂×C₄	1	12	yes	—
20	C₂×C₁₆×C₆₇	C₂×C₄	1	3	yes	—
21	C₂×C₁₆×C₆₇	C₂×C₄	1	6	yes	—
22	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
23	C₂×C₁₆×C₆₇	C₂³	1	6	yes	—
24	C₃₂×C₆₇	C₂³	1	6	yes	—
25	C₃₂×C₆₇	C₂³	1	2	yes	—
26	#41×C₆₇	C₁₆	1	4	yes	—
27	#41×C₆₇	C₂×C₈	1	4	yes	—
28	C₂×Q₈×C₆₇	C₁₆	1	1	yes	characteristic
29	C₂×D₄×C₆₇	C₁₆	1	3	yes	—
30	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
31	M₁₆×C₆₇	C₂×C₈	1	12	yes	—
32	C₂×C₈×C₆₇	C₂×C₈	1	3	yes	—
33	C₂×C₈×C₆₇	C₂×C₈	1	6	yes	—
34	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
36	C₁₆×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₁₆×C₆₇	C₂²×C₄	1	6	yes	—
38	C₁₆×C₆₇	C₂⁴	1	1	yes	characteristic
39	C₂³×C₆₇	C₂×C₁₆	1	3	yes	—
40	Q₈×C₆₇	C₂×C₁₆	1	4	yes	—
41	D₄×C₆₇	C₂×C₁₆	1	12	yes	—
42	C₂×C₄×C₆₇	C₂×C₁₆	1	3	yes	—
43	C₂×C₄×C₆₇	C₂×C₁₆	1	6	yes	—
44	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
46	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₆₇	C₂²×C₈	1	6	yes	—
48	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
49	C₂²×C₆₇	—	2	6	no	—
50	C₂²×C₆₇	C₂²×C₁₆	1	6	yes	—
51	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
52	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
53	C₄×C₆₇	C₂²×C₁₆	1	6	yes	—
54	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
55	C₂×#2258	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₆₇	—	2	6	no	—
57	C₂×C₆₇	#2258	1	2	yes	direct product factor
58	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
59	C₂×#504	C₂×C₆₇	1	1	yes	characteristic
60	#2258	C₂×C₆₇	1	8	yes	direct product factor
61	C₂×M₆₄	C₂×C₆₇	1	3	yes	complemented
62	C₂²×C₃₂	C₂×C₆₇	1	3	yes	complemented
63	C₆₇	C₂×#2258	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×#131	C₄×C₆₇	1	1	yes	characteristic
65	#504	C₄×C₆₇	1	4	yes	—
66	#504	C₂²×C₆₇	1	4	yes	—
67	C₂×M₃₂	C₄×C₆₇	1	3	yes	—
68	C₂²×C₁₆	C₂²×C₆₇	1	3	yes	—
69	M₆₄	C₂²×C₆₇	1	12	yes	complemented
70	C₂×C₃₂	C₂²×C₆₇	1	3	yes	—
71	C₂×C₃₂	C₂²×C₆₇	1	1	yes	characteristic
72	C₂×C₃₂	C₂²×C₆₇	1	12	yes	complemented
73	C₂×#41	C₈×C₆₇	1	1	yes	characteristic
74	#131	C₈×C₆₇	1	4	yes	—
75	#131	C₂×C₄×C₆₇	1	4	yes	—
76	C₂×M₁₆	C₈×C₆₇	1	3	yes	—
77	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
78	M₃₂	C₂×C₄×C₆₇	1	12	yes	—
79	C₂×C₁₆	C₂×C₄×C₆₇	1	3	yes	—
80	C₂×C₁₆	C₂×C₄×C₆₇	1	6	yes	—
81	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
82	C₂×C₁₆	C₂³×C₆₇	1	6	yes	—
83	C₃₂	C₂³×C₆₇	1	2	yes	—
84	C₃₂	C₂³×C₆₇	1	6	yes	—
85	#41	C₁₆×C₆₇	1	4	yes	—
86	#41	C₂×C₈×C₆₇	1	4	yes	—
87	C₂×Q₈	C₁₆×C₆₇	1	1	yes	characteristic
88	C₂×D₄	C₁₆×C₆₇	1	3	yes	—
89	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
90	M₁₆	C₂×C₈×C₆₇	1	12	yes	—
91	C₂×C₈	C₂×C₈×C₆₇	1	3	yes	—
92	C₂×C₈	C₂×C₈×C₆₇	1	6	yes	—
93	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
94	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
95	C₁₆	C₂²×C₄×C₆₇	1	1	yes	characteristic
96	C₁₆	C₂²×C₄×C₆₇	1	6	yes	—
97	C₁₆	C₂⁴×C₆₇	1	1	yes	characteristic
98	C₂³	C₂×C₁₆×C₆₇	1	3	yes	—
99	Q₈	C₂×C₁₆×C₆₇	1	4	yes	—
100	D₄	C₂×C₁₆×C₆₇	1	12	yes	—
101	C₂×C₄	C₂×C₁₆×C₆₇	1	3	yes	—
102	C₂×C₄	C₂×C₁₆×C₆₇	1	6	yes	—
103	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
105	C₈	C₂²×C₈×C₆₇	1	1	yes	characteristic
106	C₈	C₂²×C₈×C₆₇	1	6	yes	—
107	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
108	C₂²	—	2	6	no	—
109	C₂²	C₂²×C₁₆×C₆₇	1	6	yes	—
110	C₂²	C₂²×C₁₆×C₆₇	1	1	yes	characteristic
111	C₄	C₂²×C₁₆×C₆₇	1	1	yes	characteristic
112	C₄	C₂²×C₁₆×C₆₇	1	6	yes	—
113	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
114	C₂	—	2	6	no	—
115	C₂	C₆₇×#2258	1	2	yes	direct product factor
116	C₂	C₂³×C₁₆×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂×C₆₇	C₂²
commutator	C₂	C₂³×C₁₆×C₆₇
Frattini, cosocle	C₁₆	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₁₆
prime radical	C₂×#41×C₆₇	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2258	C₂	direct (N × H)
3. C₂×M₆₄×C₆₇	C₂	semidirect
4. C₂²×C₃₂×C₆₇	C₂	semidirect
10. M₆₄×C₆₇	C₂²	semidirect
13. C₂×C₃₂×C₆₇	C₂²	semidirect
55. C₂×#2258	C₆₇	direct (N × H)
57. C₂×C₆₇	#2258	direct (N × H)
60. #2258	C₂×C₆₇	direct (N × H)
61. C₂×M₆₄	C₂×C₆₇	semidirect
62. C₂²×C₃₂	C₂×C₆₇	semidirect
63. C₆₇	C₂×#2258	direct (N × H)
69. M₆₄	C₂²×C₆₇	semidirect
72. C₂×C₃₂	C₂²×C₆₇	semidirect
115. C₂	C₆₇×#2258	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×#504×C₆₇	2
C₆₇×#2258	2
C₂×M₆₄×C₆₇	2
C₂²×C₃₂×C₆₇	2
C₂×#2258	67

Subgroups — C2×C67×#2258 #30713224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2258 #30713224