Subgroups — C₂³×#106×C₆₇ #30739352

Order 17152.

3614 conjugacy classes of subgroups.

The subgroup lattice has 122 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal
3	C₂²×#106×C₆₇	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₆₇	—	2	2	no	—
5	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₆₇	C₄	1	16	yes	complemented
7	C₂³×C₈×C₆₇	C₂²	1	14	yes	—
8	C₂²×C₄⋊C₄×C₆₇	—	2	14	no	—
9	C₂²×C₄⋊C₄×C₆₇	C₂²	1	28	yes	—
10	C₂×#106×C₆₇	C₂²	1	112	yes	direct product factor
11	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
12	C₂³×C₄×C₆₇	—	2	28	no	—
13	C₂³×C₄×C₆₇	—	4	16	no	—
14	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
15	C₂³×C₄×C₆₇	D₄	1	7	yes	—
16	C₂³×C₄×C₆₇	Q₈	1	8	yes	—
17	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	112	yes	complemented
19	C₂²×C₈×C₆₇	C₂³	1	28	yes	—
20	C₂×C₄⋊C₄×C₆₇	—	2	84	no	—
21	C₂×C₄⋊C₄×C₆₇	C₂³	1	56	yes	—
22	#106×C₆₇	C₂³	1	64	yes	direct product factor
23	C₂⁴×C₆₇	C₄⋊C₄	1	8	yes	—
24	C₂⁴×C₆₇	SD₁₆	1	16	yes	—
25	C₂⁴×C₆₇	C₂×D₄	1	7	yes	—
26	C₂²×C₄×C₆₇	—	2	56	no	—
27	C₂²×C₄×C₆₇	—	4	112	no	—
28	C₂²×C₄×C₆₇	C₄⋊C₄	1	56	yes	—
29	C₂²×C₄×C₆₇	C₂²×C₄	1	28	yes	—
30	C₂²×C₄×C₆₇	C₂×D₄	1	21	yes	—
31	C₂²×C₄×C₆₇	C₂×Q₈	1	28	yes	—
32	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
33	C₂×C₈×C₆₇	C₂²×C₄	1	112	yes	complemented
34	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
35	C₄⋊C₄×C₆₇	—	2	56	no	—
36	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
37	C₂³×C₆₇	#106	1	64	yes	direct product factor
38	C₂³×C₆₇	C₂×C₄⋊C₄	1	28	yes	—
39	C₂³×C₆₇	C₂×SD₁₆	1	56	yes	—
40	C₂³×C₆₇	C₂²×D₄	1	7	yes	—
41	C₂×C₄×C₆₇	—	2	16	no	—
42	C₂×C₄×C₆₇	—	4	112	no	—
43	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	84	yes	—
44	C₂×C₄×C₆₇	C₂³×C₄	1	14	yes	—
45	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
46	C₂×C₄×C₆₇	C₂²×Q₈	1	14	yes	—
47	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
48	C₈×C₆₇	C₂³×C₄	1	16	yes	complemented
49	C₂²×C₆₇	C₂×#106	1	112	yes	direct product factor
50	C₂²×C₆₇	C₂²×C₄⋊C₄	1	14	yes	—
51	C₂²×C₆₇	C₂²×SD₁₆	1	28	yes	—
52	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
53	C₄×C₆₇	—	4	16	no	—
54	C₄×C₆₇	C₂²×C₄⋊C₄	1	14	yes	—
55	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
56	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
57	C₂³×#106	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂×C₆₇	C₂²×#106	1	28	yes	direct product factor
59	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₆₇	C₂³×SD₁₆	1	2	yes	—
61	C₂⁴×C₈	C₂×C₆₇	1	1	yes	characteristic
62	C₂³×C₄⋊C₄	C₂×C₆₇	1	2	yes	—
63	C₂²×#106	C₂×C₆₇	1	28	yes	direct product factor
64	C₆₇	C₂³×#106	1	1	yes	characteristic, direct product factor, Sylow
65	C₂⁴×C₄	—	2	2	no	—
66	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
67	C₂³×C₈	C₄×C₆₇	1	16	yes	complemented
68	C₂³×C₈	C₂²×C₆₇	1	14	yes	—
69	C₂²×C₄⋊C₄	—	2	14	no	—
70	C₂²×C₄⋊C₄	C₂²×C₆₇	1	28	yes	—
71	C₂×#106	C₂²×C₆₇	1	112	yes	direct product factor
72	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
73	C₂³×C₄	—	2	28	no	—
74	C₂³×C₄	—	4	16	no	—
75	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
76	C₂³×C₄	D₄×C₆₇	1	7	yes	—
77	C₂³×C₄	Q₈×C₆₇	1	8	yes	—
78	C₂³×C₄	C₂³×C₆₇	1	7	yes	—
79	C₂²×C₈	C₂×C₄×C₆₇	1	112	yes	complemented
80	C₂²×C₈	C₂³×C₆₇	1	28	yes	—
81	C₂×C₄⋊C₄	—	2	84	no	—
82	C₂×C₄⋊C₄	C₂³×C₆₇	1	56	yes	—
83	#106	C₂³×C₆₇	1	64	yes	direct product factor
84	C₂⁴	C₄⋊C₄×C₆₇	1	8	yes	—
85	C₂⁴	SD₁₆×C₆₇	1	16	yes	—
86	C₂⁴	C₂×D₄×C₆₇	1	7	yes	—
87	C₂²×C₄	—	2	56	no	—
88	C₂²×C₄	—	4	112	no	—
89	C₂²×C₄	C₄⋊C₄×C₆₇	1	56	yes	—
90	C₂²×C₄	C₂²×C₄×C₆₇	1	28	yes	—
91	C₂²×C₄	C₂×D₄×C₆₇	1	21	yes	—
92	C₂²×C₄	C₂×Q₈×C₆₇	1	28	yes	—
93	C₂²×C₄	C₂⁴×C₆₇	1	7	yes	—
94	C₂×C₈	C₂²×C₄×C₆₇	1	112	yes	complemented
95	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
96	C₄⋊C₄	—	2	56	no	—
97	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
98	C₂³	#106×C₆₇	1	64	yes	direct product factor
99	C₂³	C₂×C₄⋊C₄×C₆₇	1	28	yes	—
100	C₂³	C₂×SD₁₆×C₆₇	1	56	yes	—
101	C₂³	C₂²×D₄×C₆₇	1	7	yes	—
102	C₂×C₄	—	2	16	no	—
103	C₂×C₄	—	4	112	no	—
104	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	84	yes	—
105	C₂×C₄	C₂³×C₄×C₆₇	1	14	yes	—
106	C₂×C₄	C₂²×D₄×C₆₇	1	7	yes	—
107	C₂×C₄	C₂²×Q₈×C₆₇	1	14	yes	—
108	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
109	C₈	C₂³×C₄×C₆₇	1	16	yes	complemented
110	C₂²	C₂×#106×C₆₇	1	112	yes	direct product factor
111	C₂²	C₂²×C₄⋊C₄×C₆₇	1	14	yes	—
112	C₂²	C₂²×SD₁₆×C₆₇	1	28	yes	—
113	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
114	C₄	—	4	16	no	—
115	C₄	C₂²×C₄⋊C₄×C₆₇	1	14	yes	—
116	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
117	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
118	C₂	C₂²×#106×C₆₇	1	28	yes	direct product factor
119	C₂	C₂³×C₄⋊C₄×C₆₇	1	1	yes	characteristic
120	C₂	C₂³×SD₁₆×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₆₇	D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×#106×C₆₇	C₂	direct (N × H)
6. C₂³×C₈×C₆₇	C₄	semidirect
10. C₂×#106×C₆₇	C₂²	direct (N × H)
18. C₂²×C₈×C₆₇	C₂×C₄	semidirect
22. #106×C₆₇	C₂³	direct (N × H)
33. C₂×C₈×C₆₇	C₂²×C₄	semidirect
37. C₂³×C₆₇	#106	direct (N × H)
48. C₈×C₆₇	C₂³×C₄	semidirect
49. C₂²×C₆₇	C₂×#106	direct (N × H)
57. C₂³×#106	C₆₇	direct (N × H)
58. C₂×C₆₇	C₂²×#106	direct (N × H)
63. C₂²×#106	C₂×C₆₇	direct (N × H)
64. C₆₇	C₂³×#106	direct (N × H)
67. C₂³×C₈	C₄×C₆₇	semidirect
71. C₂×#106	C₂²×C₆₇	direct (N × H)
79. C₂²×C₈	C₂×C₄×C₆₇	semidirect
83. #106	C₂³×C₆₇	direct (N × H)
94. C₂×C₈	C₂²×C₄×C₆₇	semidirect
98. C₂³	#106×C₆₇	direct (N × H)
109. C₈	C₂³×C₄×C₆₇	semidirect
110. C₂²	C₂×#106×C₆₇	direct (N × H)
118. C₂	C₂²×#106×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
C₂⁴×C₈×C₆₇	2
C₂³×C₄⋊C₄×C₆₇	2
C₂²×#106×C₆₇	2
C₂³×#106	67

Subgroups — C23×#106×C67 #30739352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#106×C₆₇ #30739352